Identifying bound states in the continuum by their boundary sensitivity

TL;DR

The paper tackles the challenge of identifying bound states in the continuum (BICs) in open-wave systems without computing the imaginary part of eigenfrequencies, noting that BICs have infinite quality factor $Q=\Re(f)/[2\Im(f)]$. It introduces a boundary-variation approach that closes the computational domain with different external boundaries and builds spectral histograms of normal-mode frequencies $h(\omega)$ (with Gaussian smoothing) to reveal BICs, supported by integral reciprocity relations. The method is validated on two setups: a periodic chain supporting Rayleigh–Bloch waves and a circular whispering-gallery resonator, showing accumulation of modes at BIC points and agreement with conventional QNM analysis using PMLs. This approach offers a computation-efficient, boundary-insensitive means of detecting BICs across wave systems and clarifies the link between boundary sensitivity and radiation loss, with broad applicability to acoustic, optical, and quantum waves.

Abstract

We introduce a method for effectively identifying bound states in the continuum (BICs) - notably without computing the imaginary part of the eigenvalues - thereby simplifying the modeling and potentially reducing computation time. In real, open, physical systems, wave decay must be taken into account. This phenomenon is captured by complex-valued solutions of the harmonic wave equation, the so-called quasi-normal modes (QNMs). BICs, however, constitute a limiting class of solutions that do not radiate energy to infinity and are therefore, by their very nature, insensitive to the region surrounding the physical structure. Building on this observation, we identify BICs by varying the external boundary conditions that close the computational domain; the resulting behavior is displayed in the form of spectral histograms. We demonstrate the effectiveness of this procedure by comparing it with conventional QNM analysis employing perfectly matched layers. Two representative examples are considered: a periodic system of permeable inclusions supporting guided Rayleigh-Bloch waves, and a whispering-gallery resonator constructed from this configuration. Finally, we provide a mathematical explanation for the method's validity by deriving integral reciprocity statements.

Figures (4)

• Figure 1: A linear periodic chain of inclusions in a host material supporting the propagation of Rayleigh-Bloch waves laudePRB2025. The inclusion diameter is $d$ and $a$ denotes the lattice constant. Inclusions have an elastic modulus contrast $B_2/B_1 = 1/16$ relative to the surrounding material and the same mass density. (a) Surrounding the domain of computation with a perfectly matched layer (PML) enables the computation of quasi-normal modes. (b) The resolvent band structure computed using the stochastic excitation method laudePRB2018 then gives a map of the complex dispersion relation in the first Brillouin zone. (c) Enclosing the same domain as in panel (a) using specific boundary conditions, here of the Neumann type, leads to the definition of normal modes that depend on the position of the boundary closing the waveguide. (d) The spectral histogram is then obtained by computing classical band structures for $H=101$ cell heights ranging from $4a$ to $8a$.
• Figure 2: Whispering gallery modes as quasi-normal modes (QNMs). (a) A sequence of eight inclusions ($N=8$), each with a diameter of $0.6a$, forms a whispering gallery resonator subject to wave radiation. A perfectly matched layer (PML) emulates radiation and allows for the computation of QNMs of the resonator. (b) The local density of states (LDOS) computed using the stochastic excitation method, as a function of frequency and the elastic contrast between the inclusions and the host medium shows groups of QNMs that can be labeled with two indices $mn$. The azimuthal index $m$ varies from 0 to 3, and the radial index $n=0$ is fixed. The vertical dotted line is placed at $B_2/B_1 = 1/16$. (c) Enclosing the same domain as in panel (a) using a Neumann boundary condition leads to the definition of normal modes that depend on the position of the boundary closing the computation domain. (d) The local density of states of quasi-normal modes is shown with a magenta solid line for contrast $B_2/B_1 = 1/16$ and is compared with the spectral histogram of normal modes, shown with a green solid line, obtained by the sum of normal distributions of Eq. (\ref{['eq:gaussian']}).
• Figure 3: Quasi-normal, whispering-gallery, modes of a circular sequence of 8 slow inclusions with diameters of $0.6a$ for a modulus ratio of $B_2/B_1 = 1/16$. The symmetry of each mode is classified by $mn$, where $m$ denotes the azimuthal index and $n$ the radial number.
• Figure 4: Evolution of both the frequency and the quality factor $Q$ of quasi-normal mode $mn=10$ as a function of the elastic modulus ratio $B_2/B_1$.