Identifying bound states in the continuum by their boundary sensitivity

TL;DR

This paper presents a boundary sensitivity method to identify bound states in the continuum (BICs) without calculating their imaginary eigenvalues, simplifying analysis in physical systems.

Contribution

The authors introduce a novel boundary variation technique to detect BICs efficiently, avoiding complex eigenvalue computations and validated through examples and mathematical proofs.

Findings

The method accurately identifies BICs by boundary variation.

It reduces computational complexity compared to traditional QNM analysis.

Validated on periodic systems and whispering-gallery resonators.

Abstract

We introduce a method for effectively identifying bound states in the continuum (BICs) - notably without computing the imaginary part of the eigenvalues - thereby simplifying the modeling and potentially reducing computation time. In real, open, physical systems, wave decay must be taken into account. This phenomenon is captured by complex-valued solutions of the harmonic wave equation, the so-called quasi-normal modes (QNMs). BICs, however, constitute a limiting class of solutions that do not radiate energy to infinity and are therefore, by their very nature, insensitive to the region surrounding the physical structure. Building on this observation, we identify BICs by varying the external boundary conditions that close the computational domain; the resulting behavior is displayed in the form of spectral histograms. We demonstrate the effectiveness of this procedure by comparing it with conventional QNM analysis employing perfectly matched layers. Two representative examples are considered: a periodic system of permeable inclusions supporting guided Rayleigh-Bloch waves, and a whispering-gallery resonator constructed from this configuration. Finally, we provide a mathematical explanation for the method's validity by deriving integral reciprocity statements.