Theory of Four Wave Mixing-Based Parametric Amplification of Spin-Orbit Modes

We study the generation of spin-orbit (SO) modes via four-wave mixing (FWM)-based parametric amplification. SO modes carry quantized total angular momentum (TAM), and we show that FWM processes that generate new signals conserve TAM. This is a generalization of prior research which operated in a regime where FWM processes conserved spin and orbital angular momenta independently. We calculate the growth rates of new modes for both degenerate and nondegenerate pump configurations. Our theory is validated against numerical simulations for the cases where the generated signals are in the same SO mode(s) as the pump(s). We also calculate the growth rates of signals in SO modes other than the pumps.