.

††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Acoustic Bound States in the Continuum and Exceptional Points in Coupled
Helmholtz Resonators

Mariia Krasikova<EMAIL_ADDRESS>School of Physics and
Engineering, ITMO University, St. Petersburg 197101, Russia  Chair of
Vibroacoustics of Vehicles and Machines, Technical University of Munich,
Garching b. München 85748, Germany  Felix Kronowetter  Chair of Vibroacoustics
of Vehicles and Machines, Technical University of Munich, Garching b. München
85748, Germany  Mikhail Kuzmin  School of Physics and Engineering, ITMO
University, St. Petersburg 197101, Russia  Sergey Krasikov  School of Physics
and Engineering, ITMO University, St. Petersburg 197101, Russia  Marcus Maeder
Chair of Vibroacoustics of Vehicles and Machines, Technical University of
Munich, Garching b. München 85748, Germany  Tao Yang  Chair of Vibroacoustics
of Vehicles and Machines, Technical University of Munich, Garching b. München
85748, Germany  Anton Melnikov  Chair of Vibroacoustics of Vehicles and
Machines, Technical University of Munich, Garching b. München 85748, Germany
Steffen Marburg  Chair of Vibroacoustics of Vehicles and Machines, Technical
University of Munich, Garching b. München 85748, Germany  Andrey Bogdanov
<EMAIL_ADDRESS>School of Physics and Engineering, ITMO University,
St. Petersburg 197101, Russia  Harbin Engineering University, Harbin 150001,
Heilongjiang , Peoples R China

(August 28, 2024)

###### Abstract

Resonant states underlie a variety of metastructures that exhibit remarkable
capabilities for effective control of acoustic waves at subwavelength scales.
The development of metamaterials relies on the rigorous mode engineering
providing the implementation of the desired properties. At the same time, the
application of metamaterials is still limited as their building blocks are
frequently characterized by complicated geometry and can’t be tuned easily. In
this work, we consider a simple system of coupled Helmholtz resonators and
study their properties associated with the tuning of coupling strength and
symmetry breaking. We numerically and experimentally demonstrate the
excitation of quasi-bound state in the continuum in the resonators placed in a
free space and in a rectangular cavity. It is also shown that tuning the
intrinsic losses via introducing porous inserts can lead to spectral splitting
or merging of quasi-bound states in the continuum and occurrence of
exceptional points. The obtained results will open new opportunities for the
development of simple and easy-tunable metastructures based on Helmholtz
resonances.

##  I Introduction

Acoustic resonances play a pivotal role in a variety of practical applications
covering all aspects of modern society, including art and entertainment [1, 2,
3], medicine [4, 5, 6], industrial applications [7, 8], and even cosmology
[9]. The importance of resonances has become more prominent with the rise of
metamaterials [10, 11, 12], allowing effective control over acoustic fields at
the subwavelength scale. Such remarkable properties as, for instance, negative
mass density [13, 14], hyperlensing [15], or subwavelength perfect absorption
[16] arise from the resonant states and interplay between them. Further
progress in the field of metamaterials nowadays can be associated with
resonant state engineering, involving such concepts as topological states [17,
18], non-locality [19], and symmetry breaking [20].

Of particular interest in this case are the so-called bound states in the
continuum [21, 22] (BIC), also known previously as trapped waves, which are
characterized by infinitely large radiative quality factors and the associated
decoupling from the far field, usually originating from the interference of
several leaky modes. Firstly proposed in the field of quantum mechanics [23],
BIC nowadays are actively studied in a variety of other systems, including
electrical circuits [24], photonics [22, 25], and acoustics [26]. For
instance, BIC can be utilized for sound confinement [27, 28], acoustic
radiation enhancement [29], or improvement of noise-insulating systems [30,
31]. Since genuine BIC cannot be excited from the far field due to the
decoupling, practical applications utilize the so-called quasi-BIC, which have
a large but finite radiative quality factor. The excitation of quasi-BIC is
associated with the symmetry breaking allowing weak coupling with the far
field [32].

