Approximating transmission and reflection spectra near isolated nondegenerate resonances

TL;DR

This paper develops an approximation method for transmission and reflection spectra near isolated resonances in scattering problems, simplifying calculations by using the scattering matrix at a specific frequency and resonant mode information.

Contribution

It introduces a new approximation approach for spectra near isolated resonances and revises temporal coupled-mode theory for better accuracy.

Findings

Accurate approximation of spectra near resonances using the scattering matrix at a single frequency.

Validation of the theory through numerical examples involving diffraction by periodic structures.

Revised coupled-mode theory aligns with the proposed spectral approximation.

Abstract

A linear scattering problem for which incoming and outgoing waves are restricted to a finite number of radiation channels can be precisely described by a frequency-dependent scattering matrix. The entries of the scattering matrix, as functions of the frequency, give rise to the transmission and reflection spectra. To find the scattering matrix rigorously, it is necessary to solve numerically the partial differential equations governing the relevant waves. In this paper, we consider resonant structures with an isolated nondegenerate resonant mode of complex frequency $\omega_\star$, and show that for real frequencies near $\omega_0 = \mbox{Re}(\omega_\star)$, the transmission and reflection spectra can be approximated using only the scattering matrix at $\omega_0$ and information about the resonant mode. We also present a revised temporal coupled-mode theory that produces the same approximate formulas for the transmission and reflection spectra. Numerical examples for diffraction of plane waves by periodic structures are presented to validate our theory.