Parametric photoluminescence of spatially confined polaritons in patterned microcavities

We model the parametric photoluminescence process of laterally confined polaritons in a mesa structure, patterned on a semiconductor microcavity. Parametric processes are described within the lowest order in the polariton-polariton interaction. We show how a plane-wave pump produces simultaneously several signal-idler pairs, due to the broken translational symmetry, the finite polariton linewidths, and the presence of a continuum of polariton states above the shallow mesa barrier. We discuss under which conditions of pump and deluning, the simple three-mode picture for the parametric process is recovered.