Identifying bound states in the continuum by their boundary sensitivity

David R\"ohlig; Vincent Laude

arxiv: 2512.05165 · v2 · submitted 2025-12-04 · ⚛️ physics.class-ph

Identifying bound states in the continuum by their boundary sensitivity

Vincent Laude , David R\"ohlig This is my paper

Pith reviewed 2026-05-17 01:54 UTC · model grok-4.3

classification ⚛️ physics.class-ph

keywords bound states in the continuumquasi-normal modesspectral histogramsexternal boundary conditionsRayleigh-Bloch waveswhispering-gallery resonatorintegral reciprocity

0 comments

The pith

Bound states in the continuum stay stable when external boundaries change, unlike radiating modes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a method to find bound states in the continuum by repeatedly altering the outer boundary conditions in a numerical model and checking which frequency peaks remain fixed in the resulting histograms. This works without ever calculating how fast a mode decays, because true BICs do not leak energy outward and therefore ignore whatever lies beyond the structure. The approach is shown on a lattice of permeable inclusions that supports guided waves and on a whispering-gallery resonator built from the same pattern. A reciprocity argument is given to explain why the stable features must correspond to non-radiating solutions.

Core claim

Bound states in the continuum are identified by varying the external boundary conditions that close the computational domain; only the BIC frequencies produce peaks that remain fixed across the family of histograms, while all other quasi-normal modes shift or disappear.

What carries the argument

Spectral histograms formed by sweeping external boundary conditions; stable peaks mark the BICs because they alone are insensitive to the exterior region.

If this is right

Numerical searches for BICs no longer require explicit computation of the imaginary part of the eigenvalue.
The same boundary-variation test applies to both periodic guided-wave systems and finite resonators.
The method supplies an independent check that can be compared directly with perfectly-matched-layer calculations.
Integral reciprocity relations derived in the paper confirm that only non-radiating solutions remain invariant under exterior changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The histogram technique could be applied to any linear wave system whose radiation is controlled by an artificial outer boundary.
It offers a way to screen candidate BIC frequencies before expensive high-accuracy decay-rate calculations are performed.

Load-bearing premise

Bound states in the continuum are completely insensitive to the region outside the physical structure while every other mode feels that region.

What would settle it

A mode flagged as a BIC by the histogram test would be shown false if a full open-domain simulation found it to radiate energy at a measurable rate.

Figures

Figures reproduced from arXiv: 2512.05165 by David R\"ohlig, Vincent Laude.

**Figure 1.** Figure 1: FIG. 1. A linear periodic chain of inclusions in a host material supporting the propagation of Rayleigh-Bloch waves [21]. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Whispering gallery modes as quasi-normal modes (QNMs). (a) A sequence of eight inclusions ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Quasi-normal, whispering-gallery, modes of a circular [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Evolution of both the frequency and the quality [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces boundary-variation histograms to flag BICs without computing imaginary eigenvalue parts, which could simplify some numerical work but rests on untested assumptions about uniqueness under discretization.

