Inverse design of multiresonance filters via quasi-normal mode theory

TL;DR

The paper addresses inverse design of compact high-order multiresonant 2-port filters in linear passive scattering systems. Grounded in quasi-normal mode theory, it decomposes the scattering matrix S(ω) into a pole-dominated part governed by complex resonances and port couplings, and a slowly varying background, enabling direct mapping from target standard filters to design parameters. A differentiable, pole-preserving optimization enforces resonance and background constraints without pole tracking, using time-reversal zeros for poles and a Levenberg–Marquardt-based solver, with optional topology optimization or few-parameter designs. Demonstrations include 3rd- and 4th-order elliptic and Chebyshev filters for photonic metasurfaces, multilayer films, and LC ladders, achieving close spectral matches and flexible background control across both optical and electronic platforms.

Abstract

We present a practical methodology for inverse design of compact high-order/multiresonance filters in linear passive 2-port wave-scattering systems, targeting any desired transmission spectrum (such as standard pass/stop-band filters). Our formulation allows for both large-scale topology optimization and few-variable parametrized-geometry optimization. It is an extension of a quasi-normal mode theory and analytical filter-design criteria (on the system resonances and background response) derived in our previous work. Our new optimization-oriented formulation relies solely on a scattering solver and imposes these design criteria as equality constraints with easily calculated (via the adjoint method) derivatives, so that our algorithm is numerically tractable, robust, and well-suited for large-scale inverse design. We demonstrate its effectiveness by designing 3rd- and 4th-order elliptic and Chebyshev filters for photonic metasurfaces, multilayer films, and electrical LC-ladder circuits.

Figures (7)

• Figure 1: A schematic of a 2-port system, with input and output wave amplitudes $s_{\pm p}$ respectively, related at frequency $\omega$ by a scattering matrix $S(\omega)$, which is computed by QNMT via the high-$Q$ modes with frequencies $\omega_n$ and port-coupling ratios $\sigma_n$, plus an effective background response $C$.
• Figure 2: TopOpt setup: The design region of width $d$ and height $a/2$ is bounded vertically by Neumann boundary conditions, thus forming a photonic metasurface (infinite vertically with period $a$). The two input/output port modes are normally incident plane waves, while both PML and absorbing boundary conditions are imposed at the left/right boundaries of the computation cell to emulate free space.
• Figure 3: Designed photonic 3rd-order elliptic bandpass filter: (a) The structure is shown on the top left, with the design region boxed inside the dashed orange line. Its transmission spectrum $|S_{21}(f)|^2$ is plotted as the orange curve, with an enlargement of the passband on the top right. Its background transmission $|C_{21}(f)|^2$ is below $-70$dB (and hence omitted from the plot). Therefore, the response of our design matches perfectly the target standard filter (black curve). The traditional naive-optimization method performs very poorly in comparison (green curve). (b) Target and designed poles and their coupling ratios $\{\omega_n, \sigma_n\}$.
• Figure 4: Designed photonic 4th-order elliptic bandpass filter: (a) The structure is shown on the top left. Its transmission spectrum $|S_{21}(f)|^2$ is plotted in orange, with an enlargement of the passband on the inset. Its background transmission $|C_{21}(f)|^2$ is steady around the specified $-25$dB. Again, our design exhibits perfect match to the target standard filter (black curve). (b) Poles of the designed structure: the high-Q poles and their coupling ratios $\{\omega_n, \sigma_n\}$ shown in the enlarged inset match their targets, while the low-Q modes collectively account for the desired background transmission.
• Figure 5: Designed 1D-layered photonic 3rd-order Chebyshev bandpass filter: The dielectric-layer stack is shown on the left, where black is Si and grey is SiO2, so light is normally incident from the top air and transmits into the bottom silica substrate; the $28$ layer thicknesses (from top to bottom, alternating SiO2--Si) are $d_k/\lambda=(0.3528,$$0.07358,$$0.1787,$$0.07361,$$0.3449,$$0.08524,$$0.1795,$$0.07385,$$0.1793,$$0.07383,$$0.1794,$$0.07391,$$0.1804,$$0.03658,$$0.04277,$$0.07453,$$0.1794,$$0.07382,$$0.1792,$$0.07380,$$0.1793,$$0.07385,$$0.1797,$$0.1212,$$0.2876,$$0.07501,$$0.1854,$$0.2154)$. The transmission spectrum $|S_{21}(f)|^2$ (orange) matches the target filter (black) for a wide frequency range around the passband (enlarged in the inset). The background transmission $|C_{21}(f)|^2$ is below $-58$dB for the entire range.