  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.
