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Second-order NLO Materials

2009-12-28T23:46:40Z

128.95.39.42: /* Linear and Non Linear Electro-optic Effect */

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Amorphous polymers are desirable materials for several reasons however in order to take full advantage of the EO potential of the polymer embedded chomophores these molecules must be well organized in the bulk material. Poling is the process of bringing chromophores into a desired alignment in a material.

=== Linear and Non Linear Electro-optic Effect ===
'''Linear'''
In linear optics an applied field results in a linear polarization response where the refractive index n and therefore the susceptibility Χ(1) is not dependent on the the electric field. When the field is low higher order terms are insignificant and can be ignored.

:<math>P_i= \Chi^{(1)}_{ij}E_j\,\!</math>

'''Non-linear'''
When a nonlinear material is used with a powerful laser creating a high electric field the susceptibility is dependent on E due to higher order nonlinear terms.

The polarization density can be written as a Taylor series

:<math>P_{ind} = \chi^{(1)} E_1 + 1/2 \chi^{(2)} E_1 E_2 + 1/6 \chi^{(3)} E_1 E_2 E_3 + ... \,\!</math>

Index of refraction is related to susceptibility (and therefore ultimately back to polarizability) by:

:<math>n^2 = 1 + 4\pi \chi\,\!</math>

'''Electro-optic effect'''
Consider the case where one field is E1cos[ωt] and second field is a static DC electric field (or very low frequency DC field) of magnitude E2. The second-order term of the induced polarization simply becomes:

:<math>\chi^{(2)} E_1 E_2 cos[\omega t]\,\!</math>

and therefore:

:<math>P = ( \chi^{(1)} + 1/2\chi^{(2)} E_2 ) E_1 [cos \omega t]\,\!</math>

Notice that the the second term χ(2) is field dependent.

Thus, when an electric field is applied to such a poled polymer, the index of refraction of the material will change. This sensitivity of the material to having its index of refraction changed by application of an external electric field is characterized by its so-called electro-optic coefficient, coefficient (r33).

For many years it had been suggested that electro-optic polymers could exhibit large r33 greatly surpassing that of a technologically important crystal, lithium niobate (LiNb03). It is transparent, it is inexpensive, it can be grown in large sheets, and it has an r33 of 30.5 pm/V in the "telecommunications" wavelengths of 1.3 μm. This is a reasonable r33 that allows you to build devices. When we build organics we have synthesize new compounds, worry about stability, worry about orientation, and how to keep them poled. But there are a significant advantage to organics.

While there are both electronic and vibrational components to the non-linearity in lithium niobate, a large component is vibrational. This puts an upper limit on the frequency with which that LiNbO3 can be modulated. If a driving field approaches the vibrational frequency then the molecules can not respond quickly to the applied field. The response trails off and there is only the electronic contribution for the non-linearity.

Organic materials are completely dominated by the electronic contributions and therefore can be modulated at very high frequencies.

It is also possible to get organics with an r33 that is much higher (for example 55 pm/V) than LiNbO3. This means that the Vπ (the voltage required for a π phase shift) can be less to get the same change in optical property.

Finally it possible to integrate electronic with optical parts right on a chip. Polymer technology is going to make it much easier to combine these materials on a chip.

In the field of silicon photonics there are waveguides built with ebeam lithography on which the polymer can be spin coated and fill the channels. Organics are amorphous whereas LiNbO3 is a single crystal material which makes it very challenging to apply to chip-scale device.

see [[Optical_Networks#Technological_Applications_Of_The_Pockels_Effect | Applications of Pockels Effect ]]

=== NLO materials in EO-Modulators ===
[[Image:Machzehnder.png|thumb|400px|A Mach Zehnder Interferometer]]
An electrode is located on either side of the light path which passes through an electro optical material. An opposite field can be applied to each path slowing the light on one side and speeding the light on the other. There are many factors that contribute to the figure of merit Vpi (the voltage that is required to shift the wave one half of a wavelength). A longer optical wavelength is better. A long interaction length gives more time for the material to respond to the field, but this means the optical material must be very clear so that optical intensity is not lost. Light intensity can be lost by absorption, diffraction and scattering. The 1.31 microns and 1.55 microns are popular telecommunication wavelengths because they are easily propagated through optical fiber. This is the typical wavelength that the modulators need to work with. Many NLO materials absorb in the 650- 750 nm range. 640 nm is about 2 eV (1ev = 1240/ wavelength nm); 1.55 microns is about .8 eV. A lot of the molecules are absorbing at about 700nm (1.7 eV). The absorption tail of a molecule can go out a long ways. At some point you have to worry about C-H bonds because of overtones due the C-H stretching frequency that occur at 3000 cm-1 (3333 nm). This means there could be some absorption at 1600 nm (1/2 of 3333 nm) due to C-H overtones. This is a problem.

Another problem is that the high concentrations of chromophores in these systems results in chromophore-chromophore interactions, inhomogeneities of the chromophore concentrations, and chromophore polymer interactions. Each of these factors leads to a variation of the refractive index which can lead to the scattering of light at some level. If the interaction length of the light in the Mach Zehnder devices needs to be long then it is important to keep the scattering in the organic component to a minimum.

It is important to have a high index of refraction and high electro-optic coefficient. Index of refraction relates to dielectric constant which relates to susceptibility, which relates to polarizability (α), which relates to the transition dipole moment within the context of a two level model. Change of dipole of moment is the state dipole moment for the excited state minus the state dipole moment for the ground state. Transition dipole moment is an integral of the complex conjugate of the excited state wave function, the transition dipole moment operator (which scales like R) and the ground state wave function Dτ. The transition dipole moment relates to the oscillator strength (the area under the absorption curve) which is related to the extinction coefficient. Organic dyes that are used in second order NLO systems are materials that have a high refractive index and strong low energy absorptions.

=== Poling of Chromophores in Polymers ===
[[Image:NLOmaterial.png|thumb|300px|A NLO material has a bulk material with a suspension of chromophores]]
The general approach that has been employed to create a second-order nonlinear optical material has been to assemble
a bulk material using an inert “Host” material loaded with active “Chromophores”, molecules that have a high first hyperpolarizability (beta). Then get those molecules oriented in the same direction. (For now we will examine orientation for high EO coefficient and will ignore the orientation required for second harmonic generation which require a different orientation)

Orientation can be achieved with crystalline materials. The Marder group was able to develop crystals of DAST. (They were working with a Japanese group at the same time who succeeded in making a crystal of the hydrate of the material due to their high local humidity. Marder’s group in California was able to make the DAST crystal due to the low local humidity. One crystal was centro symmetric and one was non-centrosymmetric.) Growing crystals is very difficult. Another group spent 10 years developing the method to grow one crystal. (Rainbow Photonics)

The most widely used method for aligning molecules is the poled polymer method. Recall that dipole moment is a vector, and first hyperpolarizability beta is a tensor. Assume that the molecule has one axis (the long axis) along which the hyperpolarizability beta dominates. If the dipole moment is aligned along the long axis then an applied electric field will cause a torque to reorient the molecules. (Boltzman will be working against this attempting to randomize the orientation) There will be a distribution of orientations but if you sum over all the vectors there will be a net orientation. This in turn aligns the hyperpolarizability tensor. In the limiting case in which the hyperpolarizability is perpendicular to the dipole moment there will be an alignment of the hyperpolarizability tensor in a plane but within that plane the tensors will be oriented at all angles. This results in a zero overall second order optical activity. Therefore we need molecules with large dipole moments that are oriented along the same axis as the hyperpolarizability tensor.

[[Image:Poling.png|thumb|200px|Sequence of a poling a chromophore in a glassy polymer]]
*Assume you are start with molecules that have high nonlinearities but they are arranged in a random orientation. An NLO chromophore with a dipole moment (such as paranitroaniline) and a glassy polymer are dissolved in an organic solvent and spin-coated onto a substrate. The film is then heated above or around its glass transition temperature (Tg). The glass transition temperature at which there is free motion of the main chain of the polymer.

*At this temperature, the molecules can rotate in the rubbery matrix. An intense DC electric field as high as 106 V/cm is applied and this field creates a force on the dipole moments of the chromophores, aligning them.

*Thus, the polymers and chromophores must be nonconducting to support these large fields. With the electric field still applied, the film is cooled below the Tg of the polymer, restricting the motions of the chromophores.

In this manner you can produce a material that has a second order nonlinear optical effect using molecules that have a second order nonlinear optical effect.

The resulting noncentrosymmetric structure contains the molecular units oriented on the average normal to the film (C∞v symmetry). In this state, the electric field of an optical beam propagating through the film can be maximally modified when the field is parallel to the orientated molecular units.
 
[[Image:poling_electrode.png|thumb|400px|A DC electric field is a applied to a polymer as it passes through its glass stage.]]
The goal of "poling" is to get all the chromophores to orient the same direction in a macroscopic sample. There are several ways to do this. We can place the chomophore in a host matrix like a commercial polymer, heat it up to the glass transition temperature, and apply an electric field. The dielectric moment of the chromophore interacting with the field will cause some ordering.

The order parameter will be the dipole moment interaction divided by the 5 times the thermal energy.

[[Image:poling_centric.JPG|thumb|400px|Unfortunately chromophores have strong intermolecular electrostatic interactions such as dipole-dipole interactions. So this going to influence what we see.]]
 

<div id="Flash">Poling Animation</div>

<swf width="600" height="500" >http://depts.washington.edu/cmditr/media/chromophoreload.swf</swf>

In this Flash Animation various competing factors in NLO materials design for poling are illustrated. You can control the temperature, the poling field strength with the slide bars. Also the dipole moment of the chromophore and the chromophore loading can be input. The <cos3θ> term (the order parameter) is calculated based on the average angle of the molecules with respect to 0 ° (vertical, pointing towards the top). (This is an instructional interaction, not a full scale scientific simulation). Try these virtual experiments?
*What temperature is Tg for this polymer?
*What effect does having a higher molecular dipole have on the ease of poling?
*What happens to the order when you increase the loading?
*What combination of loading and dipole leads to the highest r(33)?

=== Second-order susceptiblity ===

The magnitude of the second-order NLO effect in a poled polymer film is related to several factors.

Using a non-interacting oriented molecular gas model, the follow equation has been derived to describe the second-order susceptibility in the direction of the poling field (χ (2)333). It is calculated:

:<math>\Chi^{(2)}_{{333}} \propto N F \beta_{333} <cos^3\theta>\,\!</math>

Where:

:<math>N\,\!</math> is the number density of the chromophores in the material

:<math>F\,\!</math> is a constant

:<math>\beta _{333}\,\!</math> is the component of beta that is aligned along the axis of the dipole moment (eg not 90degrees)

and the order parameter is determined by :

:<math><cos^3 \theta> = \mu_z E_p /cKT\,\!</math>

:<math>\mu_z\,\!</math> is the dipole moment of the molecule (the larger the dipole, the more it will respond to the field)

:<math>E_p\,\!</math> is the strength of the applied electric field

:<math>k\,\!</math> is the Boltzman constant, T is temperature in Kelvin

:<math>c\,\!</math> is a system dependent factor equal to 5 for isotropic systems, and 1 for ideal liquid crystalline systems (Ising systems)

In liquid crystals the orientation of one molecule can affect neighboring molecules (cooperativity)
<cos3θ> is the induced polar order which is an average angular orientation where theta is the angle between the poling axis and the molecular dipole moment of the chromophores. Molecules with large dipole moments will tend to repel and that interferes with poling especially when they get too close (high concentration).

This equation is valid a low field, :<math>\mu_zE_p < kT\,\!</math>.

When there is centric ordering

:<math><cos^2 \theta > = (\mu E/5kT) [1- L^2(W/kT)]\,\!</math>

where:
:<math>W\,\!</math> is the intermolecular electrostatic potential
:<math>L\,\!</math> is the Langevin function
:<math>E\,\!</math> is the poling field felt by the chromophore

<div id="Flash2">Interactions</div>

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

This simulation shows what happens when molecules with dipoles get close to each other. Click on either of the two chromophores on the right and drag them next to the one in the middle. The "plump" chromophore has been designed to have bulky side groups that prevent the central chain from getting too close to adjacent dipoles.

====Key Molecular Properties for Poled Polymers====
*large optical nonlinearities (large χ(2) - large r33)leading to low drive voltages.
*thermal stability - high temperatures in fabrication (want to pole 150 - 200°C above Tg) and high operating temperatures –very few organic dyes are are stable above 250°C- molecules can be engineered to remove weak points.
*photochemical stability - materials undergo intensive irradiation during applications
*stable against oxidation- and reduction processes due to high electrical fields in fabrication and exposure to air (oxygen)
*minimized self aggregation - aggregation causes scattering loss and aggregation makes molecules harder to orient - mitigated by covalent linkage to polymer-backbone
*long-term orientational stability of devices- for devices that are deployed in telecommunication (underground) they should last at least 75 years.

==== Schematic of Poled Polymers Strategies ====
[[Image:Poling_linkage.png|thumb|300px|Various strategies for linking chromophores to polymers]]
The poled orientation is thermodynamically unstable and therefore the orientation quickly decays in low Tg polymers such as poly(methylmethacrylate), resulting in a greatly reduced nonlinearity. The diagram show several strategies for linking chromophores to polymers so they are more constrained.

The motion of chromophores in polymer matrices might be further restricted if the chromophore is covalently attached at one or more sites to the polymer. One end of the chromophore is pinned down it may more thermally stable. This also reduces the tendency for the polymer and chromophore to separate into phases because they are essentially glued together.

Accordingly, several groups have explored covalently attached chromophores to polyimides either in the main chain or as a side chain. This approach can lead to polymers with exceptional thermal stability. Lately, Jen's group has reported a 2-step synthesis for NLO side-chain aromatic polyimides. The advantages of this procedure include the ease in controlling the loading level of the chromophore and adjusting the polymer backbone rigidity.

== Theoretical Methods for Improving NLO activity ==
An important aspect of research and development is connecting fundamental science to applied technology. Theoretical methods allow us to predict the NLO activity of materials and to rationally design materials that will have desired properties.

Quantum mechanics has helped improve non-linear optical activity (NLO). The molecular first hyperpolarizability, that is, the charge re-distribution you get from the application of an electric field, has gone up exponentially over the recent years. This improvement is even greater than Moore’s law. In addition statistical mechanics has help improve the macroscopic EO activity.

<math>r_{33} \propto \beta(\epsilon,\omega)N<cos^3\theta>\,\!</math>

where
:<math>r_{33}\,\!</math> is the electro-optic coefficient
:<math>\beta(\epsilon,\omega)\,\!</math> is the molecular first hyperpolarizability
:<math>N\,\!</math> is the number density of chromophores in the polymer
:<math><cos^3\theta>\,\!</math> is the order parameter

The [[electro-optic cofficient]] r33 is determined by the molecular property first hyperpolarizability <math>\beta(\epsilon,\omega)\,\!</math> and a number density. So for the maximum effect all the molecules must be lined up in the same direction. You can use quantum mechanics to optimize the first part (hyperpolarizability), and statistical mechanics to optimize the second part (number density), and have a feedback loop between the two approaches.

 

=== Density Functional Theory ===
[[Image:Td-dft.png|thumb|300px|TD-DFT modeling (red) accurately predicts actual observations of CHCl3 (blue)]]

Density Functional Theory (DFT) resulted in a Nobel Prize in physics in 2000. Basically you are replacing a wave function computation with a density. Density is <math>\Psi \psi*\,\!</math> the product of two wave functions. That has some computational speed advantages. But the major advance is Real time Time-Dependent Density Functional Theory (RT-DFT). This gives you a mechanism for incorporating time dependent phenomenon like applied radiation fields, optical fields, and internal fluctuation of charge fields.

For example, when you place a chromophore in a host material the host is typically a dielectric and it will exert an electric field on the chromophore. You want to be able to take both of these into account. The end result is that you are getting quantitative simulation of linear and non-linear optical properties.
[[Image:theor_first_polar.JPG|thumb|400px|Simulation for Molecular first hyperpolarizability vs dielectric constant for selected materials.
]]

 

=== Statistical mechanical calculations ===
[[Image:EOC_loading.png|thumb|300px|Number density affect electro-optic activity]]
As chromophore number density increases they begin to interact because of their polarity. At extreme close range this forces molecules to flip and this reduces the EO activity of the material. A balance must be struck between number density and beta of molecules to enhance macroscopic EO activity.

see Robinson 2000 <ref>Robinson, B.H., and Dalton, L.R., J. Phys. Chem. A, 2000, 104, 4785-4795</ref>.
 
[[Image:Montecarlo.png|thumb|300px|Monte Carlo Simulation]]
Monte Carlo simulations can be used to try different combinations of variables of number density, field strength, matrix and dipole moment. This theoretical model reveals that chromophore shape has a big effect on the loading parameter (n<cos3θ>)
 
[[Image:Loading shape.png|thumb|300px|Theoretical prediction of shape effect on EO activity]]
Statistical mechanics has helped explain experimental data. Notice that if we go from a prolate ellipsoid shape to a spherical structure the EO activity can be improved. This has been experimentally observed. This can be used to produce an analytical expression to tell where the maximum EO activity will occur. More importantly it has given us methods such as shape engineering.

 

=== Coarse and fine models ===
[[Image:atomistic_coarsegrained.JPG|thumb|400px|Here are three chromophores incorporated into a three arm dendrimer system.]]
We can even treat more complicated structures. We can use pseudo atomistic Monte Carlo calculations that treat anything that has high conjugation without internal motion as more or less a rigid object and then you correct electron distribution over that object. That will simplify a normal fully atomistic calculation. A chemical model becomes an electrostatic surface.

[[Image:r33_Numberdens.JPG|thumb|400px|Improvements in EO activity with dendrimer design.]]
The beauty of this is that it gives you that it gives you a quantitative prediction of EO activity. This led to the to the design of branched dendrimers which cause steric interactions that helps keep isolated chromophores from interacting each other so they don't line up.

Once there is a structure in mind an organic synthesis pathway can be devised.
 

=== Conclusion ===

The combination of new QM and SM methods permits quantitative prediction of macroscopic electro-optic activity--Yes, we really do need “Numbers”!

*Previous analysis has tended neglect dielectric effects, to over estimate acentric order and underestimate molecular first hyperpolarizability.
*Laser-assisted electric field poling can be used to improve acentric order.
*Binary chromophore organic glasses permit significant improvement in both electro-optic activity and optical transparency.
*Electro-optic activity of 1000 pm/V (2 orders of magnitude improvement) with optical loss of 2 dB/cm is possible by theoretically-inspired design. This is starting to approach the value of liquid crystals but with a factor of 1011 improvement in processing speed.

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Second-order Processes

2009-12-28T23:43:08Z

128.95.39.42: /* r coefficients */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

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

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E e \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the second hyperpolarizability, responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

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

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω1 and ω2 when summed results in a frequency of 2 x ω also referred to as second harmonic generation (SHG) or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage also known as optical rectification.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

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

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

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Structure-Property Relationships

2009-12-28T23:34:37Z

128.95.39.42: /* First Hyper-polarizability and BOA */

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Bond order alternation (BOA) and bond length alternation (BLA) are useful concepts that can be correlated to α the linear polarizability and β the first hyperpolarizability of molecules. The effect of applied electrical fields on BOA and BLA can be predicted by understanding the chemical structure and how charge distribution and dipole moment will shift.

=== Applying theory to relate structure to properties ===

Material chemists help bring theory to play with the rational design of molecules.
[[Image:Sumoverstate.png|thumb|900px|center|Sum over state expression for β]]
The sum over states expression for β describes the factors for building a molecule with high β. This can be translated into an optimized molecule, which can then be incorporated into an optimized material.

See Meyers 1994 <ref>F. Meyers, S. R. Marder, B. M. Pierce, J. L. Bredas
J. Am. Chem. Soc., 1994, 116 (23), pp 10703–10714 Electric Field Modulated Nonlinear Optical Properties of Donor-Acceptor Polyenes: Sum-Over-States Investigation of the Relationship between Molecular Polarizabilities (.alpha., .beta., and .gamma.) and Bond Length Alternation http://pubs.acs.org/doi/abs/10.1021/ja00102a040?journalCode=jacsat&quickLinkVolume=116&quickLinkPage=10703&volume=116</ref>

=== Bond Length Alternation ===
[[Image:Bla.png|thumb|300px|Bond Length Alternation (BLA) in a polymethine chain]]
Bond length alternation is a construct that can be used to monitor the amount of polarization in a molecule. Bond-length alternation (BLA) is defined as the average of the difference in the length between adjacent carbon-carbon bonds in a polymethine ((CH)n) chain. A polymethine dye is a series of CH units connected by double or single bonds.

