Selection rules for transverse-mode excitation in nonlinear ring and Fabry-Perot resonators

We establish that the energy transfer between transverse modes in nonlinear Fabry-Perot resonators with curved mirrors, having degenerate or nearly degenerate modes, is governed by selection rules. Although these rules are derived in a perturbative limit, assuming a thin medium centered in the resonator, we have found numerically and experimentally that they still account for the otherwise unexpected enhancement or suppression of certain modes in more realistic cases. The existence of selection rules constitutes a fundamental difference between standing wave and traveling wave cavities and therefore prevents, in principle, the possibility of describing experiments on transverse effects performed in Fabry-Perot cavities with (simpler) ring resonator models.