Frequency-dependent capacitance matrix formulation for Fabry-Perot resonances in two and three dimensional systems

Abstract

We study scattering resonances of finite and infinite periodic two- and three-dimensional systems of high-contrast resonators beyond the subwavelength regime. At each fixed admissible nonzero reference frequency, we introduce frequency-dependent capacitance matrices and derive quantitative asymptotic expansions of hybridized Fabry--Pérot resonant frequencies and their corresponding eigenmodes in terms of the material contrast parameter. We provide a partial differential equation (PDE) formulation of the frequency-dependent capacitance matrix analogous to the one for the capacitance matrix in the subwavelength regime. Based on this PDE formulation, we establish key properties of the frequency-dependent capacitance matrix and estimate its norm at high frequencies for a single smooth resonator with nontrapping exterior in three dimensions, identifying a regime in which the reduced residual remains perturbative. For infinite periodic resonator arrays, we prove bandgap opening under a uniform exterior non-resonance condition and generic Dirac cones for a honeycomb scaling family. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of finite and infinite periodic systems of high-contrast resonators at arbitrarily high frequencies and understanding their anomalous localization and transport properties arising from strong coupling to a discrete set of eigenfrequencies.

revised; 50 pages