  We propose a simple scheme of degenerate spontaneous parametric
down-conversion (SPDC) in nonlinear metasurfaces or photonic crystal slabs with
quasi-guided modes. It employs a band crossing between P- and S-polarized
quasi-guided mode bands inside the light cone and a selection rule in the
conversion efficiency of the SPDC. The efficiency can be evaluated fully
classically via the inverse process of noncollinear second-harmonic generation
(SHG). As a toy model, we study the SPDC and SHG in a monolayer of
noncentrosymmetric spheres and confirm that the scenario works well to enhance
the SPDC.