Bond-length alternation (BLA) is defined as the average of the difference in the length between adjacent carbon-carbon bonds in a polymethine ((CH)n) chain. Polyenes (such as ethylene) have alternating C-C double (1.34 Å), the carbon bond in ethane (1.54 Å) and single bonds between two sp2 hybridized carbons in a polyene is (1.45 Å). As you move from butadiene to polyacetylene the double bonds a bit longer and the single bonds become a little shorter. The average difference between single and double bond lengths is the bond length alternation (BLA), for polyenes show a high degree of BLA (+ 0.11 Å). If instead of a structural view we wish to consider an electronic or molecular orbital related parameter we can consider bond order. A double bond has a bond order of 2 and single bond has a bond order of one. The bond order alternation is -1 (1-2=-1). If you attach a donor and acceptor group to opposite ends of the molecule donor will attempt to transfer electrons to the acceptor by pushing electrons into the chain.

A simple model has been proposed recently where α, β and γ are correlated with the degree of ground-state polarization.
The degree of ground-state polarization, or in other words the degree of charge separation in the ground state, depends primarily on the chemical structure (for example, the structure of the π-conjugated system, or the strength of the donor and acceptor substituents), but also on its surroundings (for example, the polarity of the medium).
In donor-acceptor polyenes, this variable is related to a geometrical parameter, the bond-length alternation (BLA)

=== Resonance Structures and BLA ===
[[Image:Resonance_bla.png|thumb|300px|Resonance structures trends. As the two structures contribute more equally to the ground-state structure of the molecule the BLA decreases]]
It is illustrative to discuss the wave function of the ground state in terms of a linear combination of the two limiting resonance structures:
#a neutral form characterized by a positive BLA and
#a charge-separated form characterized by a negative BLA (since the double and single bond pattern is now reversed relative to the neutral form).

From an energetic standpoint the two resonance structures can be treated as having two potential wells separated by a nuclear coordinate. The first resonance structure on the left has a BLS of .11. Using the same carbon numbering scheme first molecule on the right would have a BLA of -.11 because the double bonds in the center are shifted right one carbon atom. The neutral molecule has a lower potential energy and therefore will the more favored state. The potential energy diagram shows a higher energy curve for the charge separated resonance structure and a dotted line shows the mixing state. The ground of this is a resonance structure that looks more like the neutral molecule. For substituted polyenes with weak donors and acceptors, the neutral resonance form dominates the ground-state wave function, and the molecule has a high degree of BLA.

This is visualized by the relative size of the two balloons over the equilibrium arrow, (this is not a representation of the molecule itself).

By adding a donor group on the right and acceptor on the left you decrease the energy difference between the two forms thus stabilizing the molecule. The relative contribution of the two forms to the resonance structure becomes more equal. The bond length alternation will decrease and the molecule will look less polyene like and more zwitterionic (charge separated). With stronger donors and acceptors, the contribution of the charge-separated resonance form to the ground state increases and simultaneously, BLA decreases.

In the bottom structure we replace the C=O with a C=N+ (an iminium group) which is a stronger charge acceptor. The electrons move from one side to the other. The structure on the right is an identical to the structure on the left. The charge will be equally distributed between the two nitrogens resulting in 1.5 bonds between all carbons. The two forms have the same energy. When the two resonance forms contribute equally to the ground-state structure, the molecule exhibits essentially no BLA. This zero BLA limit is the so-called cyanine limit. The position of the electron will be more sensitive to the electric field in the cyanine case; this is also known as polarizability.

Second order optical materials requires asymmetrical polarizability. Neither the first case with total transfer (polyene limit) and or the last case with equal transfer (cyanine limit) will have asymmetry; they will not have no β. The middle molecule will has some asymmetrical polarizability. Starting at the polyene limit or the cyanine limit you can add weak acceptors and thereby increase the asymmetry, and the β increases. The curve of β vs bond length alternation will start at zero, go up to a peak and then go back down to zero. For example, with stilbene and paranitroaniline and add stronger donors and acceptor groups, or make the molecules longer, you can increase the β. However, past a certain point stronger donors and acceptors simply results in a zwitterion and the polarizability decreases. The optimal point can be described and predicted this using quantum mechanics and four orbitals.

=== Electric Field Perturbation of Structure ===
[[Image:BOA_perturb.png|thumb|500px|BLA as a function of Electric Field]]
To move electrons from one end to another it requires a very large electric field (107 volts per cm) on the order of the energy that holds the molecule together. These fields are about 100 larger than are typically used in a laboratory. You can plot the BLA and the Bond order alternation (BOA) as well as the induced dipole moment. This molecule does not start at the polyene limit for BLA of .11 because there first molecule already has a donor and acceptor groups. The BLA starts at .08 and as the field is increase it goes down to zero and then becomes negative. The bond order alternative (BOA) starts negative, goes to zero and then becomes positive. The change with respect to electric field is greatest in the center of the curve near zero BLA or BOA. An electric field can increase charge separation in the ground –state of molecules. This in turn modifies the BLA, the Bond Order Alternation (BOA) and the dipole moment.

[[Image:Appliedfield_distribution.png|thumb|500px|center|Effect of Applied field on pi-Charge Distribution]]

As you move from no electric field to a large electric field (shown as 0, 5 and 8 x 107 V/cm) there is an increase in positive charge on one end (the nitrogen end) and an increase in negative charge on the other end (the oxygen end).

=== Dipole Moment vs. Field and BOA ===
[[Image:Dipole_vs_BOA.png|thumb|400px|dipole versus electric field and dipole versus BOA]]
As a consequence of the charge redistribution, the dipole moment increases as a function of the applied electric field. Since the BOA is a function of field and the dipole moment is a function of field one can now correlate the dipole with the BOA to develop our first structure (BOA)- property (dipole moment) relationship.

The dipole moment versus electric field goes up and then saturates. The mathematical definition of polarizability is
Dμ DE; the slope of any function at any given point. The slope increases and then decreases. The slope is highest around 30-40 μ

A plot of dipole vs BOA going from a polyene to a cyanine is approximately a straight line because both the dipole and the BOA are sigmoidal curves. Around zero BOA (the cyanine limit) the polarizability is largest. The second derivative of this curve is the first hyperpolizability.

=== Linear Polarizability and BOA ===
[[Image:Linearpolarize_boa.png|thumb|500px|Linear polarizability as a function of BOA]]

:<math>\alpha \propto \left ( \frac {\mu^{2}_{ge}}{E_{ge}}\right )\,\!</math>

A linear polarizablity α(alpha) vs BOA curve starts small, increases to a maximum and then goes down. A graph of polarizability vs charge is a sigmoidal curve so the maximum slope of such a curve occurs near the inflection point where BOA is zero. The polarizability is highest at the cyanine limit, and the hyperpolarizability is zero at the cyanine limit. Rate of change of polarizability is zero at the cyanine limit and from a particle- in-a-box perspective the molecule does not have an asymmetrical polarizability. Thus without any sum-over-states calculations you can draw conclusions about the linear polarizability and hyperpolarizability using only ground state dipole moment calculations.

In a sum-over-states calculation the linear polarizability is defined as the sum over the square of all of the transition dipole moments, between the ground and any of the excited states, divided by the energy gap. The result is the red curve which peaks at the cyanine limit. Spectroscopically molecules like these usually have a single strong peak which has a transition dipole moment that is larger than others and that peak tends to be the lowest energy, resulting in a larger energy gap.

In a two state model we choose to ignore the contribution from all terms except the one that give rise to the strong peak. α is the transition dipole moment squared between the ground and one excited state, over the energy gap between these two states results in the blue open circles. This matches the red curve very well so the two state model seems to capture the essence of the relationship. The question is which term is responsible for the peak; is it the transition dipole moment (blue triangle) or one over the energy gap (blue square)? Calculating and plotting these two terms shows that both peak near the cyanine limit. As a result the polarizabilty peaks at the cyanine limit.

=== First Hyper-polarizability and BOA ===

[[Image:First_hyperpole.png|thumb|400px|β versus BOA- (Oudar, Chemla, Garito and Lalama)]]
By calculating β (beta) for each electric field and differing BOA and taking into account all the states it shows that β goes up, peaks about halfway between the polyene and the cyanine limit, goes down to zero and then goes negative. In the two level model for β is related to the transition dipole moment squared, the change in the dipole moment and divided by the energy gap. Most aromatic molecules tend to be near polyene limit with a small β.

:<math>\beta \propto \left( \frac {\mu^{2} _{ge} ( \mu _ {ee} - \mu _{gg})} {E^{2}_{ge}}\right)\,\!</math>

Where:
:<math>\mu_{ge}\,\!</math> is the transition dipole moment
:<math>\mu_{ee}\,\!</math> is the dipole moment of the excited state
:<math>\mu_{gg}\,\!</math> is the dipole moment of the ground state
:<math>E_{ge}\,\!</math> is the energy gap between the states - or ωge the frequency corresponding to the excited state energy.

Considering only the two states the β model (blue circles) matches the experimental values (red dots) well. So the two state model looks good for β as well as α. The energy gap squared peaks at the cyanine limit as does the transition dipole moment.

A polyene that has no dipole moment (eg it is symmetrical) in the ground state will have no dipole moment in the excited state. The change in dipole moment (blue plus symbol) peaks closer to the polyene limit. Most aromatic molecules tend to be near polyene limit with a small beta. The beginning portion of the β curve looks almost linear, this led researchers to believe that they just had to add stronger donors and acceptors to the molecule in order to increase β. But they were not seeing the whole picture.

=== Factors Affecting Charge Separation ===
[[Image:Chargeseparaton.png|thumb|400px|Effect of aromaticity on charge distribution in resonance structures]]

The relative contribution of each limiting resonance structure to the ground-state structure of a molecule is related to their relative energies. When the two resonance forms are very different in energy the ground-state structure will be dominated by the lower energy form and the molecules will exhibit a large degree of BLA.

In organic molecules there are two factors that dominate the energetics of the resonance structures to a first approximation.

First, there is a Coulombic term that is destabilizing when charge is separated. Adding donors and acceptors does encourage separation of charge but this also costs energy. Increasing the strength of the donor and acceptor end groups, and/or placing the molecule in a more polar solvent to the molecule can each lead to stabilization of the charge separated form.

In addition, there is also an energy consideration associated with the topology of the molecule.If a molecule has 6 π-electrons in a ring the molecule has an additional resonance stabilization and is referred to as aromatic. In the case of stilbene with added donor and acceptors there is a loss of aromaticity, (a loss of 36kcal/mol). This favors the molecule that is aromatic.

For molecules whose neutral forms are aromatic, charge separation will interrupt the aromaticity and yield structure with a higher energy quinoidal resonance form.

This disruption of the aromaticity results in additional destabilization due to the loss of aromatic stabilization and in such systems the molecule will be further biased towards the neutral resonance form

If, on the other hand, the neutral form is quinoidal, then the molecule will gain aromatic stabilization in the charge-separated resonance form. Aromaticity can be used as a driving force to get past the cyanine limit. The groups can used to tune the molecules for particular desired polarization characteristics.

'''Tuning Bond Length Alternation Leads to Optimization of β'''

We can tune aromaticity and charge separation.

Therefore we can tune energetics of resonance structures.

Therefore we can tune bond length alternation across a wide range.

If βis a peaked function, we can tune the bond length alternation so as to be at the peak and optimize β.

'''Evolution of Dipole Moments'''
[[Image:Dipole_evolution.png|thumb|400px|Dipole Moments and Bond Order Alternation]]
Note that both the ground- and the excited-state dipole moments increase as BOA becomes more positive.
However note also that the past the cyanine-limit (0 BOA) the excited-state dipole moment(μee) is less than the ground-state dipole moment(μgg) so Δ μ becomes negative. The excited state dipole moment vs BOA has a flat slope, while the groundstate dipole is going up, therefore the ground state is going to exhibit more polarizability.

 
=== Manipulation of BLA Through Topology ===
[[Image:Topology.png|thumb|300px|Tuning Aromaticity to Optimize β ]]
Until recently, most molecules that had been examined for NLO, such as donor-acceptor substituted stilbenes or diphenyl polyenes, had a very large BLA, typically greater than 0.10 Å. The calculations above, for example, predict that β is maximized at a value of ~0.04 Å; thus, these molecules with large BLA are not sufficiently polarized to give the correct BLA needed to maximize β. It was hypothesized that the high magnitude of BLA observed in the central polyene bridge of donor-acceptor substituted stilbenes and related molecules is indicative of an insufficient contribution of the charge-separated resonance forms to the ground-state configurations of the molecules and is a consequence of the loss of aromatic stabilization in the charge-separated forms.

We can tune aromaticity and charge separation. There first molecule is called an phenylisoxazimone. Aromatic groups must have 4n+2 electrons in a ring. The benzene has 4n+2. In the charge separated form there is a somewhat aromatic ring which offsets the loss of aromaticity from benzene.

In the second example there single bonds that are cross conjugated to a fully conjugated form when in the charge separated form.

In the third example of heterocycles such as thiophenes we start with neutral molecules that lack the full benzene aromaticity. You still lose aromaticity on both sides but not a full benzenes’ worth.

Using these techniques we can tune energetics of resonance structures therefore we can tune bond length alternation across a wide range. If β is a peaked function, we can tune the bond length alternation so as to be at the peak and optimize β.

== References ==
<references/>

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Two Photon Absorption

2009-12-28T23:29:58Z

128.95.39.42: /* Two Photo Absorption */

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Two photon absorption (TPA)is a third order non linear optical phenonmena in which a molecule absorbs two lower energy photons at the same time. The absorption is sensitive to a threshold value of intensity. This presents a number of possibilities for biomedical applications, and microscopy and microfabrication.

== Two-Photon Excited Processes ==

=== Two Photo Absorption ===
[[Image:Tpa_energy.png|thumb|300px|Two photon absorption takes a molecule directly to the S2 state]]
Two photon absorption is related to the imaginary component of the χ(3) tensor. On a molecular level it is the imaginary component of γ. Two photons are simultaneously absorbed directly to S2 excited state without population of an intermediate eigenstate. Neither of the two photons have enough energy to get to S1 but with the sum of the two they can reach the second singlet state. The dotted line indicates a virtual state that last a certain length of time (10-15 sec) which means a very short time to get the second photon in. The closer the energy is to the normal S1 state the closer the duration of time is to that of a normal excited state (10-9 sec). As the energy different between S1 and the photon gets smaller, the longer the duration, and the greater the chance of a second photon absorption.

The two photon cross section δ has a term in the denominator referred to as the detuning term that relates to the energy difference. The smaller the difference the higher the two photon cross section.

[[Image:Tpa_centro.png|thumb|400px|Rate expression and selection rules for TPA]]
Once S2 is attained it will quickly relax down to the first excited state and then there can be fluorescence, an additional photon absorption, electron transfer to another molecule, energy transfer to another molecule or photo chemistry. All the same processes can occur that would happen with one photon absorption to the S1. The two photon cross section is very small and it can only be had with the use of intense lasers.

σ is the one photon cross section which relates to the transition dipole moment. δ is the two photon cross section.

The rate of formation of the excited state (analagous to two reactants colliding A + B = C) can be thought of as a rate expression that is dependent on concentration of the reactants, and a first order or second order rate constant.

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math> is the rate for one photon absorption

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

Where
:<math>\sigma\,\!</math> is the intrinsic probability of a collision resulting in absorption, the absorption cross section
:<math>N_GS\,\!</math> is the number of molecules in the ground state
:<math>F\,\!</math> is the flux or intensity

If a reaction requires two A and one B to all collide simultaneously to create C then the rate expression has a third order rate constant.

In two photon absorption the selection rules are the inverse of the rules for those for single photon absorption because there are two transitions, so gerade to ungrerade is forbidden and gerade to gerade, and ungerade to ungerade are allowed.
Single photon absorption is the equivalent of IR spectroscopy
Raman spectroscopy is a two photon process and therefore it is a third order non linear optical phenomena.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/08 TPA.swf</swf>

== Advantages of TPA ==

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|A sample molecule in a cuvette is bombarded with light tuned for two photon absorption above, and a different wavelength fro one photon absorption below.]]
As you shine a focused beam (bottom) into sample at a wavelength suitable for one photon absorption you get fluoresence along the entire path. A beam tuned for two photon absorption (top) focuses to a very small point. Each single photon has too little energy to be absorbed right up to the point that the intensity is sufficient to trigger TPA. The focal point of the beam varies linearly with the distance from the lens. The area (and therefore intensity) of the beam varies as the square of the radius of beam at that point (pir2), and the two photon absorption varies as the square of the intensity. So TPA varies as the forth power of the distance. Thus TPA builds up and drops off very quickly on either side of the focal point giving it a single point appearance. Thus it can be directed accurately in in three dimensions.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|A light beam from the right is quickly absorbed in single photon wavelength below. The upper two photon tuned-beam is able to penetrate without absorption until the the focus gets the intensity to a sufficient level.]]
In a concentrated solution of cumarin one photon absorption (below right) starts immediately at the interface and decays off with the (penetration depth). Because single photons are unable to excite the two photon tuned sample until they are at a critical intensity the two photon laser beam can penetrate farther into a sample. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA ideal for medical applications in which a drug can be activated by TPA at a very precise location.
 

== Calculating and Designing for TPA ==

=== Measuring the two photon cross section ===
[[Image:Tpa_measurement.png|thumb|400px|A beam splitter allows comparison of each pulse through a signal cell and a reference cell]]
To measure the TPA cross section a tunable laser is focused on a beam splitter which then goes to a reference cell with a photomultiplier tube (PMT) and to a signal cell (with unknown sample) and a PMT. A cross section is created by measuring the fluorescence of the unknown sample across a range of wavelengths produced by a tunable laser. In practice single pulses of intense laser light can vary greatly in intensity, and since TPA is exponentially related to the light intensity these random fluctuations can be exaggerated. By comparing the fluorescence of the sample with each pulse to the reference cell with a known TPA cross section, and then integrating the results you can determine the sample cross section accurately. Otherwise you would have to be able to measure the intensity of each pulse individually which is rather more complicated.

'''Measurement considerations:'''
*fs-ns pulse duration; cw
:<math>
n^{(2)}_{ph} = 1/2 \delta(\omega) N F^2(\omega)\,\!</math>

:<math>n_{fl} = \eta n^{(2)}_{ph}\,\!</math>

*only for fluorescent materials

*90º or backward collection of fluorescence

:<math>signal \propto I^2\,\!</math>
*deviations: linear absorption, stimulated emission, saturation, ...
*importance of spatial and temporal profile

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

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

The perturbative expression for gamma can be applied for two photon absorption with some changes.

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

The denominator is the difference between the state energy (Ege) and the photon energy (&hbar; ω).

There are two components, and the imaginary component accounts for the absorption. The :<math>\Gamma_{ge}\,\!</math> is the damping factor. In order to achieve TPA the sum of the energies of the two photons must equal the energy of the excited state.

For a centrosymmetric molecule :<math>M^2_{ge} \Delta \mu^2_{ge}\,\!</math> (the transition dipole) goes to zero so that entire term goes to zero. For non centrosymmetric molecules both terms are required.

The TPA cross section (δ ) can be related to gamma and several other terms.

:<math>\delta(\omega) = \frac {8\pi^2h\omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

=== Donor-Acceptor TPA Calculations ===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
This is a calculation for a simple donor/acceptor stilbene with an amino group and a formyl group. The delta for the first excited state (S0 to S1) starts out small reaches a peak and then near the cyanine limit it goes down to zero. This is because the delta mu term follows this same pattern.

The change to the second state (S0 to S2) has multiple peaks because the system is two photon resonant and one of the photons is resonant with the S1 state. This is referred to as double resonance and occurs when the transition to S1 is very close to half of the transition to S2. The scale of the cross section for the transition into S1 is on the order 1000 while the transition for S2 is on the order of 10-20000.

