Introduction to Third-order Processes and Materials

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

Nonlinear Susceptibility and Hyperpolarizability

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

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

where:

<math>P_0\,\!</math> is the polarization in the absence of a field (some materials, such as polyvinylene difluoride when poled, can have a finite bulk polarization in the absence of an applied field),

<math>\chi^{(1)}\,\!</math> is the susceptibility of the material (which is related to the dielectric constant; see [[1]]),

<math>\chi^{(2)}\,\!</math> and <math>\chi^{(3)}\,\!</math> are the second- and third-order susceptibilities, respectively,

<math>E\,\!</math> is the applied electric field,

(more generic expressions can be written instead of (1), if more than one field is present).

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

Taylor Expansion for Molecular Dipole Moment

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

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

where:

<math>\mu_0\,\!</math> is the permanent dipole of the molecule,

<math>\alpha\,\!</math> is the polarizability of the molecule (the microscopic equivalent to <math>\chi^{(1)}\,\!</math>),

<math>\beta\,\!</math> and <math>\gamma\,\!</math> are the first and second hyperpolarizabilities, respectively.

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

Under normal conditions,

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

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

Third-order Nonlinear Polarization of Matter and Third-Order NLO Effects

Deviation from simple harmonic potential with a positive or negative quartic term.

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

Non linear self focusing process

When a beam of light passes into a NLO material with a higher refractive index it will have an intensity distribution that is higher in the center than at the edge. The material that is in the highest intensity will generate a higher refractive index than the material at the edge where there is low intensity. The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. Thus an NLO material behaves like a lens that focuses light closer to the interface between materials. In a focusing beam the cross-sectional area of the beam decreases as you approach the focal point and the intensity increases (because there are more photons in a unit area). If the polarizability and susceptibility is proportional to the cube of the electric field then the refractive index will increase. So as a beam becomes focused the added intensity increases the refractive index, causing even more concentrated focus, more intensity and more change in refractive index. This process is called “non linear self focusing”.

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 Self Focusing

See also Encyclopedia of Laser Physics Self Focusing

Third order polarization

If we reconsider equation (14) for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric) we see that:

<math>\mu = \mu_0+ \alpha E_0cos(\omega t) + \gamma/6E_{0}^{3}cos3(\omega t) + ...\,\!</math> (22)

If a single field, E(omega,t), is acting on the material, we know from trigonometry that:

<math>\mu/6E_{0}^{3}cos3(\omega t) = \gamma/6E_{0}^{3}[(3/4)cos(\omega t) + (1/4)cos(3\omega t)]\,\!</math> (23)

These leads to process of frequency tripling in that you can shine light on the molecule and get light at the third harmonic.

thus,

<math>\mu = \mu_0+ \alpha E_0 cos(omega t) + \gamma /6 E03(3/4)cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (24)

or:

<math>\mu= \mu_0+ [\alpha + \gamma /6 E_{0}^{2}(3/4)]E_0cos(\omega t) + \gamma /6 E03(1/4)cos(3\omega t)\,\!</math> (25)

This is an effective polarizability that is related to E₀² (the maximum deviation of the sinusoidal electric field) and γ. E₀² is always positive. On the other hand γ can be either positive or negative. Thus by increasing the magnitude of the electric field (light) shining on the materials (with a positive γ) increase the polarizability as the square of the field or decrease the polarizability ( if the γ is negative). So due to the third order effect the linear polarizability can be changed simply by modifying the intensity of the applied light.

Third Harmonic Generation and the Optical Kerr Effect

Thus, the interaction of light with third-order NLO molecules will create a polarization component at its third harmonic.

In addition, there is a component at the fundamental, and we note that the :<math>[\alpha + \gamma /6 E_{0}^{2}(3/4)]\,\!</math> term of equation (25) is similar to the term leading to the linear electrooptic effect or the pockels effect.

Likewise the induced polarization for a bulk material, would lead to third harmonic generation through chi(3), the material susceptibility analogous to γ.

There are two kinds of Kerr effects. In an optical frequency Kerr effect a very high intensity beam is applied that changes the refractive index of a material.

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

Four Wave Mixing

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

This does not have to be case. There could be three beams with different phases at arbitrary directions, polarizations and frequency components that can all mix and give sums and differences of frequency leading to all kinds of output light.

<math>\omega 1 + \omega 2 + \omega 3\,\!</math> : this is third harmonic generation

or

<math>\omega 1 + \omega 2 - \omega 3\,\!</math> : this gives light out at the same frequency (degenerate four wave mixing) as the input leading to the self-focusing effect.

Another interesting manifestation of third-order NLO effect is degenerate four wave mixing in which two beams of light interacting within a material create an interference pattern that will lead to a spatially periodic variation in light intensity across the material. As we have noted before the induced change in refractive index of a third-order nonlinear optical material is proportional to the intensity of the applied field. Thus, if two beams are interacting with a third-order NLO material, the result will be a refractive index grating because of constructive and destructive interference. The diffraction pattern creates areas of high and low light intensity on an NLO material. The areas that are brightest will have an increased refractive index (with a positive χ⁽³⁾). At the darkest point the refractive index will have zero change. So if the intensity is changing periodically then the refractive index will have a periodic variation as well. When a third beam is incident on this grating a fourth beam, called the phase conjugate, is diffracted from the grating. This process is called four wave mixing: two writing beams and a probe beam result in a fourth phase conjugate beam.

Degenerate Four-wave Mixing

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

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.

Phase conjugate mirror animation

<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

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

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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