Mathematical Analysis of Symmetry-Protected Bound States in the Continuum in Waveguide Arrays

TL;DR

This work provides a rigorous mathematical framework for symmetry-protected Bound States in the Continuum ($\mathrm{BIC}$s) in optical waveguide arrays by developing Nonorthogonal Coupled-Mode Equations (NCME) and deriving exact overlap and coupling matrices $S_{nm}$ and $K_{nm}$ via Bessel function addition theorems. It extends the analysis to the infinite array using convolution operators on $\ell^2(\mathbb{Z})$ and Wiener’s lemma to obtain a precise dispersion relation $\widehat W(\theta)$ and the associated continuum. A strict existence proof is given for a symmetry-protected BIC in a 53-waveguide system with two vertical defects, together with a symmetry-preserving excitation strategy and a quantitative symmetry-breaking transition to a leaky mode described by a Crank–Nicolson scheme. The results deliver a high-fidelity, design-oriented mathematical framework that surpasses orthogonal approximations and has implications for robust BIC device engineering and photonic-band design.

Abstract

This paper presents a rigorous mathematical analysis for symmetry-based Bound States in the Continuum (BICs) in optical waveguide arrays. Different from existing research, we consider a finite system of horizontally and equidistantly aligned waveguides and transform the wave propagation problem into Nonorthogonal Coupled-Mode Equations (NCME), rather than adopting the tight-binding approximation or orthogonal coupled-mode equations. We derive the exact expressions of the overlap integrals and coupling coefficients by utilizing the addition theorems of Bessel functions. We then generalize the discussion to an infinite waveguide array and rigorously characterize the dispersion relation and continuum with the help of theories in harmonic analysis. In the second part of the paper, we give a strict proof of the existence of BICs in the aforementioned waveguide system with two additional identical vertical waveguides aligned symmetrically above and below the horizontal waveguide array. We further numerically demonstrate the transition from a perfect BIC to a leaky mode by introducing a symmetry-breaking refractive index perturbation and quantitatively analyze the resulting radiation losses. This work gives a comprehensive study of symmetry-protected BICs and provides an efficient and precise computational model for designing such BICs devices.

Figures (2)

• Figure 1: Waveguide array
• Figure 2: Intensity Distribution at $z=100.00mm$

Theorems & Definitions (50)

• remark 1
• remark 2
• remark 3
• proof
• proof
• proof
• proof
• Theorem 3.3
• proof
• proof