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Introduction to Third-order Processes and Materials

2010-06-11T01:28:48Z

Mrumi: /* Summary */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-rank tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization become more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensors and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arise from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. The curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest, will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index is now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in the unit area). So, as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. Usually, a material with fewer defects is less likely to be damaged (or will damage at higher intensities). When doing experiments involving frequency tripling researchers need to use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative ''γ'' and positive and negative ''χ''(3) lead to different effects. Specifically, negative ''χ''(3) leads to self-defocusing, and positive ''χ''(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02).

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be the case. There could be three beams with different phases and different directions, or different polarizations. If the frequency components of the fields are different other nonlinear effects can be observed, for example, the fields can "mix" and produce new fields with frequency equal to the sums and differences of frequencies of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, the latter situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive ''χ''(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems, as described in the next subsection.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrates the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will travel faster than others, and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the propagation direction of the beam reversed, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, portions of the beam are going to be slowed, other sped up, as previously, but now the process acts on a distorted wavefront. The result is that distortion produced by the second pass exactly cancels that of the first pass, thus the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes, you can get back the original undistorted image. This is useful for targeting applications and for looking at images of the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bonds in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''', in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and is divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles), are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta-carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strengths on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''γ'' for a series of these derivatives. It can be seen that the magnitude of ''γ'' increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in ''γ'' can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability ''γ'' or the bulk susceptibility ''χ''(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed as a function of these frequencies.
These are also complex quantities. In the case of the linear polarizability, the real part is related to how light is refracted by the material, the imaginary part is related to the absorption of light by the material. The situation is similar for ''γ''. Molecules will have both real and imaginary parts to ''γ''. The real part affects to how the refractive index is changed as light of a given intensity goes through it (the refractive index is field-dependent in these materials). The imaginary part is manifested in two-photon absorption. An intensity dependent refractive index can lead to self-focussing or self-defocussing of a beam propagating through a layer of material. Other third-order optical effects that can be exploited for applications are sum (or difference) frequency generation and wave mixing.

In order to make useful devices like the Mach Zehnder interferometer you want to use a material whose index of refraction can be changed using a light beam, but you don’t want to lose light in the material because of linear absorption or scattering. They can also lose transparency at a high intensity due to the process of two-photon absorption. The identification of materials with large ''γ''s at the desired wavelength and low loss is an active area of research. Dipolar molecules tend to have large positive ''γ'' values but also tend to have high two-photon absorption cross sections.
Recently it has been reported that a class of molecules with negative ''γ'' have very large real parts of the hyperpolarizability but, in certain spectral regions , their imaginary part is almost zero so there would be no light lost due to two-photon absorption. These are good candidates for all optical switching applications because until now molecules with high ''γ'' (''χ''(3)) have had a high a loss due to two-photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Two Photon Absorption

2010-06-11T01:24:59Z

Mrumi: /* Applications for TPA */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where ''η'' is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section ''δ''. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable only to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample needs to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability ''γ'' for the frequency component relevant to two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for ''γ'' goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (''δ'') is proportional to the imaginary component of ''γ'':

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene.]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the ground and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are known, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goeppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit is named in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481; <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352; <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536. <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity ''δ'' is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the measurement problems in the NLO field in general, as these are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a ''δ'' value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross section that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depends on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), ''δ'' was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. ''δ'' for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to the vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of ''δ''.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make the dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for ''γ'' (keeping in mind that δ is proportional to Im ''γ''):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seen that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for ''δ'' as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large ''δ'' values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for ''δ'' in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trend in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two-photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for ''δ'', detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-donating and/or electron-withdrawing functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control of the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two-photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage.]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because of the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bits in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process is with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two-photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large ''δ'' values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T01:14:40Z

Mrumi: /* Design of TPA Chromophores */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where ''η'' is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section ''δ''. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable only to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample needs to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability ''γ'' for the frequency component relevant to two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for ''γ'' goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (''δ'') is proportional to the imaginary component of ''γ'':

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene.]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the ground and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are known, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goeppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit is named in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481; <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352; <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536. <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity ''δ'' is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the measurement problems in the NLO field in general, as these are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a ''δ'' value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross section that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depends on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), ''δ'' was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. ''δ'' for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to the vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of ''δ''.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make the dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for ''γ'' (keeping in mind that δ is proportional to Im ''γ''):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seen that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for ''δ'' as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large ''δ'' values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for ''δ'' in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trend in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two-photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for ''δ'', detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-donating and/or electron-withdrawing functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T01:09:49Z

Mrumi: /* Examples of Two-Photon Absorbing Materials */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where ''η'' is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section ''δ''. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable only to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample needs to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability ''γ'' for the frequency component relevant to two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for ''γ'' goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (''δ'') is proportional to the imaginary component of ''γ'':

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene.]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the ground and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are known, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goeppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit is named in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481; <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352; <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536. <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity ''δ'' is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the measurement problems in the NLO field in general, as these are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a ''δ'' value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross section that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depends on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), ''δ'' was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. ''δ'' for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to the vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of ''δ''.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make the dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for ''γ'' (keeping in mind that δ is proportional to Im ''γ''):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seen that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T00:51:22Z

Mrumi: /* Calculations of the TPA Cross Section in a Donor-Acceptor Molecule */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where ''η'' is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section ''δ''. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable only to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample needs to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability ''γ'' for the frequency component relevant to two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for ''γ'' goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (''δ'') is proportional to the imaginary component of ''γ'':

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene.]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T00:51:07Z

Mrumi: /* Calculation of TPA Cross Sections */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where ''η'' is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section ''δ''. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable only to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample needs to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability ''γ'' for the frequency component relevant to two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for ''γ'' goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (''δ'') is proportional to the imaginary component of ''γ'':

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T00:46:45Z

Mrumi: /* Measuring the Two-Photon Absorption Cross Section */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where ''η'' is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section ''δ''. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable only to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample needs to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T00:42:57Z

Mrumi: /* Advantages of TPA */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two-photon absorption is considered to be a nuisance in the case of all-optical switching using ''χ''(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-11T00:38:22Z

Mrumi: /* Two-Photon Excited Processes */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the ''χ''(3) tensor (or, on a molecular level, to the imaginary component of ''γ'') and depends on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same energy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons has enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two corresponds to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order to observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math>

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Introduction to Third-order Processes and Materials

2010-06-11T00:30:52Z

Mrumi: /* Second Hyperpolarizability and BOA */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-rank tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization become more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensors and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arise from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. The curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest, will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index is now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in the unit area). So, as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. Usually, a material with fewer defects is less likely to be damaged (or will damage at higher intensities). When doing experiments involving frequency tripling researchers need to use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative ''γ'' and positive and negative ''χ''(3) lead to different effects. Specifically, negative ''χ''(3) leads to self-defocusing, and positive ''χ''(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02).

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be the case. There could be three beams with different phases and different directions, or different polarizations. If the frequency components of the fields are different other nonlinear effects can be observed, for example, the fields can "mix" and produce new fields with frequency equal to the sums and differences of frequencies of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, the latter situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive ''χ''(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems, as described in the next subsection.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrates the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will travel faster than others, and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the propagation direction of the beam reversed, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, portions of the beam are going to be slowed, other sped up, as previously, but now the process acts on a distorted wavefront. The result is that distortion produced by the second pass exactly cancels that of the first pass, thus the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes, you can get back the original undistorted image. This is useful for targeting applications and for looking at images of the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bonds in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''', in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and is divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles), are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta-carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strengths on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''γ'' for a series of these derivatives. It can be seen that the magnitude of ''γ'' increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in ''γ'' can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability ''γ'' or the bulk susceptibility ''χ''(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed as a function of these frequencies.
These are also complex quantities. In the case of the linear polarizability, the real part is related to how light is refracted by the material, the imaginary part is related to the absorption of light by the material. The situation is similar for ''γ''. Molecules will have both real and imaginary parts to ''γ''. The real part affects to how the refractive index is changed as light of a given intensity goes through it (the refractive index is field-dependent in these materials). The imaginary part is manifested in two-photon absorption. An intensity dependent refractive index can lead to self-focussing or self-defocussing of a beam propagating through a layer of material. Other third-order optical effects that can be exploited for applications are sum (or difference) frequency generation and wave mixing.

In order to make useful devices like the Mach Zehnder interferometer you want to use a material whose index of refraction can be changed using a light beam, but you don’t want to lose light in the material because of linear absorption or scattering. They can also lose transparency at a high intensity due to the process of two-photon absorption. The identification of materials with large ''γ''s at the desired wavelength and low loss is an active area of research. Dipolar molecules tend to have large positive ''γ'' values but also tend to have high two-photon absorption cross sections.
Recently we have discovered that a class of molecules with negative ''γ'' have very large real parts of the hyperpolarizability but, in certain spectral regions , their imaginary part is almost zero so there would be no light lost due to two-photon absorption. These are good candidates for all optical switching applications because until now molecules with high γ (''χ''(3)) have had a high a loss due to two-photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-11T00:25:11Z

Mrumi: /* Four Wave Mixing */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-rank tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization become more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensors and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arise from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. The curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest, will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index is now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in the unit area). So, as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. Usually, a material with fewer defects is less likely to be damaged (or will damage at higher intensities). When doing experiments involving frequency tripling researchers need to use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative ''γ'' and positive and negative ''χ''(3) lead to different effects. Specifically, negative ''χ''(3) leads to self-defocusing, and positive ''χ''(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02).