Figure 1: Illustration of the work concept. Bound states in the continuum
(BIC) occur at the anti-crossings formed by real parts of symmetric and anti-
symmetric modes in parametric space. Exceptional points (EP) represent a
special case of coupling regime when the eigenmodes coalesce in the real and
imaginary planes. In the considered system of two coupled Helmholtz resonators
these regimes can be achieved by tuning the distance between the resonators
and tuning their intrinsic losses. Both of the resonators are characterized by
the inner radius $r$, the outer radius $R$ and the slit width $w$. In
addition, the volume of the resonators is supplemented by absorbing inserts.

Symmetry breaking plays a key role not only in the engineering of high-Q
quasi-BICs but also in bianisotropic [33, 34, 35, 36, 37] and non-Hermitian
systems [38, 19, 29]. In particular, exceptional point (EP), which denotes a
condition where two or more eigenmodes and their complex eigenfrequencies
coalesce, is achieved via breaking of parity (P-symmetry) and time-reversal
symmetry (T-symmetry) [39, 40, 41]. EPs find today a variety of applications
in photonics and acoustics [42, 29]. Practical implementations of EP in
acoustic systems have enabled asymmetric field propagation [43], mode
stabilization [44], and enhanced absorption [45].

Therefore, it can be inferred that such singularities as BIC and EP are
pivotal for the engineering of acoustics metamaterials and devices on their
base with unique functionality and performance. However, their practical
implementations are still limited by the typically nontrivial geometry of unit
cells and the narrow operational frequency range. Producing metastructures
with easily tunable designs and acoustic properties is also a difficult task,
sometimes requiring the use of non-trivial algorithms of numerical
optimization. Therefore, the creation of resonant meta-atoms with
straightforward designs that facilitate the easy adjustment of resonant and
singular states, including BIC and EP, could greatly enhance the adoption of
metamaterials in real-world applications.

In this study, we suggest and explore a straightforward design of meta-atom
with easily tuned acoustic properties. The suggested meta-atom represents a
pair of two-dimensional coupled C-shaped Helmholtz resonators. Such a system
can be tuned via mechanical displacement of the resonators to each other as
well as via the adjustment of their intrinsic losses using absorbing inserts
made of conventional porous materials. We demonstrate experimentally that such
a meta-atom supports quasi-BIC that could be merged into an EP via the tuning
of intrinsic losses.

##  II Results

###  II.1 System description

The considered system consists of coupled 2D Helmholtz resonators with
circular cross-sections, as shown in Fig. 1. The coupling can be tuned via the
change of distance between the resonators and their mutual orientation, while
the intrinsic losses can be controlled by absorbing inserts. Resonators of
similar shapes were considered, for instance, in Refs. [35, 46, 47, 48, 49].
Normal modes of such a system are formed by the superposition of the modes of
solitary resonators. In-phase excitation of the resonators corresponds to a
symmetric mode, which in optics is usually called a bright one since it can be
directly excited and observed from the far field. On the other hand, the
resonators can be excited out-of-phase, forming an anti-symmetric mode.

When an antisymmetric mode is localized entirely inside the resonators, its
radiative quality factor is infinite. Such states are genuine BIC, whose
coupling with the far field is prohibited by symmetry reasons. Therefore, the
excitation of BIC implies a symmetry break, allowing the transformation of BIC
into a quasi-BIC, whose radiative quality factor is still large but finite.
Regarding eigenmodes, BIC occur in the strong mode coupling regime when the
dispersion curves (real parts of eigenfrequencies) repel. For the weak
coupling, the situation is reversed and the real parts of modes are crossed.
In between these regimes, achieving an exceptional point (EP) in which both
real and imaginary parts of the modes coalesce and the eigenmodes become
linearly dependent. In the considered system, the coupling between the modes
can be easily tuned via simple displacement of the resonators, affecting the
radiative losses. At the same time, the intrinsic losses and coupling between
the resonators can be tuned via the introduction of porous inserts into their
volume. Hence, it might be expected that a simple variation of several
parameters is enough to achieve the considered coupling regimes.

