Existence of Friedrich-Wintgen Bound States in the Continuum: Cavity with a Thin Waveguide Opening

TL;DR

This paper rigorously proves the existence of Friedrich-Wintgen bound states in the continuum within electromagnetic cavities coupled to thin waveguides, expanding understanding of BICs through mathematical analysis of mode interactions and perturbations.

Contribution

It introduces a rigorous mathematical framework for establishing FW-BICs in various cavity geometries using mode-matching and eigenvalue analysis, extending prior numerical findings.

Findings

Existence of FW-BICs is guaranteed for small waveguide widths.

BICs occur at intersections of implicit curves derived from the governing equations.

Eigenvalue crossings and non-vanishing coupling are key conditions for BIC emergence.

Abstract

Bound states in the continuum (BICs) are localized states embedded within a continuum of propagating waves. Perturbations that disrupt BICs typically induce ultra-strong resonances, a phenomenon enabling diverse applications in photonics. This work investigates the existence of BICs in two-dimensional electromagnetic cavities coupled to thin waveguides for H-polarized waves. Our focus is on Friedrich-Wintgen BICs (FW-BICs), which arise from destructive interference between two resonant modes and were identified numerically in rectangular cavities with waveguide openings by Lyapina et al. [J. Fluid Mech., 780 (2015), pp. 370--387]. Here, we rigorously establish the existence of FW-BICs in a broader class of cavity geometries by introducing perturbations to the refractive index under regularity constraints. We show that BICs correspond to intersections of two curves derived implicitly from the governing equations constructed via the mode-matching method. Crucially, we prove that such intersections are guaranteed for sufficiently small waveguide widths, provided that two eigenvalues of the cavity cross and the associated eigenfunctions exhibit non-vanishing coupling to the radiation channel at the cavity-waveguide interface. Furthermore, our approach remains applicable for studying the emergence of FW-BICs under parameter-dependent boundary perturbations to the cavity.