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

=== Two-photon absorption example ===
[[Image:Tpa_spectra.png|thumb|400px|Two photon and linear absorption spectra compared]]

Here is a centrosymmetric molecule with donor groups on both ends. It has a long conjugation path. There can be excitation into S1, TPA into S2, rapid relaxation by internal conversion back to S1 and then fluorescence coming out of S2. There is no fluorescence from S2 because in most cases the rate of relaxation from S2 to S1 is much faster than the fluorescence lifetime. In fact in centrosymmetric molecules the transition from S2 to S1 is symmetry forbidden for one photon. Therefore the transition dipole moment is close to zero and the coupling between the grounds and the excited state is close to zero resulting in a long radiative lifetime of the excited state. If there is strong transition dipole going up then there will be a short radiative lifetime in the excited state which in turn leads to a high quantum yield for fluorescence. This is because the rate term for this step dominates the other terms in the expression for quantum yield.

However even if the molecule was not centrosymmetric and both one and two photon transitions are possible the internal conversion relaxation is so fast that there still would not be fluorescence from S2. This leads to the rule coined by Michael Kasha that says that no matter what state you excite into fluorescence will only occur from the lowest lying excited state. There are molecules that are exceptions to Kashsa rule such as azuylene but these are very few. The fluorescence from both one and two photon absorption will be from S1.

In the graph note that the energy of the TPA is lower (longer wavelength- red line) than the single photon absorption (blue line) and of the fluorescence (green line). The TPA peak is around 720, dividing this by two the equivalent single photon energy for this would be 360nm. The peak for linear absorption to S1 is 430nm. This shows that each photon in TPA is absorbing to a state that is higher than the S1 state and that there is very little single photon absorption in this zone (the small absorption seen at 360nm is vibrational rather than electronic).

In non linear optics studies it is important to do the measurements at a variety of wavelengths so that you can get a spectrum. You can’t compare molecules by looking at a single wavelength or using the wrong pulse length.

see Rumi 2000 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

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

Xu and Webb measured the cross section for laser dyes a goupert mayer. One dye had cross section of 300 goupert meyer (two-photon absorptivity, δ, is expressed in Goppert-Mayer units (GM), with 1 GM=1×10-50 cm4 s molecules−1 photon−1.). The process of two photon absorption was predicted by a German physicist Maria Goeppert Mayer in 1929. TPA was not actually observed until the early 60s when lasers were developed that had sufficient intensity to cause it.
Most molecules have GM < 1. These dye have very high two photon cross sections..

See Xu and Webb <ref>C. Xu, JOSA B, 1996;</ref> Albota 1998 <ref>M. Albota, Appl. Opt., 1998;</ref>Fisher 1998 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of measurement: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Various reported measurements of δ for AF-50]]

The chart shows various measurements for the molecule AF-50. This is indicative of the problems of measurement in the NLO field. The measurements were made using various techniques and conditions. The experiment with a delta of 11560 was conducted with a timescale of nanoseconds. This is duration is long enough for there to be a convolution of TPA and single photon absorptions that can appear to give a high TPA cross section. This high value could very well be useful for example in making coatings for safety glasses that could exclude high intensity laser light. But it is not useful for the scientific measurement of the molecules.

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Two photon fluoresence vs non linear transmission ]]

This molecule behaves as an optical limiter. The curve of two photon fluoresence (TPF shown in red) and non linear transmission (NLT shown in blue) are similar except their scales are different. The NLT scale has units of 10-46 and TPF is on the scale of 200, there is a factor of 100 difference. TPF is measure of the cross section, NLT is an effective cross section that involves two photon absorption followed by excited state absorption.
 

=== Initial Observations on Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|bis-Donor Stilbene]]

Bis Donor Stilbene (BDAS) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*The molecule is very electron rich
*Strong nonlinear transmission
*Strong blue fluorescence when pumped with orange light
*Fluorescence depends on I2
*Two-photon cross section, δ = 210 x 10-50 cm4 s/photon, for stilbene δ = 12 x 10-50 cm4 s/photon

The cross section is about 20 times that for the molecule without the electron donor groups. This is very high value and it is useful to learn why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+

It 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 this molecule’s properties.

BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) that is associated with large transition dipole moment between S1 and S2. (μ ee’2 )

These results suggest that a large change in quadrupole moment between S0 and S1 is leads to enhanced delta. 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.

 
'''Bis Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px|]]
This suggests certain design strategies for increasing the dipole or quandrupole moment. The transition dipole moment between the ground and first excited state is 7.2 for stilbene and 8.9 for BDAS. Between S1 and S2 the dipole goes from 3.1 D stilbene to 7.4 D for BDAS. This term is squared and this is the reason there such a high TPA cross section for BDAS.

:<math>\delta_{S_0 \rightarrow S_2} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_1 - E_0 - \hbar \omega ) \eta} \,\!</math>

'''Transition Dipole Moments and BOA'''
[[Image:Tpa_transdip_densities.png|thumb|300px|]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule there will be higher transition dipoles. In DBAS large coefficients on nitrogens coupled with extended distance from the molecular origin result in large excited state coupling.

[[Image:Tpa_transdip_boa.png|thumb|300px|Transition energy in eV for various conditions of charge transfer.]]

There is an optimal amount of charge transfer that is required to get the largest cross section.
 

== Design of TPA chromophores ==
[[Image:Tpa_chromophores.png|thumb|300px|Chromophore variations and δ in 10-50 cm4 s/photon]]

To get large quadrupole moment and large quadrupole moment changes you can do the following;
*Increase the distance between the donors so that you increase the distance over which you transfer the charge.
*Increase the strength of the donors so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule so that that section attracts charge.
*Flip the relation with donors in the center and acceptors on the end.

The diagram shows the effect on delta of different donors (shown in blue) and acceptors (shown in red) configurations and longer molecular chain lengths. The combination of these techniques achieves cross-sections in the thousands.

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|Two Photon Fluorescence by wavelength for various length molecules]]
The graph shows that as the molecule gets longer the energy of the two photon absorption cross section decreases so the wavelength of the peak increases and the magnitude of the TPA increases due to the increasing transition dipole moments
 

=== Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

Here is an expression for delta that explores the change in transition dipole moment between the ground state and the excited state and the photon energy. The oscillator strength of the curve is calculated by integrating the absorption spectra plotted in energy and measuring the area under the curve. From this you can extract the transition dipole moment.

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

where:

:<math>\delta_{max}\,\!</math> is the cross sections that was measured.

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

:<math>E_{\alpha\beta}\,\!</math> is the energy for absorption to for two levels α,β

:<math>\hbar \omega\,\!</math> is the photon energy that was used to get the TPA.

:<math>\Tau_ge^{\prime}\,\!</math> is the damping term which is estimated

In summary

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

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for dipole moments of quadrupolar molecules]]

By making these molecules quadrupolar we increase the transition dipole moment between the first and second excited state (ee and ee’) The data supports this strategy.
 

== 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 polymerization on a precise microscopic scale. A beam can be focused at 3D coordinates in a polymer causing cross linking. Then non-crosslinked portions are washed away. This can be used for microfabrication. The model of the bull is about 5 microns wide and has some very fine features.

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

and Maruo 1997 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

[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, thus increasing the possible resolution]]

Regular photo-initiators are not excellent two photon absorbers. However a femptosecond layer can apply very large power. The goal is strike a power balance between the minimum writing range and the maximum destructive level. For conventional initiators there is only a factor of 2.5 in the power writing range. Newly developed two photon dyes expand this to range to a factor of 50. This lets you write more accurately and faster because the beam does not have remain in the same place as long.
 

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

This technique can be used to convert non fluorescent polymer into a fluorescent form that has a higher density due to cross linking. If the density goes up and the polarizability stays the same the susceptibility goes up and the refractive index goes up. Peter Renzepus at the UC Urvine is using this method to create 3D optical memory. The ability to write on a 100 different planes increases the amount of information that can be stored (gigabits or terabits of data per cm3)

=== 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 dye connected to a photoactive group will absorb the two photons to 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 Nanofabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|Examples of applications for nanofabrication]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to free-form fabricate 3 dimensional structures
*Increasing need for ability 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. The photo on the left is a chain link fence is twice the thickness of a human hair. In tissue engineering cells grow better on certain topologies (ie scaffolds).

==== Media: Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Exposed areas remain after development in the positive tone resist process]]
This is an example of a two photon dye that creates a radical initiator. After absorption the radicals initiate polymerization. The polymer is less soluble and the nonpolymerized material thus this is a known as a negative photo resist.

==== Sub-diffraction limited resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are shown are 170nm.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can do this resolution with 3D resolution was well.
 

==== Media: Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|In positive tone resist the exposed areas are removed during development]]

A positive tone resist becomes more soluble in portions that are exposed to light. This allows you to etch away material by exposing it with light.

Tetrahydropyran can be protonated on the oxygen creating a carbo cation thus converting the esther to a carboxylic acid. Esthers are not soluble in basic water but acids are so by dipping the material in an alkaline solution you can dissolve away the exposed material. A proton is needed.

A photo acid generator (PAG) will absorb light and then transfer an electron to the carbon sulfur sigma bond orbital, cause the bond to break homolytically yielding a methyl radical which will attack benzene and create a proton. The proton then starts the esther to acid reaction.

=== 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.

Two Photon Absorption

2009-12-28T23:27:01Z

128.95.39.42: /* Two Photo Absorption */

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Two photon absorption (TPA)is a third order non linear optical phenonmena in which a molecule absorbs two lower energy photons at the same time. The absorption is sensitive to a threshold value of intensity. This presents a number of possibilities for biomedical applications, and microscopy and microfabrication.

== Two-Photon Excited Processes ==

=== Two Photo Absorption ===
[[Image:Tpa_energy.png|thumb|300px|Two photon absorption takes a molecule directly to the S2 state]]
Two photon absorption is related to the imaginary component of the χ(3) tensor. On a molecular level it is the imaginary component of γ. Two photons are simultaneously absorbed directly to S2 excited state without population of an intermediate eigenstate. Neither of the two photons have enough energy to get to S1 but with the sum of the two they can reach the second singlet state. The dotted line indicates a virtual state that last a certain length of time (10-15 sec) which means a very short time to get the second photon in. The closer the energy is to the normal S1 state the closer the duration of time is to that of a normal excited state (10-9 sec). As the energy different between S1 and the photon gets smaller, the longer the duration, and the greater the chance of a second photon absorption.

The two photon cross section δ has a term in the denominator referred to as the detuning term that relates to the energy difference. The smaller the difference the higher the two photon cross section.

[[Image:Tpa_centro.png|thumb|400px|Rate expression and selection rules for TPA]]
Once S2 is attained it will quickly relax down to the first excited state and then there can be fluorescence, an additional photon absorption, electron transfer to another molecule, energy transfer to another molecule or photo chemistry. All the same processes can occur that would happen with one photon absorption to the S1. The two photon cross section is very small and it can only be had with the use of intense lasers.

σ is the one photon cross section which relates to the transition dipole moment. Delta is the two photon cross section.

The rate of formation of the excited state (analagous to two reactants colliding A + B = C) can be thought of as a rate expression that is dependent on concentration of the reactants, and a first order or second order rate constant.

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math> is the rate for one photon absorption

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

Where
:<math>\sigma\,\!</math> is the intrinsic probability of a collision resulting in absorption, the absorption cross section
:<math>N_GS\,\!</math> is the number of molecules in the ground state
:<math>F\,\!</math> is the flux or intensity

If a reaction requires two A and one B to all collide simultaneously to create C then the rate expression has a third order rate constant.

In two photon absorption the selection rules are the inverse of the rules for those for single photon absorption because there are two transitions, so gerade to ungrerade is forbidden and gerade to gerade, and ungerade to ungerade are allowed.
Single photon absorption is the equivalent of IR spectroscopy
Raman spectroscopy is a two photon process and therefore it is a third order non linear optical phenomena.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/08 TPA.swf</swf>

== Advantages of TPA ==

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|A sample molecule in a cuvette is bombarded with light tuned for two photon absorption above, and a different wavelength fro one photon absorption below.]]
As you shine a focused beam (bottom) into sample at a wavelength suitable for one photon absorption you get fluoresence along the entire path. A beam tuned for two photon absorption (top) focuses to a very small point. Each single photon has too little energy to be absorbed right up to the point that the intensity is sufficient to trigger TPA. The focal point of the beam varies linearly with the distance from the lens. The area (and therefore intensity) of the beam varies as the square of the radius of beam at that point (pir2), and the two photon absorption varies as the square of the intensity. So TPA varies as the forth power of the distance. Thus TPA builds up and drops off very quickly on either side of the focal point giving it a single point appearance. Thus it can be directed accurately in in three dimensions.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|A light beam from the right is quickly absorbed in single photon wavelength below. The upper two photon tuned-beam is able to penetrate without absorption until the the focus gets the intensity to a sufficient level.]]
In a concentrated solution of cumarin one photon absorption (below right) starts immediately at the interface and decays off with the (penetration depth). Because single photons are unable to excite the two photon tuned sample until they are at a critical intensity the two photon laser beam can penetrate farther into a sample. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA ideal for medical applications in which a drug can be activated by TPA at a very precise location.
 

== Calculating and Designing for TPA ==

=== Measuring the two photon cross section ===
[[Image:Tpa_measurement.png|thumb|400px|A beam splitter allows comparison of each pulse through a signal cell and a reference cell]]
To measure the TPA cross section a tunable laser is focused on a beam splitter which then goes to a reference cell with a photomultiplier tube (PMT) and to a signal cell (with unknown sample) and a PMT. A cross section is created by measuring the fluorescence of the unknown sample across a range of wavelengths produced by a tunable laser. In practice single pulses of intense laser light can vary greatly in intensity, and since TPA is exponentially related to the light intensity these random fluctuations can be exaggerated. By comparing the fluorescence of the sample with each pulse to the reference cell with a known TPA cross section, and then integrating the results you can determine the sample cross section accurately. Otherwise you would have to be able to measure the intensity of each pulse individually which is rather more complicated.

'''Measurement considerations:'''
*fs-ns pulse duration; cw
:<math>
n^{(2)}_{ph} = 1/2 \delta(\omega) N F^2(\omega)\,\!</math>

:<math>n_{fl} = \eta n^{(2)}_{ph}\,\!</math>

*only for fluorescent materials

*90º or backward collection of fluorescence

:<math>signal \propto I^2\,\!</math>
*deviations: linear absorption, stimulated emission, saturation, ...
*importance of spatial and temporal profile

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

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

The perturbative expression for gamma can be applied for two photon absorption with some changes.

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

The denominator is the difference between the state energy (Ege) and the photon energy (&hbar; ω).

There are two components, and the imaginary component accounts for the absorption. The :<math>\Gamma_{ge}\,\!</math> is the damping factor. In order to achieve TPA the sum of the energies of the two photons must equal the energy of the excited state.

For a centrosymmetric molecule :<math>M^2_{ge} \Delta \mu^2_{ge}\,\!</math> (the transition dipole) goes to zero so that entire term goes to zero. For non centrosymmetric molecules both terms are required.

The TPA cross section (δ ) can be related to gamma and several other terms.

:<math>\delta(\omega) = \frac {8\pi^2h\omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

=== Donor-Acceptor TPA Calculations ===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
This is a calculation for a simple donor/acceptor stilbene with an amino group and a formyl group. The delta for the first excited state (S0 to S1) starts out small reaches a peak and then near the cyanine limit it goes down to zero. This is because the delta mu term follows this same pattern.

The change to the second state (S0 to S2) has multiple peaks because the system is two photon resonant and one of the photons is resonant with the S1 state. This is referred to as double resonance and occurs when the transition to S1 is very close to half of the transition to S2. The scale of the cross section for the transition into S1 is on the order 1000 while the transition for S2 is on the order of 10-20000.

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

=== Two-photon absorption example ===
[[Image:Tpa_spectra.png|thumb|400px|Two photon and linear absorption spectra compared]]

Here is a centrosymmetric molecule with donor groups on both ends. It has a long conjugation path. There can be excitation into S1, TPA into S2, rapid relaxation by internal conversion back to S1 and then fluorescence coming out of S2. There is no fluorescence from S2 because in most cases the rate of relaxation from S2 to S1 is much faster than the fluorescence lifetime. In fact in centrosymmetric molecules the transition from S2 to S1 is symmetry forbidden for one photon. Therefore the transition dipole moment is close to zero and the coupling between the grounds and the excited state is close to zero resulting in a long radiative lifetime of the excited state. If there is strong transition dipole going up then there will be a short radiative lifetime in the excited state which in turn leads to a high quantum yield for fluorescence. This is because the rate term for this step dominates the other terms in the expression for quantum yield.

However even if the molecule was not centrosymmetric and both one and two photon transitions are possible the internal conversion relaxation is so fast that there still would not be fluorescence from S2. This leads to the rule coined by Michael Kasha that says that no matter what state you excite into fluorescence will only occur from the lowest lying excited state. There are molecules that are exceptions to Kashsa rule such as azuylene but these are very few. The fluorescence from both one and two photon absorption will be from S1.

In the graph note that the energy of the TPA is lower (longer wavelength- red line) than the single photon absorption (blue line) and of the fluorescence (green line). The TPA peak is around 720, dividing this by two the equivalent single photon energy for this would be 360nm. The peak for linear absorption to S1 is 430nm. This shows that each photon in TPA is absorbing to a state that is higher than the S1 state and that there is very little single photon absorption in this zone (the small absorption seen at 360nm is vibrational rather than electronic).

In non linear optics studies it is important to do the measurements at a variety of wavelengths so that you can get a spectrum. You can’t compare molecules by looking at a single wavelength or using the wrong pulse length.

see Rumi 2000 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

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

Xu and Webb measured the cross section for laser dyes a goupert mayer. One dye had cross section of 300 goupert meyer (two-photon absorptivity, δ, is expressed in Goppert-Mayer units (GM), with 1 GM=1×10-50 cm4 s molecules−1 photon−1.). The process of two photon absorption was predicted by a German physicist Maria Goeppert Mayer in 1929. TPA was not actually observed until the early 60s when lasers were developed that had sufficient intensity to cause it.
Most molecules have GM < 1. These dye have very high two photon cross sections..

See Xu and Webb <ref>C. Xu, JOSA B, 1996;</ref> Albota 1998 <ref>M. Albota, Appl. Opt., 1998;</ref>Fisher 1998 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of measurement: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Various reported measurements of δ for AF-50]]

The chart shows various measurements for the molecule AF-50. This is indicative of the problems of measurement in the NLO field. The measurements were made using various techniques and conditions. The experiment with a delta of 11560 was conducted with a timescale of nanoseconds. This is duration is long enough for there to be a convolution of TPA and single photon absorptions that can appear to give a high TPA cross section. This high value could very well be useful for example in making coatings for safety glasses that could exclude high intensity laser light. But it is not useful for the scientific measurement of the molecules.

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Two photon fluoresence vs non linear transmission ]]

This molecule behaves as an optical limiter. The curve of two photon fluoresence (TPF shown in red) and non linear transmission (NLT shown in blue) are similar except their scales are different. The NLT scale has units of 10-46 and TPF is on the scale of 200, there is a factor of 100 difference. TPF is measure of the cross section, NLT is an effective cross section that involves two photon absorption followed by excited state absorption.
 

=== Initial Observations on Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|bis-Donor Stilbene]]

Bis Donor Stilbene (BDAS) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*The molecule is very electron rich
*Strong nonlinear transmission
*Strong blue fluorescence when pumped with orange light
*Fluorescence depends on I2
*Two-photon cross section, δ = 210 x 10-50 cm4 s/photon, for stilbene δ = 12 x 10-50 cm4 s/photon

The cross section is about 20 times that for the molecule without the electron donor groups. This is very high value and it is useful to learn why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+

It 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 this molecule’s properties.

BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) that is associated with large transition dipole moment between S1 and S2. (μ ee’2 )

These results suggest that a large change in quadrupole moment between S0 and S1 is leads to enhanced delta. 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.

 
'''Bis Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px|]]
This suggests certain design strategies for increasing the dipole or quandrupole moment. The transition dipole moment between the ground and first excited state is 7.2 for stilbene and 8.9 for BDAS. Between S1 and S2 the dipole goes from 3.1 D stilbene to 7.4 D for BDAS. This term is squared and this is the reason there such a high TPA cross section for BDAS.

:<math>\delta_{S_0 \rightarrow S_2} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_1 - E_0 - \hbar \omega ) \eta} \,\!</math>

'''Transition Dipole Moments and BOA'''
[[Image:Tpa_transdip_densities.png|thumb|300px|]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule there will be higher transition dipoles. In DBAS large coefficients on nitrogens coupled with extended distance from the molecular origin result in large excited state coupling.

[[Image:Tpa_transdip_boa.png|thumb|300px|Transition energy in eV for various conditions of charge transfer.]]

