Quantum-optical Description of Phase Conjugation of Vector Vortex Beams in Stimulated Parametric Down Conversion

We present a quantum optics approach for describing stimulated parametric down conversion in the two type-I crystal sandwich configuration, which allows for parametric interaction of vector vortex beams. We analyze the conditions for which phase conjugation of the seed vector beam occurs. We then use two strategies for defining generalized Stokes parameters to describe phase conjugation of vector vortex beams. These allow for geometrical representations, such as higher-order Poincare spheres. Our results are useful for description and design of stimulated and spontaneous parametric down conversion experiments with vector vortex beams.