  Second order nonlinearities in whispering gallery mode resonators are highly
investigated for their many applications such as wavelength converters,
entangled photon sources and generation of frequency combs. In such systems,
depending on the material under scrutiny, the derivation of quasi-phase
matching equations can lead to the appearance of additional quanta in the
selection rule on the azimuthal confinement order. Here, we demonstrate that
these additional quanta show up due to the Berry phase experienced by the
transverse spin angular momentum components of the whispering gallery modes
during their circulation within the resonator. We first detail the case of
Zinc-blende materials and then generalize this theory to other crystal
symmetries relevant for integrated photonics.