There is an optimal amount of charge transfer that is required to get the largest cross section.
 

== Design of TPA chromophores ==
[[Image:Tpa_chromophores.png|thumb|300px|Chromophore variations and δ in 10-50 cm4 s/photon]]

To get large quadrupole moment and large quadrupole moment changes you can do the following;
*Increase the distance between the donors so that you increase the distance over which you transfer the charge.
*Increase the strength of the donors so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule so that that section attracts charge.
*Flip the relation with donors in the center and acceptors on the end.

The diagram shows the effect on delta of different donors (shown in blue) and acceptors (shown in red) configurations and longer molecular chain lengths. The combination of these techniques achieves cross-sections in the thousands.

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|Two Photon Fluorescence by wavelength for various length molecules]]
The graph shows that as the molecule gets longer the energy of the two photon absorption cross section decreases so the wavelength of the peak increases and the magnitude of the TPA increases due to the increasing transition dipole moments
 

=== Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

Here is an expression for delta that explores the change in transition dipole moment between the ground state and the excited state and the photon energy. The oscillator strength of the curve is calculated by integrating the absorption spectra plotted in energy and measuring the area under the curve. From this you can extract the transition dipole moment.

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

where:

:<math>\delta_{max}\,\!</math> is the cross sections that was measured.

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

:<math>E_{\alpha\beta}\,\!</math> is the energy for absorption to for two levels α,β

:<math>\hbar \omega\,\!</math> is the photon energy that was used to get the TPA.

:<math>\Tau_ge^{\prime}\,\!</math> is the damping term which is estimated

In summary

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

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for dipole moments of quadrupolar molecules]]

By making these molecules quadrupolar we increase the transition dipole moment between the first and second excited state (ee and ee’) The data supports this strategy.
 

== 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 polymerization on a precise microscopic scale. A beam can be focused at 3D coordinates in a polymer causing cross linking. Then non-crosslinked portions are washed away. This can be used for microfabrication. The model of the bull is about 5 microns wide and has some very fine features.

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

and Maruo 1997 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

[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, thus increasing the possible resolution]]

Regular photo-initiators are not excellent two photon absorbers. However a femptosecond layer can apply very large power. The goal is strike a power balance between the minimum writing range and the maximum destructive level. For conventional initiators there is only a factor of 2.5 in the power writing range. Newly developed two photon dyes expand this to range to a factor of 50. This lets you write more accurately and faster because the beam does not have remain in the same place as long.
 

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

This technique can be used to convert non fluorescent polymer into a fluorescent form that has a higher density due to cross linking. If the density goes up and the polarizability stays the same the susceptibility goes up and the refractive index goes up. Peter Renzepus at the UC Urvine is using this method to create 3D optical memory. The ability to write on a 100 different planes increases the amount of information that can be stored (gigabits or terabits of data per cm3)

=== 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 dye connected to a photoactive group will absorb the two photons to 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 Nanofabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|Examples of applications for nanofabrication]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to free-form fabricate 3 dimensional structures
*Increasing need for ability 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. The photo on the left is a chain link fence is twice the thickness of a human hair. In tissue engineering cells grow better on certain topologies (ie scaffolds).

==== Media: Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Exposed areas remain after development in the positive tone resist process]]
This is an example of a two photon dye that creates a radical initiator. After absorption the radicals initiate polymerization. The polymer is less soluble and the nonpolymerized material thus this is a known as a negative photo resist.

==== Sub-diffraction limited resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are shown are 170nm.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can do this resolution with 3D resolution was well.
 

==== Media: Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|In positive tone resist the exposed areas are removed during development]]

A positive tone resist becomes more soluble in portions that are exposed to light. This allows you to etch away material by exposing it with light.

Tetrahydropyran can be protonated on the oxygen creating a carbo cation thus converting the esther to a carboxylic acid. Esthers are not soluble in basic water but acids are so by dipping the material in an alkaline solution you can dissolve away the exposed material. A proton is needed.

A photo acid generator (PAG) will absorb light and then transfer an electron to the carbon sulfur sigma bond orbital, cause the bond to break homolytically yielding a methyl radical which will attack benzene and create a proton. The proton then starts the esther to acid reaction.

=== 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.

Two Photon Absorption

2009-12-28T23:22:49Z

128.95.39.42: /* Two Photo Absorption */

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Two photon absorption (TPA)is a third order non linear optical phenonmena in which a molecule absorbs two lower energy photons at the same time. The absorption is sensitive to a threshold value of intensity. This presents a number of possibilities for biomedical applications, and microscopy and microfabrication.

== Two-Photon Excited Processes ==

=== Two Photo Absorption ===
[[Image:Tpa_energy.png|thumb|300px|Two photon absorption takes a molecule directly to the S2 state]]
Two photon absorption is related to the imaginary component of the chi(3) tensor. On a molecular level it is the imaginary component of gamma. Two photons are simultaneously absorbed directly to S2 excited state without population of an intermediate eigenstate. Neither of the two photons have enough energy to get to S1 but with the sum of the two they can reach the second singlet state. The dotted line indicates a virtual state that last a certain length of time (10-15 sec) which means a very short time to get the second photon in. The closer the energy is to the normal S1 state the closer the duration of time is to that of a normal excited state (10-9 sec). As the energy different between S1 and the photon gets smaller, the longer the duration , and the greater the chance of a second photon absorption.

The two photon cross section delta has a term in the denominator referred to as the detuning term that relates to the energy difference. The smaller the difference the higher the two photon cross section.

[[Image:Tpa_centro.png|thumb|400px|Rate expression and selection rules for TPA]]
Once S2 is attained it will quickly relax down to the first excited state and then there can be fluorescence, an additional photon absorption, electron transfer to another molecule, energy transfer to another molecule or photo chemistry. All the same processes can occur that would happen with one photon absorption to the S1. The two photon cross section is very small and it can only be had with the use of intense lasers.

σ is the one photon cross section which relates to the transition dipole moment. Delta is the two photon cross section.

The rate of formation of the excited state (analagous to two reactants colliding A + B = C) can be thought of as a rate expression that is dependent on concentration of the reactants, and a first order or second order rate constant.

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math> is the rate for one photon absorption

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

Where
:<math>\sigma\,\!</math> is the intrinsic probability of a collision resulting in absorption, the absorption cross section
:<math>N_GS\,\!</math> is the number of molecules in the ground state
:<math>F\,\!</math> is the flux or intensity

If a reaction requires two A and one B to all collide simultaneously to create C then the rate expression has a third order rate constant.

In two photon absorption the selection rules are the inverse of the rules for those for single photon absorption because there are two transitions, so gerade to ungrerade is forbidden and gerade to gerade, and ungerade to ungerade are allowed.
Single photon absorption is the equivalent of IR spectroscopy
Raman spectroscopy is a two photon process and therefore it is a third order non linear optical phenomena.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/08 TPA.swf</swf>

== Advantages of TPA ==

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|A sample molecule in a cuvette is bombarded with light tuned for two photon absorption above, and a different wavelength fro one photon absorption below.]]
As you shine a focused beam (bottom) into sample at a wavelength suitable for one photon absorption you get fluoresence along the entire path. A beam tuned for two photon absorption (top) focuses to a very small point. Each single photon has too little energy to be absorbed right up to the point that the intensity is sufficient to trigger TPA. The focal point of the beam varies linearly with the distance from the lens. The area (and therefore intensity) of the beam varies as the square of the radius of beam at that point (pir2), and the two photon absorption varies as the square of the intensity. So TPA varies as the forth power of the distance. Thus TPA builds up and drops off very quickly on either side of the focal point giving it a single point appearance. Thus it can be directed accurately in in three dimensions.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|A light beam from the right is quickly absorbed in single photon wavelength below. The upper two photon tuned-beam is able to penetrate without absorption until the the focus gets the intensity to a sufficient level.]]
In a concentrated solution of cumarin one photon absorption (below right) starts immediately at the interface and decays off with the (penetration depth). Because single photons are unable to excite the two photon tuned sample until they are at a critical intensity the two photon laser beam can penetrate farther into a sample. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA ideal for medical applications in which a drug can be activated by TPA at a very precise location.
 

== Calculating and Designing for TPA ==

=== Measuring the two photon cross section ===
[[Image:Tpa_measurement.png|thumb|400px|A beam splitter allows comparison of each pulse through a signal cell and a reference cell]]
To measure the TPA cross section a tunable laser is focused on a beam splitter which then goes to a reference cell with a photomultiplier tube (PMT) and to a signal cell (with unknown sample) and a PMT. A cross section is created by measuring the fluorescence of the unknown sample across a range of wavelengths produced by a tunable laser. In practice single pulses of intense laser light can vary greatly in intensity, and since TPA is exponentially related to the light intensity these random fluctuations can be exaggerated. By comparing the fluorescence of the sample with each pulse to the reference cell with a known TPA cross section, and then integrating the results you can determine the sample cross section accurately. Otherwise you would have to be able to measure the intensity of each pulse individually which is rather more complicated.

'''Measurement considerations:'''
*fs-ns pulse duration; cw
:<math>
n^{(2)}_{ph} = 1/2 \delta(\omega) N F^2(\omega)\,\!</math>

:<math>n_{fl} = \eta n^{(2)}_{ph}\,\!</math>

*only for fluorescent materials

*90º or backward collection of fluorescence

:<math>signal \propto I^2\,\!</math>
*deviations: linear absorption, stimulated emission, saturation, ...
*importance of spatial and temporal profile

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

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

The perturbative expression for gamma can be applied for two photon absorption with some changes.

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

The denominator is the difference between the state energy (Ege) and the photon energy (&hbar; ω).

There are two components, and the imaginary component accounts for the absorption. The :<math>\Gamma_{ge}\,\!</math> is the damping factor. In order to achieve TPA the sum of the energies of the two photons must equal the energy of the excited state.

For a centrosymmetric molecule :<math>M^2_{ge} \Delta \mu^2_{ge}\,\!</math> (the transition dipole) goes to zero so that entire term goes to zero. For non centrosymmetric molecules both terms are required.

The TPA cross section (δ ) can be related to gamma and several other terms.

:<math>\delta(\omega) = \frac {8\pi^2h\omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

=== Donor-Acceptor TPA Calculations ===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
This is a calculation for a simple donor/acceptor stilbene with an amino group and a formyl group. The delta for the first excited state (S0 to S1) starts out small reaches a peak and then near the cyanine limit it goes down to zero. This is because the delta mu term follows this same pattern.

The change to the second state (S0 to S2) has multiple peaks because the system is two photon resonant and one of the photons is resonant with the S1 state. This is referred to as double resonance and occurs when the transition to S1 is very close to half of the transition to S2. The scale of the cross section for the transition into S1 is on the order 1000 while the transition for S2 is on the order of 10-20000.

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

=== Two-photon absorption example ===
[[Image:Tpa_spectra.png|thumb|400px|Two photon and linear absorption spectra compared]]

Here is a centrosymmetric molecule with donor groups on both ends. It has a long conjugation path. There can be excitation into S1, TPA into S2, rapid relaxation by internal conversion back to S1 and then fluorescence coming out of S2. There is no fluorescence from S2 because in most cases the rate of relaxation from S2 to S1 is much faster than the fluorescence lifetime. In fact in centrosymmetric molecules the transition from S2 to S1 is symmetry forbidden for one photon. Therefore the transition dipole moment is close to zero and the coupling between the grounds and the excited state is close to zero resulting in a long radiative lifetime of the excited state. If there is strong transition dipole going up then there will be a short radiative lifetime in the excited state which in turn leads to a high quantum yield for fluorescence. This is because the rate term for this step dominates the other terms in the expression for quantum yield.

However even if the molecule was not centrosymmetric and both one and two photon transitions are possible the internal conversion relaxation is so fast that there still would not be fluorescence from S2. This leads to the rule coined by Michael Kasha that says that no matter what state you excite into fluorescence will only occur from the lowest lying excited state. There are molecules that are exceptions to Kashsa rule such as azuylene but these are very few. The fluorescence from both one and two photon absorption will be from S1.

In the graph note that the energy of the TPA is lower (longer wavelength- red line) than the single photon absorption (blue line) and of the fluorescence (green line). The TPA peak is around 720, dividing this by two the equivalent single photon energy for this would be 360nm. The peak for linear absorption to S1 is 430nm. This shows that each photon in TPA is absorbing to a state that is higher than the S1 state and that there is very little single photon absorption in this zone (the small absorption seen at 360nm is vibrational rather than electronic).

In non linear optics studies it is important to do the measurements at a variety of wavelengths so that you can get a spectrum. You can’t compare molecules by looking at a single wavelength or using the wrong pulse length.

see Rumi 2000 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

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

Xu and Webb measured the cross section for laser dyes a goupert mayer. One dye had cross section of 300 goupert meyer (two-photon absorptivity, δ, is expressed in Goppert-Mayer units (GM), with 1 GM=1×10-50 cm4 s molecules−1 photon−1.). The process of two photon absorption was predicted by a German physicist Maria Goeppert Mayer in 1929. TPA was not actually observed until the early 60s when lasers were developed that had sufficient intensity to cause it.
Most molecules have GM < 1. These dye have very high two photon cross sections..

See Xu and Webb <ref>C. Xu, JOSA B, 1996;</ref> Albota 1998 <ref>M. Albota, Appl. Opt., 1998;</ref>Fisher 1998 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of measurement: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Various reported measurements of δ for AF-50]]

The chart shows various measurements for the molecule AF-50. This is indicative of the problems of measurement in the NLO field. The measurements were made using various techniques and conditions. The experiment with a delta of 11560 was conducted with a timescale of nanoseconds. This is duration is long enough for there to be a convolution of TPA and single photon absorptions that can appear to give a high TPA cross section. This high value could very well be useful for example in making coatings for safety glasses that could exclude high intensity laser light. But it is not useful for the scientific measurement of the molecules.

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Two photon fluoresence vs non linear transmission ]]

This molecule behaves as an optical limiter. The curve of two photon fluoresence (TPF shown in red) and non linear transmission (NLT shown in blue) are similar except their scales are different. The NLT scale has units of 10-46 and TPF is on the scale of 200, there is a factor of 100 difference. TPF is measure of the cross section, NLT is an effective cross section that involves two photon absorption followed by excited state absorption.
 

=== Initial Observations on Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|bis-Donor Stilbene]]

Bis Donor Stilbene (BDAS) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*The molecule is very electron rich
*Strong nonlinear transmission
*Strong blue fluorescence when pumped with orange light
*Fluorescence depends on I2
*Two-photon cross section, δ = 210 x 10-50 cm4 s/photon, for stilbene δ = 12 x 10-50 cm4 s/photon

The cross section is about 20 times that for the molecule without the electron donor groups. This is very high value and it is useful to learn why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+

It 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 this molecule’s properties.

BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) that is associated with large transition dipole moment between S1 and S2. (μ ee’2 )

These results suggest that a large change in quadrupole moment between S0 and S1 is leads to enhanced delta. 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.

 
'''Bis Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px|]]
This suggests certain design strategies for increasing the dipole or quandrupole moment. The transition dipole moment between the ground and first excited state is 7.2 for stilbene and 8.9 for BDAS. Between S1 and S2 the dipole goes from 3.1 D stilbene to 7.4 D for BDAS. This term is squared and this is the reason there such a high TPA cross section for BDAS.

:<math>\delta_{S_0 \rightarrow S_2} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_1 - E_0 - \hbar \omega ) \eta} \,\!</math>

'''Transition Dipole Moments and BOA'''
[[Image:Tpa_transdip_densities.png|thumb|300px|]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule there will be higher transition dipoles. In DBAS large coefficients on nitrogens coupled with extended distance from the molecular origin result in large excited state coupling.

[[Image:Tpa_transdip_boa.png|thumb|300px|Transition energy in eV for various conditions of charge transfer.]]

There is an optimal amount of charge transfer that is required to get the largest cross section.
 

== Design of TPA chromophores ==
[[Image:Tpa_chromophores.png|thumb|300px|Chromophore variations and δ in 10-50 cm4 s/photon]]

To get large quadrupole moment and large quadrupole moment changes you can do the following;
*Increase the distance between the donors so that you increase the distance over which you transfer the charge.
*Increase the strength of the donors so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule so that that section attracts charge.
*Flip the relation with donors in the center and acceptors on the end.

The diagram shows the effect on delta of different donors (shown in blue) and acceptors (shown in red) configurations and longer molecular chain lengths. The combination of these techniques achieves cross-sections in the thousands.

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|Two Photon Fluorescence by wavelength for various length molecules]]
The graph shows that as the molecule gets longer the energy of the two photon absorption cross section decreases so the wavelength of the peak increases and the magnitude of the TPA increases due to the increasing transition dipole moments
 

=== Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

Here is an expression for delta that explores the change in transition dipole moment between the ground state and the excited state and the photon energy. The oscillator strength of the curve is calculated by integrating the absorption spectra plotted in energy and measuring the area under the curve. From this you can extract the transition dipole moment.

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

where:

:<math>\delta_{max}\,\!</math> is the cross sections that was measured.

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

:<math>E_{\alpha\beta}\,\!</math> is the energy for absorption to for two levels α,β

:<math>\hbar \omega\,\!</math> is the photon energy that was used to get the TPA.

:<math>\Tau_ge^{\prime}\,\!</math> is the damping term which is estimated

In summary

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

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for dipole moments of quadrupolar molecules]]

By making these molecules quadrupolar we increase the transition dipole moment between the first and second excited state (ee and ee’) The data supports this strategy.
 

== 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 polymerization on a precise microscopic scale. A beam can be focused at 3D coordinates in a polymer causing cross linking. Then non-crosslinked portions are washed away. This can be used for microfabrication. The model of the bull is about 5 microns wide and has some very fine features.

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

and Maruo 1997 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

[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, thus increasing the possible resolution]]

Regular photo-initiators are not excellent two photon absorbers. However a femptosecond layer can apply very large power. The goal is strike a power balance between the minimum writing range and the maximum destructive level. For conventional initiators there is only a factor of 2.5 in the power writing range. Newly developed two photon dyes expand this to range to a factor of 50. This lets you write more accurately and faster because the beam does not have remain in the same place as long.
 

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

This technique can be used to convert non fluorescent polymer into a fluorescent form that has a higher density due to cross linking. If the density goes up and the polarizability stays the same the susceptibility goes up and the refractive index goes up. Peter Renzepus at the UC Urvine is using this method to create 3D optical memory. The ability to write on a 100 different planes increases the amount of information that can be stored (gigabits or terabits of data per cm3)

=== 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 dye connected to a photoactive group will absorb the two photons to 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 Nanofabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|Examples of applications for nanofabrication]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to free-form fabricate 3 dimensional structures
*Increasing need for ability 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. The photo on the left is a chain link fence is twice the thickness of a human hair. In tissue engineering cells grow better on certain topologies (ie scaffolds).

==== Media: Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Exposed areas remain after development in the positive tone resist process]]
This is an example of a two photon dye that creates a radical initiator. After absorption the radicals initiate polymerization. The polymer is less soluble and the nonpolymerized material thus this is a known as a negative photo resist.

==== Sub-diffraction limited resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are shown are 170nm.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can do this resolution with 3D resolution was well.
 

==== Media: Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|In positive tone resist the exposed areas are removed during development]]

A positive tone resist becomes more soluble in portions that are exposed to light. This allows you to etch away material by exposing it with light.

Tetrahydropyran can be protonated on the oxygen creating a carbo cation thus converting the esther to a carboxylic acid. Esthers are not soluble in basic water but acids are so by dipping the material in an alkaline solution you can dissolve away the exposed material. A proton is needed.

A photo acid generator (PAG) will absorb light and then transfer an electron to the carbon sulfur sigma bond orbital, cause the bond to break homolytically yielding a methyl radical which will attack benzene and create a proton. The proton then starts the esther to acid reaction.

=== 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.

Two Photon Absorption

2009-12-28T23:22:34Z

128.95.39.42: /* Two Photo Absorption */

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Two photon absorption (TPA)is a third order non linear optical phenonmena in which a molecule absorbs two lower energy photons at the same time. The absorption is sensitive to a threshold value of intensity. This presents a number of possibilities for biomedical applications, and microscopy and microfabrication.

