Effect of Molecular Redistribution on the Nonlinear Refractive Index of Liquids

An expression for the static nonlinear dielectric constant of a nonpolar liquid is derived which takes into account the reorientation and local spatial redistribution of molecules in the presence of a strong electric field. This result is used to calculate the nonlinear index of refraction of various lossless liquids at "optical" frequencies which are much higher than molecular reorientation rates. From this nonlinear index, we obtain estimates of the optical power required to initiate self-focusing in a liquid over periods so short that macroscopic density changes do not have time to take place. This situation is commonly approached in experiment. The theory predicts that the effects of molecular redistribution, hitherto not considered, will be generally as important as the widely considered effects of molecular reorientation, and will in fact dominate the nonlinear index of many symmetric and nearly symmetric molecules. Furthermore, the results suggest that some purely symmetric molecules, such as Si${\mathrm{Br}}_4$, may exhibit self-focusing in liquid as readily as do some commonly studied asymmetric molecules, such as nitrobenzene. The calculation proceeds from classical statistical mechanics with the aid of a variational principle that is valid for arbitrary density. In the low-density limit our results reflect only molecular reorientation and reduce to those of Debye and others. The accuracy of the results depends mainly on the accuracy of Kirkwood's "superposition approximation" in representing three- and four-particle correlation functions in liquids. Since the accuracy of this approximation is at present unknown, nonlinear index data may prove useful in checking it. As a by-product of our investigation, we have proven that the Clausius-Mossotti expression gives a lower bound for the dielectric constant of a fluid in which the two-particle correlation function is a function only of the interparticle spacing and approaches a constant at large spacing. Molecular redistribution must also play a role in induced birefringence (ac and dc Kerr effects), especially for symmetric molecules. However, the present treatment is limited to waves of a single linear polarization, and does not cover the Kerr effect.