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

Jiaxin Zhou, Wangtao Lu, Ya Yan Lu

TL;DR

This work addresses the existence of Friedrich-Wintgen bound states in the continuum (FW-BICs) in two-dimensional cavities coupled to thin waveguides for $H$-polarized waves. It develops a mode-matching framework that reduces the infinite-dimensional system to two implicit curves $\mu(\delta)$ whose intersections yield FW-BICs when two cavity eigenvalues cross transversally and their eigenfunctions couple nontrivially to the radiation channel; the analysis covers small waveguide width $h$ and analytic refractive-index perturbations in $\delta$. The main contribution is a rigorous existence proof showing that FW-BICs persist under broad cavity geometries and parameter-dependent boundary perturbations, with a precise reduction to a finite-dimensional system and explicit conditions for the intersection that yields a BIC. Numerical experiments validate the theory, demonstrating FW-BICs in asymmetric cavities for small $h$ and modest refractive-index perturbations, and highlighting robustness under changes in $h$ and perturbation magnitude, with implications for photonic devices relying on ultra-high-Q resonances.

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.

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

TL;DR

This work addresses the existence of Friedrich-Wintgen bound states in the continuum (FW-BICs) in two-dimensional cavities coupled to thin waveguides for -polarized waves. It develops a mode-matching framework that reduces the infinite-dimensional system to two implicit curves whose intersections yield FW-BICs when two cavity eigenvalues cross transversally and their eigenfunctions couple nontrivially to the radiation channel; the analysis covers small waveguide width and analytic refractive-index perturbations in . The main contribution is a rigorous existence proof showing that FW-BICs persist under broad cavity geometries and parameter-dependent boundary perturbations, with a precise reduction to a finite-dimensional system and explicit conditions for the intersection that yields a BIC. Numerical experiments validate the theory, demonstrating FW-BICs in asymmetric cavities for small and modest refractive-index perturbations, and highlighting robustness under changes in and perturbation magnitude, with implications for photonic devices relying on ultra-high-Q resonances.

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.
Paper Structure (18 sections, 13 theorems, 115 equations, 7 figures)

This paper contains 18 sections, 13 theorems, 115 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and key requirements
  3. The mode-matching method
  4. Eigenfunction expansion in the cavity
  5. Mode-expansion in the waveguide
  6. Mode-matching
  7. Reduction of the system
  8. Decomposition into real and imaginary components
  9. First reduction
  10. Further reduction
  11. Existence of BICs under analytic medium perturbations
  12. Uniform convergence of $f_{0}$ and $f_{1}$ to $0$ as $h\to0$
  13. Continuity in $\delta$ and Lipschitz continuity in $\mu$ for $f_{0}$ and $f_{1}$
  14. Existence of Friedrich-Wintgen BICs
  15. Numerical experiments
  16. ...and 3 more sections

Key Result

Lemma 3.1

\newlabellem:sec3:eigen:continuity0 Under Condition A.1, we can choose a set of normalized eigenfunctions $\{\psi_{m}(\cdot,\delta)\}$ such that the mapping $\delta\to\psi_{m}(\cdot,\delta)$ is continuous from $[-\delta_{0},\delta_{0}]$ to $H^{1}(\Omega_{\mathrm{in}})$ for each $m\in\mathbb{N}$, i Furthermore, the associated eigenvalue satisfies where $C_{0}>0$ is a constant independent of $m$.

Figures (7)

  • Figure 1: A cavity with a thin waveguide opening is represented as $\Omega:=\Omega_{\mathrm{in}}\cup\Gamma_{h}\cup\Omega_{\mathrm{out}}$, where $\Omega_{\mathrm{in}}$ represents the cavity region featuring a line segment on its boundary, $\Omega_{\mathrm{out}}$ denotes the waveguide region of width $h$ extended from the line segment vertically, and $\Gamma_{h}$ is the common boundary between these two regions. Additionally, $n(\mathbf{x})$ denotes the refractive index in $\Omega$, and $\Omega_{B}$ denotes a rectangular region in $\Omega_{\mathrm{in}}$ that $\Gamma_{h}\subset\partial\Omega_{B}$.
  • Figure 1: Schematic of a rectangular cavity with a waveguide opening. The computational region is defined as $\Omega:=\Omega_{\mathrm{in}}\cup\Gamma_{h}\cup\Omega_{\mathrm{out}}$ comprises a rectangular cavity $\Omega_{\mathrm{in}}$ spanning between coordinates $\mathbf{o}_{1}=(0,-4\pi/3)$ and $\mathbf{o}_{2}=(-\pi,2\pi/3)$, a semi-infinite waveguide $\Omega_{\mathrm{out}}$ of width $h=2\pi/9$, and their shared interface $\Gamma_{h}$. A rectangular coordinate system is centered at $\mathbf{o}$, the midpoint of $\Gamma_{h}$. Six identical circular elements (radius 0.48) with refractive index $n(\cdot)$ are symmetrically embedded in $\Omega_{\mathrm{in}}$, centered at positions $(-\pi/2\pm0.8,\pi/3)$, $(-\pi/2\pm0.8,-\pi/3)$ and $(-\pi/2\pm0.8,-\pi)$.
  • Figure 2: (a). Computational mesh used for the eigenvalue problem \ref{['eq:sec2:eigen:goveq']}--\ref{['eq:sec2:eigen:bc']} in $\Omega_{\mathrm{in}}$. (b) Computational mesh used for computing the resonances of the system governed by \ref{['eq:sec2:goveq']}--\ref{['eq:sec2:bc']} in $\Omega$ with $h=2\pi/9$.
  • Figure 3: First eight eigenvalues of the problem \ref{['eq:sec2:eigen:goveq']}--\ref{['eq:sec2:eigen:bc']} in $\Omega_{\mathrm{in}}$ as the refractive index of the elements varies from $1$ to $2$.
  • Figure 4: Resonances of the problem \ref{['eq:sec2:goveq']}--\ref{['eq:sec2:bc']} in $\Omega$ approximating the eigenvalues $\lambda_{6}$ and $\lambda_{7}$ as the refractive index of the elements varies. A BIC emerges at $n\approx1.442$ and $\mu=\widetilde{\lambda}_{6}\approx1.718$.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Lemma 3.1
  • Proof 1
  • Lemma 4.1
  • Proof 2
  • Lemma 4.2
  • Proof 3
  • Theorem 4.3
  • Proof 4
  • Lemma 4.4
  • Proof 5
  • ...and 17 more