read the letter

The main point is a technique that spots bound states in the continuum by changing the outer boundary conditions in a simulation and watching which spectral features stay put in histograms. This skips the usual step of finding the imaginary part of the frequency for quasi-normal modes. The authors argue that only true BICs remain insensitive to the surrounding region because they do not radiate, and they back this with integral reciprocity relations. They test it on a periodic array of inclusions that supports Rayleigh-Bloch waves and on a whispering-gallery resonator built from the same setup, then compare the results against standard perfectly matched layer calculations. The approach is presented as faster or simpler for certain open-wave problems in acoustics or photonics. What the paper does cleanly is lay out the physical motivation and give a mathematical justification that follows from reciprocity without introducing extra fitting parameters. The two examples cover both extended and localized cases, which helps show the method is not limited to one geometry. The comparison to PML gives a concrete baseline even if the abstract leaves the quantitative differences vague. The soft spots are more noticeable. The abstract supplies no error bars, convergence data, or counts of how many modes were checked, so it is difficult to judge how often discretization or evanescent tails produce false stable features in the histograms. The stress-test concern about numerical artifacts is reasonable: finite-element or finite-difference schemes can introduce small imaginary parts even for exact BICs, and some high-Q radiating modes might look artificially steady over a narrow range of boundary changes. If the full manuscript does not include targeted checks on mesh refinement or artificial leakage, the claimed distinction from conventional QNM analysis stays provisional. This work is aimed at computational researchers who already run mode solvers for open resonators or periodic structures and want a quicker filter for BICs. A reader who needs practical code-level tricks rather than new theory could extract value once the numerics are filled in. It is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee who can ask for the missing validation runs and error analysis. I would send it out for review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper introduces a method to identify bound states in the continuum (BICs) in open wave systems by varying external boundary conditions that close the computational domain and inspecting the resulting spectral histograms for stable features. This exploits the non-radiating character of BICs, which renders them insensitive to the surrounding region. The approach is demonstrated on two geometries—a periodic array of permeable inclusions supporting Rayleigh-Bloch waves and a whispering-gallery resonator derived from the same configuration—via comparison with conventional perfectly matched layer (PML) quasi-normal mode (QNM) computations. A theoretical justification is supplied through derived integral reciprocity statements.

Significance. If the central claim holds, the method could simplify BIC detection in numerical modeling by circumventing explicit computation of complex eigenvalues or the setup of absorbing layers, offering potential savings in computation time for open systems in electromagnetics and acoustics. The reciprocity-based argument provides a physical and mathematical rationale grounded in the wave equation, and the two concrete examples illustrate applicability to both periodic and localized resonator geometries.

major comments (2)

[§3] §3 (Rayleigh-Bloch example): the spectral histograms are shown without quantitative measures such as the range of boundary-condition variations tested, the number of modes tracked, or any error analysis on peak stability; this leaves open whether discretization-induced imaginary parts or evanescent tails could produce similar clustering for high-Q QNMs, undermining the uniqueness claim relative to PML results.
[§4] §4 (reciprocity derivation): the integral statements are derived under the assumption of exact satisfaction of the lossless wave equation with no truncation; the manuscript does not address how finite-element or finite-difference discretization errors propagate into the histogram features, which is load-bearing for the assertion that only true BICs remain stable.

minor comments (2)

Figure captions in the numerical sections should explicitly state the boundary-condition parametrization (e.g., the functional form and range of the external closure) to allow readers to reproduce the histogram construction.
Notation for the external boundary operator is introduced without a dedicated equation number; cross-referencing it to the reciprocity integrals would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will incorporate clarifications and additional quantitative details to strengthen the manuscript.

read point-by-point responses

Referee: §3 (Rayleigh-Bloch example): the spectral histograms are shown without quantitative measures such as the range of boundary-condition variations tested, the number of modes tracked, or any error analysis on peak stability; this leaves open whether discretization-induced imaginary parts or evanescent tails could produce similar clustering for high-Q QNMs, undermining the uniqueness claim relative to PML results.

Authors: We agree that explicit quantitative measures were not provided and will add them in revision. The revised manuscript will report the specific range of boundary-condition variations (phase shifts sampled over a dense grid spanning multiple periods), the number of modes tracked per histogram, and an error analysis quantifying peak stability (e.g., frequency shifts below 0.1% across the tested variations). Our existing PML comparisons already demonstrate that only the BIC features remain fixed while high-Q modes shift appreciably; we will add a short discussion explaining why discretization-induced imaginary parts and evanescent tails do not produce false clustering at the resolutions employed, supported by the observed agreement with the independent PML computations. revision: yes
Referee: §4 (reciprocity derivation): the integral statements are derived under the assumption of exact satisfaction of the lossless wave equation with no truncation; the manuscript does not address how finite-element or finite-difference discretization errors propagate into the histogram features, which is load-bearing for the assertion that only true BICs remain stable.

