Multi-photon schemes for mid-infrared detection

We calculate the theoretical non-degenerate two photon absorption and three color injected current response tensors for bulk GaAs and Ge$_{1-x}$Sn$_x$ for a range of alloy compositions. In particular, by including a ''pump'' beam we compare two ''schemes'' that are sensitive to mid-infrared photons. In ''scheme I'' we consider GaAs and a pump photon with energy greater than half the band gap, and in ''scheme II'' we consider Ge$_{1-x}$Sn$_x$ with a pump photon with energy less than half the band gap. We find that for certain pump and alloy concentrations Ge$_{1-x}$Sn$_x$ has a substantially larger nonlinear response and three-color injected current than GaAs in the mid-infrared frequency window where both materials can absorb photons via non-degenerate two-photon absorption.