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be the case. There could be three beams with different phases and different directions, or different polarizations. If the frequency components of the fields are different other nonlinear effects can be observed, for example, the fields can "mix" and produce new fields with frequency equal to the sums and differences of frequencies of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, the latter situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive ''χ''(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems, as described in the next subsection.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrates the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will travel faster than others, and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the propagation direction of the beam reversed, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, portions of the beam are going to be slowed, other sped up, as previously, but now the process acts on a distorted wavefront. The result is that distortion produced by the second pass exactly cancels that of the first pass, thus the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes, you can get back the original undistorted image. This is useful for targeting applications and for looking at images of the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strength on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''&gamma''; for a series of these derivatives. It can be seen that the magnitude of ''&gamma''; increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in γ can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by a an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability ''γ'' or the bulk susceptibility ''χ''(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed as a function of these frequencies.
These are also complex quantities. In the case of the linear polarizability, the real part is related to how light is refracted by the material, the imaginary part is related to the absorption of light by the material. The situation is similar for ''γ''. Molecules will have both real and imaginary parts to ''γ''. The real part affects to how the refractive index is changed as light of a given intensity goes through it (the refractive index is field-dependent in these materials). The imaginary part is manifested in two-photon absorption. An intensity dependent refractive index can lead to self-focussing or self-defocussing of a beam propagating through a layer of material. Other third-order optical effects that can be exploited for applications are sum (or difference) frequency generation and wave mixing.

In order to make useful devices like the Mach Zehnder interferometer you want to use a material whose index of refraction can be changed using a light beam, but you don’t want to lose light in the material because of linear absorption or scattering. They can also lose transparency at a high intensity due to the process of two-photon absorption. The identification of materials with large ''γ''s at the desired wavelength and low loss is an active area of research. Dipolar molecules tend to have large positive ''γ'' values but also tend to have high two-photon absorption cross sections.
Recently we have discovered that a class of molecules with negative ''γ'' have very large real parts of the hyperpolarizability but, in certain spectral regions , their imaginary part is almost zero so there would be no light lost due to two-photon absorption. These are good candidates for all optical switching applications because until now molecules with high γ (''χ''(3)) have had a high a loss due to two-photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-11T00:11:33Z

Mrumi: /* Third-Order NLO Effects */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-rank tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization become more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensors and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arise from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. The curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest, will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index is now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in the unit area). So, as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. Usually, a material with fewer defects is less likely to be damaged (or will damage at higher intensities). When doing experiments involving frequency tripling researchers need to use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative ''γ'' and positive and negative ''χ''(3) lead to different effects. Specifically, negative ''χ''(3) leads to self-defocusing, and positive ''χ''(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02).

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strength on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''&gamma''; for a series of these derivatives. It can be seen that the magnitude of ''&gamma''; increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in γ can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by a an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability ''γ'' or the bulk susceptibility ''χ''(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed as a function of these frequencies.
These are also complex quantities. In the case of the linear polarizability, the real part is related to how light is refracted by the material, the imaginary part is related to the absorption of light by the material. The situation is similar for ''γ''. Molecules will have both real and imaginary parts to ''γ''. The real part affects to how the refractive index is changed as light of a given intensity goes through it (the refractive index is field-dependent in these materials). The imaginary part is manifested in two-photon absorption. An intensity dependent refractive index can lead to self-focussing or self-defocussing of a beam propagating through a layer of material. Other third-order optical effects that can be exploited for applications are sum (or difference) frequency generation and wave mixing.

In order to make useful devices like the Mach Zehnder interferometer you want to use a material whose index of refraction can be changed using a light beam, but you don’t want to lose light in the material because of linear absorption or scattering. They can also lose transparency at a high intensity due to the process of two-photon absorption. The identification of materials with large ''γ''s at the desired wavelength and low loss is an active area of research. Dipolar molecules tend to have large positive ''γ'' values but also tend to have high two-photon absorption cross sections.
Recently we have discovered that a class of molecules with negative ''γ'' have very large real parts of the hyperpolarizability but, in certain spectral regions , their imaginary part is almost zero so there would be no light lost due to two-photon absorption. These are good candidates for all optical switching applications because until now molecules with high γ (''χ''(3)) have had a high a loss due to two-photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-10T23:58:51Z

Mrumi: /* Nonlinear Susceptibility and Hyperpolarizability */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-rank tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization become more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensors and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arise from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. The curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strength on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''&gamma''; for a series of these derivatives. It can be seen that the magnitude of ''&gamma''; increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in γ can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by a an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability ''γ'' or the bulk susceptibility ''χ''(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed as a function of these frequencies.
These are also complex quantities. In the case of the linear polarizability, the real part is related to how light is refracted by the material, the imaginary part is related to the absorption of light by the material. The situation is similar for ''γ''. Molecules will have both real and imaginary parts to ''γ''. The real part affects to how the refractive index is changed as light of a given intensity goes through it (the refractive index is field-dependent in these materials). The imaginary part is manifested in two-photon absorption. An intensity dependent refractive index can lead to self-focussing or self-defocussing of a beam propagating through a layer of material. Other third-order optical effects that can be exploited for applications are sum (or difference) frequency generation and wave mixing.

In order to make useful devices like the Mach Zehnder interferometer you want to use a material whose index of refraction can be changed using a light beam, but you don’t want to lose light in the material because of linear absorption or scattering. They can also lose transparency at a high intensity due to the process of two-photon absorption. The identification of materials with large ''γ''s at the desired wavelength and low loss is an active area of research. Dipolar molecules tend to have large positive ''γ'' values but also tend to have high two-photon absorption cross sections.
Recently we have discovered that a class of molecules with negative ''γ'' have very large real parts of the hyperpolarizability but, in certain spectral regions , their imaginary part is almost zero so there would be no light lost due to two-photon absorption. These are good candidates for all optical switching applications because until now molecules with high γ (''χ''(3)) have had a high a loss due to two-photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T22:09:56Z

Mrumi: /* Summary */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strength on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''&gamma''; for a series of these derivatives. It can be seen that the magnitude of ''&gamma''; increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in γ can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by a an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability ''γ'' or the bulk susceptibility ''χ''(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed as a function of these frequencies.
These are also complex quantities. In the case of the linear polarizability, the real part is related to how light is refracted by the material, the imaginary part is related to the absorption of light by the material. The situation is similar for ''γ''. Molecules will have both real and imaginary parts to ''γ''. The real part affects to how the refractive index is changed as light of a given intensity goes through it (the refractive index is field-dependent in these materials). The imaginary part is manifested in two-photon absorption. An intensity dependent refractive index can lead to self-focussing or self-defocussing of a beam propagating through a layer of material. Other third-order optical effects that can be exploited for applications are sum (or difference) frequency generation and wave mixing.

In order to make useful devices like the Mach Zehnder interferometer you want to use a material whose index of refraction can be changed using a light beam, but you don’t want to lose light in the material because of linear absorption or scattering. They can also lose transparency at a high intensity due to the process of two-photon absorption. The identification of materials with large ''γ''s at the desired wavelength and low loss is an active area of research. Dipolar molecules tend to have large positive ''γ'' values but also tend to have high two-photon absorption cross sections.
Recently we have discovered that a class of molecules with negative ''γ'' have very large real parts of the hyperpolarizability but, in certain spectral regions , their imaginary part is almost zero so there would be no light lost due to two-photon absorption. These are good candidates for all optical switching applications because until now molecules with high γ (''χ''(3)) have had a high a loss due to two-photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T21:47:22Z

Mrumi: /* Third-order Nonlinear Optical Properties of Polarized Polyenes */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strength on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''&gamma''; for a series of these derivatives. It can be seen that the magnitude of ''&gamma''; increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in γ can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by a an increase in ''γ''. 

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability γ or the bulk susceptibility χ(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T21:27:16Z

Mrumi: /* Third-order Nonlinear Optical Properties of Polarized Polyenes */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on ''γ'' when various acceptors are added to beta-carotene.]]
Beta carotene is an organic compound with a long polyenic chain, it is orange-red and it is found in many fruits and vegetables.
Derivatives of beta-carotene have been synthesized with acceptor groups of various strength on one end of the chain, to test the effect on the polarizability and hyperpolarizability of the molecule. This table reports the values of ''&gamma''; for a series of these derivatives. It can be seen that the magnitude of ''&gamma''; increases by a factor of 45 when the strength of the acceptor is increased. This is also accompanied by a red-shift in the position of the absorption maximum. The increase in γ can be explained by a change in the BOA value in the presence of acceptor groups, as BOA becomes less negative going from the initial polyene (symmetric structure, no acceptor; see top row) to a molecule with strong acceptor on one side (bottom row). As seen in the plot in the previous subsection, this type of change in BOA is accompanied by a an increase in ''γ''.

== Summary ==

Third-order nonlinear effects can be described in terms of the molecular parameter hyperpolarizability γ or the bulk susceptibility χ(3). These quantities depend on the frequency of the electromagnetic waves interacting in a material and different effects can be observed

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T21:01:51Z

Mrumi: /* Second Hyperpolarizability and BOA */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to ''γ'' from various terms as a function of bond-order-alternation, BOA.]]
The figure at right displays how the second hyperpolarizability ''γ'' varies as a function of bond-order alternation (BOA) in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule (the other components are negligible for these systems). As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Structure-Property_Relationships&action=edit&section=3]].
The left side of the plot corresponds to the polyene limit (large and negative BOA), the cyanine-like limit corresponds to BOA = 0, and the zwitterionic limit to large and positive BOA.
''γ''xxxx (red dots) is calculated using perturbation theory. It can be noted that ''γ'' starts positive in the polyene limit, goes through a maximum, then through zero before assuming a large negative value at the cyanine-like limit; the behavior is symmetrical on the BOA > 0 side of the plot .

If only few states contribute to the perturbation expression for ''γ'', it is possible to write a simplified equation that contains three terms, dubbed '''n''' (negative), '''tp''' (two-photon) and '''d''' (dipolar, because it only comes into effect when there is a change in dipole moment between the ground and the excited state):

:<math>\gamma_{model} \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

where:

''g'' is the ground state, ''e'' the lowest excited state. The summation in the middle term is over two-photon allowed states, ''e''' in the system.
:<math>\mu_{ab}\,\!</math> is the transition dipole moment between the states ''a'' and ''b''.
:<math> E_{ab}\,\!</math> is the energy difference between the states ''a'' and ''b''.