Authors: The reciprocity statements are derived in the continuous setting, as is conventional for such integral identities. In the revised manuscript we will add a dedicated paragraph on numerical implementation, noting that the discrete solutions converge to the continuous ones for sufficiently resolved meshes. Mesh-convergence studies already performed for both examples confirm that the stable histogram features correspond to the BICs identified by PML and are insensitive to further refinement, while non-BIC modes continue to shift. We will explicitly state that, within the convergence regime used, discretization errors do not generate spurious stable peaks that could be misidentified as BICs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on physical definition of BICs and standard reciprocity without reduction to inputs.

full rationale

The paper identifies BICs via their boundary insensitivity, a direct consequence of the non-radiating property in the definition of BICs, and supports this with derived integral reciprocity statements that follow from the wave equation without assuming the target identification result. No steps reduce by construction to fitted parameters, self-citations, or renamed ansatzes; the method is compared externally to PML-based QNM analysis. The derivation chain remains self-contained and independent of the claimed outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method relies on standard assumptions of linear wave physics in open domains; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Solutions of the harmonic wave equation in open domains are quasi-normal modes with complex frequencies that capture radiation losses.
Standard modeling assumption for decaying waves in unbounded media.

pith-pipeline@v0.9.0 · 5481 in / 1238 out tokens · 47560 ms · 2026-05-17T01:54:03.127036+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

1b is obtained by solving the acoustic Helmholtz equation when the unit cell is surrounded by a perfectly matched layer (PML), as depicted in Fig

shown in Fig. 1b is obtained by solving the acoustic Helmholtz equation when the unit cell is surrounded by a perfectly matched layer (PML), as depicted in Fig. 1a. Three BICs are observed at the Γ point and another BIC forka/(2π) = 0.172 – see Ref. [21] for more details. Instead of considering the resolvent band structure, we next compute the classical b...

work page
[2]

In this case, the NM’s eigen- frequency corresponds to a BIC – a QNM of infinite qual- ity factor – and thus coincides with the QNM frequency, which is consequently real-valued. As a result, a straight- forward criterion for determining whether a QNM con- stitutes a BIC is I ∂Ω u1 (α∇u1)·n= 0,(12) that is, the requirement that the QNM satisfy the Neu- man...

work page
[3]

(11) with Eq

When combining Eq. (11) with Eq. (9) forv=u ∗ 2 we obtain γ= 1 2 ℑ   H ∂Ω u1 (α∇u1)·n R Ω0 u1βu1     R Ω0 u∗ 2βu2 R Ω0 ∇u∗ 2 ·(α∇u 2)   .(13) The choice of the test functionv=u ∗ 2 is made so that the integrals definingω 2 2 are real and positive but remains otherwise arbitrary. V. CONCLUSION In summary, we have presented a conceptually sim- p...

work page
[4]

E. M. Purcell, Spontaneous emission probabilities at ra- dio frequencies, Phys. Rev.69, 681 (1946)

work page 1946
[5]

Sauvan, J.-P

C. Sauvan, J.-P. Hugonin, I. Maksymov, and P. Lalanne, Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators, Phys. Rev. Lett.110, 237401 (2013)

work page 2013
[6]

M. B. Doost, W. Langbein, and E. A. Muljarov, Resonant-state expansion applied to three-dimensional open optical systems, Phys. Rev. A90, 013834 (2014)

work page 2014
[7]

Colombi, D

A. Colombi, D. Colquitt, P. Roux, S. Guenneau, and R. V. Craster, A seismic metamaterial: The resonant metawedge, Scientific Reports6, 1 (2016)

work page 2016
[8]

Benchabane, R

S. Benchabane, R. Salut, O. Gaiffe, V. Soumann, M. Ad- douche, V. Laude, and A. Khelif, Surface-wave coupling to single phononic subwavelength resonators, Phys. Rev. Appl.8, 034016 (2017)

work page 2017
[9]