###  II.2 Quasi-BIC

####  II.2.1 Numerical Simulations

The considerations start with the resonators without absorbing inserts. For
this case, the geometric parameters of the resonators were fixed to have the
outer radius of the resonators $R=$30\text{\,}\mathrm{mm}$$, the inner radius
$r=$23\text{\,}\mathrm{mm}$$, and the slit width $w=$14\text{\,}\mathrm{mm}$$,
which correspond to the scaled geometry considered in Refs. [48, 49]. At
first, the resonators placed in a free space are considered [see Fig. 2(a)].
Such a system is characterized by two eigenmodes with symmetric and anti-
symmetric field distributions [Fig. 2(b)], which correspond to the resonances
of pressure spectra inside one of the resonators [Fig. 2(c)]. When the
distance between the resonators becomes large enough, they almost do not
interact and their eigenfrequency become almost identical corresponding to the
frequency of a single Helmholtz resonator. On the contrary, the spectral width
of the mode with the anti-symmetric distribution vanishes for small $d_{x}$
when resonators are close to each other, which is associated with the
vanishing of the imaginary parts of eigenmodes [Fig. 2(d)]. Therefore, the
real parts of the modes merge with the increase of $d_{x}$, demonstrating the
independent behavior of two non interacting resonators, while the imaginary
parts diverge and vice versa. This is a manifestation of a symmetry-protected
quasi-BIC, which becomes a genuine BIC when the imaginary part of the mode is
equal to zero.

Figure 2: Quasi-BIC in the resonators placed in a free space. (a) Schematic
illustration of the considered system. (b) Eigenmodes field distributions at
$d_{x}=$100\text{\,}\mathrm{mm}$$ and their quality factor $Q$. (c) Pressure
spectra inside one of the resonators as a function of the distance $d_{x}$
between the resonators. The dashed lines indicate the corresponding eigenmodes
with symmetric ($S$) and anti-symmetric ($AS$) field distributions. (d)
Imaginary part of the eigenfrequencies as a function of $d_{x}$. (e) Measured
pressure spectra inside one of the resonators. The values of $Q$ indicate the
corresponding quality factors of the resonances. (f) Measured spectra of phase
difference between the fields inside the resonators. Figure 3: Transmission
tube measurements of quasi-BIC. (a) Schematic illustration of the system
consisting of the box with the resonators embedded into the transmission tube.
(b) Transmission loss spectra. (c) Measured spectra of phase difference
between the fields inside the resonators. (d) Experimentally obtained
reflection, transmission and absorption coefficients. (e) Pressure spectra
calculated inside the resonators. (f) Field distributions corresponding to the
transmission loss resonances occurring near $929\text{\,}\mathrm{Hz}$ and
$987\text{\,}\mathrm{Hz}$.

####  II.2.2 Anechoic Chamber Measurements

The numerical calculations are verified by the measurements of pressure inside
the resonators placed in the 2D anechoic chamber (see Methods). As it is shown
in [Fig. 2(e)], the quasi-BIC manifests as a peak in the pressure spectra,
occurring within the range $900--$1200\text{\,}\mathrm{Hz}$$, depending on the
distance between the resonators. For instance, when
$d_{x}=$100\text{\,}\mathrm{mm}$$ the quasi-BIC is excited at approximately
$1025\text{\,}\mathrm{Hz}$ and its quality factor is about $100$. In
accordance with the numerical calculations, the quasi-BIC shifts towards the
higher frequencies with a decrease in the distance between the resonators
since the coupling between the resonators becomes stronger. By definition, the
anti-symmetric mode is characterized by the $\pi$ phase difference between the
fields inside the resonators, which can be seen from the spectra in [Fig.
2(f)], demonstrating the conventional smoothstep-like shapes. Note that for
the case of $d_{x}=$70\text{\,}\mathrm{mm}$$, the phase difference does not
reach the value of $\pi$, which might be explained by the weak coupling of the
antisymmetric mode with the far field. In addition, it should be noted that
the resonances of the measured spectra are shifted from the numerically
calculated ones by approximately
$30\text{\,}\mathrm{Hz}$–$50\text{\,}\mathrm{Hz}$, which arises from
manufacturing errors as sub-millimeter changes in the resonators sizes affect
the position of the resonances (see Supplementary Information). However, the
experimental results are in good agreement with those from numerical
simulations, cf. Figs. 2(c) and 2(e) as well as the Supplementary Information.