In the '''n''' term (leftmost term in the equation), the transition dipole moment between the ground and the initial state comes in at the 4th power and divided by the energy gap between those two states to the third power. This term is always negative and it assumes the largest absolute value at the cyanine limit.

The '''tp''' term (middle term in the equation) depends on the transition dipole moment between the ground and the excited state, and between the excited state and a higher lying two-photon state, both squared. This term is positive, with a maximum for BOA = 0.

The '''d''' term (rightmost term in the equation) is similar to that that appears in simplified expressions for the hyperpolarizability ''β''. The difference in dipole moment is squared so that it always positive, the energy term is cubed. The two positive maxima in ''γ'' can be traced mostly to this term.

The value of each of these terms as a function of BOA and their sum, ''γ''model (open circles) are displayed in the plot. The values of ''γ''model are similar to those of ''γ''xxxx, indicating that the simplified model is sufficient to account for the main changes in the magnitude and sign of the hyperpolarizability for this class of compounds.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T20:10:01Z

Mrumi: /* Second Hyper-polarizability and BOA */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyperpolarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms as a function of bond-order-alternation.]]
The figure at right displays how the second hyperpolarizability γ varies as a function of bond-order alternation in a one-dimensional polymethine structure. The tensor component included in the plot is xxxx, where x represents the long axis of the molecule. As introduced elsewhere, BOA is the average value of the difference between the length of adjacent CC bond in the chain.
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T20:05:41Z

Mrumi: /* Phase Conjugation */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase conjugate optics.]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T20:04:43Z

Mrumi: /* Phase Conjugate Mirror */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase Conjugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from a phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate mirror in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities in the atmosphere. This is a third order nonlinear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T20:03:09Z

Mrumi: /* Phase Conjugation */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase Conjugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.
 

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T20:01:55Z

Mrumi: /* Four Wave Mixing */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situation yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' (DFWM) process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase Conjugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

A beam of light has a momentum determined by the direction it is traveling. If beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter-propagating beams with the same phase have a momentum sum of zero.
If the probe beam is counter-propagating with respect to one of the writing beams in a DFWM configuration, then the diffracted beam exactly retraces the path of the other writing beam.
Phase conjugate optics takes advantage of this special feature of the diffracted beam.
As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".
In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

One consequence of this is that distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate mirror retraces exactly the same path and alterations as the incoming wave.]]
The figure at right illustrate the operation of a phase conjugate mirror. A planar wave (a) passes through a distorting material (rectangle), which introduces an aberration in the wavefront. Then the light interacts with a phase conjugate mirror (gray area) creating the phase conjugate wavefront (c). Then the phase conjugate wave passes through the distorting material on the reverse path, canceling the original aberration thus producing an undistorted wavefront (d).
Aberrations in the wavefront can be caused by the medium not having a uniform refractive index, so that portions of the light will faster than other and leading to a deformation in the original wavefront. When the wavefront hits the phase conjugate mirror, the beam reversed the propagation direction, but the shape of the wavefront is not changed: the part of the beam that comes into the mirror first ends up leaving last and viceversa. When the reversed beam travels back and encounters the original material, the portions of the beam are going to be slowed, other sped up, as previously, but now acting on a distorted wavefront. The result is that the aberration is removed and the original wavefront is obtained.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T16:51:12Z

Mrumi: /* Four Wave Mixing */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process: three waves (electric fields 1, 2 and 3 at frequency ''ω'') interact in a material to create a fourth wave (at frequency 3''ω''). In the case of third harmonic generation with a single beam of light the three fields are degenerate: electric field 1 has the same frequency, phase and momentum ('''k''' vector) as electric field 2 and 3.

This does not have to be case. There could be three beams with different phases and different directions, polarizations. If the frequency components of the fields are different other nonlinear effects cam be observed, for example, the field can "mix" and produce new fields with frequency equal to the sums and differences of frequency of the input fields (four wave mixing):

:<math>\omega_1 + \omega_2 + \omega_3\,\!</math> (this is third harmonic generation if ''ω''1 = ''ω''2 = ''ω''3)

or

:<math>\omega_1 + \omega_2 - \omega_3\,\!</math>

If the frequencies are the same, this situations yields light out at the same frequency as the input and is called degenerate four wave mixing (in the case of a single beam, this leads to the self-focusing effect discussed earlier).

Another way to describe the '''degenerate four wave mixing''' process is to consider two beams of light (at the same frequency) interacting within a material to create an interference pattern and thus a spatially periodic variation in light intensity in the material. As we have noted before, the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interfering in a third-order NLO material, the result will be a refractive index grating, that is a periodic modulation of the refractive index in the areas of constructive and destructive interference: the areas that are brightest will have an increased refractive index (with a positive χ(3)); at the darkest point the refractive index will have zero change. When a third beam is incident on this grating, the beam is diffracted from the grating, generating a fourth beam, called the phase conjugate. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase-conjugate beam.

=== Phase Conjugation ===
[[Image:4wavemixing.png|thumb|200px|Phase Conjugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T16:12:49Z

Mrumi: /* Third Harmonic Generation and the Optical Kerr Effect */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field, assuming that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency ''ω'':

:<math>E = E_0 cos(\omega t) \,\!</math>

the dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on ''E''02 (the maximum deviation of the sinusoidal electric field) and ''γ''. This is called Kerr effect.
''E''02 is always positive. On the other hand ''γ'' can be either positive or negative. Thus, by increasing the magnitude of the electric field, the polarizability of the material increases if ''γ'' is positive and decreases if ''γ'' is negative. If ''E'' is the electric field of an electromagnetic wave, due to this third-order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to ''E''02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ''ω''. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that equation (5) also contain a term at frequency 3''ω'', the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on ''γ''.
Thus, the interaction of light with a third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ''ω'' the bulk material will have an induced polarization at 3''ω'', due to ''χ''(3). This process is called third-harmonic generation.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T01:37:40Z

Mrumi: /* Third Harmonic Generation and the Optical Kerr Effect */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (2) for the expansion of dipole moment of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric), and we apply an oscillating field at frequency \omega;:

:<math>E = E_0 cos(\omega t) \,\!</math>

The dipole moment of the molecule becomes:

:<math>\mu = \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3cos^3 (\omega t) + ...\,\!</math> (3)

From trigonometry it can be shown that:

:<math>cos^3(\omega t) = (3/4) cos(\omega t) + (1/4) cos (3\omega t)\,\!</math> (4)

Substituting (4) in (3), we obtain

:<math>\mu = \mu_0+ \alpha E_0 cos(\omega t) + (1/6) \gamma E_0^3 (3/4) cos(\omega t) + (1/6) \gamma E_0^3 (1/4) cos (3\omega t)\,\!</math>

or, equivalently:

:<math>\mu= \mu_0+ [\alpha +(1/6) \gamma E_{0}^{2}(3/4)]E_0cos(\omega t) + (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math> (5)

Thus, the material has an effective polarizability that depends on E02 (the maximum deviation of the sinusoidal electric field) and γ. This is called Kerr effect.
E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (i. e. the intensity of the light, as this is proportional to E02 ) in the materials the polarizability of the material increases if γ is positive and decreases if γ is negative). If E is the electric field of an electromagnetic wave, due to a third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light (because the light intensity is proportional to E02.

This component of the dipole moment or polarization oscillates at the same frequency of the input field, ω. We note that the :<math>[\alpha + (1/6) \gamma E_{0}^{2}(3/4)]\,\!</math> term of equation (5) is similar to the term leading to the linear electrooptic effect or the Pockels effect.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is used to changed the refractive index of a material at the same frequency the beam.

The DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an oscillating applied voltage that can modulate the refractive index.

It can be seen that eq (5) also contain a term at frequency 3&omega, the third harmonic of the incident field:

:<math> (1/6) \gamma E_0^3(1/4)cos(3\omega t)\,\!</math>

The magnitude of this component depends on &gamma.
Thus, the interaction of light with third-order NLO material will create a polarization component at its third harmonic.
Likewise, at the macroscopic level, in the presence of a strong laser beam at frequency ω the bulk material will have an induced polarization at 3&omega, due to χ^(3). This process is called third-harmonic generation

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T00:42:19Z

Mrumi: /* Third-Order NLO Effects */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third Harmonic Generation and the Optical Kerr Effect ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T00:39:33Z

Mrumi: /* Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

== Third-Order NLO Effects ==

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T00:31:28Z

Mrumi: /* Nonlinear Self-Focusing */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

Positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-09T00:20:16Z

Mrumi: /* Non linear self focusing process */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

=== Nonlinear Self-Focusing ===

Let's consider a beam of light propagating into a NLO material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, where the intensity is highest will have a higher refractive index than the material at the edge of the beam profile, where there is low intensity. The refractive index changes because in this NLO material the polarizability (and susceptibility) depends in the intensity of the light, and thus the refractive index is also intensity dependent. As the refractive index in now different across the beam profile, the different portions of the beam will be refractive to different degrees, in particular the rays near the optical axis are refractive more than rays farther away from the axis. Thus this NLO material behaves like a lens that focuses light that propagates through it. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). So as a beam becomes focused the added intensity increases the refractive index further, causing even more focussing, higher intensity at the center of the beam and more change in refractive index. This process is called “'''nonlinear self-focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light, which is largest at the center of the beam, thus leading to focusing, even higher intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the material when the intensity becomes too high. Usually, a material with fewer defects is the less likely it is be damaged (or will damage at higher intensities). Catastrophic self-focusing thus poses limits to the maximum intensities within optical materials, for example those used in lasers. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals.

In an NLO material in which polarization decreases with the light intensity (a material with a negative nonlinear refractive index), the opposite effect is observed: the refractive index encountered by the beam while propagating in the medium is smaller at the center of the beam and larger near the edges, leading to '''self-defocusing''' of the beam. Thus, the beam passing through this material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T23:41:58Z

Mrumi: /* Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.
 