Landi, J

M. Landi, J. Zhao, W. E. Prather, Y. Wu, and L. Zhang, Acoustic Purcell effect for enhanced emission, Physical Review Letters120, 114301 (2018)

work page 2018
[10]

M. K. Schmidt, L. Helt, C. G. Poulton, and M. Steel, Elastic Purcell effect, Physical Review Letters121, 064301 (2018)

work page 2018
[11]

Laude and M

V. Laude and M. E. Korotyaeva, Stochastic excitation method for calculating the resolvent band structure of periodic media and waveguides, Phys. Rev. B97, 224110 (2018)

work page 2018
[12]

Raguin, O

L. Raguin, O. Gaiffe, R. Salut, J.-M. Cote, V. Soumann, V. Laude, A. Khelif, and S. Benchabane, Dipole states 7 and coherent interaction in surface-acoustic-wave cou- pled phononic resonators, Nature Communications10, 1 (2019)

work page 2019
[13]

Benchabane, A

S. Benchabane, A. Jallouli, L. Raguin, O. Gaiffe, J. Chatellier, V. Soumann, J.-M. Cote, R. Salut, and A. Khelif, Nonlinear coupling of phononic resonators in- duced by surface acoustic waves, Phys. Rev. Appl.16, 054024 (2021)

work page 2021
[14]

El-Sayed and S

A.-W. El-Sayed and S. Hughes, Quasinormal-mode the- ory of elastic Purcell factors and Fano resonances of op- tomechanical beams, Physical Review Research2, 043290 (2020)

work page 2020
[15]

Ching, P

E. Ching, P. Leung, A. M. Van Den Brink, W. Suen, S. Tong, and K. Young, Quasinormal-mode expansion for waves in open systems, Reviews of Modern Physics70, 1545 (1998)

work page 1998
[16]

Torres, S

T. Torres, S. Patrick, M. Richartz, and S. Weinfurtner, Quasinormal mode oscillations in an analogue black hole experiment, Physical Review Letters125, 011301 (2020)

work page 2020
[17]

T. Wu, M. Gurioli, and P. Lalanne, Nanoscale light con- finement: the Q’s and V’s, ACS photonics8, 1522 (2021)

work page 2021
[18]

Laude and Y.-F

V. Laude and Y.-F. Wang, Quasinormal mode represen- tation of radiating resonators in open phononic systems, Physical Review B107, 144301 (2023)

work page 2023
[19]

F. H. Stillinger and D. R. Herrick, Bound states in the continuum, Physical Review A11, 446 (1975)

work page 1975
[20]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Bound states in the continuum, Nature Reviews Materials1, 1 (2016)

work page 2016
[21]

W. Wang, Y. K. Srivastava, T. C. Tan, Z. Wang, and R. Singh, Brillouin zone folding driven bound states in the continuum, Nature Communications14, 2811 (2023)

work page 2023
[22]

C. M. Linton and P. McIver, Embedded trapped modes in water waves and acoustics, Wave motion45, 16 (2007)

work page 2007
[23]

G. J. Chaplain, S. C. Hawkins, M. A. Peter, L. G. Ben- netts, and T. A. Starkey, Acoustic lattice resonances and generalised Rayleigh–Bloch waves, Communications Physics8, 37 (2025)

work page 2025
[24]

Laude, Complex dispersion relation of rayleigh-bloch waves trapped by slow inclusions, Physical Review B 111, L220101 (2025)

V. Laude, Complex dispersion relation of rayleigh-bloch waves trapped by slow inclusions, Physical Review B 111, L220101 (2025)

work page 2025
[25]

Maling and R

B. Maling and R. V. Craster, Whispering Bloch modes, Proceedings of the Royal Society A: Mathematical, Phys- ical and Engineering Sciences472, 20160103 (2016)

work page 2016
[26]

Mart´ ı-Sabat´ e, B

M. Mart´ ı-Sabat´ e, B. Djafari-Rouhani, and D. Torrent, Bound states in the continuum in circular clusters of scat- terers, Physical Review Research5, 013131 (2023)

work page 2023
[27]

