Nonlinear refractive index of optical crystals

Abstract

The nonlinear refractive indices ($n_2$) of a large number of optical crystals have been measured at a wavelength near one micrometer with use of nearly degenerate three-wave mixing. The measurements are compared with the predictions of an empirical formula derived by Boling, Glass, and Owyoung. This formula, which relates $n_2$ to the linear refractive index and its dispersion, is shown to be accurate to within about 30% for materials with nonlinear indices ranging over 3 orders of magnitude. Measurements for a number of binary oxide and fluoride crystals have been analyzed under the assumption that the hyperpolarizability of the anion is much larger than that of the cation. It is found that the hyperpolarizability of oxygen varies by a factor of 10, and that of fluorine varies by a factor of 7, depending on the size of the coordinating cation. This behavior is similar to that of the linear polarizability, although the hyperpolarizability is much more sensitive than the linear polarizability to the identity of the cation. The measured halide ion hyperpolarizabilities for several alkali-halide crystals are in reasonable agreement with recent self-consistent calculations. A semiempirical model was proposed by Wilson and Curtis to account for the dependence of the linear anionic polarizability on the radius of the cation. This model also accounts quite well for the variation of the hyperpolarizability of both fluorine and oxygen, except for cation partners that have filled or unfilled d-electron shells. The nonlinear indices of a number of complex oxides (i.e., those with more than one cation) have been calculated from the partial hyperpolarizabilities deduced from the data for the binary oxides. The calculated and measured values of $n_2$ agree to within an average error of 13%.