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T23:41:37Z

Mrumi: /* Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). If, as you polarize this material more and more it becomes harder to polarize, its susceptibility decreases with the field, as does its refractive index.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T23:27:20Z

Mrumi: /* Taylor Expansion for Molecular Dipole Moment */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank, are frequency dependent, and each component can be a complex quantity.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize, the refractive index decreases with the field.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T20:29:27Z

Mrumi: /* Third-order Nonlinear Polarization of Matter and Third-order NLO Effects */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank and are frequency dependent.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic potential with a positive or negative quartic term.]]

It can be shown that the description of Eq. (2) above corresponds to the behavior of a molecule in an anharmonic potential well. Second-order NLO effects arose from a cubic component in the potential as a function of displacement from the equilibrium position. A quartic term in the potential is at the origin of third-order effects. This is illustrated in the graph at right. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. The change is symmetric with respect to x = 0. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there a deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from that characteristic of the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize, the refractive index decreases with the field.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T19:59:53Z

Mrumi: /* Nonlinear Susceptibility and Hyperpolarizability */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Molecular Dipole Moment ===

In a manner similar to (1), at the molecular level, the dipole moment of a molecule, ''μ'', is affected by an external electric field and it can be expanded in a Taylor series as a function of the field:

:<math>\mu = \mu_0 + \alpha E + (1/2)\beta ·· E^2 + (1/6)\gamma ···E^3+ ...\,\!</math> (2)

where:

:<math>\mu_0\,\!</math> is the permanent dipole of the molecule,
:<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),
:<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

As for the susceptibilities, <math>\alpha\,\!</math>, <math>\beta\,\!</math>, and <math>\gamma\,\!</math> are tensors of progressively higher rank and are frequency dependent.
<math>\beta\,\!</math> is zero for centrosymmetric molecules.

Under normal conditions,

:<math>\alpha_{ij} E_j > \beta_{ijk} E_j · E_k > \gamma_{ijkl} E_j · E_k · E_l \,\!</math>

(here we have introduced the subscript to identify a specific component of the tensor and vectors; ''i'', ''j'', ''k'', ''l'' = ''x'', ''y'', ''z'').
However, for field strengths large enough or for specific frequencies of the electromagnetic field, one of the nonlinear terms may become the dominant contribution to the dipole moment.
Typically, high intensity laser beams are need for the effects of the hyperpolarizabilities to become observable.

=== Third-order Nonlinear Polarization of Matter and Third-order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic plot with + or - quartic terms]]

Remember that in χ(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With χ(3) we add a restoring force that scales as a displacement to the 4th power. This is an even function. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize and the refractive index will decrease.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
<table id="toc" style="width: 100%">
<tr>

<td style="text-align: center; width: 33%">[[Main_Page#Third-order Processes, Materials & Characterization |Return to Third-order Processes, Materials & Characterization Menu]]</td>
<td style="text-align: right; width: 33%">[[Two Photon Absorption | Next Topic]]</td>
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</table>

Introduction to Third-order Processes and Materials

2010-06-08T19:19:42Z

Mrumi: /* Hyperpolarizability */

<table id="toc" style="width: 100%">
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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Nonlinear Susceptibility and Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, <math>\chi^{(2)}\,\!</math> is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to <math>\chi^{(2)}\,\!</math> = 0). Similarly, higher order <math>\chi^{(n)}\,\!</math> with even ''n'' are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on <math>\chi^{(n)}\,\!</math> when ''n'' is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
''P'' and ''E'' are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; the susceptibilities <math>\chi^{(n)}\,\!</math> are higher-order tensors (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency/frequencies (wavelength/wavelengths) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Polarization ===
Under normal conditions,

:<math>\alpha_{ij}E .> \beta_{ijk}/2 E·E > \gamma_{ijkl} /6 E·E·E.\,\!</math>

Where
:<math>j\,\!</math> is the coordinate system for the applied field

:<math>i\,\!</math> is the coordinate system for the induced polarization in the molecule

:<math>\alpha\,\!</math> is 3 x 3 tensor

:<math>\beta\,\!</math> is 3 x 3 x 3 tensor with 27 components

:<math>\gamma\,\!</math> is a 3 x 3 x 3 x 3 tensor with 81 components

Just as α is the linear polarizability, the higher order terms β and γ are called the first and second hyperpolarizabilities respectively. γ is the second hyperpolarizability which is a molecular property. It scales as the cube of the electric field.

There were few observations of NLO effects before the invention of the laser with its associated large electric fields.

=== Third-order Nonlinear Polarization of Matter and Third-order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic plot with + or - quartic terms]]

Remember that in χ(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With χ(3) we add a restoring force that scales as a displacement to the 4th power. This is an even function. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize and the refractive index will decrease.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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2010-06-08T19:13:51Z

Mrumi: /* Third-order Processes, Materials & Characterization */

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Two Photon Absorption

2010-06-08T19:13:17Z

Mrumi:

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Introduction to Third-order Processes and Materials

2010-06-08T19:12:05Z

Mrumi: /* Hyperpolarizability */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[http://depts.washington.edu/cmditr/mediawiki/index.php?title=Quantum-Mechanical_Theory_of_Molecular_Polarizabilities]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, χ(2) is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to χ(2) = 0). Similarly, higher order χ(n) with even n are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on χ(n) when n is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
P and E are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; The susceptibilities χ(n) are higher-order tensor (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency (wavelength) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Polarization ===
Under normal conditions,

:<math>\alpha_{ij}E .> \beta_{ijk}/2 E·E > \gamma_{ijkl} /6 E·E·E.\,\!</math>

Where
:<math>j\,\!</math> is the coordinate system for the applied field

:<math>i\,\!</math> is the coordinate system for the induced polarization in the molecule

:<math>\alpha\,\!</math> is 3 x 3 tensor

:<math>\beta\,\!</math> is 3 x 3 x 3 tensor with 27 components

:<math>\gamma\,\!</math> is a 3 x 3 x 3 x 3 tensor with 81 components

Just as α is the linear polarizability, the higher order terms β and γ are called the first and second hyperpolarizabilities respectively. γ is the second hyperpolarizability which is a molecular property. It scales as the cube of the electric field.

There were few observations of NLO effects before the invention of the laser with its associated large electric fields.

=== Third-order Nonlinear Polarization of Matter and Third-order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic plot with + or - quartic terms]]

Remember that in χ(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With χ(3) we add a restoring force that scales as a displacement to the 4th power. This is an even function. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize and the refractive index will decrease.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T19:10:27Z

Mrumi: /* Hyperpolarizability */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Hyperpolarizability ==

For a material in an electric field, the bulk polarization, ''P'', can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[Link title]]),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expressions can be written instead of (1), if more than one field is present).

Because of symmetry, χ(2) is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules leads to χ(2) = 0). Similarly, higher order χ(n) with even n are zero in centrosymmetric materials. There are no symmetry restrictions, instead, on χ(n) when n is odd (that is, these susceptibilities can be finite in centrosymmetric materials).
P and E are vectors and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3 x 3 tensor; The susceptibilities χ(n) are higher-order tensor (3 x 3 x 3 for n = 2, 3 x 3 x 3 x 3 for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g., <math>\chi_{ijk}^{(2)}\,\!</math>, where ''i'', ''j'', ''k'' are one of the cartesian coordinates). The linear and nonlinear susceptibilities depend on the frequency (wavelength) of the electromagnetic field and are material's properties.
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.

=== Taylor Expansion for Polarization ===
Under normal conditions,

:<math>\alpha_{ij}E .> \beta_{ijk}/2 E·E > \gamma_{ijkl} /6 E·E·E.\,\!</math>

Where
:<math>j\,\!</math> is the coordinate system for the applied field

:<math>i\,\!</math> is the coordinate system for the induced polarization in the molecule

:<math>\alpha\,\!</math> is 3 x 3 tensor

:<math>\beta\,\!</math> is 3 x 3 x 3 tensor with 27 components

:<math>\gamma\,\!</math> is a 3 x 3 x 3 x 3 tensor with 81 components

Just as α is the linear polarizability, the higher order terms β and γ are called the first and second hyperpolarizabilities respectively. γ is the second hyperpolarizability which is a molecular property. It scales as the cube of the electric field.

There were few observations of NLO effects before the invention of the laser with its associated large electric fields.

=== Third-order Nonlinear Polarization of Matter and Third-order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic plot with + or - quartic terms]]

Remember that in χ(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With χ(3) we add a restoring force that scales as a displacement to the 4th power. This is an even function. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize and the refractive index will decrease.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T18:06:39Z

Mrumi: /* Hyperpolarizability */

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Hyperpolarizability ==

For a material in an electric field, the bulk polarization, P, can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,
:<math>E\,\!</math> is the applied electric field,
(more generic expression can be written instead of (1) if more than one field is present).

Because of symmetry, χ(2) is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero χ(2)). Similarly, higher order χ(n) with even n are zero in centrosymmetric materials. There are no symmetry restrictions on χ(n) when n is odd (that is, these hyperpolarizabilities can be finite in centrosymmetric materials).
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.
P and E are vectors, and the linear susceptibility <math>\chi^{(1)}\,\!</math> is a 3x3 tensor, The hyperpolarizabilities χ(n) are higher order tensor (third order for n = 2, fourth order for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g. , :<math>\chi_{ijk}^{(2)}\,\!</math>, were i, j, k are one of the cartesian coordinates).

=== Taylor Expansion for Polarization ===
Under normal conditions,

:<math>\alpha_{ij}E .> \beta_{ijk}/2 E·E > \gamma_{ijkl} /6 E·E·E.\,\!</math>

Where
:<math>j\,\!</math> is the coordinate system for the applied field

:<math>i\,\!</math> is the coordinate system for the induced polarization in the molecule

:<math>\alpha\,\!</math> is 3 x 3 tensor

:<math>\beta\,\!</math> is 3 x 3 x 3 tensor with 27 components

:<math>\gamma\,\!</math> is a 3 x 3 x 3 x 3 tensor with 81 components

Just as α is the linear polarizability, the higher order terms β and γ are called the first and second hyperpolarizabilities respectively. γ is the second hyperpolarizability which is a molecular property. It scales as the cube of the electric field.

