Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption

Measurements of the nonlinear refractive index of several semiconductors using beam-distortion methods and four-wave mixing show a strong systematic dispersion in the bound-electronic nonlinearity (electronic Kerr effect ${\mathit{n}}_2$) near the two-photon-absorption edge. This eventually turns from positive to negative at higher frequencies. We find that by using the two-photon-absorption spectrum as predicted by a two-parabolic-band model, we can predict the observed universal dispersion, scaling, and values of ${\mathit{n}}_2$ that range over 4 orders of magnitude and change sign, using a simple Kramers-Kronig analysis (i.e., relating the real and imaginary parts of the third-order susceptibility). The resulting scaling rule correctly predicts the value of ${\mathit{n}}_2$ for all the 26 different materials we have examined. This includes wide-gap dielectrics which have 3 to 4 orders of magnitude smaller values of ${\mathit{n}}_2$ than semiconductors.