Theory of the dispersion of ultrafast nonlinear refraction in zinc-blende semiconductors below the band edge

The dispersion of the nonlinear refractive index of direct-gap semiconductors, over the full sub-band-gap spectral region, is determined within the four-band Kane model. The divergences of individual terms in the nonlinear susceptibility, which have prevented previous full calculations, are avoided by selecting suitable combinations of terms and by a consistent application of the analytic continuation technique. The origins of different forms of divergence are analyzed. Results are presented for InSb, InAs, GaAs, CdTe, ZnTe, and ZnSe; these are compared with previous model calculations based on a Kramers-Kr\"onig transformation of the nonlinear absorptive susceptibilities. The present model gives excellent agreement, with no fitting parameters, to spectral midgap experimental results for the alloy Al0.18{Ga}0.82$As.