There were few observations of NLO effects before the invention of the laser with its associated large electric fields.

=== Third-order Nonlinear Polarization of Matter and Third-order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic plot with + or - quartic terms]]

Remember that in χ(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With χ(3) we add a restoring force that scales as a displacement to the 4th power. This is an even function. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize and the refractive index will decrease.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Introduction to Third-order Processes and Materials

2010-06-08T15:45:58Z

Mrumi:

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The term "third order nonlinear optical (NLO) materials" refers to materials whose polarization depends on the intensity of an applied electromagnetic field. This intensity dependence gives rise to a variety of useful properties such as self-focusing, two-photon absorption, and third harmonic generation.

== Hyperpolarizability ==

For a material in an electric field, the bulk polarization, P, can be expanded as follows :

:<math>P = P_0 + \chi^{(1)}·E + (1/2)\chi^{(2)}·· E^2 + (1/6)\chi^{(3)}···E^3+ ...\,\!</math> (1)

where:

:<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),
:<math>\chi^{(1)}\,\!</math> is the polarizability of the material (which is related to the dielectric constant),
:<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the first and second hyperpolarizabilities,
:<math>E\,\!</math> is the applied electric field,
(more generic expression can be written instead of (1) if more than one field is present).

Because of symmetry, χ(2) is non-zero only if the material is not centrosymmetric overall (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero χ(2)). Similarly, higher order χ(n) with even n are zero in centrosymmetric materials. There are no symmetry restrictions on χ(n) when n is odd (that is, these hyperpolarizabilities can be finite in centrosymmetric materials).
Nonlinear contributions to the material's polarization becomes more important with increasing field strength, since they scale with higher powers of the field.
P and E are vectors, and the polarizability is a 3x3 tensor, The hyperpolarizabilities χ(n) are higher order tensor (third order for n = 2, fourth order for n = 3, etc.). The specific component of the relevant tensor is usually represented by subscripted indices (e.g. , :<math>\chi_{ijk}^{(2)}\,\!</math>, were i, j, k are one of the cartesian coordinates).

=== Taylor Expansion for Polarization ===
Under normal conditions,

:<math>\alpha_{ij}E .> \beta_{ijk}/2 E·E > \gamma_{ijkl} /6 E·E·E.\,\!</math>

Where
:<math>j\,\!</math> is the coordinate system for the applied field

:<math>i\,\!</math> is the coordinate system for the induced polarization in the molecule

:<math>\alpha\,\!</math> is 3 x 3 tensor

:<math>\beta\,\!</math> is 3 x 3 x 3 tensor with 27 components

:<math>\gamma\,\!</math> is a 3 x 3 x 3 x 3 tensor with 81 components

Just as α is the linear polarizability, the higher order terms β and γ are called the first and second hyperpolarizabilities respectively. γ is the second hyperpolarizability which is a molecular property. It scales as the cube of the electric field.

There were few observations of NLO effects before the invention of the laser with its associated large electric fields.

=== Third-order Nonlinear Polarization of Matter and Third-order NLO Effects ===
[[Image:Harmonic_quartic.png|thumb|300px|Deviation from simple harmonic plot with + or - quartic terms]]

Remember that in χ(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With χ(3) we add a restoring force that scales as a displacement to the 4th power. This is an even function. If the correction is added in a positive way the well becomes steeper, adding the correction in a negative way the potential well is more shallow. These curves shown are greatly exaggerated, in reality the deviation would be less than the thickness of the lines as they are drawn. For the most part during normal oscillations the electrons are held within a quadratic potential. Only when there is a large electric field is there deviation of the electron from their resting position to the point where these terms (terms which account for anharmonicity) are manifested in any significant way. When a restoring force of x4 is added to a molecule the polarization deviates from the harmonic potential. A greater displacement means that it is getting harder to polarize the molecule and the greater the difference between the harmonic potential and the quartic potential. A material with a greater susceptibility has a higher refractive index (and a higher dielectric constant). As you polarize this material more and more it becomes harder to polarize and its susceptibility decreases while its refractive index decreases. If when you polarize a material it becomes easier to polarize and the refractive index will decrease.

=== Non linear self focusing process ===

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “'''non linear self focusing'''”.

[[Image:Grin-lens.png|thumb|300px|A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. A non-linear material acts like a graded index because the index changes with the intensity of the light being absorbed, thus leading to more focusing, more intensity and so on.]]

All materials (including glass and air) have third order non-linear optical effects. Sometimes these effects can lead to catastrophic self-focusing, leading to the destruction of the materials. This can cause an extremely high intensity of light that can actually damage a laser (it will blow apart). The more perfect the material the less likely it is to blow it apart. When are doing experiments involving frequency tripling researchers use perfect defect-free crystals. In laser fusion crystals are used that are as big as a person.

In a material in which polarization decreases with intensity the condition is called '''self-defocusing'''. The beam passing through a material has a tendency to spread out.

A molecule with a negative β or a negative χ(2) has an axis or plane of the molecule that has been flipped so that the donor and acceptors are opposite. There will still be an asymmetric polarizability in response to a static electric field. Positive and negative β lead to the same effects but with opposite signs. However positive and negative γ and positive and negative χ(3) lead to different effects. Specifically, negative χ(3) leads to self-defocusing, and positive χ(3) leads to self-focusing.

The quartic contribution to the potential has mirror symmetry with respect to the distortion coordinate; as a result both centrosymmetric and noncentrosymmetric materials will have third-order optical nonlinearities.

See Wikipedia [http://en.wikipedia.org/wiki/Self-focusing Self Focusing]

See also Encyclopedia of Laser Physics [http://www.rp-photonics.com/self_focusing.html Self Focusing]

=== Third order polarization ===

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

:<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

:<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

:<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)
or:

:<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E02 (the maximum deviation of the sinusoidal electric field) and γ. E02 is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

=== Third Harmonic Generation and the Optical Kerr Effect ===

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects.
In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

In the DC Kerr effect or the quadratic electro-optic effect involves a low intensity beam combined with an applied voltage that can modulate the refractive index.

== Four Wave Mixing ==

Third harmonic generation is a four wave mixing process. Three waves (electric 1, 2 and 3) interact in a material to create a fourth wave. In the case of third harmonic generation with single beam of light the three fields are degenerate; electric field 1 has the same frequency, phase and momentum (k-vect) as electric field 2 and three.

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

:<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

:<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ(3)). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

=== Degenerate Four-wave Mixing ===
[[Image:4wavemixing.png|thumb|200px|Phase Congugate Optics]]
A potential use of Degenerate Four-wave Mixing (DFWM) is in phase conjugate optics.

If two beams are directed on a material they create a diffractive index grating. A beam of light has a momentum determined by the direction it is traveling. If the beams of light mix and do not transfer energy to the material the momentum must be conserved. Two counter propagating beams (with the same phase) have a momentum sum of zero.

Phase conjugate optics takes advantage of a special feature of the diffracted beam: its path exactly retraces the path of one of the writing beams.

*As a result, a pair of diverging beams impinging on a phase conjugate mirror will converge after "reflection".

*In contrast, a pair of diverging beams reflected from an ordinary mirror will continue to diverge.

Thus, distorted optical wavefronts can be reconstructed using phase conjugate optical systems.

=== Phase Conjugation ===
A diverging set of beams reflected off of a normal mirror continues to diverge. (left)
A diverging set of beams reflected off of a phase conjugate mirror exactly retrace their original path and are recombined at their point of origin. (right)

=== Phase Conjugate Mirror ===
[[Image:Phaseconjugate_mirror.png|thumb|300px|Reflection from phase conjugate retraces exactly same path and alterations as incoming wave.]]
A planar wave (a) passes through a distorting material (b) that introduces an aberration and the light interacts with a phase conjugate mirror (c) creating the phase conjugate wavefront. (d)
Phase conjugate wave passes through the distorting material on the reverse path canceling the original aberration thus producing an undistorted wavefront.

A wavefront is made up a lot of beams traveling in the same direction a through a medium. Some aberration (with lower refractive index) in the material allows a portion of the light to go faster causing a bump in the wavefront. When the wavefront hits the phase conjugate mirror all parts are reversed. The part of the beam that comes into the mirror first ends up leaving last; there is a time reversal. When the reversed beam travels back and encounters the original aberration the distortion is removed.

In the following Flash animation a wavefront of light passes through a material with uneven index of refraction. Select either "normal mirror" or "phase conjugate mirror" to see the effect on the final wavefront after passing through the medium twice.

<div id="Flash">Phase conjugate mirror animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/conjugatemirror.swf</swf>

There are applications for this when looking at distant objects that have passed through a material that is scattering. If you bounce the light off a phase conjugate in two passes and you can get back the original undistorted image. This is useful for targeting applications and for looking at images on the Earth from a satellite where there are distortions due to inhomogeneities the atmosphere. This is a third order non linear optical effect.

See wikipedia http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation

== Second Hyper-polarizability and BOA ==
[[Image:Secondpolarizability_boa.png|thumb|500px|Contributions to γ from various terms]]
The curve in red shows γ as a function of BOA as it goes from a polyene limit, through cyanine-like limit, up to a zwitterionic polyene limit. γ is calculated using perturbation theory. It starts positive, goes up, goes through zero and has negative peak at the cyanine-like limit and then comes back up and is positive.

The simplified perturbation expression for γ that involves three expressions, dubbed '''n''' (negative), '''tp''' (two photon) and '''d''' (dipolar because it only comes into effect when there is a change in dipole between the ground and the excited state.)

:<math>\gamma \propto - \left ( \frac {\mu^{4}_{ge}} {E^{3}_{ge}} \right) + \sum_{e^\prime} \left( \frac {\mu^{2}_{ge} \mu^{2}_{ee^\prime}} {E^{2}_{ge} E_{ge^\prime}} \right ) + \left ( \frac {\mu^{2}_{ge} (\mu_{ee} - \mu_{gg})^{2}} {E^{3}_{ge}} \right )\,\!</math>

N the transition dipole moment between the ground and the initial site (coming in at the 4th power) divided by the energy gap between those two states.

