Bound states in the continuum in circular clusters of scatterers Marc Martí-Sabaté1Bahram Djafari-Rouhani2and Dani Torrent1 1GROC UJI Institut de Noves Tecnologies de la Imatge INIT

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Bound states in the continuum in circular clusters of scatterers

Marc Martí-Sabaté,1Bahram Djafari-Rouhani,2and Dani Torrent1, ∗

1GROC, UJI, Institut de Noves Tecnologies de la Imatge (INIT),

Universitat Jaume I, 12071, Castelló de la Plana, Spain

2IEMN, University of Lille, Cité scientiﬁque, 59650 Villeneuve d’Ascq, France

(Dated: October 19, 2022)

In this work, we study the localization of ﬂexural waves in highly symmetric clusters of scatterers. It is

shown that when the scatterers are placed regularly in the perimeter of a circumference the quality factor of the

resonances strongly increases with the number of scatterers in the cluster. It is also found that in the continuous

limit, that is to say, when the number of scatterers tends to inﬁnite, the quality factor is inﬁnite so that the modes

belong to the class of the so called bound states in the continuum or BICs, and an analytical expression for the

resonant frequency is found. These modes have different multipolar symmetries, and we show that for high

multipolar orders the modes tend to localize at the border of the circumference, forming therefore a whishpering

gallery mode with an extraordinarily high quality factor. Numerical experiments are performed to check the

robustness of these modes under different types of disorder and also to study their excitation from the far ﬁeld.

Although we have focused our study to ﬂexural waves, the methodology presented in this work can be applied

to other classical waves, like electromagnetic or acoustic waves, being therefore a promissing approach for the

design of high quality resonators based on ﬁnite clusters of scatterers.

I. INTRODUCTION

Bound states in the continuum (BICs) are eigenmodes of a

system whose energy lies in the radiation part of the spectrum

while remaining localized in a ﬁnite part of the system and

with an inﬁnite lifetime. These states were ﬁrst mathemati-

cally proposed in 1929 by von Neumann and Wigner in the

framework of quantum mechanics [1], although the concept

has been extended to classical waves, like acoustics [2–6], mi-

crowaves [7, 8] or optics [9–11].

Despite the fact that the practical realization of BICs is a

challenging problem, structures based on them present sharp

resonances with extremely high quality factors, which have

as well the advantage, unlike ideal BICs, that can be ex-

cited with external radiative ﬁelds. Also named quasi-BICs

(or QBICs), these modes have been widely used in sensing

applications[12–14].

Among the wide variety of geometries and structures used

to ﬁnd BICs[15], those based on ﬁnite structures are specially

interesting for practical applications, since periodic or waveg-

uide BICs will always present ﬁnite-size effects which will de-

crease their efﬁciency. For instance, circular clusters of scat-

terers studied in some recent works[14, 16] are extraordinarily

convenient from the practical point of view. In this work, we

will generalize the study of these circular clusters of scatter-

ers to provide a general schema for the realization of QBICs

based on this geometry.

The manuscript is organized as follows: After this intro-

duction, in section II we study the formation of bound states

in the continuum in open systems by attaching a cluster of

mass-spring resonators to a thin elastic plate. We will ﬁnd that

when the scatterers in the cluster are arranged in the corners of

a regular polygon the quality factor of the resonances quickly

increases with the number of scatterers in the cluster. In sec-

tion III we perform several numerical experiments to check

∗dtorrent@uji.es

the robustness of these modes, and in section IV their excita-

tion from the far ﬁeld will be considered. Finally, section V

sumarizes the work.

II. EIGENMODES OF A POLYGONAL CLUSTER OF

SCATTERERS

The propagation of ﬂexural waves in thin elastic plates

where a cluster of Npoint-like resonators has been attached

at positions Rαis described by means of the inhomogeneous

Kirchhoff[17] equation

(∇4−k4

0)ψ(r) =

∑

α=1

tαδ(r−Rα)ψ(r)(1)

where ψ(r)is the spatial part of the vertical displacement of

the plate, which is assumed to be harmonic and of the form

W(r,t) = ψ(r)e−iωt.(2)

Also, the free space wavenumber k0is given by

0=ρh

Dω2,(3)

with ρ,hand Dbeing the plate’s mass density, height and

rigidity, respectively. The response of each resonator is given

by the tαcoefﬁcient, which is a resonant quantity whose prop-

erties depend on the geometry of the scatterer attached to the

plate[18]. However, for the purposes of the present work, it

will be assumed that it can take any real value in the range

tα∈(−∞,∞).

