Introduction to Third-order Processes and Materials

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Hyperpolarizability

Gamma is the second hyperpolarizability which is a molecular property. It scales as the cube of the electric field

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field. This relates to chi(3), which is a materials property. Chi (3) and gamma can exist in all materials and all molecules, even those which are centrosymmetric materials. (chi(2) can only happen in non centrosymmetric materials).

Taylor Expansion for Polarization

Under normal conditions, aijE .> ßijk/2 E·E > gijkl /6 E·E·E.

J is the coordinate system for the applied field I is the coordinate system for the induced polarization in the molecule Alpha is 3 x 3 tensor Beta is 3 x 3 x 3 tensor with 27 components Gamma is a 3 x 3 x 3 x 3 tensor with 81 components

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

Just as alpha is the linear polarizability, the higher order terms beta and gamma (equation (7)) are called the first and second hyperpolarizabilities respectively.

Taylor Expansion for Polarization

Or when all the fields are identical there is a Taylor expansion for bulk polarization:

P = Po + chi (1)·E + 1/2chi(2)·· E2 + 1/6chi(3)···E3+ ... (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 b if it is noncentrosymmetric, a material can only have a chi(2) if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero chi(2)) .

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

Remember that in chi(2) NLO the harmonic potential has a cubic term that makes one side of the potential somewhat more steep and other side flattened.

With chi(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 and its refractive index decreases. If when you polarize a material it becomes easier to polarize then the refractive index will decrease.

When a beam of light passes into a material with a higher refractive index it will bend so it focuses closer to the interface between materials (by Snell’s law). The refractive index changes because the intensity of light changes the polarizability, the susceptibility, and therefore the refractive index. 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”.

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 you are 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 the case where the polarization decreases with intensity the condition is called self-defocusing. The beam passing through a material has a tendency to spread.

A molecule with a negative beta or a negative chi(2) has a axis or the plane of the molecule 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 beta lead to the same effects but with opposite signs. However positive and negative gamma and positive and negative chi(3) lead to different effects. Specifically, negative chi(3) leads to self-defocusing, and positive chi(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.

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