Local Robustness of Bound States in the Continuum through Scattering-Matrix Eigenvector Continuation

TL;DR

This paper presents a topological framework for understanding the local robustness of bound states in the continuum (BICs) in periodic structures, using scattering-matrix eigenvector continuation and mapping degree theory.

Contribution

It introduces a novel topological interpretation of BICs' robustness and provides a practical numerical method for their detection and verification.

Findings

BICs correspond to zeros of a parameter-to-coefficient mapping.

The robustness of BICs is explained via the mapping degree of this function.

A numerical criterion for BIC detection based on the mapping degree is proposed.

Abstract

We consider the diffraction of time-harmonic plane waves by a periodic structure, governed by the Helmholtz equation. Bound states in the continuum (BICs) are quasi-periodic fields that remain $L^{2}$-bounded over one period and occur at frequencies embedded in the continuous spectrum. Perturbations that break a BIC can lead to ultra-strong resonances, enabling various applications in photonics. Employing the implicit function theorem, we demonstrate how a simple BIC continuously deforms into a propagating field as system parameters vary in a neighborhood, with the frequency adjusting accordingly. In this setting, the incident coefficients of the field persist as an eigenvector of the scattering matrix with a fixed eigenvalue. By introducing a mapping $\mathcal{P}$ from the parameters to these coefficients, the zeros of $\mathcal{P}$ correspond precisely to BICs. When such a zero is isolated and the dimensions of the domain and range coincide, the BIC can be related to the mapping degree of $\mathcal{P}$ in a small neighborhood. This perspective clarifies the phase singularity associated with BICs and provides a general topological interpretation of their local robustness with respect to the given parameters. Moreover, it yields a practical numerical criterion for detecting and verifying BICs via computation of the mapping degree of $\mathcal{P}$.