== Two-Photon Excited Processes ==

=== Two Photo Absorption ===
[[Image:Tpa_energy.png|thumb|300px|Two photon absorption takes a molecule directly to the S2 state]]
Two photon absorption is related to the imaginary component of the chi(3) tensor. On a molecular level it is the imaginary component of gamma. Two photons are simultaneously absorbed directly to S2 excited state without population of an intermediate eigenstate. Neither of the two photons have enough energy to get to S1 but with the sum of the two they can reach the second singlet state. The dotted line indicates a virtual state that last a certain length of time (10-15 sec) which means a very short time to get the second photon in. The closer the energy is to the normal S1 state the closer the duration of time is to that of a normal excited state (10-9 sec). As the energy different between S1 and the photon gets smaller, the longer the duration , and the greater the chance of a second photon absorption.

The two photon cross section delta has a term in the denominator referred to as the detuning term that relates to the energy difference. The smaller the difference the higher the two photon cross section.

[[Image:Tpa_centro.png|thumb|400px|Rate expression and selection rules for TPA]]
Once S2 is attained it will quickly relax down to the first excited state and then there can be fluorescence, an additional photon absorption, electron transfer to another molecule, energy transfer to another molecule or photo chemistry. All the same processes can occur that would happen with one photon absorption to the S1. The two photon cross section is very small and it can only be had with the use of intense lasers.

Σ is the one photon cross section which relates to the transition dipole moment. Delta is the two photon cross section.

The rate of formation of the excited state (analagous to two reactants colliding A + B = C) can be thought of as a rate expression that is dependent on concentration of the reactants, and a first order or second order rate constant.

:<math>\frac {dN_{OP}} {dt} = \sigma N_{GS} F\,\!</math> is the rate for one photon absorption

:<math>\frac {dN_{TP}} {dt} = \frac {1}{ 2} \delta N_{GS} F^2\,\!</math> is the rate for two photon absorption

Where
:<math>\sigma\,\!</math> is the intrinsic probability of a collision resulting in absorption, the absorption cross section
:<math>N_GS\,\!</math> is the number of molecules in the ground state
:<math>F\,\!</math> is the flux or intensity

If a reaction requires two A and one B to all collide simultaneously to create C then the rate expression has a third order rate constant.

In two photon absorption the selection rules are the inverse of the rules for those for single photon absorption because there are two transitions, so gerade to ungrerade is forbidden and gerade to gerade, and ungerade to ungerade are allowed.
Single photon absorption is the equivalent of IR spectroscopy
Raman spectroscopy is a two photon process and therefore it is a third order non linear optical phenomena.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/08 TPA.swf</swf>

== Advantages of TPA ==

 
=== Two-Photon Processes Provide 3-D Resolution ===
[[Image:Tpa_cuvette_3D.png|thumb|400px|A sample molecule in a cuvette is bombarded with light tuned for two photon absorption above, and a different wavelength fro one photon absorption below.]]
As you shine a focused beam (bottom) into sample at a wavelength suitable for one photon absorption you get fluoresence along the entire path. A beam tuned for two photon absorption (top) focuses to a very small point. Each single photon has too little energy to be absorbed right up to the point that the intensity is sufficient to trigger TPA. The focal point of the beam varies linearly with the distance from the lens. The area (and therefore intensity) of the beam varies as the square of the radius of beam at that point (pir2), and the two photon absorption varies as the square of the intensity. So TPA varies as the forth power of the distance. Thus TPA builds up and drops off very quickly on either side of the focal point giving it a single point appearance. Thus it can be directed accurately in in three dimensions.

=== TPA Processes Provide Improved Penetration of Light Into Absorbing Materials ===
[[Image:Tpa_cuvette_penetrate.png|thumb|400px|A light beam from the right is quickly absorbed in single photon wavelength below. The upper two photon tuned-beam is able to penetrate without absorption until the the focus gets the intensity to a sufficient level.]]
In a concentrated solution of cumarin one photon absorption (below right) starts immediately at the interface and decays off with the (penetration depth). Because single photons are unable to excite the two photon tuned sample until they are at a critical intensity the two photon laser beam can penetrate farther into a sample. The ability to penetrate a material and to be focused accurately in three dimensions make the TPA ideal for medical applications in which a drug can be activated by TPA at a very precise location.
 

== Calculating and Designing for TPA ==

=== Measuring the two photon cross section ===
[[Image:Tpa_measurement.png|thumb|400px|A beam splitter allows comparison of each pulse through a signal cell and a reference cell]]
To measure the TPA cross section a tunable laser is focused on a beam splitter which then goes to a reference cell with a photomultiplier tube (PMT) and to a signal cell (with unknown sample) and a PMT. A cross section is created by measuring the fluorescence of the unknown sample across a range of wavelengths produced by a tunable laser. In practice single pulses of intense laser light can vary greatly in intensity, and since TPA is exponentially related to the light intensity these random fluctuations can be exaggerated. By comparing the fluorescence of the sample with each pulse to the reference cell with a known TPA cross section, and then integrating the results you can determine the sample cross section accurately. Otherwise you would have to be able to measure the intensity of each pulse individually which is rather more complicated.

'''Measurement considerations:'''
*fs-ns pulse duration; cw
:<math>
n^{(2)}_{ph} = 1/2 \delta(\omega) N F^2(\omega)\,\!</math>

:<math>n_{fl} = \eta n^{(2)}_{ph}\,\!</math>

*only for fluorescent materials

*90º or backward collection of fluorescence

:<math>signal \propto I^2\,\!</math>
*deviations: linear absorption, stimulated emission, saturation, ...
*importance of spatial and temporal profile

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

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

The perturbative expression for gamma can be applied for two photon absorption with some changes.

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

The denominator is the difference between the state energy (Ege) and the photon energy (&hbar; ω).

There are two components, and the imaginary component accounts for the absorption. The :<math>\Gamma_{ge}\,\!</math> is the damping factor. In order to achieve TPA the sum of the energies of the two photons must equal the energy of the excited state.

For a centrosymmetric molecule :<math>M^2_{ge} \Delta \mu^2_{ge}\,\!</math> (the transition dipole) goes to zero so that entire term goes to zero. For non centrosymmetric molecules both terms are required.

The TPA cross section (δ ) can be related to gamma and several other terms.

:<math>\delta(\omega) = \frac {8\pi^2h\omega^2} {n^2c^2} L^4 Im \gamma( -\omega; \omega, -\omega, \omega)\,\!</math>

=== Donor-Acceptor TPA Calculations ===
[[Image:Tpa_donaracceptor.png|thumb|300px|TPA calculations for stilbene]]
This is a calculation for a simple donor/acceptor stilbene with an amino group and a formyl group. The delta for the first excited state (S0 to S1) starts out small reaches a peak and then near the cyanine limit it goes down to zero. This is because the delta mu term follows this same pattern.

The change to the second state (S0 to S2) has multiple peaks because the system is two photon resonant and one of the photons is resonant with the S1 state. This is referred to as double resonance and occurs when the transition to S1 is very close to half of the transition to S2. The scale of the cross section for the transition into S1 is on the order 1000 while the transition for S2 is on the order of 10-20000.

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

=== Two-photon absorption example ===
[[Image:Tpa_spectra.png|thumb|400px|Two photon and linear absorption spectra compared]]

Here is a centrosymmetric molecule with donor groups on both ends. It has a long conjugation path. There can be excitation into S1, TPA into S2, rapid relaxation by internal conversion back to S1 and then fluorescence coming out of S2. There is no fluorescence from S2 because in most cases the rate of relaxation from S2 to S1 is much faster than the fluorescence lifetime. In fact in centrosymmetric molecules the transition from S2 to S1 is symmetry forbidden for one photon. Therefore the transition dipole moment is close to zero and the coupling between the grounds and the excited state is close to zero resulting in a long radiative lifetime of the excited state. If there is strong transition dipole going up then there will be a short radiative lifetime in the excited state which in turn leads to a high quantum yield for fluorescence. This is because the rate term for this step dominates the other terms in the expression for quantum yield.

However even if the molecule was not centrosymmetric and both one and two photon transitions are possible the internal conversion relaxation is so fast that there still would not be fluorescence from S2. This leads to the rule coined by Michael Kasha that says that no matter what state you excite into fluorescence will only occur from the lowest lying excited state. There are molecules that are exceptions to Kashsa rule such as azuylene but these are very few. The fluorescence from both one and two photon absorption will be from S1.

In the graph note that the energy of the TPA is lower (longer wavelength- red line) than the single photon absorption (blue line) and of the fluorescence (green line). The TPA peak is around 720, dividing this by two the equivalent single photon energy for this would be 360nm. The peak for linear absorption to S1 is 430nm. This shows that each photon in TPA is absorbing to a state that is higher than the S1 state and that there is very little single photon absorption in this zone (the small absorption seen at 360nm is vibrational rather than electronic).

In non linear optics studies it is important to do the measurements at a variety of wavelengths so that you can get a spectrum. You can’t compare molecules by looking at a single wavelength or using the wrong pulse length.

see Rumi 2000 <ref>Rumi et al., JACS 122, 9500, 2000</ref>

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

Xu and Webb measured the cross section for laser dyes a goupert mayer. One dye had cross section of 300 goupert meyer (two-photon absorptivity, δ, is expressed in Goppert-Mayer units (GM), with 1 GM=1×10-50 cm4 s molecules−1 photon−1.). The process of two photon absorption was predicted by a German physicist Maria Goeppert Mayer in 1929. TPA was not actually observed until the early 60s when lasers were developed that had sufficient intensity to cause it.
Most molecules have GM < 1. These dye have very high two photon cross sections..

See Xu and Webb <ref>C. Xu, JOSA B, 1996;</ref> Albota 1998 <ref>M. Albota, Appl. Opt., 1998;</ref>Fisher 1998 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
 

=== Vagaries of measurement: The “famous” AF-50 ===
[[Image:Tpa_af50.png|thumb|500px|Various reported measurements of δ for AF-50]]

The chart shows various measurements for the molecule AF-50. This is indicative of the problems of measurement in the NLO field. The measurements were made using various techniques and conditions. The experiment with a delta of 11560 was conducted with a timescale of nanoseconds. This is duration is long enough for there to be a convolution of TPA and single photon absorptions that can appear to give a high TPA cross section. This high value could very well be useful for example in making coatings for safety glasses that could exclude high intensity laser light. But it is not useful for the scientific measurement of the molecules.

 

=== Optical Limiting via Two-Photon Absorption in bis-Donor Stilbene ===
[[Image:Tpn_limiting_bisdonorstilbene.png|thumb|500px|Two photon fluoresence vs non linear transmission ]]

This molecule behaves as an optical limiter. The curve of two photon fluoresence (TPF shown in red) and non linear transmission (NLT shown in blue) are similar except their scales are different. The NLT scale has units of 10-46 and TPF is on the scale of 200, there is a factor of 100 difference. TPF is measure of the cross section, NLT is an effective cross section that involves two photon absorption followed by excited state absorption.
 

=== Initial Observations on Bis-Donor Stilbene ===
[[Image:Tpn_bisdonorstilbene.png|thumb|300px|bis-Donor Stilbene]]

Bis Donor Stilbene (BDAS) is an instructive molecule to study.

'''Evidence for two-photon absorption'''
*The molecule is very electron rich
*Strong nonlinear transmission
*Strong blue fluorescence when pumped with orange light
*Fluorescence depends on I2
*Two-photon cross section, δ = 210 x 10-50 cm4 s/photon, for stilbene δ = 12 x 10-50 cm4 s/photon

The cross section is about 20 times that for the molecule without the electron donor groups. This is very high value and it is useful to learn why the donors have this effect.

'''Interesting features for two-photon applications'''
*High fluorescence quantum yield, φfl ~ 0.9
*High optical transmission
*Low oxidation potential, ED+/D = + 0.035 V vs. Fc/Fc+

It 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 this molecule’s properties.

BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to central vinyl group in the middle (becoming more negative) that is associated with large transition dipole moment between S1 and S2. (μ ee’2 )

These results suggest that a large change in quadrupole moment between S0 and S1 is leads to enhanced delta. 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.

 
'''Bis Donor Substitution'''

[[Image:Tpn_bisdonorstilbene_subst.png|thumb|300px|]]
This suggests certain design strategies for increasing the dipole or quandrupole moment. The transition dipole moment between the ground and first excited state is 7.2 for stilbene and 8.9 for BDAS. Between S1 and S2 the dipole goes from 3.1 D stilbene to 7.4 D for BDAS. This term is squared and this is the reason there such a high TPA cross section for BDAS.

:<math>\delta_{S_0 \rightarrow S_2} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_1 - E_0 - \hbar \omega ) \eta} \,\!</math>

'''Transition Dipole Moments and BOA'''
[[Image:Tpa_transdip_densities.png|thumb|300px|]]
If the components that contribute to transition dipole moments are located farther from the middle of the molecule there will be higher transition dipoles. In DBAS large coefficients on nitrogens coupled with extended distance from the molecular origin result in large excited state coupling.

[[Image:Tpa_transdip_boa.png|thumb|300px|Transition energy in eV for various conditions of charge transfer.]]

There is an optimal amount of charge transfer that is required to get the largest cross section.
 

== Design of TPA chromophores ==
[[Image:Tpa_chromophores.png|thumb|300px|Chromophore variations and δ in 10-50 cm4 s/photon]]

To get large quadrupole moment and large quadrupole moment changes you can do the following;
*Increase the distance between the donors so that you increase the distance over which you transfer the charge.
*Increase the strength of the donors so as to increase the amount of charge that is transferred.
*Add acceptors to the middle of molecule so that that section attracts charge.
*Flip the relation with donors in the center and acceptors on the end.

The diagram shows the effect on delta of different donors (shown in blue) and acceptors (shown in red) configurations and longer molecular chain lengths. The combination of these techniques achieves cross-sections in the thousands.

=== Chain-Length Dependence ===
[[Image:Tpa_chainlength.png|thumb|300px|Two Photon Fluorescence by wavelength for various length molecules]]
The graph shows that as the molecule gets longer the energy of the two photon absorption cross section decreases so the wavelength of the peak increases and the magnitude of the TPA increases due to the increasing transition dipole moments
 

=== Transition Dipole Moments ===
[[Image:Tpa_trans_dipole.png|thumb|400px|]]

Here is an expression for delta that explores the change in transition dipole moment between the ground state and the excited state and the photon energy. The oscillator strength of the curve is calculated by integrating the absorption spectra plotted in energy and measuring the area under the curve. From this you can extract the transition dipole moment.

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

where:

:<math>\delta_{max}\,\!</math> is the cross sections that was measured.

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

:<math>E_{\alpha\beta}\,\!</math> is the energy for absorption to for two levels α,β

:<math>\hbar \omega\,\!</math> is the photon energy that was used to get the TPA.

:<math>\Tau_ge^{\prime}\,\!</math> is the damping term which is estimated

In summary

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

=== Effect of D/A Substitution ===
[[Image:Tpa_donaracceptor_substitute.png|thumb|300px|Trends for dipole moments of quadrupolar molecules]]

By making these molecules quadrupolar we increase the transition dipole moment between the first and second excited state (ee and ee’) The data supports this strategy.
 

== 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 polymerization on a precise microscopic scale. A beam can be focused at 3D coordinates in a polymer causing cross linking. Then non-crosslinked portions are washed away. This can be used for microfabrication. The model of the bull is about 5 microns wide and has some very fine features.

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

and Maruo 1997 <ref>S. Maruo, O. Nakamura & S. Kawata, Opt. Lett. 22, 132 (1997)</ref>

[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, thus increasing the possible resolution]]

Regular photo-initiators are not excellent two photon absorbers. However a femptosecond layer can apply very large power. The goal is strike a power balance between the minimum writing range and the maximum destructive level. For conventional initiators there is only a factor of 2.5 in the power writing range. Newly developed two photon dyes expand this to range to a factor of 50. This lets you write more accurately and faster because the beam does not have remain in the same place as long.
 

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

This technique can be used to convert non fluorescent polymer into a fluorescent form that has a higher density due to cross linking. If the density goes up and the polarizability stays the same the susceptibility goes up and the refractive index goes up. Peter Renzepus at the UC Urvine is using this method to create 3D optical memory. The ability to write on a 100 different planes increases the amount of information that can be stored (gigabits or terabits of data per cm3)

=== 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 dye connected to a photoactive group will absorb the two photons to 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 Nanofabrication ===
[[Image:Tpa_nanofab.png|thumb|300px|Examples of applications for nanofabrication]]
There is a technology pull towards miniaturization of devices and patterned materials.
*Need to free-form fabricate 3 dimensional structures
*Increasing need for ability 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. The photo on the left is a chain link fence is twice the thickness of a human hair. In tissue engineering cells grow better on certain topologies (ie scaffolds).

==== Media: Negative Tone Resist ====
[[Image:Tpa_neg_tone.png|thumb|300px|Exposed areas remain after development in the positive tone resist process]]
This is an example of a two photon dye that creates a radical initiator. After absorption the radicals initiate polymerization. The polymer is less soluble and the nonpolymerized material thus this is a known as a negative photo resist.

==== Sub-diffraction limited resolution ====
[[Image:Tpa_subdiffraction.png|thumb|300px|TPA microfabrication of subdiffraction scale features. The lines are shown are 170nm.]]
Current lithography techniques are able to make structures at 60 nm. However TPA microfabrication can do this resolution with 3D resolution was well.
 

==== Media: Positive Tone Resist ====
[[Image:Tpa_positivetone.png|thumb|300px|In positive tone resist the exposed areas are removed during development]]

A positive tone resist becomes more soluble in portions that are exposed to light. This allows you to etch away material by exposing it with light.

Tetrahydropyran can be protonated on the oxygen creating a carbo cation thus converting the esther to a carboxylic acid. Esthers are not soluble in basic water but acids are so by dipping the material in an alkaline solution you can dissolve away the exposed material. A proton is needed.

A photo acid generator (PAG) will absorb light and then transfer an electron to the carbon sulfur sigma bond orbital, cause the bond to break homolytically yielding a methyl radical which will attack benzene and create a proton. The proton then starts the esther to acid reaction.

=== 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.

Second-order Processes

2009-12-28T22:05:01Z

128.95.39.42: /* Frequency Doubling and Sum-Frequency Generation */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

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

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E e \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the second hyperpolarizability, responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

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

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω1 and ω2 when summed results in a frequency of 2 x ω also referred to as second harmonic generation (SHG) or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage also known as optical rectification.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

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

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. If you map the index of refraction with respect to each polarization of light you end up with a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of alpha).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

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Second-order Processes

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128.95.39.42: /* Frequency Doubling and Sum-Frequency Generation */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

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

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E e \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the second hyperpolarizability, responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

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

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω1 and ω2 when summed results in a frequency of 2 x &omega also referred to as second harmonic generation (SHG); or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

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

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. If you map the index of refraction with respect to each polarization of light you end up with a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of alpha).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

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128.95.39.42: /* Frequency Doubling and Sum-Frequency Generation */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

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

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E e \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the second hyperpolarizability, responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

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

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω 1 and ω 2 when summed results in a frequency of 2 x &omega also referred to as second harmonic generation (SHG); or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

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

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. If you map the index of refraction with respect to each polarization of light you end up with a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of alpha).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

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Second-order Processes

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

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

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E e \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the second hyperpolarizability, responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

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

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω 1 and ω 2 when summed you get a sum of 2 x ω or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

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

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. If you map the index of refraction with respect to each polarization of light you end up with a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of alpha).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Second-order Processes, Materials & Characterization |Return to Second-order Processes Menu]]</td>
<td style="text-align: right; width: 33%">[[Structure-Property Relationships| Next Topic]]</td>
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</table>

OLED Device Applications

2009-12-21T17:53:10Z

128.95.39.42:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: center; width: 33%">[[Main_Page#Organic_Light_Emitting_Diodes|Return to OLED Menu]]</td>
<td style="text-align: right; width: 33%">[[Light Emitting Electrochemical Processes|Next Topic]]</td>
</tr>
</table>
[[Image:OLED_EarlyProduct.JPG|thumb|300px|]]
Organic Light Emitting Diodes (OLEDs)are semiconductor devices that use organic compounds instead of silicon. Electricity is used to create an excited state in the compound which then loses energy in the form a photon emission as it returns to the ground state.

These products represent the fruition of 50 years of research, building first on the principles of silicon LEDS.

OLEDs are just are just beginning to appear in the commercial market. The first OLED devices include TVs, computer monitors, electronic control displays, cameras, phones, and lighting.

