Phase conjugation and mode conversion in stimulated parametric down-conversion with orbital angular momentum: a geometrical interpretation

We report on an experiment that investigates the spatial mode conversion in the process of parametric down-conversion seeded by a light beam in a superposition of orbital angular momentum modes. This process is interpreted in terms of a geometric representation of first-order spatial modes in a Poincar\'e sphere, providing an intuitive image of the phase conjugation and the topological charge conservation. We also make a comparison with the analogous phenomenon for optical parametric oscillators.