Optical parametric oscillation under injection of orbital angular momentum

We present a theoretical model for the spatial mode dynamics of an optical parametric oscillator under injection of orbital angular momentum. This process is then interpreted in terms of an interesting picture based on a Poincare representation of first order spatial modes. The spatial properties of the down-converted fields can be easily understood from their symmetries in this geometric representation. By considering the adiabatic mode conversion of the injected signal, we also predict the occurrence of a geometric phase conjugation in the down-converted beams. An experimental setup to measure this effect is proposed.