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

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.