[http://depts.washington.edu/cmditr/media/OLEDs.html OLED Concept Map]

===Advantages of OLEDs===
*Superior viewing angle- Monitors and TV screens are visible from side angles, unlike many LCD monitors.
*Color Rendition- New dopants and dyes are being developed to give OLEDs a larger range and flexibility of color rendition.
*Brightness- OLED pixels produce light rather than block light with polarizers as an LCD display does (100,000 cd/m2).
*Faster Response- OLED devices have a typical response time of .01 ms compared to 2.0 ms for LEDs.
*Energy Efficiency- The OLED is an efficient, low power consumpton, low heat light source.
*Low cd drive voltage
*Cost- New polymers and coatings will allow LEDs to be produced by printing and spin-coating techniques.
*Flexibility- Polymer backing and thin coatings permit OLEDs to flex without breaking.
*Thin / lightweight- An OLED display can be thin as a sheet of paper(< 1μm) .

===Device Construction===
An OLED consists of a thin transparent electrode, two or more organic transport/emitting layers, and a metal cathode. When power is applied to the electrodes light is emitted from the central layer.

Individual red, green and blue emitting OLEDs are arranged in a grid with individual power supplies for each pixel. This is called a passive display. This is being replaced with active thin film transistor displays that use a transistor to control each pixel. This is called an active matrix display.

===Design Challenges===
These are some of the challenges that have been undertaken in current research:

*Improve efficiency
*Increase stability and lifetime by excluding oxygen and water
*Demonstrate manufacturability
*Improve color purity
*Demonstrate compatibility with electronic drivers
*Explore OLEDs for white light sources

=== External Links ===

[http://techtv.mit.edu/genres/19-engineering/videos/3175-vladimir-bulovic-on-oled-displays MIT Electric Pickle OLED movie]

[http://en.wikipedia.org/wiki/Oled Wikepedia on OLED]

===Commercial OLED Products===
[http://www.sonystyle.com/webapp/wcs/stores/servlet/CategoryDisplay?catalogId=10551&storeId=10151&langId=-1&categoryId=8198552921644539854| Sony OLED TV]

http://www.universaldisplay.com/

http://www.kodak.com/eknec/PageQuerier.jhtml?pq-path=1473&pq-locale=en_US&_requestid=204

http://www.cdtltd.co.uk/

http://www.novaled.com/

[http://www.ewh.ieee.org/soc/cpmt/presentations/cpmt0401a.pdf Osram Opto Semiconductors]
[[category:organic LED]]
<table id="toc" style="width: 100%">
<tr>

<td style="text-align: center; width: 33%">[[Main_Page#Organic_Light_Emitting_Diodes|Return to OLED Menu]]</td>
<td style="text-align: right; width: 33%">[[Light Emitting Electrochemical Processes|Next Topic]]</td>
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</table>

Scanning Electron Microscope

2009-12-21T17:21:58Z

128.95.39.42: /* Overview */

=== Overview ===
[[Image:Sirion_sem.png|thumb|300px|]]
The scanning electron microscope is used to image the surface of a conducting sample by scanning it with a high energy beam of electrons. Some SEMs have additional software enhancements than enable them to focus the beam on a photomask for [[E-beam lithography]] or are equipped for focused ion beam (FIB) milling. The SEM is a useful tool for photonics research because it reveals nano-scale surface features and topography that is critical to the performance of multi-layer devices.

See Wikipedia on [http://en.wikipedia.org/wiki/Scanning_electron_microscope Scanning Electron Microscope]

=== Operation ===
Part 1 Tour and Sample Preparation
{{#ev:youtube|3oSbtV4BRRM}}

Part 2 Loading the Sample
{{#ev:youtube|8LKnSZfnuIY}}

Part 3 Setting the Working Distance
{{#ev:youtube|-0a0xWxtMTE}}

Part 4 Lens Alignment and Stigmation
{{#ev:youtube|QXm11ERNXUQ}}

Part 5 Moving the Stage and Imaging
{{#ev:youtube|-8FH1I_2IcU}}

Part 6 Changing the Sample and Shutdown
{{#ev:youtube|s1OabL2sLBo}}

Training Manual for Sirion SEM[http://depts.washington.edu/cmditr/media/siriontraining_rev8_04223.pdf]

Training Video on [http://grover.mirc.gatech.edu/training/viewVideo.php?video=sem-high&size=0 Hitachi 3500H SEM at GT MiRC]

=== Significance ===

Scanning Electron Microscope

2009-12-21T17:19:28Z

128.95.39.42: /* Operation */

=== Overview ===
[[Image:Sirion_sem.png|thumb|300px|]]
The scanning electron microscope is used to image the surface of a conducting sample by scanning it with a high energy beam of electrons. Some SEMs have additional software enhancements than enable them to focus the beam on a photomask for [[E-beam lithography]] or are equipped for focused ion beam (FIB) milling.

See Wikipedia on [http://en.wikipedia.org/wiki/Scanning_electron_microscope Scanning Electron Microscope]

=== Operation ===
Part 1 Tour and Sample Preparation
{{#ev:youtube|3oSbtV4BRRM}}

Part 2 Loading the Sample
{{#ev:youtube|8LKnSZfnuIY}}

Part 3 Setting the Working Distance
{{#ev:youtube|-0a0xWxtMTE}}

Part 4 Lens Alignment and Stigmation
{{#ev:youtube|QXm11ERNXUQ}}

Part 5 Moving the Stage and Imaging
{{#ev:youtube|-8FH1I_2IcU}}

Part 6 Changing the Sample and Shutdown
{{#ev:youtube|s1OabL2sLBo}}

Training Manual for Sirion SEM[http://depts.washington.edu/cmditr/media/siriontraining_rev8_04223.pdf]

Training Video on [http://grover.mirc.gatech.edu/training/viewVideo.php?video=sem-high&size=0 Hitachi 3500H SEM at GT MiRC]

=== Significance ===

Scanning Electron Microscope

2009-12-21T17:18:55Z

128.95.39.42: /* Operation */

=== Overview ===
[[Image:Sirion_sem.png|thumb|300px|]]
The scanning electron microscope is used to image the surface of a conducting sample by scanning it with a high energy beam of electrons. Some SEMs have additional software enhancements than enable them to focus the beam on a photomask for [[E-beam lithography]] or are equipped for focused ion beam (FIB) milling.

See Wikipedia on [http://en.wikipedia.org/wiki/Scanning_electron_microscope Scanning Electron Microscope]

=== Operation ===
Part 1 Tour and Sample Preparation
{{#ev:youtube|3oSbtV4BRRM}}

Part 2 Loading the Sample
{{#ev:youtube|8LKnSZfnuIY}}

Part 3 Setting the Working Distance
{{#ev:youtube|-0a0xWxtMTE}}

Part 4 Lens Alignment and Stigmation
{{#ev:youtube|QXm11ERNXUQ}}

Part 5 Moving the Stage and Imaging
{{#ev:youtube|-8FH1I_2IcU}}

Part 6 Changing the Sample and Shutdown
{{#ev:youtube|s1OabL2sLBo}}

Basic tour

(Remaining videos in production)

Training Manual for Sirion SEM[http://depts.washington.edu/cmditr/media/siriontraining_rev8_04223.pdf]

Training Video on [http://grover.mirc.gatech.edu/training/viewVideo.php?video=sem-high&size=0 Hitachi 3500H SEM at GT MiRC]

=== Significance ===

Scanning Electron Microscope

2009-12-21T17:16:57Z

128.95.39.42: /* Operation */

=== Overview ===
[[Image:Sirion_sem.png|thumb|300px|]]
The scanning electron microscope is used to image the surface of a conducting sample by scanning it with a high energy beam of electrons. Some SEMs have additional software enhancements than enable them to focus the beam on a photomask for [[E-beam lithography]] or are equipped for focused ion beam (FIB) milling.

See Wikipedia on [http://en.wikipedia.org/wiki/Scanning_electron_microscope Scanning Electron Microscope]

=== Operation ===
{{#ev:youtube|3oSbtV4BRRM}}

{{#ev:youtube|8LKnSZfnuIY}}

{{#ev:youtube|-0a0xWxtMTE}}

{{#ev:youtube|QXm11ERNXUQ}}

{{#ev:youtube|-8FH1I_2IcU}}

{{#ev:youtube|s1OabL2sLBo}}

Basic tour

(Remaining videos in production)

Training Manual for Sirion SEM[http://depts.washington.edu/cmditr/media/siriontraining_rev8_04223.pdf]

Training Video on [http://grover.mirc.gatech.edu/training/viewVideo.php?video=sem-high&size=0 Hitachi 3500H SEM at GT MiRC]

=== Significance ===

Scanning Electron Microscope

2009-12-21T17:15:24Z

128.95.39.42: /* Operation */

=== Overview ===
[[Image:Sirion_sem.png|thumb|300px|]]
The scanning electron microscope is used to image the surface of a conducting sample by scanning it with a high energy beam of electrons. Some SEMs have additional software enhancements than enable them to focus the beam on a photomask for [[E-beam lithography]] or are equipped for focused ion beam (FIB) milling.

See Wikipedia on [http://en.wikipedia.org/wiki/Scanning_electron_microscope Scanning Electron Microscope]

=== Operation ===
{{#ev:youtube|3oSbtV4BRRM}}

{{#ev:youtube|8LKnSZfnuIY}}

{{#ev:youtube|0a0xWxtMTE}}

{{#ev:youtube|QXm11ERNXUQ}}

{{#ev:youtube|8FH1I_2IcU}}

{{#ev:youtube|s1OabL2sLBo}}

Basic tour

(Remaining videos in production)

Training Manual for Sirion SEM[http://depts.washington.edu/cmditr/media/siriontraining_rev8_04223.pdf]

Training Video on [http://grover.mirc.gatech.edu/training/viewVideo.php?video=sem-high&size=0 Hitachi 3500H SEM at GT MiRC]

=== Significance ===

Fluorescent/Phosphorescent Dopants

2009-12-17T21:29:06Z

128.95.39.42: /* Increasing Performance */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Organic Heterojunctions|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Organic_Light_Emitting_Diodes|Return to OLED Menu]]</td>
<td style="text-align: right; width: 33%">[[Metal Complex Dopants|Next Topic]]</td>
</tr>
</table>

==Example OLED applications==
The Pioneer car stereo was one of the first green and blue OLEDs to reach the market.

Polymer OLEDs have an emissive layer of polyphenylenevinylene derivatives which are spin-cast or deposited with an inkjet printing process. The first polymer OLEDs consisted of two different polymer materials with different ionization potentials and electron affinities to create a heterojunction. These were capped with calcium or magnesium, and more recently with aluminum or other more stable metals.
==Jablonski Diagram==
[[Image:Oled1 19 jablonski ratio.png|thumb|400px]]
The Jablonski diagram shows a dopant molecule that can be excited from its singlet state to its first excited singlet state by overlap of its absorption with the emission from the host dye such as aluminum quinolate. The first dopants were primarily fluorescent dopants which shift the energy of the emission of the device slightly more to the red and improve the efficiency and stability of the device. The holy grail is to identify dopants which can capture the energy in the triplet state, which is 75% of the energy in the system.

 
==Device Efficiency==
[[Image:Device Efficiency.JPG|thumb|400px|External efficiency is a measure of how much light is produced given the power supplied.]]
External power efficiency is a product of the light out-coupling efficiency and the internal efficiency. Typical OLEDs only emit about 20% their light in the forward direction. The remaining energy is lost in substrate or waveguide modes which go to the side, which opens intriguing possibilities for sensors. Internal efficiency is related to the efficiency of all the components in the OLED.

:<math>\eta _\mathrm{ext} = \eta_\mathrm{ph} \eta_\mathrm{int} = \eta_\mathrm{ph} \gamma \phi \eta_\mathrm{ex}\,\!</math>

where

:<math>\eta _\mathrm{ext}\,\!</math> = external efficiency

:<math>\eta _\mathrm{ph}\,\!</math> = light out-coupling efficiency (~ 20 %)

:<math>\eta _\mathrm{int}\,\!</math> = internal efficiency

:<math>\gamma\,\!</math> = ratio of electrons to holes typically ≤ 1. There is an energy loss due this imbalance.

:<math>\phi\,\!</math> = quantum efficiency of the emitting molecule. Optimal molecules will have the highest possible quantum efficiency for luminescence.

:<math>\eta _\mathrm{ex}\,\!</math> = fraction of luminescent excitations based on spin statistics. Fluorescent OLEDs have external efficiency of about 5%, while phosphorescent OLEDs have up to 20% efficiency.

The quantum efficiency can also be written

:<math>\phi = \phi_R \cdot \phi_S \cdot \phi_F \cdot \phi_E\,\!</math>

where

:<math>\phi_R\,\!</math> is Langevin Recombination efficiency

:<math>\phi_S\,\!</math> is Singlet Triplet Branching efficiency

:<math>\phi_F\,\!</math> is Fluorescent efficiency

:<math>\phi_R\,\!</math> is External Coupling efficiency

==Increasing Performance==
You can increase device performance by balancing charge injection (γ). Single anthracene crystal devices had an efficiency < 1%. The first double-layer devices increased efficiency to 1%. The first doped devices used DCM2- a cumarin dye used in some dye lasers. It was doped at a high level of 10% works as a energy acceptor from the Alq3 host and it boosted the external efficiency to 2.5%. It is an orange–red emitter which moves the emission from green to the red part of the spectrum.

See Tang <ref>Tang et al. Appl. Phys. Lett. 1987, 51, 913-915</ref>

[[Image:Forstertransfer.JPG|thumb|400px|Förster is a non-radiative transfer of energy from a donor to an acceptor.]]
You can also increase device performance by increasing Φ using energy transfer. Consider the dynamics of Förster energy transfer. Spin statistics dictate that 75% of the electron transfer is in the triplet state. In the case of emission, the singlet state gives you fluorescence, and phosphorescence comes from the decay of the triplets state back to the ground state. It would be best to capture the energy from both states. Energy efficiency is strongly dependent on the overlap of donor emission with acceptor absorbance.

See Wikipedia on Förster transfer<ref>http://en.wikipedia.org/wiki/Fluorescence_resonance_energy_transfer</ref>
 
[[Image:Phosphorescent dopants.JPG|thumb|400px|Phosphorescent doping increases the amount of intersystem crossing]]

Because of the weak probability of intersystem crossing from fluorescent to the phosphorescent triplet state these molecules have lifetimes that are milliseconds long. Display devices such as OLEDs need to have lifetimes that are shorter.

Phosphorescent dopants such as the heavy metal porphyrin PtOEP can increase efficiency to 5%. The heavy metal increases the probability of intersystem crossing.

See Baldo 1998 <ref>Baldo et al. Nature 1998, 395, 151</ref>

The mechanism of triplet energy transfer, called Dexter energy transfer, involves the exchange of an electron. This has a much closer distance-dependence than Forster transfer. The right phosphorescent dyes create triplet states at relatively low concentrations of these energy acceptors. Efficiency is strongly dependent on the orbital overlap between the donor and acceptor system. Heavy metal atoms such as platinum and iridium help to mix the singlet and triplet states by spin-orbit coupling.

See Tang 1989 <ref>Tang et al. J. Appl. Phys. 1989, 9, 3610-3616</ref>

==Excited States of Organometallic Complexes==
[[Image:organometalliccomplex.JPG|thumb|400px|Ligand to Metal Charge Transfer (LMCT) compared to Metal to Ligand Charge Transfer (MLCT) with Metal Centered (MC) and Ligand Centered (LC)]]
There are a couple of considerations when you work with organometallic complexes. There are ligand to metal transitions in the absorption of these molecules. There are ligand centered transitions, metal to ligand centered, and metal centered transitions. One must understand which of the molecular orbitals you are going to use to maximize the triplet state and the phosphorescence.

See Balzani 1991 <ref>Balzani, V. and Scandola, F. Supramolecular Photochemistry, Ellis Harwood, England 1991</ref>, for a review of the rules for design of phosphorescent dopants.

Platinum octoethyl porphyrin (Pt OEP) was the first phosphorescent dopant. The graph shows the host emission from an aluminum quinolate diode. At 20% PtOEP there is virtually no emission from the Alq3 and nice red emission from the PtOEP.

[[Image:PtOEP.png|thumb|300px|Platinum octaethylporphyrin|http://www.chemspider.com/ImageView.aspx?id=21169877&mode=3d]]

 
[[Image:OLED9_ptoep_efficiency.JPG|thumb|400px|Quantum efficiency drops off at higher current density.]]
It is important to look at the quantum efficiency for these devices as a function of the applied current density. The optimal system will maximize light output while minimizing the current injected into the system. At a threshold luminance of 100 candelas per square meter (about the about the intensity of a computer screen display), PtOEP has a quantum efficiency of 1% and current density of 10 mA per square centimeter. This is not bad but could be better. PtOEP might not be the best choice for an emitter because of its long lifetime of 100 microseconds. Some energy will be lost through this phosphorescence because it lasts so long and there are other non-radiative decay processes that suck energy away.
[[category:organic LED]]
 

== References ==

<references/>
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Organic Heterojunctions|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Organic_Light_Emitting_Diodes|Return to OLED Menu]]</td>
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Concept Map

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This concept map shows the linkage between concepts and knowledge domains that reflect the interdisciplinary approach of the STC.

Concept Map

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This concept map shows the linkage between concepts and knowledge domains that reflect the interdisciplinary approach of the STC.

__NOINDEX__

singularity

Concept Map

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This concept map shows the linkage between concepts and knowledge domains that reflect the interdisciplinary approach of the STC.

Main Page

2009-12-15T17:08:29Z

128.95.39.42: /* Concept Map */

<big>'''Center for Materials and Devices for Information Technology Research (CMDITR) Wiki'''</big> 
This wiki is a reference collection on photonics. Most of the text has been captured from a series of lectures recorded in 2005-2008 by Center faculty Jean-Luc Bredas (Georgia Tech), Neal Armstrong (University of Arizona) and Seth Marder (Georgia Tech). You may also want to search the
[http://depts.washington.edu/cmditr/cwis/SPT--Home.php CMDITR Photonics Digital Libary] for individual learning objects.

