Singular value decomposition to describe bound states in the continuum in periodic metasurfaces

TL;DR

This paper introduces a singular value decomposition (SVD) framework to analyze and predict bound states in the continuum (BICs) in periodic metasurfaces, providing a systematic way to engineer high-Q resonances.

Contribution

The paper develops a general SVD-based method to identify and characterize BICs in metasurfaces, linking singular values and vectors to BIC formation and properties.

Findings

SVD predicts BIC spectral positions accurately.

The approach distinguishes symmetry-protected and accidental BICs.

It enables analysis of BIC behavior under perturbations.

Abstract

Understanding how bound states in the continuum (BICs) emerge in periodic metasurfaces is essential for the controlled design of high-Q resonances and their systematic manipulation. Here, we investigate the singular value decomposition (SVD) of the effective transition matrix and the scattering matrix of periodic metasurfaces within a parameter range where the metasurface sustains a BIC. Our analysis yields general and practically applicable conditions on the singular values and singular vectors that enable BIC formation. At the BIC eigenfrequency, the inverse of the largest singular value of both matrices vanishes, and the corresponding left (right) singular vector is orthogonal to outgoing (incoming) plane waves that propagate in the directions of open diffraction orders. Our SVD-based approach predicts the spectral position of the BIC and provides detailed information about its properties, including the expansion coefficients in the multipole and plane-wave bases, as well as its behavior under perturbations that transform the BIC into a quasi-BIC. The approach is numerically validated by considering both symmetry-protected and accidental BICs in arrays of scatterers supporting electromagnetic or acoustic multipole resonances. The presented SVD framework offers a broadly applicable foundation for engineering BICs and quasi-BICs in complex metasurfaces, potentially enabling new routes for wave-based devices with tailored radiative properties.