Theory of ultrafast nonlinear refraction in zinc-blende semiconductors

The ultrafast nonlinear refractive index n₂ has been of recent interest, particularly in semiconductors, due to the possibility of fabricating compact, integrated all-optical switching elements. It has shown that n₂ can be obtained by a nonlinear Kramers-Kronig transform of the (nondegenerate) nonlinear absorption (e.g. two-photon absorption). Applying a two parabolic band model for a semiconductor provides the material scaling and approximate dispersion of n₂, although the resulting quantity has to be scaled by a constant factor to fit experimental data. As has been shown for two-photon absorption, this difference is probably due to the neglection of the multiple valence bands near the centre of the Brillouin zone. In this paper, the more realistic bandstructure model of Kane (consisting of a conduction band and heavy-hole, light-hole and split-off valence bands) will be employed in the determination of n₂. Rather than use a nonlinear Kramers-Kronig transform, instead a direct calculation of n₂ will be performed which is numerically simpler and also ensures that all nonresonant terms are properly accounted for