Quasi-normal mode theory of the scattering matrix, enforcing fundamental constraints for truncated expansions

Phys. Rev. Research 3, 033228 (2021) We develop a quasi-normal mode theory (QNMT) to calculate a system's scattering $S$ matrix, simultaneously satisfying both energy conservation and reciprocity even for a small truncated set of resonances. It is a practical reduced-order (few-parameter) model based on the resonant frequencies and constant mode-to-port coupling coefficients, easily computed from an eigensolver without the need for QNM normalization. Furthermore, we show how low-$Q$ modes can be separated into an effective slowly varying background response $C$, giving an additional approximate formula for $S$, which is useful to describe general Fano-resonant phenomena. We demonstrate our formulation for both normal and fixed-angle oblique plane-wave incidence on various electromagnetic metasurfaces.