Scattering matrix for chiral harmonic generation and frequency mixing in nonlinear metasurfaces

We generalize the concept of optical scattering matrix ($S$-matrix) to characterize harmonic generation and frequency mixing in planar metasurfaces in the limit of undepleted pump approximation. We show that the symmetry properties of such nonlinear $S$-matrix are determined by the microscopic and macroscopic symmetries of the metasurface. We demonstrate that for description of degenerate frequency mixing processes such as optical harmonic generation, the multidimensional $S$-matrix can be replaced with a reduced two-dimensional $S$-matrix. We show that for metasurfaces possessing specific point group symmetries, the selection rules determining the transformation of the reduced nonlinear $S$-matrix are simplified substantially and can be expressed in a compact form. We apply the developed approach to analyse chiral harmonic generation in nonlinear metasurfaces with various symmetries including rotational, in-plane mirror, and out-of-plane mirror symmetries. For each of those symmetries, we confirm the results of the developed analysis by full-wave numerical calculations. We believe our results provide a new paradigm for engineering nonlinear optical properties of metasurfaces which may find applications in active and nonlinear optics, biosensing, and quantum information processing.