Guided modes and resonant transmission in periodic structures

Abstract

We analyze resonant scattering phenomena of scalar fields in periodic slab and pillar structures that are related to the interaction between guided modes of the structure and plane waves emanating from the exterior. The mechanism for the resonance is the nonrobust nature of the guided modes with respect to perturbations of the wavenumber, which reflects the fact that the frequency of the mode is embedded in the continuous spectrum of the pseudo-periodic Helmholtz equation. We extend previous complex perturbation analysis of transmission anomalies to structures whose coefficients are only required to be measurable and bounded from above and below, and we establish sufficient conditions involving structural symmetry that guarantee that the transmission coefficient reach 0% and 100% at nearby frequencies close to those of the guided modes. Our analysis demonstrates a few more patterns of anomalies in nongeneric cases, including anomalies of two peaks and one dip on the transmission graph with total background transmission, anomalies of one peak and two dips with total background reflection, and multiple anomalies, and we also prove sufficient conditions for these transmission coefficients to reach 0% and 100%. For pillar structures, we establish a fundamental framework using Bessel functions for the analysis of guided modes, and prove the existence and nonexistence in structures in analogy to results for slabs. We provide a new existence result of nontrivial embedded guided modes, which are stable with respect to the wavenumber and nonrobust under perturbations of the structural geometry, in periodic pillars with smaller periodic cells.