Inverse design of multiresonance filters via quasi-normal mode theory

We present a practical methodology for the 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). It can deliver filters sized only a few wavelengths by incorporating overlapping resonances, in contrast to traditional designs via coupled-mode theory leading to rather large devices with spatially separated resonators. It is particularly suited for frequency spectra with sharp (passband-to-stopband) transitions, where previous "brute-force" design attempts faced severe difficulties. 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 present 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 <i>LC</i>-ladder circuits.