(The sections below with ** asterisks are currently in development, the rest are in draft form)

== Photonics Core Concepts and Applications ==

 

===Basics of Light ===
[[Image:Snells_law_wavefronts.gif|thumb|150px|]]
*[[Propagation, Reflection and Refraction]]
*[[Dispersion and Scattering of Light]]
*[[Diffraction of Light]]
*[[Electromagnetic Radiation]]
*[[Luminescence Phenomena]]
*[[Color and Chromaticity]]

 

=== Optical Fibers, Waveguides, and Lasers ===
[[Image:800px-Military_laser_experiment.jpg|thumb|200px|]]

*[[Optical Fibers]]
*[[Total Internal Reflection]]
*[[Planar Dielectric Waveguides]]
*[[Optical Fiber Waveguides]]
*[[Dispersion and Attenuation Phenomena]]
*[[Lasers]]

===Molecular Orbitals===
[[Image:HAtomOrbitals.png|thumb|150px|]]
*[[Atomic Orbitals and Nodes]]
*[[Electronegativity and Bonding Between Atoms]]
*[[Sigma and pi Orbitals|Sigma and Pi Orbitals]]
*[[Polarization and Polarizability]]
*[[Electronic Coupling Between Orbitals]]
*[[Donors and Acceptors]]
 

===Electronic Band Structure of Organic Materials===
[[Image:Ethylene.JPG|thumb|200px|]]
*[[Introduction to Band Structure]]
*[[Electronic Structure of Hydrogen]]
*[[The Polyene Series]]
*[[Bloch's Theorem]]
*[[Electrical Properties]]
*[[Electronic States vs Molecular Levels]]
 

===Absorption and Emission of Light===
[[Image:Abs Emis stokes.png|thumb|200px|]]
*[[Introduction to Absorption]]
*[[Changes in Absorption Spectra]]
*[[Jablonksi Diagram]]
*[[Fluorescence Process]]
*[[Transition Dipole Moment]]
*[[Absorption and Emission]]
*[[Photochromism]]
*[[Interchain Interactions]]

 

===Transport Properties===
[[Image:rubrene.png|thumb|150px|]]
*[[Charge Carrier Mobility]]
*[[Band Regime versus Hopping Regime]]
*[[Electronic Coupling]]
*[[Model Calculations of Electronic Coupling]]
*[[Marcus Theory and Reorganization Energy]]
*[[Electron-Phonon Coupling]]

 

===Liquid Crystals and Displays===
[[Image:smectic_C.jpg|thumb|200px|]]
*[[Liquid Crystals]]
*[[Double Refraction and Birefringence]]
*[[Director – Degrees of Order in Liquid Crystals]]
*[[Classification and Examples of Liquid Crystals]]
*[[Alignment]]
*[[Freederickz Transition and Dielectric Anisotropy]]
*[[Liquid Crystal Displays]]

 

===Organic Light Emitting Diodes===
[[Image:PNNL_Light_Lab_041.jpg|thumb|200px|Blue phosphorescent OLED developed by Pacific Northwest National Laboratory.]]
*[[OLED Device Applications]]
*[[Light Emitting Electrochemical Processes]]
*[[The OLED Test Cell]]
*[[What is a Light Emitting Diode?]]
*[[The First OLEDs]]
*[[Organic/Organic Heterojunctions in OLEDs]]
*[[OLED Charge Mobilities]]
*[[Organic Heterojunctions]]
*[[Fluorescent/Phosphorescent Dopants]]
*[[Metal Complex Dopants]]
 

===Organic Solar Cells===
[[Image:Opvtestcells.png|thumb|200px|OPV Test Cells]]
*[[Organic Solar Cells|OPV Introduction]]
*[[Solar Technologies]]
*[[Major Processes in Organic Solar Cells]]
*[[Organic Heterojunctions in Solar Cells]]
*[[Physics of Solar Cells]]
*[[Energy vs Charge Transfer at Heterojunctions]]
*[[Current OPV Research Directions]]
 

===Organic Electronics===
*[[Organic Electronics Overview]]
*[[Synthesis of Organic Semiconductors]](In progress)
*[[field effect transistors]]
*Design of n-type Semiconductors for Organic Electronic Applications

==Non linear Optics and Devices==

===Quantum Mechanical and Perturbation Theory of Polarizability===
*[[Quantum-Mechanical Theory of Molecular Polarizabilities]]

===Second-order Processes, Materials & Characterization ===
[[Image:MachZehnder.gif|thumb|200px]]
*[[Second-order Processes]]
*[[Structure-Property Relationships]]
*[[Second-order NLO Materials]]
*[[Second-order Material Design]]
*[[Terahertz Radiation]]
*[[Second-order Material Characterization]]

 

===Third-order Processes, Materials & Characterization ===
[[Image:Tpa_concentrated.png|thumb|100px|]]
*[[Introduction to Third-order Processes and Materials]]
*[[Two Photon Absorption]]
*Advanced Concepts in Third-order Processes
*Characterization of Third-order Materials (Perry)
 

===**Techniques for Fundamental Processes (Ginger) ===

===Organic Photonics Applications in Information Technology ===
[[Image:Dualmz packaged.png|thumb|200px|]]
*[[Optical Networks]]
*[[Passive Optical Polymers]]
*[[Electro-optic Polymers and Devices]]
*[[Materials Processing and Fabrication]]
 

===Photonics Integration===
[[Image:Si_waveguide_em.jpg‎|thumb|200px|]]
*[[The Need for Photonic Integration]]
*Integrated Si Photonics (Hochberg)
 

== Research Equipment, Devices and Techniques ==
 
[[Image:PES.jpg|thumb|200px|]]
'''Characterization'''
*[[Photoelectron Spectrometer XPS and UPS]]
*[[Conducting Tip Atomic Force Microscopy]]
*[[Organic Photovoltaic Fabrication and Test Apparatus]]
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*[[Hyper Rayleigh Scattering]]

'''In Development'''
*[[Scanning Electron Microscope]]
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'''Fabrication'''
*[[E-beam Lithography]]
*Reactive ion etcher
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*Atomic layer deposition
*[[Spin coater]]
*Sputter coater

==Acronyms and Unit Abbreviations==
*[[Acronyms]]
*[[Variables and Constants]]
*[[Units]]

== General Research Best Practices ==
*[[How to Keep a Lab Notebook]]
*[[How to Give a Research Presentation]]
*[[Writing a Scientific Paper]]
*[[Writing a Successful Proposal]]
*[[Mentoring]]

==[[External Photonics Education Links]]==

==K-12 Outreach Kits==
[[Image:AssembledCell_small.JPG|thumb|200px|]]
*[[K-12 Outreach Introduction]]
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Main Page

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Introduction to Third-order Processes and Materials

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<table id="toc" style="width: 100%">
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<td style="text-align: center; width: 33%">[[Main_Page#Third-order Processes, Materials & Characterization |Return to Third-order Processes, Materials & Characterization Menu]]</td>
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Third order non linear optical (NLO) materials change their polarization based on the intensity of the applied light. This gives rise to a variety of useful properties such as self-focusing, two photon absorption, and third harmonic generation.

== Hyperpolarizability ==

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field. This relates to χ(3), which is a materials property. χ (3) and γ can exist in all materials and all molecules, even those which are centrosymmetric materials. (χ(2) can only happen in non centrosymmetric materials).

=== 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.

=== Taylor Expansion for Polarization ===

When all the fields are identical there is a Taylor expansion for bulk polarization:

:<math>P = P_0 + \chi^{(1)}·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math> (9)

Some materials such as polyvinylene difluoride when polled can have a bulk polarization in the absence of an applied field.

Just as a molecule can only have a β if it is noncentrosymmetric, a material can only have a χ(2) if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero χ(2)) .

=== 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.

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.

<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]]

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Second-order Processes

2009-12-14T16:44:05Z

128.95.39.42: /* EO Materials have a voltage-controlled index of refraction. */

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Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

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

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E e \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the second hyperpolarizability, responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

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

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω 1 and ω 2 when summed you get a sum of 2 x ω or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

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

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. If you map the index of refraction with respect to each polarization of light you end up with a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of alpha).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

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Organic Heterojunctions in Solar Cells

2009-12-14T16:40:22Z

128.95.39.42: /* Excitonic Solar Cells */

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[[category:organic solar cell]]

[[Image:Opv_flexible.jpg|thumb|300px|A flexible organic solar cell prototype.]]

A heterojunction is the contact of two materials with different electrical properties. These materials could be polymers or small molecules and both types are used in our research. The donor is more electron rich than the acceptor. When the donor is excited by light it causes a negative charge in the acceptor while the donor becomes positively charged. The positive charges move by hole hopping, and the negative charges by electron hopping via charge transfer between molecules. Thus charges move in the organic layers which force charge to move in the electrodes.

== Materials used in Organic Heterojunctions ==
<embed_document width="55%" height="400">http://depts.washington.edu/cmditr/media/type2-heterojunctionOPV.pdf</embed_document>

See review article Gunes 2007 <ref>Serap Gunes, Conjugated Polymer-Based Organic Solar Cells, Chem. Rev 2007, 107, 1324-1338</ref>

 

[[Image:OPV21_materials.JPG|thumb|400px|As an example Poly(3-hexylthiophene) a serves as a donor which absorbs in the blue spectrum, PCBM is an acceptor based on a functionalized buckball (C60)]]

For organic solar cells and organics light emitting diodes, the materials belong to the same two types of schools. One type uses small molecules that you can vapor deposit in a vacuum. Another consists of uses polymers can be processed through wet chemistry. A ubiquitous component as an electron acceptor is Fullerene or C60. In particular, this derivative of C60 provides for a much better compatibility with the other components. Therefore c60 is an excellent electron acceptor and what is left needed is an electron donor. It is found that if you just use C60 it will usually face segregate. Where as if C60 or C70 is prepared as a PCBM there that provides for a much better compatibility and the ability to have better quality blends with a donor polymer such as polyphenyl vinylene (PPV) or a polythiophene such as the regioregular poly(3-hexylthiophene) (P3HT)

So PPV and P3HT in the polymer form or in the oligomer form are frequently used as electron donors. C60 is the electron acceptor but there are other systems. For instance Bernard Kippelin and his group have been able to design and fabricate excellent solar cells based on C60 as the acceptor and pentacene as the donor.

 

== Energy levels in heterojunctions==

[[Image:Organicheterojunctions.JPG|thumb|400px| An energy diagram includes the WA work function of the anode, WC the work function of the cathode; similar to the diagram for light emitting diodes.]]
These are the levels or the states. It is important to understand the differences between the levels that refer to one electron level picture and states that refer to the whole molecule or system. The ionization potential of a molecule can be approximated as the HOMO, the energy or the inverse of the energy of the HOMO level. The electron affinity can be related to the energy of the LUMO level. But one must not forget that the ionization potential and electron affinity are characteristics of the whole molecule.

The energy state diagram refer to states, the difference in energy between the ground state and the ionized state of the whole molecule. Where as HOMOS and LUMOS refer to electron levels. When organic solar cells are discussed, we must pay attention to the electron affinities and ionization potentials of the donor and acceptor components. The ionization potential of the donor is the difference between the neutral molecule and the energy it takes to take away an electron from that molecule. HOMO and LUMO levels are often seen in the literature but these are only approximations. We need to take into account are the ionization potentials and the electron affinities of both the donor and the acceptor. In order for this concept of the donor and acceptor to be valid, the ionization potential of the donor should be lower than that of the acceptor because the donor is the species that most easily gives away an electron. In this diagram the ionization energy of the donor is the distance between to top of the donor band and the vacuum level band above. Conversely, the electron affinity of the acceptor should be larger than the electron affinity of the donor.

The photovoltage (Voc) is dictated by the relative HOMO (ionization potential) energy level of the donor and the LUMO (electron affinity) of the acceptor.

<math>V_{oc}= (\frac {1}{ e})(|E^{Donor}HOMO |-|E^{PCBM}LUMO|)-).3V\,\!</math>

Voltages with OPVs are typically very low so many layers in series may be required. Heeger group recently developed a repeated multilayer that effectively double the voltage and allowed for a second donor pigment that could harvest a different part of the spectrum.

See Heeger 2007 <ref>Heegar, A. Efficient Tandem Polymer Solar Cells Fabricated by All-Solution Processing, SCIENC, Vol 317 July 13 2007
</ref>

See [[Work Function of Metals]]
[[Image:Wrongmatchenergy.jpg|thumb|300px|Consider a system with two compounds, A and B. If compound B has lower Lumo and a higher Homo than compound A you would not expect separation of charges. If there is an excess electron it will go the lowest Lumo which is compound B. If there is an excess hole it will go to the highest homo which is also compound B. The excess electron and the hole will want to be in this compound B so there won’t be any charge separation; it will not be favored. ]]

== Selecting Donor and Acceptor energy levels ==

A favored situation is shown where the LUMO of the donor is higher than the LUMO of the acceptor, and the HOMO of the donor is higher than that of the homo of the acceptor. This is a situation where charge separation might be possible but it doesn’t guarantee that it will happen. It is a necessary but it not sufficient condition. For charge separation to occur the combination of these differences in electron affinities and ionization potentials must have the ability to overcome the exciton binding energy. For example in the case of ppv and cyano ppv the levels that are aligned like this but still there is no charge separation. There is energy transfer but no charge separation.

Consider these structures in the absence of any external field. There are two semi conductors, with a work function of the anode, and the work function of the cathode. The work function refers to the Fermi level that is the chemical potential of the two electrodes. When a system is in equilibrium, the system tends to get the chemical potentials to be the same across the entire system. In order to reach an equilibrium, the system will tend to have some charge redistribution that will increase chemical potential on the anode side and decrease the potential energy on the cathode side so that the two chemical potentials can try to align.

Since the chemical potential of the anode and the cathode are different, there will be some charge redistribution to increase the energy on the anode side and decrease the chemical potential on the cathode side. This will create a bias of the levels within the organic semiconductor or insulators, which is called a built in potential or built in electrical field. In the physics literature, a description of what is happening can be found in the context of metal insulators and metal structures. If there is an elimination and an electron hole pair is formed at the interface, the electron will travel to the acceptor side, and a hole will be left on the donor side. This built-in potential makes the electron drift towards the cathode and the hole drift towards the anode. Without that built-in field, there will be no incentive for the electron to drift one way or the other; the electron and hole would diffuse. However, the built in potential gives a direction for the motion of the separated charge carriers.

== Excitonic Solar Cells ==
[[Image:Hjsteps.JPG|thumb|400px|Five steps of the excitonic process]]
There are five different steps in excitonic solar cells. These 5 steps should also be valid for the organic light emitting diodes-- the fifth step would be the out coupling. The slide shows the energy scheme.

'''Step A photon absorption''' What is most favorable is to have a material that absorbs as intensely and broadly as possible across the solar spectrum.

'''Step B Exciton formation''' This whole process occurs in excitonic solar cells. On the other hand, in organic semi conductors, this exciton part is not dealt with. Actually the very first organic solar cells that has been described in the literature in 1959 by Martin Pope and his workers used a single component called anthracene. The number of charges that were generated was extremely low because when the exciton was formed, it remained bound. Thus, not many charges were separated and little current was generated which leads to a poor efficiency. That is particularly why Ching Tang <ref>C.W. Tang, APL 48, 193 (1985)</ref> made a major advance using heterojunctions that consists of a donor and an acceptor. At step B, the exciton diffuses.

:<math>L_D = \sqrt{D\tau}\,\!</math>

where:
:<math>L_D\,\!</math> is the diffusion length
:<math>D\,\!</math> the diffusion coefficient
:<math>\tau\,\!</math> is the lifetime of the exciton.

As you can see, the equation is an important parameter. The diffusion coefficient D represents how fast the exciton can diffuse within the material or how fast the exciton can hop from molecule to molecule, or from chain segment to chain segment. The larger the diffusion coefficient, the longer the lifetime, which means the further the exciton can go before it decays back to the ground state. It is better if the exciton has a long diffusion length LD because it allows the exciton to have a higher chance of reaching the interface with the acceptor within its lifetime. If this process does not occur, no charge will generate and the exciton will decay back to the ground state. This decay will either produce heat, vibrations, or release the photons it once absorbed-- also called photoluminescence. But photoluminescence is not what you want in solar cell.

:<math>\alpha L\,\!</math> is the figure of merit. It is the product of the diffusion length and the absorption coefficient or the absorbance of the material. The most efficient organic layer would be very absorbing, have a large alpha, and the excitons would have a large diffusion length so that the exciton will have a large probability of finding the interface during its lifetime.

In the following Flash simulation the path of an exciton is visualized as you control the thickness of the electon donor layer. You can also experiment with the use of projecting "fingers", a dendritic surface morphology that is being attempted. Note that exciton migration is random diffusion process that does not have a directionality as shown here for instructive purposes. We are only showing the excitons that migrate in a productive direction.

[[Image:opvdesignimprove.jpg|thumb|400px|Light absorption and exciton diffusion. The red line shows the intensity of light as a function of thickness of donor layer. It shows an exponential drop indicating that a thicker layer could absorb all the light. The dashed yellow line indicates the exciton migration before it dies. A thicker layer absorbs more light but the excitons do not make it to the interface to generate a charge. As consequence organic dye layers must be really thin. ]]

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/Excitonmigration.swf</swf>
'''Step C Charge Separation''' Disassociation, or charge separation, occurs at the interface. Once the electron and the hole have separated, the built in potential cause the electron to drift toward the cathode and the hole to drift toward the anode. If there was no built in potential, the electron and the hole will simply diffuse without any directionality to their motion.

'''Step D Charge Migration''' the charge move away from the heterojunction interface toward the electrodes.

'''Step E Charge collection''', The electron and the hole will be collected efficiently at their respective electrodes, generate a current, and enter the external circuit

 

== Geometries of excitonic solar cells ==
[[Image:Opvgeometry.JPG|thumb|400px|Multilayer thin films vs bulk heterojunction]]
Recently, there were steady improvements in efficiencies of organic solar cells. Whether it is based on multilayer thin film geometry or intepenerating polymer blend, the efficient light harvesting relies on donor-acceptor heterojunction in most cases. Therefore, it is our purpose to establish the model that characterizes the photoelectrical response of the cellls based on those donor-acceptor heterojunctions.
These are the two major geometries that have been considered. For example, from Chin Tang’s paper in 1986, one of the geometries included a bi-layer: it had an anode, cathode, donor component, and an acceptor component. The idea of Chin Tang is to move away from a homojunction that uses a single component, and go to a two component system with a donor and an acceptor. However, an issue with this type of system is that the diffusion length must be quite large in order for the exciton to reach the interface and disassociate. It is possible to make the layer thinner so that the distance to the interface is shorter, but in doing so, the layer will be less absorbing and fewer photons will be absorbed.
 
[[Image:opvmultilayer.jpg|thumb|400px|Typical multilayer OPV with associated band gaps.]]
A typical multilayer device is built on glass or plastic substrate. First a layer of transparent, conductive indium tin oxide (ITO) as an electrode, A thalociline donor molecule, and then an acceptor layer containing buckyballs and lastly an opaque aluminum electrode. Light shines through the glass and the ITO, is absorbed by the donor, charge is passed to the acceptorand then finally to the aluminum electrode.

See review by Forrest <ref>Forrest,Steve MRS Bulletin 2005</ref>
 

== Bulk heterojunctions ==
[[Image:Bulk_heterojunction.png|thumb|200px|A bulk heterojunction mixes the donor material with the acceptor material]]
So the idea in the mid 90’s, from the group of Allen Heegger in Santa Barbara, was to build a bulk heterojunction. Bulk heterojunctions are made by creating interpenetrating networks of the donor and the acceptor components. With the two components interpenetrating one another, an exciton will never be far from the interface. In addition, with respect to the volume, the surface represented by the interface will be larger. The interfacial area will be large and the diffusion length of the excitons can be short. This puts many constraints on how to manage the morphology of the interpenetrating network. This is due to the fact that once the disassociation has taken place, the electron and the hole need to have a clear path to get to the cathode and the anode respectively. If the morphology is such that the path closes somewhere along the way, the electron and the hole will be lost. Bulk heterojunctions, are described as systems of interpenetrating “fingers” of the donor component and the acceptor component sandwiched between the anode and cathode. This system has a large interfacial area, the interface is not far from where the exciton is generated, and once the exciton disassociates, the electron and the hole will have a clear path to their respective electrodes. Creating perfecting aligned interdigitating fingers is a challenge for material designers. In practice the interface is much more uneven and the path the exciton must take is perilously long.

You are often are limited by the diffusion length of the exciton. All these factors contribute to why people have spent so much time trying to design efficient bulk heterojunctions. The morphology must be controlled so that the electron and the hole can always go back to the cathode and anode respectively. Also there should not be many places where the hole and the electron see each other and can recombine.

Another problem is that of phase separation of the heterojunction blend, which decreases performance.

 

== Pioneers in organic photovoltaics ==
[[Image:Opvpioneers.JPG|thumb|400px|]]
There are a few people who have advanced the field. Chin Tang used to work for the Kodak research center but moved to become a full professor at University of Rochester in 1986. There have also been major advances from Steve Forrest who was at Princeton for quite some time and had recently transferred to his alma meter as VP for research at Michigan. In terms of polymers, a major step forward in understanding was made when Serdar Sariciftci was working as a research scientist in Allen Heeger’s groups in the early 90’s. They discovered an ultra-fast charge separation at the interface between a PPV derivative and C60. The charge separation occurred within about 30 – 40 femptoseconds. Since the charge separation is so much faster than any other process, it leads to a charge separation of unity. That is why there has been so much work on phenyline, vinylines with c60 or c60 derivatives as well as more recently on polythiophenes or derivatives thereof and C60 and derivates thereof.

== References ==
<references/>

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Major Processes in Organic Solar Cells|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Organic Solar Cells|Return to OPV Menu]]</td>
<td style="text-align: right; width: 33%">[[Physics of Solar Cells|Next Topic]]</td>

</tr>
</table>

External quantum efficiency

2009-12-10T18:51:44Z

128.95.39.42: New page: === Overview === External Quantum Efficiency is a measure of.. === Operation === === Significance === <ref>article</ref>

=== Overview ===

External Quantum Efficiency is a measure of..

=== Operation ===

=== Significance ===

<ref>article</ref>

Main Page

2009-12-10T18:50:37Z

128.95.39.42: /* Research Equipment, Devices and Techniques */

Lasers and Telecommunication- Outreach Kit

2009-11-12T17:21:36Z

128.95.39.42: /* Sources for Building your own kit */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photovoltaics- Outreach Kit|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#K-12 Outreach Kits|K-12 Outreach Kits]]</td>

</tr>
</table>
=== Overview ===
One of the major research thrusts of CMDITR is organic electronics that can be used in information technology and telecommunications. At the heart of this is the modulation of light using new organic electro-optical materials. Students need to understand how the system currently uses light and optics to carry information. Electro-optic materials can change their index of refraction in the presence of an electric field. This property combined with wave interference makes high speed switching possible. Finally, electro-optical and all optical switches can be miniaturized to the nano-scale to take advantage of other unique properties in device design.