Ge is the transition dipole moment between and the excited state squared, and between the excited state and a higher lying excited state squared.

Two energy terms goes between the ground and the excited state squared and the other between the ground and the higher excited state.

The final term should look a lot like β. The difference in dipole moment is squared so that it always positive, the energy term is cubed. It starts at the zero, increases to maximum and then return to zero.

The calculation gives γ using this model which is plotted as open blue circle. These look a lot like the red dots.

Each term contributes to the resulting curve for γ.

=== Third-order Nonlinear Optical Properties of Polarized Polyenes ===
[[Image:Betacarotene_NLO.png|thumb|400px|Effect on γ when various acceptors are added to beta-carotene]]
Beta carotene is the pigment found in margarine. By adding stronger and stronger acceptors it is polarized. λ max increases by a factor of 45.

The real part of the refractive index is related to how light is diffracted, the imaginary part is related to absorption of light.

The same is true about γ. Molecules will have both real and imaginary parts to γ. The real part refers to how the refractive index is changed as light of a given intensity goes through it. The imaginary part is related to two photon absorption.

In order to make useful devices like the Mach Zehnder interferometer you want the index of refraction to change but don’t want to lose light in the material. ELO materials can lose transparency due to absorption or scattering. They can also lose transparency at a high intensity due to the process of two photon absorption. Dipolar molecules tend to have large positive γ but also tend to have high two photon absorption cross sections.

Recently we have discovered that molecules with negative γ that have verge large real parts that lead to interesting optical effects; in certain spectral regions their imaginary part is almost zero so there is no light lost due to two photon absorption. These are good candidates for all optical switching applications because until now molecules with high χ(3) have had a high a loss due to two photon absorption.

see also [[All Optical Switching]]

[[category:third order NLO]]
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Two Photon Absorption

2010-06-05T01:04:38Z

Mrumi: /* Design of TPA chromophores */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA Chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T01:03:16Z

Mrumi: /* Examples of two-photon absorbing materials */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of Two-Photon Absorbing Materials ==

=== Two-Photon Absorption Spectrum of a Centrosymmetric Molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser Dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA Measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in Bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T01:02:19Z

Mrumi: /* Calculation of TPA cross sections */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA Cross Sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of two-photon absorbing materials ==

=== Two-photon absorption spectrum of a centrosymmetric molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T01:01:23Z

Mrumi: /* Measuring the two-photon absorption cross section */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the Two-Photon Absorption Cross Section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement Considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA cross sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA cross section in a donor-acceptor molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of two-photon absorbing materials ==

=== Two-photon absorption spectrum of a centrosymmetric molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T00:58:35Z

Mrumi: /* Summary */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the two-photon absorption cross section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA cross sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA cross section in a donor-acceptor molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of two-photon absorbing materials ==

=== Two-photon absorption spectrum of a centrosymmetric molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory can been used to express the TPA cross section as a function of molecular parameters, such as state energies and transition dipole moments. This information and knowledge on how these parameters change with molecular structure have been used to predict which molecules have large TPA cross sections. One such class of compounds are linear conjugated molecules with a symmetric arrangement of electron donating or withdrawing groups, as these molecule have a large change in the quadrupolar charge distribution from the ground to the excited state and large coupling between the relevant excited states.
 
For the measurement of TPA cross sections and the observation of TPA induced effects short pulses and large photon fluxes need to be used. Experiments need to be carried out with great care, to ensure that results are not affected by processes other than TPA, that sources of noise and error are minimized, and that the experimental conditions are consistent with the assumptions inherent in the models used to analyze the data.
 
TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T00:34:20Z

Mrumi: /* Microscopic Imaging */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the two-photon absorption cross section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA cross sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA cross section in a donor-acceptor molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of two-photon absorbing materials ==

=== Two-photon absorption spectrum of a centrosymmetric molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== 3D Microscopy Imaging ===

If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area. 
A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.

== Summary ==

Perturbation theory predicts which molecules will have large two photon cross sections. Molecules with symmetrical quadrupolar charge transfer lead to large TPA cross sections because they have strong coupling between different excited states. Measurements need to be done very carefully using very short pulses and done over many wavelengths. There are many applications for TPA including microfabrication, optical limiting, and 3D microscopic imaging.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T00:17:59Z

Mrumi: /* Why 3D Micro- and Nano-fabrication */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the two-photon absorption cross section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA cross sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA cross section in a donor-acceptor molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of two-photon absorbing materials ==

=== Two-photon absorption spectrum of a centrosymmetric molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for two-photon induced microfabrication, as in the microchannel structure shown here. Notice that the two "pools" at the top of the structure are connected by a series of very fine tubes below the surface.
[[Image:Tpa_microchannels.png|thumb|300px|Top: Schematic of microchannel structure. Bottom: Two-photon induced fluorescence images of the structure, fabricated in a positive tone resist, at various depths in the material (the central image is a cross section halfway along the channel lengths).]]
 
==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

=== Microscopic Imaging ===

If you attach a two photon dye to a particular organelle and then scan the cell in 3D with a precise laser beam to build a detailed microscopic 3D model of the structure with submicron resolution. All of this technology begins with the design of molecules that are able absorb light effectively which goes back to third order nonlinear optics, polarizability and hyperpolarizability.

== Summary ==

Perturbation theory predicts which molecules will have large two photon cross sections. Molecules with symmetrical quadrupolar charge transfer lead to large TPA cross sections because they have strong coupling between different excited states. Measurements need to be done very carefully using very short pulses and done over many wavelengths. There are many applications for TPA including microfabrication, optical limiting, and 3D microscopic imaging.

[[category:third order NLO]]

Two Photon Absorption

2010-06-05T00:05:38Z

Mrumi: /* Micro-electromechanical Systems (MEMS) Applications */

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Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the energies of the two photons absorbed. The probability that a molecules undergoes two-photon absorption depends on the square of the intensity of the incident light, if the two photons are from the same beam (in the case of two incident beams, the transition probability depends on the product of the beams intensities). This intensity dependence is at the origin of the interest in two-photon absorbing materials for use in microscopy and microfabrication applications.

== Two-Photon Excited Processes ==

=== Two-Photon Absorption ===
[[Image:Tpa_energy.png|thumb|300px|A molecule can be excited to state S2 by absorption of two photons of energy hνA.]]
Two-photon absorption in a material can be quantified by the two-photon absorption cross section, a quantity that is proportional to the imaginary component of the χ(3) tensor (or, on a molecular level, to the imaginary component of γ) and depend on the photon energy/energies. In the following we will discuss in detail the case of degenerate (or one-color) two-photon absorption, that is the case where the two photons have the same enrgy. The figure at right schematically illustrates the degenerate two-photon absorption process, in which two photons, each of energy hνA, are simultaneously absorbed and the molecule is excited directly from the ground (S0) state to an excited state (state S2 in the figure) without the formation of an intermediate eigenstate. Neither of the two photons have enough energy to excite the molecule to S2 by itself, but the sum of the energies of the two correspond to the energy of state S2. The dotted line represents a virtual state (as opposed to an actual eigenstate of the system) that can be thought of as created by the absorption of the first photon and having a very short lifetime (on the order of 10-15 sec if the photon energy is not in resonance with any electronic transition). If a second photon is absorbed within this lifetime, the two-photon absorption transition occurs and the molecule is excited to S2.

[[Image:Tpa_centro.png|thumb|400px|Rate equations and selection rules for two-photon and one-photon absorption.]]
Once the molecule is in state S2, it quickly relaxes by internal conversion to the lowest excited state (S1 in the figure). From this state, the system can return to the ground state S0 by emission of fluorescence or by nonradiative decay. Alternatively, an additional photon can be absorbed (bringing the molecule to a higher-lying excited state, Sn), or electron or energy transfer to another molecule can take place. Typically, once the molecule is in state S1, the same set of processes will take place whether the molecule was initially excited into S2 by two-photon absorption or directly into S1 by one-photon absorption. For most known materials the two-photon absorption cross section, δ, is small and it is necessary to use intense laser beams in order observe the effects of two-photon absorption.

The rate equation that describes the formation of the excited state of a molecule by one-photon absorption can we written as follows:

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math>

where
:<math>\sigma\,\!</math> is the one-photon absorption cross section (and is related to the transition dipole moment of the molecule between the initial and final states of the transition)
:<math>N_{GS}\,\!</math> is the number of molecules per unit volume in the ground state
:<math>N_{OP}\,\!</math> is the number of molecules per unit volume in the excited state due to one-photon absorption
:<math>F\,\!</math> is the photon flux (number of photons per unit area and time)
:<math>t\,\!</math> is the time

(this is analogous to the equation to describe the rate of a reaction between two reactants, A and B, to give a product C: A + B = C, and that is first-order with respect to each of the reactants).

In a similar manner, the rate equation for two-photon absorption can be written as follows, keeping in mind that two photons are needed to produce one excited molecule (in the reaction analogy, the reaction is now second-order in the photon flux, but still first-order in the concentration of molecules):

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

where
:<math>\delta\,\!</math> is the two-photon absorption cross section
:<math>N_{TP}\,\!</math> is the number of molecules per unit volume in the excited state due to two-photon absorption

This shows that the probability of two-photon absorption depends on the square of the photon flux (or, equivalently, to the square of the light intensity).
The selection rules for the two-photon absorption process are different from those for one-photon absorption (in analogy to the situation for infrared and Raman spectroscopies). In particular, a two-photon absorption transition is allowed only between two states that have the same parity. Thus, in molecules with an inversion center, transitions are two-photon allowed from a gerade (''g'') state to another gerade state or from an underage (''u'') to another ungerade state, but not between ''g'' and ''u'' states. In the case of one-photon absorption, instead, transitions are allowed between an initial state and final state with opposite parity (from ''g'' to ''u'', or vice versa). This implies that, in centrosymmetric molecules, if the transition from the ground state to a given excited state is one-photon allowed it is not two-photon allowed and, vice versa, if it is two-photon allowed it is not one-photon allowed. For molecules without inversion center, certain transitions can be both one-photon and two-photon allowed.
In the scheme above, the blue arrow represents the excitation of a centrosymmetric molecule to the lowest excited state, S1, by the absorption of one photon (with one-photon absorption cross section σ); this transition is not two-photon allowed. The molecule can be excited by two-photon absorption (red arrows) into a higher-lying state, S2 (with two-photon absorption cross section δ).

