Analytical criteria for designing multiresonance filters in scattering systems, with application to microwave metasurfaces

We present general analytical criteria for the design of lossless reciprocal two-port systems, which exhibit prescribed scattering spectra $S(\omega)$ satisfying $S_{22}(\omega)=e^{i\varphi}S_{11}(\omega)$, including symmetric ($S_{22}=S_{11}$) or "antimetric" ($S_{22}=-S_{11}$) responses, such as standard filters (Butterworth, Chebyshev, elliptic, etc.). We show that the non-normalized resonant (quasi-normal) modes (QNMs) of all such two-port systems couple to the input and output ports with specific unitary ratios, whose relative signs determine the position of the scattering zeros on the real frequency axis. This allows us to obtain design criteria assigning values to the poles, background response, and QNM-to-ports coupling coefficients. Filter devices can then be designed via a well-conditioned nonlinear optimization (or root-finding) problem using a numerical eigensolver. As an application, we design multiple microwave metasurfaces configured for polarization-preserving transmission, reflective polarization conversion, or diffractive "perfect anomalous reflection", to realize filters that precisely match standard bandpass or bandstop filters of various types, orders and bandwidths, with focus on the best-performing elliptic filters.