=== Materials in the kit ===
*laser pointer
*plastic tub
*plastic tubing
*Optic fiber receiver and transmitter
*Michelson interferometer

=== User Guide ===
[[Image:120px-Digital.signal.png|thumb|300px|]]
1) '''Information can be carried by modulating electrical signals either analog (continuously changing) or digital (timed pulses either on or off).''' Show or draw pictures of waveforms of analog and digital signals.

2) '''Information can be carried by light.''' Bounce a laser off of mirror attached to a balloon stretched on a can. Tap the rubber membrane or speak into it. Brainstorm the advantages of using light as carrier for information?

*CMDITR is working on ways to modulate light by controlling the optical properties of the materials it passes through. This may lead to improved efficiency of switching light in communication systems. It also may lead to all-optical switching (AOS) in which light from one channel controls light in another channel.

3) '''Optic fiber transmits light due to total internal reflection.''' Demonstrate light reflecting inside of plastic hose filled with water. Fill the plastic chamber with water and add a pinch of dry milk powder to reveal the light rays. Demonstrate the critical angle where there is total internal reflection. Place a wood plug in the hole in the plastic bottle and fill the bottle with water, place it on top of the plastic case. Position the laser in the support so that it points directly at the end of the plug. Place the larger plastic tub below the stream path. Turn on the laser and pull the plug. Use a white card to interrupt the stream and demonstrate that light is internally reflected in the stream.

4) '''Light is refracted and reflected when it encounters a material with a different index of refraction.''' Place a straw in the plastic tub with water (add a little milk to visualize laser) and observe that it appears to bend when placed at an angle. Repeat the demonstration this time with a laser pointed down from above. Index of refraction is the ratio of the speed of light in a vacuum over to that in the material. Air and water have very different indexes of refraction.

'''Solids and liquids can have different indexes of refraction.''' Carefully pour a layer of 80% sugar solution or corn syrup into the bottom of the plastic container. Tell students that sugar water has index of refraction of 1.49 and pure water is 1.33. Repeat the refraction demo. The light should bend a second time when it reaches the sugar water layer. Another variation is to carefully mix a series of sugar solutions and layer them so the solution is progressively more dense. This will result in a smoothly bending straw or light beam.

*This property is used in optic fiber in which the core and cladding have different IOR. Some fibers use the graded index fiber with special glass or polymers with progressively higher IOR. This decreases the dispersion of light that is caused by light at different angles passing through the fiber at different speeds (modal dispersion).

5) '''Fiber optics are used to transmit signals over long distances''' for phone and computer networks, or short distances between computer servers where high speed connections are needed. This usually involves transferring from electrical to optical and back to electrical signals. The optical fiber communications demonstration kit includes a transmitter and a receiver. Apply the 5V power supply to the demonstration device. This demo shows one way communication. (for two way communication the system would have a receiver and transmitter at both ends). The transmitter has a simple oscillator that controls an LED at the point where the optical fiber enters the device. Light passes along the fiber to the receiver where a photo detector senses the light and the circuitry coverts the light back into electricity at the Data output lines. Connect EN and EXT on the transmitter to the +5V positive terminal. An LED will flash on the Data lines on the receiver, proving that the signal has made it all the way back to electricity again. Disconnect the optical fiber from the receiver end and show that that the fiber end is flashing, the LED on the data will stop flashing when it stops getting its optical signal. As you bring the fiber back into the receiver the LED will begin flashing again.

The circuit is controlled with the following logic on the transmitter side. 0 means the line is connected to the ground, 1 means it is connected to the +5V line.

Mode EN EXT LED State

1 0 1 ON

2 0 0 OFF

3 1 1 OSCILLATING

4 1 0 OFF

*CMDITR researchers are concerned with various aspects of the optical network. An electro-optical switch can built using polymers. A key problem is switching speed. New organic molecules may be able to operate at higher speeds than current materials.
 
6) '''Telecommunications depend on the ability to turn light on and off quickly (modulation).''' Show how the polarizing filter can pass or block light from the laser pointer.

 
[[Image:MachZehnder.gif|thumb|400px|]]

7) '''Coherent light can be modulated by using interference of light waves.''' The Mach-Zehnder (MZ) interferometer is a device used to modulate light. In the MZ device a light beam is divided into two paths and one path goes through some electro-optic material. Changing the electric field on the EO material changes its index of refraction. When the two paths converge again destructive interference cancel the output light creating a signal.

 
[[Image:600px-Interferometre Michelson pattern.png|thumb|300px|]]
The Michelson Interferometer demonstrates this kind of interference with a more complicated path. One path of the split beam goes through the beam splitter, reflects off a mirror, then reflects off the back side of the image splitter and finally reflect off a second mirror before exiting the beam splitter along the original path. The two beams then pass through a diverging lens to spread the beam out into a wider area to reveal a series of bands or circles where there has been interference. This device is extremely sensitive to distance (for visible light it is 1/100th the thickness of human hair. This makes it sensitive to minute vibrations and minute changes in the index of refraction of material placed in the path.
 

[[Image:Interferometer_labeled.jpg|thumb|400px|]]
Follow the manufacturer’s instructions for arranging the pieces for the interferometer demo. Once you get a somewhat stable interference pattern then experiment with placing glass in the path. By turning the glass slightly you will alter its effective index of refraction and cause a shift in the interference pattern. Similarly, when an electro-optical material is placed in the beam path it will alter the interference pattern. A variation on this setup can be used to measure the electro-optic coefficient of materials.

*CMDITR research is building materials that can be used in telecom and all optical switching.
 
8) '''New materials can be fashioned into extremely small nano-scale devices integrated right on to the chip.''' Show photos, discuss micro-electronic trends, fabrication and scale.
 

=== Links ===

=== Sources for Building your own kit ===
[http://www.i-fiberoptics.com/laser-kits-projects-detail.php?id=2110 Michelson interferometer with pointer $109]

[http://www.i-fiberoptics.com/educational-detail.php?id=14200 Educational Communication Kit $18 includes fiber, LED photodetector]

Lasers and Telecommunication- Outreach Kit

2009-11-12T17:21:24Z

128.95.39.42: /* Sources for Building your own kit */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photovoltaics- Outreach Kit|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#K-12 Outreach Kits|K-12 Outreach Kits]]</td>

</tr>
</table>
=== Overview ===
One of the major research thrusts of CMDITR is organic electronics that can be used in information technology and telecommunications. At the heart of this is the modulation of light using new organic electro-optical materials. Students need to understand how the system currently uses light and optics to carry information. Electro-optic materials can change their index of refraction in the presence of an electric field. This property combined with wave interference makes high speed switching possible. Finally, electro-optical and all optical switches can be miniaturized to the nano-scale to take advantage of other unique properties in device design.

=== Materials in the kit ===
*laser pointer
*plastic tub
*plastic tubing
*Optic fiber receiver and transmitter
*Michelson interferometer

=== User Guide ===
[[Image:120px-Digital.signal.png|thumb|300px|]]
1) '''Information can be carried by modulating electrical signals either analog (continuously changing) or digital (timed pulses either on or off).''' Show or draw pictures of waveforms of analog and digital signals.

2) '''Information can be carried by light.''' Bounce a laser off of mirror attached to a balloon stretched on a can. Tap the rubber membrane or speak into it. Brainstorm the advantages of using light as carrier for information?

*CMDITR is working on ways to modulate light by controlling the optical properties of the materials it passes through. This may lead to improved efficiency of switching light in communication systems. It also may lead to all-optical switching (AOS) in which light from one channel controls light in another channel.

3) '''Optic fiber transmits light due to total internal reflection.''' Demonstrate light reflecting inside of plastic hose filled with water. Fill the plastic chamber with water and add a pinch of dry milk powder to reveal the light rays. Demonstrate the critical angle where there is total internal reflection. Place a wood plug in the hole in the plastic bottle and fill the bottle with water, place it on top of the plastic case. Position the laser in the support so that it points directly at the end of the plug. Place the larger plastic tub below the stream path. Turn on the laser and pull the plug. Use a white card to interrupt the stream and demonstrate that light is internally reflected in the stream.

4) '''Light is refracted and reflected when it encounters a material with a different index of refraction.''' Place a straw in the plastic tub with water (add a little milk to visualize laser) and observe that it appears to bend when placed at an angle. Repeat the demonstration this time with a laser pointed down from above. Index of refraction is the ratio of the speed of light in a vacuum over to that in the material. Air and water have very different indexes of refraction.

'''Solids and liquids can have different indexes of refraction.''' Carefully pour a layer of 80% sugar solution or corn syrup into the bottom of the plastic container. Tell students that sugar water has index of refraction of 1.49 and pure water is 1.33. Repeat the refraction demo. The light should bend a second time when it reaches the sugar water layer. Another variation is to carefully mix a series of sugar solutions and layer them so the solution is progressively more dense. This will result in a smoothly bending straw or light beam.

*This property is used in optic fiber in which the core and cladding have different IOR. Some fibers use the graded index fiber with special glass or polymers with progressively higher IOR. This decreases the dispersion of light that is caused by light at different angles passing through the fiber at different speeds (modal dispersion).

5) '''Fiber optics are used to transmit signals over long distances''' for phone and computer networks, or short distances between computer servers where high speed connections are needed. This usually involves transferring from electrical to optical and back to electrical signals. The optical fiber communications demonstration kit includes a transmitter and a receiver. Apply the 5V power supply to the demonstration device. This demo shows one way communication. (for two way communication the system would have a receiver and transmitter at both ends). The transmitter has a simple oscillator that controls an LED at the point where the optical fiber enters the device. Light passes along the fiber to the receiver where a photo detector senses the light and the circuitry coverts the light back into electricity at the Data output lines. Connect EN and EXT on the transmitter to the +5V positive terminal. An LED will flash on the Data lines on the receiver, proving that the signal has made it all the way back to electricity again. Disconnect the optical fiber from the receiver end and show that that the fiber end is flashing, the LED on the data will stop flashing when it stops getting its optical signal. As you bring the fiber back into the receiver the LED will begin flashing again.

The circuit is controlled with the following logic on the transmitter side. 0 means the line is connected to the ground, 1 means it is connected to the +5V line.

Mode EN EXT LED State

1 0 1 ON

2 0 0 OFF

3 1 1 OSCILLATING

4 1 0 OFF

*CMDITR researchers are concerned with various aspects of the optical network. An electro-optical switch can built using polymers. A key problem is switching speed. New organic molecules may be able to operate at higher speeds than current materials.
 
6) '''Telecommunications depend on the ability to turn light on and off quickly (modulation).''' Show how the polarizing filter can pass or block light from the laser pointer.

 
[[Image:MachZehnder.gif|thumb|400px|]]

7) '''Coherent light can be modulated by using interference of light waves.''' The Mach-Zehnder (MZ) interferometer is a device used to modulate light. In the MZ device a light beam is divided into two paths and one path goes through some electro-optic material. Changing the electric field on the EO material changes its index of refraction. When the two paths converge again destructive interference cancel the output light creating a signal.

 
[[Image:600px-Interferometre Michelson pattern.png|thumb|300px|]]
The Michelson Interferometer demonstrates this kind of interference with a more complicated path. One path of the split beam goes through the beam splitter, reflects off a mirror, then reflects off the back side of the image splitter and finally reflect off a second mirror before exiting the beam splitter along the original path. The two beams then pass through a diverging lens to spread the beam out into a wider area to reveal a series of bands or circles where there has been interference. This device is extremely sensitive to distance (for visible light it is 1/100th the thickness of human hair. This makes it sensitive to minute vibrations and minute changes in the index of refraction of material placed in the path.
 

[[Image:Interferometer_labeled.jpg|thumb|400px|]]
Follow the manufacturer’s instructions for arranging the pieces for the interferometer demo. Once you get a somewhat stable interference pattern then experiment with placing glass in the path. By turning the glass slightly you will alter its effective index of refraction and cause a shift in the interference pattern. Similarly, when an electro-optical material is placed in the beam path it will alter the interference pattern. A variation on this setup can be used to measure the electro-optic coefficient of materials.

*CMDITR research is building materials that can be used in telecom and all optical switching.
 
8) '''New materials can be fashioned into extremely small nano-scale devices integrated right on to the chip.''' Show photos, discuss micro-electronic trends, fabrication and scale.
 

=== Links ===

=== Sources for Building your own kit ===
[http://www.i-fiberoptics.com/laser-kits-projects-detail.php?id=2110 Michelson interferometer with pointer $109]
[http://www.i-fiberoptics.com/educational-detail.php?id=14200 Educational Communication Kit $18 includes fiber, LED photodetector]

2009-10-01T21:30:18Z

128.95.39.42: New page: Spin coating is method to apply uniform thin coatings to substrates. It is widely used in microfabrication for applying photoresist. In photonics research it is used to directly fabricate ...

Main Page

2009-10-01T21:21:51Z

128.95.39.42: /* Research Equipment, Devices and Techniques */

<big>'''Center for Materials and Devices for Information Technology Research (CMDITR) Wiki'''</big> 
This wiki is a reference collection on photonics. Most of the text has been captured from a series of lectures recorded in 2005-2008 by Center faculty Jean-Luc Bredas (Georgia Tech), Neal Armstrong (University of Arizona) and Seth Marder (Georgia Tech). You may also want to search the
[http://depts.washington.edu/cmditr/cwis/SPT--Home.php CMDITR Photonics Digital Libary] for individual learning objects.

(The sections below with ** asterisks are currently in development, the rest are in draft form)

== Photonics Core Concepts and Applications ==

 

===Basics of Light ===
[[Image:Snells_law_wavefronts.gif|thumb|150px|]]
*[[Propagation of Light]]
*[[Reflection and Refraction ]]
*[[Dispersion and Scattering of Light]]
*[[Diffraction of Light]]
*[[Electromagnetic Radiation]]
*[[Luminescence Phenomena]]
*[[Color and Chromaticity]]

 

=== Optical Fibers, Waveguides, and Lasers ===
[[Image:800px-Military_laser_experiment.jpg|thumb|200px|]]

*[[Optical Fibers]]
*[[Total Internal Reflection]]
*[[Planar Dielectric Waveguides]]
*[[Optical Fiber Waveguides]]
*[[Dispersion and Attenuation Phenomena]]
*[[Lasers]]

===Molecular Orbitals===
[[Image:HAtomOrbitals.png|thumb|150px|]]
*[[Atomic Orbitals and Nodes]]
*[[Electronegativity and Bonding Between Atoms]]
*[[Sigma and pi Orbitals|Sigma and Pi Orbitals]]
*[[Polarization and Polarizability]]
*[[Electronic Coupling Between Orbitals]]
*[[Donors and Acceptors]]
 

===Electronic Band Structure of Organic Materials===
[[Image:Ethylene.JPG|thumb|200px|]]
*[[Introduction to Band Structure]]
*[[Electronic Structure of Hydrogen]]
*[[The Polyene Series]]
*[[Bloch's Theorem]]
*[[Electrical Properties]]
*[[Electronic States vs Molecular Levels]]
 

===Absorption and Emission of Light===
[[Image:Abs Emis stokes.png|thumb|200px|]]
*[[Introduction to Absorption]]
*[[Changes in Absorption Spectra]]
*[[Jablonksi Diagram]]
*[[Fluorescence Process]]
*[[Transition Dipole Moment]]
*[[Absorption and Emission]]
*[[Photochromism]]
*[[Interchain Interactions]]

 

===Transport Properties===
[[Image:rubrene.png|thumb|150px|]]
*[[Charge Carrier Mobility]]
*[[Band Regime versus Hopping Regime]]
*[[Electronic Coupling]]
*[[Model Calculations of Electronic Coupling]]
*[[Marcus Theory and Reorganization Energy]]
*[[Electron-Phonon Coupling]]

 

===Liquid Crystals and Displays===
[[Image:smectic_C.jpg|thumb|200px|]]
*[[Liquid Crystals]]
*[[Double Refraction and Birefringence]]
*[[Director – Degrees of Order in Liquid Crystals]]
*[[Classification and Examples of Liquid Crystals]]
*[[Alignment]]
*[[Freederickz Transition and Dielectric Anisotropy]]
*[[Liquid Crystal Displays]]

 

===Organic Light Emitting Diodes===
[[Image:PNNL_Light_Lab_041.jpg|thumb|200px|Blue phosphorescent OLED developed by Pacific Northwest National Laboratory.]]
*[[OLED Device Applications]]
*[[Light Emitting Electrochemical Processes]]
*[[The OLED Test Cell]]
*[[What is a Light Emitting Diode?]]
*[[The First OLEDs]]
*[[Organic/Organic Heterojunctions in OLEDs]]
*[[OLED Charge Mobilities]]
*[[Organic Heterojunctions]]
*[[Fluorescent/Phosphorescent Dopants]]
*[[Metal Complex Dopants]]
 

===Organic Solar Cells===
[[Image:Opvtestcells.png|thumb|200px|OPV Test Cells]]
*[[Organic Solar Cells|OPV Introduction]]
*[[Solar Technologies]]
*[[Major Processes in Organic Solar Cells]]
*[[Organic Heterojunctions in Solar Cells]]
*[[Physics of Solar Cells]]
*[[Energy vs Charge Transfer at Heterojunctions]]
*[[Current OPV Research Directions]]
 

===Organic Electronics===
*[[Organic Electronics Overview]]
*[[Synthesis of Organic Semiconductors]](In progress)
*[[field effect transistors]]
*Design of n-type Semiconductors for Organic Electronic Applications

==Non linear Optics and Devices==

===**Quantum Mechanical and Perturbation Theory of Polarizability (Brédas, Robinson, Rehr)===

===Second-order Processes, Materials & Characterization ===
[[Image:MachZehnder.gif|thumb|200px|]]
*[[Second-order Processes]]
*[[Structure-Property Relationships]]
*[[Second-order NLO Materials]]
*[[Second-order Material Design]]
*[[Terahertz Radiation]]
*[[Second-order Material Characterization]]

 

===Third-order Processes, Materials & Characterization ===
[[Image:Tpa_concentrated.png|thumb|100px|]]
*[[Introduction to Third-order Processes and Materials]]
*[[Two Photon Absorption]]
*Advanced Concepts in Third-order Processes
*Characterization of Third-order Materials (Perry)
 

===**Techniques for Fundamental Processes (Ginger) ===

===Organic Photonics Applications in Information Technology ===
[[Image:Dualmz packaged.png|thumb|200px|]]
*[[Optical Networks]]
*[[Passive Optical Polymers]]
*[[Electro-optic Polymers and Devices]]
*[[Materials Processing and Fabrication]]
 

===Photonics Integration===
[[Image:Si_waveguide_em.jpg‎|thumb|200px|]]
*[[The Need for Photonic Integration]]
*Integrated Si Photonics (Hochberg)
 

== Research Equipment, Devices and Techniques ==
 
[[Image:PES.jpg|thumb|200px|]]
*[[Photoelectron Spectrometer XPS and UPS]]
*[[Conducting Tip Atomic Force Microscopy]]
*[[Organic Photovoltaic Fabrication and Test Apparatus]]
*[[Two-Photon Spectroscopy]]

'''In Development'''

'''Characterization'''

*UV/VIS/NIR spectrometer
*[[Teng-Mann Method for Measuring Electro-optic coefficient]]
*[[Profilometer]]
*[[Ellipsometer]]
*[[Hyper Rayleigh Scattering]]
*Fluorometer
*NMR spectrometer
*External quantum efficiency/yield
*[[Scanning Electron Microscope]]
*TEM
*SPM
*Raman microscope
*[[confocal microsope]]

'''Fabrication'''
*[[E-beam Lithography]]
*Reactive ion etcher
*Plasma etcher
*Atomic layer deposition
*[[Spin coater]]
*Sputter coater

==Acronyms and Unit Abbreviations==
*[[Acronyms]]
*[[Variables and Constants]]
*[[Units]]

== General Research Best Practices ==
*[[How to Keep a Lab Notebook]]
*[[How to Give a Research Presentation]]
*[[Writing a Scientific Paper]]
*[[Writing a Successful Proposal]]
*[[Mentoring]]

==[[External Photonics Education Links]]==

==K-12 Outreach Kits ==
[[Image:AssembledCell_small.JPG|thumb|200px|]]
*[[Nanocrystalline - Dye Solar Cell Kit]]

 

==[[Credits and Reviewers]]==

==[[Suggested Wiki Sequence By Audience]]==