== Advantages of TPA ==
Two photon absorption is considered to be a nuisance in the case of all-optical switching using χ(3) materials because it causes attenuation of the light beam and damage to the material can result from severe heating of samples. However the two-photon absorption process can be exploited in another contexts, apart from being of interest in the study of fundamental spectroscopic properties of materials.

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|Two laser beams are focused in a sample molecule in solution inside a cuvette; the top beam is tuned at a wavelength at which two-photon absorption can take place, the bottom one at a different wavelength that can excite the molecules by one-photon absorption. The blue light visible in the photograph is the fluorescence emitted by the molecules after excitation.]]
If a light beam is focused into sample at a wavelength suitable for one-photon absorption, molecules are excited throughout the beam path in the sample, as evidenced by the fluorescence emission in the bottom part of the cuvette at right. If the beam is tuned at a wavelength at which the material exhibits two-photon absorption, only the molecules located very close to the focus of the laser beam are excited (top beam in the photograph). This is related to the fact that the excitation rate depends on the square of the light intensity, as discussed above, and that in a focussed beam the light intensity is maximum at the focal plane and decreases approximately with the square of the distance from the focal plane, ''z'', along the propagation direction (because the area of the beam increases moving away from the focus). Overall, the excitation rate for TPA and the intensity of the two-photon induced fluorescence decrease as the forth power of the distance from the focal plane. Consequently, in the solution in the figure the two-photon induced fluorescence is strongest at the beam focus and its intensity drops off very quickly on either side of the focal plane, resulting in what looks like emission from a "single point" (or small volume) in the solution. At the diffraction limit, the TPA excitation is confined to a volume on the order of the cube of the wavelength of the excitation light. Most of the applications of TPA are based on this ability to confine and control the excitation volume in a material with good resolution in three dimensions, as any excited state process that can take place in the material after two-photon excitation (such as fluorescence emission or energy transfer) will be confined to the same small volume.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|One-photon absorption (bottom): a light beam from the right is quickly absorbed by a concentrated solution of a fluorescent material. Two-photon absorption (top) the beam is able to penetrate the solution without being attenuated until the focus of the beam; only near the focus the light intensity is high enough to activate two-photon induced fluorescence.]]
In the figure at right the same two beams as in the case above are focused in a solution of a fluorescent compound, but now the solution is much more concentrated. It can be seen that the for the bottom beam (one-photon absorption case) fluorescence emission can be seen only close to the right wall of the cuvette, as the beam is strongly absorbed and attenuated by the solution (the beam is propagating from right to left). Thus the penetration depth of the beam in the solution is small. In the case of the top beam, there is no one-photon absorption and two-photon absorption only occurs near the focus of the beam (in this case the center of the cuvette), so that there is no beam attenuation before the beam reaches the focus and the laser beam can penetrate farther into a sample with respect to the one-photon absorption situation. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA process ideal for fluorescence imaging of thick samples (even in vivo) or in medical applications in which, for example, a drug can be activated by TPA at a very precise location without affecting the tissue above and below the focal plane of the excitation beam.
 

== Measuring the two-photon absorption cross section ==
[[Image:Tpa_measurement.png|thumb|400px|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focussed or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measure by a detector (for example a photomultiplier tube, PMT). As seen above, the number number of molecules excited by two-photon absorption per unit time and volume, ''N''TP, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

:<math>n_{fl} = \eta N_{TP}\,\!</math>,

where η is the fluorescence quantum yield of the material.

Thus, the intensity of the two-photon induced fluorescence is proportional to the TPA cross section δ. This technique can be used for absolute or relative measurements of TPA cross sections. In the absolute case, all the relevant parameters (such as the instantaneous photon flux and the detection efficiency of the optical setup) need to be measured independently. In relative measurements, an unknown compound is analyzed under the same conditions as a material of known TPA cross section.
The figure at right shows an example of optical setup that can be used for TPA cross section measurements. A tunable laser is needed to measure the TPA spectrum, as the TPA cross section depends on the excitation wavelength. In this example, the laser beam is split in two by a beam splitter and one of arm is used as reference for the intensity of the beam, to correct for fluctuations in intensity from pulse to pulse during the course of the measurement.

'''Measurement considerations:'''
* A pulsed laser is needed (pulse durations in the ns, ps, fs range can be used).

* The laser need to be tunable to obtain spectral information on the material.

* The two-photon fluorescence method is applicable to fluorescent materials (the z-scan technique can be used for fluorescent and non-fluorescent materials).

* The detection of the fluorescence emission can be done in various configurations (90º collection, backward scattering, forward scattering, ...); scattering of the excitation beam by the sample need to be removed from the fluorescence signal before detection (for example using filters or monochromators).

* The dependence of the two-photon induced fluorescence signal on the square of the laser beam intensity needs to be tested for the actual experimental conditions used, to exclude contributions from other effects.

* Possible reasons for deviation from the dependence on the square of the laser intensity: linear absorption, stimulated emission, ground state depletion, ...

* Importance of spatial and temporal profile of the excitation beam.

See equipment video on [[Two-Photon Spectroscopy]]

== Calculation of TPA cross sections ==

=== Perturbative Expression for γ, as Relevent to Two-Photon Absorption ===

The perturbative expression for the hyperpolarizability γ for the frequency component relevant to for two-photon absorption can be written as follows (under the assumption that the system can be described by the so-called "three level model", in which only the ground state, ''g'', the lowest excited state, ''e'', and a higher-lying two-photon allowed state, ''e''', are considered):

:<math>\gamma( -\omega; \omega, -\omega, \omega) \propto \frac{ M^2_{ge} \Delta \mu^2_{ge}} {(E_{ge} - \hbar \omega - i \Gamma_{ge})^2(E_{ge}-2\hbar \omega -i \Gamma_{ge})} + \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega- i \Gamma_{ge})^2 (E_{ge^{\prime}} - 2 \hbar \omega - i\Gamma_{ge^{\prime}})}\,\!</math>

where:

:<math>M_{ge} \,\!</math> is the transition dipole moment between states ''g'' and ''e''
:<math>M_{ge^{\prime}} \,\!</math> is the transition dipole moment between states ''g'' and ''e'''
:<math> \Delta \mu_{ge} \,\!</math> is the difference between the dipole moment of states ''g'' and ''e''
:<math>E_{ge} \,\!</math> and <math>E_{ge^{\prime}} \,\!</math> are the energies for the transitions between the subscripted states
:<math> \omega \,\!</math> is the angular frequency of the excitation beam and
:<math> \Gamma \,\!</math> are damping terms.

For a centrosymmetric molecule <math>\Delta \mu_{ge}\,\!</math> goes to zero (the dipole moment is zero in both ''g'' and ''e'' states) so that the first term in the equation for γ goes to zero. For non centrosymmetric molecules both terms contribute to the hyperpolarizability.
From the equation above it can be seen that two-photon resonances can occur when the photon energy is such that: <math> 2 \hbar \omega = E_{ge} \,\!</math> or <math> 2 \hbar \omega = E_{ge^{\prime}} \,\!</math>

The TPA cross section (δ ) is proportional to the imaginary component of γ:

:<math>\delta(\omega) = \frac {4\pi^2 \hbar \omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

The above equation is valid in the cgs system of units; <math> n\,\!</math> is the refractive index of the material, <math> L\,\!</math> is the local field factor (which depends on the refractive index), and <math> c\,\!</math> is the speed of light.

=== Calculations of the TPA cross section in a donor-acceptor molecule===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of δ for the transition from the ground to the first excited state (S0 to S1, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
The TPA cross section for the transition to the second excited state (S0 to S2, right plot) exhibits a more complicated behavior as a function of BOA and with multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S1 is very close to half of the energy for the transition to S2, the detuning term becomes small and δ increase; this situation is referred to a "double resonance".

See T. Kogej et.al. Chem. Phys. Lett. 1998, vol. 298, p. 1 <ref>T. Kogej et.al. Chem. Phys. Lett. 1998, 298, 1.</ref>

== Examples of two-photon absorbing materials ==

=== Two-photon absorption spectrum of a centrosymmetric molecule ===
[[Image:Tpa_spectra.png|thumb|400px|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S1 (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S2 (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S1 and then fluorescence emission from S1. There is no fluorescence emission from S2 because, in most cases, the relaxation from S2 to S1 is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S2 to S0 is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the grounds and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S2 (or Sn). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S1). Most molecules behave according to Kasha's rule, but a few exceptions are know, such as azulene.

The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
In the graph note that the photon energy of the TPA band is lower (longer wavelength; the horizontal axis represent the wavelength of the excitation beam) than the one-photon absorption band (blue line) and of the fluorescence (green line). The TPA peak is around 720 nm, for this molecule. As two photons are involved in the transition, a TPA peak at 720 nm corresponds to a "transition wavelength" of 720 nm / 2 = 360 nm. The peak for linear absorption to S1 is 430 nm. This shows that the state reached by two-photon absorption is higher in energy than the S1 state and that there is very little one-photon absorption in this range (the small absorption seen at 360 nm is due to vibronic sub-levels of S1), consistent with the selection rules described earlier for centrosymmetric molecules.

see Rumi et al., J. Am. Chem. Soc. 2000, vol. 122, p. 9500 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

=== Laser dyes ===
[[Image:Tpa_laserdyes.png|thumb|300px|]]

Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goppert-mayer (GM) units: 1 GM = 1 × 10-50 cm4 s molecules−1 photon−1. The unit are names in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of TPA measurements: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems in measurement in the NLO field in general, as this are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S1 to a higher state Sn, if the population of S1 becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross sections that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depend on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

 

=== Initial Observations on a Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|Bis-donor substituted stilbene.]]

