We investigate the transverse mode structure of the down-converted beams generated by a type-II optical parametric oscillator (OPO) driven by a structured pump. Our analysis focuses on the selection rules imposed by the spatial overlap between the transverse modes of the three fields involved in the nonlinear interaction. These rules imply a hierarchy of oscillation thresholds that determine the possible transverse modes generated by the OPO, as remarkably confirmed with experimental results.
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