Generalized Orbital Angular Momentum Symmetry in Parametric Amplification

We investigate interesting symmetry properties verified by the down-converted beams produced in optical parametric amplification with structured light. We show that the Poincar\'e sphere symmetry, previously demonstrated for first-order spatial modes, translates to a multiple Poincar\'e sphere structure for higher orders. Each one of these multiple spheres is associated with a two-dimensional subspace defined by a different value of the orbital angular momentum. Therefore, the symmetry verified by first order modes is reproduced independently in each subspace. This effect can be useful for parallel control of independently correlated beams.