The pseudo-spin dynamics of propagating exciton-polaritons in semiconductor microcavities are known to be strongly influenced by TE-TM splitting. As a vivid consequence, in the Rayleigh scattering regime, the TE-TM splitting gives rise to the optical spin Hall effect (OSHE). Much less is known about its role in the nonlinear optical regime in which four-wave mixing for example allows the formation of spatial patterns in the polariton density, such that hexagons and two-spot patterns are observable in the far field. Here we present a detailed analysis of spin-dependent four-wave mixing processes, by combining the (linear) physics of TE-TM splitting with spin-dependent nonlinear processes, i.e., exciton-exciton interaction and fermionic phase-space filling. Our combined theoretical and experimental study elucidates the complex physics of the four-wave mixing processes that govern polarization and orientation of off-axis modes.
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