A self-consistent multiple scattering solution can be found

for the above equation as

ψ(r) = ψ0(r) +

∑

α=1

BαG(r−Rα)(4)

arXiv:2210.09833v1 [physics.class-ph] 17 Oct 2022

where ψ0(r)is the external incident ﬁeld on the cluster of

scatterers, G(r)is the Green’s function of Kirchhoff equation,

G(r) = i

8k2

(H0(k0r)−H0(ik0r)) (5)

with H0(·)being Hankel’s function of ﬁrst class. The multiple

scattering coefﬁcients Bαcan be obtained by means of the

self-consistent system of equations

∑

β=1

Mαβ Bβ=ψ(Rα),(6)

where

Mαβ =t−1

αδαβ −G(Rαβ )(7)

is the multiple scattering matrix M.

The eigenmodes of a cluster of Nscatterers attached to a

thin elastic plate can be found assuming that there is no in-

cident ﬁeld, so that the total ﬁeld excited in the plate is due

only to the scattered ﬁeld by all the particles[19, 20]. Under

these conditions equation (6) becomes a homogeneous system

of equations with non-trivial solutions only for those frequen-

cies satisfying

detM(ω) = 0.(8)

For ﬁnite clusters of scatterers the above condition can be

satisﬁed only for complex frequencies, being the inverse of

the imaginary part of this frequency the quality factor of the

resonance. Those conﬁgurations in which the imaginary part

of the resonant frequency is extraordinarily small (hence the

quality factor extraordinarily big) receive the name of quasi-

BIC or QBIC modes. In the following lines it will be shown

that arranging the scatterers in the vertices of regular polygons

we can obtain resonances whose quality factor diverges as the

number of scatterers approaches to inﬁnite.

Then, if the scatterers are all identical with impedance t0

and they are regularly arranged in a circumference of radius

R0and placed at angular positions 2πα/N, for α=0,...,N−

1, (as shown in Figure 9 in Appendix A) the Hamiltonian of

the system commutes with the rotation operator RN, whose

eigenvalues are λ`=exp(i2π`/N), with `=0,...,N−1, and

this implies a relationship between the coefﬁcients of the

form[16]

α=e2iπ`α/NB`

0,(9)

thus equation (6) becomes

(1−t0∑

G(R0β)e2iπ`β/N)B`

0=0.(10)

It is more suitable now to deﬁne the Green’s function as

G(r)≡ig0ξ(r)(11)

where

g0=1

8k2

(12)

and

ξ(r) = H0(k0r)−H0(ik0r),(13)

so that ξ(0) = 1 and γ0=t0g0is a real quantity. The eigen-

modes of the system are found as the non-trivial solutions of

equation (10), thus for the `-th mode we need to solve

1−iγ0∑

ξ(Rβ)e2iπ`β/N=0.(14)

This equation will give us a set of complex free-space

wavenumbers kn

0from which we can obtain the eigenfrequen-

cies ωnby means of the plate’s dispersion relation. The imag-

inary part of these eigenfrequencies is related with the quality

factor of the mode: the lower the imaginary part the larger

the quality factor, thus a BIC will be found if we can obtain

a real wavenumber kn

0satisfying the above equation. Thus,

assuming this wavenumber exists, we deﬁne

S`=∑

ξβe2iπ`β/N=S`

R+iS`

I,(15)

and the secular equation can be divided in real and imaginary

parts as

R(k0) = 0 (16)

1+γ0S`

I(k0) = 0.(17)

The second of these equations will always be satisﬁed, since

γ0is a resonant factor that can be selected to run from −∞to

∞. Therefore, we have to ﬁnd the conditions for which the

ﬁrst of the equations can be satisﬁed.

FIG. 1. SRsummation for different situations. In panel athe differ-

ent lines correspond to different number of scatterers in the cluster,

and the resonance index is ﬁxed at l=0. In panel b, the number of

scatterers in the cluster is ﬁxed (N=10) and the evolution of SR/N

as a function of k0is shown for different resonant index.

Figure 1, panel a, shows the evolution of S`

R(in logarithmic

scale, for clarity) as a function of k0R0for `=0 and for differ-

ent number of scatterers Nin the cluster. As can be seen, for

a small number of scatterers the function does not approach

zero, so that no BIC can be found, although for a relatively

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BoundstatesinthecontinuumincircularclustersofscatterersMarcMartí-Sabaté,1BahramDjafari-Rouhani,2andDaniTorrent1,1GROC,UJI,InstitutdeNovesTecnologiesdelaImatge(INIT),UniversitatJaumeI,12071,CastellódelaPlana,Spain2IEMN,UniversityofLille,Citéscientique,59650Villeneuved'Ascq,France(Dated:October19,20...

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Bound states in the continuum in circular clusters of scatterers Marc Martí-Sabaté1Bahram Djafari-Rouhani2and Dani Torrent1 1GROC UJI Institut de Noves Tecnologies de la Imatge INIT

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