A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects

TL;DR

This paper introduces a tight-binding method for accurately computing subwavelength guided modes in high-contrast periodic media with line defects, leveraging capacitance matrix decay properties.

Contribution

It provides a rigorous proof of exponential decay in capacitance matrices, enabling efficient spectral computations for defect modes in complex media.

Findings

Validated the decay properties through numerical experiments

Demonstrated accurate computation of topological interface modes

Established error bounds for matrix approximations

Abstract

In this paper, we develop an accurate and efficient framework for computing subwavelength guided modes in high-contrast periodic media with line defects, based on a tight-binding approximation. The physical problem is formulated as an eigenvalue problem for the Helmholtz equation with high-contrast parameters. By employing layer potential theory on unbounded domains, we characterize the subwavelength frequencies via the quasi-periodic capacitance matrix. Our main contribution is the proof of exponential decay of the off-diagonal elements of the associated full and quasi-periodic capacitance matrices. These decay properties provide error bounds for the banded approximation of the capacitance matrices, thereby enabling a tight-binding approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are presented to validate the theoretical results, including applications to topological interface modes.