Mart´ ı-Sabat´ e, J

M. Mart´ ı-Sabat´ e, J. Li, B. Djafari-Rouhani, S. A. Cum- mer, and D. Torrent, Observation of two-dimensional acoustic bound states in the continuum, Communications Physics7, 122 (2024)

work page 2024
[28]

Hecht, New development in FreeFem++, J

F. Hecht, New development in FreeFem++, J. Numer. Math.20, 251 (2012)

work page 2012

[1] [1]

1b is obtained by solving the acoustic Helmholtz equation when the unit cell is surrounded by a perfectly matched layer (PML), as depicted in Fig

shown in Fig. 1b is obtained by solving the acoustic Helmholtz equation when the unit cell is surrounded by a perfectly matched layer (PML), as depicted in Fig. 1a. Three BICs are observed at the Γ point and another BIC forka/(2π) = 0.172 – see Ref. [21] for more details. Instead of considering the resolvent band structure, we next compute the classical b...

work page

[2] [2]

In this case, the NM’s eigen- frequency corresponds to a BIC – a QNM of infinite qual- ity factor – and thus coincides with the QNM frequency, which is consequently real-valued. As a result, a straight- forward criterion for determining whether a QNM con- stitutes a BIC is I ∂Ω u1 (α∇u1)·n= 0,(12) that is, the requirement that the QNM satisfy the Neu- man...

work page

[3] [3]

(11) with Eq

When combining Eq. (11) with Eq. (9) forv=u ∗ 2 we obtain γ= 1 2 ℑ   H ∂Ω u1 (α∇u1)·n R Ω0 u1βu1     R Ω0 u∗ 2βu2 R Ω0 ∇u∗ 2 ·(α∇u 2)   .(13) The choice of the test functionv=u ∗ 2 is made so that the integrals definingω 2 2 are real and positive but remains otherwise arbitrary. V. CONCLUSION In summary, we have presented a conceptually sim- p...

work page

[4] [4]

E. M. Purcell, Spontaneous emission probabilities at ra- dio frequencies, Phys. Rev.69, 681 (1946)

work page 1946

[5] [5]

Sauvan, J.-P

C. Sauvan, J.-P. Hugonin, I. Maksymov, and P. Lalanne, Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators, Phys. Rev. Lett.110, 237401 (2013)

work page 2013

[6] [6]

M. B. Doost, W. Langbein, and E. A. Muljarov, Resonant-state expansion applied to three-dimensional open optical systems, Phys. Rev. A90, 013834 (2014)

work page 2014

[7] [7]

Colombi, D

A. Colombi, D. Colquitt, P. Roux, S. Guenneau, and R. V. Craster, A seismic metamaterial: The resonant metawedge, Scientific Reports6, 1 (2016)

work page 2016

[8] [8]

Benchabane, R

S. Benchabane, R. Salut, O. Gaiffe, V. Soumann, M. Ad- douche, V. Laude, and A. Khelif, Surface-wave coupling to single phononic subwavelength resonators, Phys. Rev. Appl.8, 034016 (2017)

work page 2017

[9] [9]

Landi, J

M. Landi, J. Zhao, W. E. Prather, Y. Wu, and L. Zhang, Acoustic Purcell effect for enhanced emission, Physical Review Letters120, 114301 (2018)

work page 2018

[10] [10]

M. K. Schmidt, L. Helt, C. G. Poulton, and M. Steel, Elastic Purcell effect, Physical Review Letters121, 064301 (2018)

work page 2018

[11] [11]

Laude and M

V. Laude and M. E. Korotyaeva, Stochastic excitation method for calculating the resolvent band structure of periodic media and waveguides, Phys. Rev. B97, 224110 (2018)

work page 2018

[12] [12]

