  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 description and design of stimulated and spontaneous parametric down
conversion experiments with vector vortex beams.
