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

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