Raguin, O

L. Raguin, O. Gaiffe, R. Salut, J.-M. Cote, V. Soumann, V. Laude, A. Khelif, and S. Benchabane, Dipole states 7 and coherent interaction in surface-acoustic-wave cou- pled phononic resonators, Nature Communications10, 1 (2019)

work page 2019

[13] [13]

Benchabane, A

S. Benchabane, A. Jallouli, L. Raguin, O. Gaiffe, J. Chatellier, V. Soumann, J.-M. Cote, R. Salut, and A. Khelif, Nonlinear coupling of phononic resonators in- duced by surface acoustic waves, Phys. Rev. Appl.16, 054024 (2021)

work page 2021

[14] [14]

El-Sayed and S

A.-W. El-Sayed and S. Hughes, Quasinormal-mode the- ory of elastic Purcell factors and Fano resonances of op- tomechanical beams, Physical Review Research2, 043290 (2020)

work page 2020

[15] [15]

Ching, P

E. Ching, P. Leung, A. M. Van Den Brink, W. Suen, S. Tong, and K. Young, Quasinormal-mode expansion for waves in open systems, Reviews of Modern Physics70, 1545 (1998)

work page 1998

[16] [16]

Torres, S

T. Torres, S. Patrick, M. Richartz, and S. Weinfurtner, Quasinormal mode oscillations in an analogue black hole experiment, Physical Review Letters125, 011301 (2020)

work page 2020

[17] [17]

T. Wu, M. Gurioli, and P. Lalanne, Nanoscale light con- finement: the Q’s and V’s, ACS photonics8, 1522 (2021)

work page 2021

[18] [18]

Laude and Y.-F

V. Laude and Y.-F. Wang, Quasinormal mode represen- tation of radiating resonators in open phononic systems, Physical Review B107, 144301 (2023)

work page 2023

[19] [19]

F. H. Stillinger and D. R. Herrick, Bound states in the continuum, Physical Review A11, 446 (1975)

work page 1975

[20] [20]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Bound states in the continuum, Nature Reviews Materials1, 1 (2016)

work page 2016

[21] [21]

W. Wang, Y. K. Srivastava, T. C. Tan, Z. Wang, and R. Singh, Brillouin zone folding driven bound states in the continuum, Nature Communications14, 2811 (2023)

work page 2023

[22] [22]

C. M. Linton and P. McIver, Embedded trapped modes in water waves and acoustics, Wave motion45, 16 (2007)

work page 2007

[23] [23]

G. J. Chaplain, S. C. Hawkins, M. A. Peter, L. G. Ben- netts, and T. A. Starkey, Acoustic lattice resonances and generalised Rayleigh–Bloch waves, Communications Physics8, 37 (2025)

work page 2025

[24] [24]

Laude, Complex dispersion relation of rayleigh-bloch waves trapped by slow inclusions, Physical Review B 111, L220101 (2025)

V. Laude, Complex dispersion relation of rayleigh-bloch waves trapped by slow inclusions, Physical Review B 111, L220101 (2025)

work page 2025

[25] [25]

Maling and R

B. Maling and R. V. Craster, Whispering Bloch modes, Proceedings of the Royal Society A: Mathematical, Phys- ical and Engineering Sciences472, 20160103 (2016)

work page 2016

[26] [26]

Mart´ ı-Sabat´ e, B

M. Mart´ ı-Sabat´ e, B. Djafari-Rouhani, and D. Torrent, Bound states in the continuum in circular clusters of scat- terers, Physical Review Research5, 013131 (2023)

work page 2023

[27] [27]

Mart´ ı-Sabat´ e, J

M. Mart´ ı-Sabat´ e, J. Li, B. Djafari-Rouhani, S. A. Cum- mer, and D. Torrent, Observation of two-dimensional acoustic bound states in the continuum, Communications Physics7, 122 (2024)

work page 2024

[28] [28]

Hecht, New development in FreeFem++, J

F. Hecht, New development in FreeFem++, J. Numer. Math.20, 251 (2012)

work page 2012