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

Abstract

We present general analytical criteria for the precise design of symmetric or "antimetric" 2-port systems, including ideal standard filters (Chebyshev, elliptic, etc.), based on the non-normalized resonant (quasi-normal) modes of the system. We first develop a quasi-normal mode theory (QNMT) to calculate a system's scattering S matrix, satisfying both energy conservation and reciprocity, and describe how low-Q modes can be combined into an effective slowly varying background response C. We then show that the resonant QNM fields of all lossless reciprocal 2-port systems with a symmetric (S22 = S11) or "antimetric" (S22 = −S11) response couple to the input and output ports with specific unitary ratios, whose relative signs determine the position of the scattering zeros. This allows us to obtain design criteria assigning values to the poles, background response, and QNMto-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 example application, we design microwave metasurface filters that precisely match standard amplitude filters of various orders and bandwidths, with focus on the best-performing elliptic filters.

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

“…( 4) then implies that C(ω) = C e i2τ ω , D(ω) = D e iτ ω and K(ω) = K e iτ ω , where C is a diagonal constant phase matrix that can be taken equal to −I (as we justify later and is often used in CMT [17]) without loss of generality, and D , K are now constant matrices. This is the case for CPMs with fields transverse to their direction of propagation (φ p •k p = 0), such as plane waves or dual-conductor TEM microwave modes, which will be n , we can also construct a slowly-varying background matrix C, which can give a physical intuition about the scattering response and help in specific scattering designs [7].…”

“…Thus, in all structures simulated in this article, we remove this linear phase to compute S , referenced at the new port cross-sections z p on the scatterer boundary. In practice, τ pp may be slightly larger from (z p − z p )/c p , adding a small constant group delay just to the phases of S, so it is of no concern for applications dependent only on their amplitudes, such as amplitude filters [7].…”

“…Therefore, as promised, S in Eq. (7) does not require computing the cumbersome volume-integral norms of the QNMs. Note also that, as more modes are included, the residue coefficients automatically update themselves through M in order to satisfy energy conservation for the entire set, in contrast to other formulations based on the exact field equations, where these residues are cosntant [4,5].…”

“…Weak absorption or gain can then be easily incorporated as a perturbation. Furthermore, by explicitly separating a slowly varying effective-background response C, we provide a novel additional formula for S, approximate but very convenient to design Fano-scattering systems [6] such as even-order elliptic filters [7]. This C is calculated without resorting to any type of fitting [4,8,9] and without having to choose a specific background scattering medium [5]: we simply use a subset of low-Q modes of the entire system.…”

“…The final result is a simple equation for S [Eq. (7)] using only the eigenfrequencies and the fine-tuned D [Eq. (11)] (Sec.…”