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 Poincar\'e spheres. Our results are useful for the description and design of stimulated and spontaneous parametric down-conversion experiments with vector vortex beams.