A stilbene derivative (BDAS) with two π-donor groups (dibutyl amino group) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*Strong blue fluorescence was observed when pumped with orange laser light
*Fluorescence intensity for pumping with orange light depends on I2
*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10-50 cm4 s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10-50 cm4 s/photon.
Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission at low intensity
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+ (this compound is very easy to oxidize in the ground state and is a powerful reducing agent in the excited state)

=== Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules ===
[[Image:Tpn_bisdonorstilbene_symm.png|thumb|300px|]]
Theoretical calculations can help to explain the properties of the BDAS molecule.

Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) when the molecule is excited from S0 (''g'') to S1 (''e'') and to S2 (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S1 and S2 (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is instead much smaller in the case of stilbene.

These results suggest that a large change in quadrupole moment between S0 and S2 can lead to large values of δ.

 
'''Effect of Bis-Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):

:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>

The term <math>M_{ee^{\prime}}\,\!</math> is squared in the equation and it can be seen that an increase in the value of this parameter from stilbene with respect to BDAS is the reason for the very high TPA cross section for BDAS.

'''Transition Dipole Moments'''
[[Image:Tpa_transdip_densities.png|thumb|300px| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seems that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Left: Output energy vs input energy for two solutions of BDAS excited at 600 nm with nanosecond pulses. Right: Spectra obtained from two-photon fluoresence with ps pulses (ps-TPF) and nonlinear transmission with nanosecond pulses (ns-NLT). ]]

It was mentioned above that nonlinear transmission was observed in BDAS. This material behaves as a good optical limiter for nanosecond pulses (see plot on the left side of the figure, for two different concentrations). This is due to the fact that the TPA cross section of this molecule is large, as seen above, and that this molecule also has a large excited state absorption cross section; when the molecule is excited with nanosecond pulses at a wavelength between 550 and 650 nm, a cascade of TPA and ESA processes takes place, resulting in large attenuation of the incident beam (optical limiting). The plot on the right side of the figure displays the TPA cross section of the compound obtained from a two-photon induced fluorescence measurement (ps-TPF, shown in red) and the nonlinear transmission spectrum (ns-NLT, shown in blue), that is the values of "effective" TPA cross section obtained from a NLT experiment with ns pulses: the two curves are similar in shape, but the scales of cross sections are different. The ns-NLT scale has units of 10-46 cm4 s/photon and the ps-TPF spectrum is on the scale of 10-48 cm4 s/photon, a factor of 100 difference. This is another example of the fact that NLT measurement performed with long pulse durations yield only the "effective" TPA cross section, which involves two-photon absorption followed by excited state absorption, as opposed to the "intrinsic" TPA cross section, which can be obtained from two-photon induced fluorescence measurements or other techniques that allow to exclude the effect of ESA (such as z-scan measurements with fs pulses at moderate peak intensities of the laser beam).

 

== Design of TPA chromophores ==

=== Three-Level Model and Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:

:<math>\delta _{max} = f(\omega, n) \frac {M^2_{ge} M^2 _{ee^{\prime}}} {(E_{ge} - \hbar\omega)^2 \Gamma_{ge^\prime}}\,\!</math>

where:

:<math>\delta_{max}\,\!</math> is the cross sections at the peak of the TPA band (i.e. for <math>\hbar \omega\,\!</math> = <math> (1/2) E_{ge^{\prime}}\,\!</math>).

:<math>M_{\alpha\beta}\,\!</math> is the transition dipole moment for two levels α and β

:<math>E_{\alpha\beta}\,\!</math> is the energy difference between levels α and β

:<math>\hbar \omega\,\!</math> is the photon energy.

:<math>\Gamma_{ge^{\prime}}\,\!</math> is the damping term (typically on the order of 0.1 eV)

In summary:

:<math>\delta_{max} \propto M^2_{ge} M^2_{ee^\prime} \Delta E^{-2}\,\!</math>

where <math>\Delta E\,\!</math> is the detuning energy: <math>\Delta E\ = E_{ge} - \hbar \omega \,\!</math>

[[Image:Tpa_chromophores.png|thumb|300px|Molecular structure and value of δ (in GM units) for series of compounds with different substitution patters.]]

This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
*Increase the strength of the donors, so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule, to further increase the amount of charge transfer.
*Flip the relative position of donors and acceptors in the molecule, using donors in the center and acceptors on the ends.

The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trendd in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.

 

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|TPA spectra for molecules with different conjugation lengths.]]
The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
 

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

The table shows that by making these molecules quadrupolar, that is attaching electron-rich and/or electron-poor functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
 

== Applications for TPA ==

=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
[[Image:Tpa_crosslinked.png|thumb|300px|TPA can be used to stimulate cross linking in a polymer.]]
Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:

Wu et al., Proc. SPIE, 1992, vol. 1674, p. 776 <ref>E. S. Wu, J. H. Strickler, W. R. Harrell & W. W. Webb, SPIE Proc. 1674, 776 (1992)</ref>

Maruo et al., Opt. Lett., 1997, vol. 22, p. 132 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

Cumpston et al., Nature 1999, vol. 398, p. 51 [http://www.nature.com/nature/journal/v398/n6722/full/398051a0.html]

Kawata et al., Nature 2001, vol. 412, p. 697 [http://www.nature.com/nature/journal/v412/n6848/full/412697a0.html "Micro bull"]

[http://spie.org/x19493.xml?ArticleID=x19493 "Thinking man"]

=== Two-Photon Initiators with Enhanced Sensitivity ===
[[Image:Tpa initiators.png|thumb|300px|New dyes increase the effective range of power where 3D "writing" can occur.]]

Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
 

=== Fluorescent and Refractive Bit Optical Data Storage ===
[[Image:Tpa_optical_storage.png|thumb|300px|Fluorescent and Refractive Bit Optical Data Storage]]

This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bit in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm3 are achievable).
 

=== Photochemistry Generated via an Intramolecular Electron Transfer ===
[[Image:Tpa_photochemistry.png|thumb|300px|]]
Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.
 

=== Why 3D Micro- and Nano-fabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|SEM images of microstructures obtained by two-photon induced polymerization.]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to fabricate free-form structures in three dimensions with micron and sub-micron feature sizes
*Increasing need to pattern a variety of materials
*Need to couple nano-scale object with micro-scale objects
*Areas impacted by 3D micro- and nano-fabrication include MEMs, microfluidics, photonics and tissue engineering.

Two-photon induced polymerization (or other two-photon induced reaction) provides a means to fabricate three dimensional structures with a high degree of complexity and good fidelity, structures that can be difficult or time-consuming to make by other fabrication methods. A few examples of structures obtained by this fabrication technique in a polymer resin are shown in the figure: the photo on the left is a "chain link fence" that is twice the thickness of a human hair and has free-moving but interconnected parts. The second figure shows a series of channels; the third is a photonic crystal structure. The image on the right shows a series of criss-cross lines that could potentially be used as a scaffold for tissue engineering, as cells grow better on certain topologies.
 

==== Sub-Diffraction Limited Resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are 170 nm wide.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can produce features close to this size and with 3D resolution. In this example, the width of the lines is about 170 nm and they were obtained by two-photon induced polymerization with 730 nm laser light.

 

==== Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Top: In a negative tone resist, exposed areas (yellow) remain after development. Bottom: example of a negative tone resist containing a two-photon initiator.]]
This is an example of a TPA dye that creates a radical initiator after excitation. The radicals generated can then initiate polymerization in a resist (for example an acrylate). The polymer is less soluble than the starting resin and the nonpolymerized material can be removed by immersion in the solvent. Only the exposed areas that are attached to the substrate (or linked to each other) remain after development in the solvent. This is known as a negative tone resist.
 

==== Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|Top: In a positive tone resist the exposed areas (yellow) are removed (grey) during development. Bottom: example of a positive tone resist containing a two-photon photoacid generator.]]

A positive tone resist becomes more soluble in the areas that are exposed to light. This allows you to carve out parts of the material by exposure to light and then immersion in a suitable solvent.

Tetrahydropyran can be protonated on the oxygen creating a carbocation thus converting the ester to a carboxylic acid. Esters are not soluble in basic water but acids are. So you can dissolve away selectively the converted material by dipping the sample in an alkaline solution; this is an example of a positive tone resist. A proton is needed to activate the conversion.
A photoacid generator (PAG) is a compound that after absorption of light releases an acid. For example, in the structure shown, the excited compound can transfer an electron to the carbon-sulfur sigma orbital, causing the bond to break homolytically and yielding a methyl radical, which will attack the benzene and create a proton. The proton then starts the ester-to-acid reaction in the positive tone resist. The PAG shown here has a large TPA cross section and it has been used for microfabrication by two-photon induced polymerization.

==== Micro-electromechanical Systems (MEMS) Applications ====
Micro-electro mechanical systems are used for sensors, actuators, micromachines and optical switches. Inkjet heads and disc drive heads are MEMS. These can be fabricated with negative and positive tone materials.

[http://www.memx.com/image_gallery.htm MEMS image gallery]

[[Image:Tpa_microchannels.png|thumb|300px|This example is drawn with a positive tone resist. The two pools are connected by a series of very fine tubes.]]

=== Microscopic Imaging ===

If you attach a two photon dye to a particular organelle and then scan the cell in 3D with a precise laser beam to build a detailed microscopic 3D model of the structure with submicron resolution. All of this technology begins with the design of molecules that are able absorb light effectively which goes back to third order nonlinear optics, polarizability and hyperpolarizability.

== Summary ==

Perturbation theory predicts which molecules will have large two photon cross sections. Molecules with symmetrical quadrupolar charge transfer lead to large TPA cross sections because they have strong coupling between different excited states. Measurements need to be done very carefully using very short pulses and done over many wavelengths. There are many applications for TPA including microfabrication, optical limiting, and 3D microscopic imaging.

[[category:third order NLO]]