Figure 4: Splitting of quasi-BIC (a) Schematic illustration of the considered
system consisting of two pairs of coupled resonators placed in a free space.
One of the pairs is supplemented by porous inserts with the flow resistivity
$R_{f_{2}}$. (b) Eigenmodes field distributions for the case of
$R_{f_{2}}=$1\text{\,}\mathrm{Pa}\cdot\mathrm{s}\mathrm{/}\mathrm{m}^{2}$$ and
(c) real and imaginary parts of eigenmodes for different values of
$R_{f_{2}}$. (d) Photo of the samples. Experimentally obtained pressure
spectra measured inside (e) the resonator I and (f) the resonator IV.

####  II.2.3 Transmission Tube Measurements

While the quasi-BIC mode is weakly coupled to the far field, it is also almost
decoupled from the other modes of the system. Hence, it can be expected that
the quasi-BIC can still be excited even if the environment of the resonators
changes. For instance, the resonators can be placed inside a rectangular
cavity [see Fig. 3(a)], which actually by itself supports quasi-BIC [50]. When
the anti-symmetric modes of the cavity and the resonators do not interact, the
transmission loss spectra of such a system should also demonstrate a high-Q
peak near the frequency corresponding to the quasi-BIC occurring for the case
of the resonators in a free space. Indeed, Fig. 3(b) shows the presence of the
two pronounced peaks near $940\text{\,}\mathrm{Hz}$ and
$1000\text{\,}\mathrm{Hz}$, obtained via the impedance tube measurements (see
Methods). These peaks expected correspond to the peaks of the pressure spectra
calculated inside the resonators [see Fig. 3(e)]. Again, the phase difference
between the fields inside the resonators represents the smooth step-like curve
[see Fig. 3(c)], demonstrating that the phase difference is nearly
$\Delta\varphi=\pi$ at the quasi-BIC frequency. The peak of the transmission
loss spectrum occurring near $940\text{\,}\mathrm{Hz}$ can be associated with
the symmetric excitation of Helmholtz resonators [see Fig. 3(f)], leading to
the decrease of the transmission coefficient [Fig. 3(d)]. At the same time,
the second peak in the transmission spectra, occurring near
$1000\text{\,}\mathrm{Hz}$, corresponds to the quasi-BIC characterized by the
anti-symmetric field distribution, as previously [see Fig. 3(f)]. The
resonance in the transmission loss, in this case, is related to the increase
of the absorption coefficient [see Fig. 3(d)]. However, it should be noted
that the transmission loss in the considered case is probably associated with
energy leakage into the mechanical vibrations of the box walls, which are not
taken into account in the numerical calculations.

Figure 5: EP in the system of four resonators. (a) Schematic illustration of
the considered system consisting of two pairs of coupled resonators, both
supplemented by porous inserts with the flow resistivity $R_{f_{1}}$ and
$R_{f_{2}}$, respectively. (b) The real and (d) imaginary parts of eigenmodes
for the different values of $R_{f_{2}}$. Experimentally obtained pressure
spectra measured inside (c) the resonator I and (e) the resonator IV. The
results shwon in panels (b)–(e) are obtained for
$R_{f_{1}}=$1\text{\,}\mathrm{Pa}\cdot\mathrm{s}\mathrm{/}\mathrm{m}^{2}$$.

###  II.3 Tuning of Intrinsic Losses

Apart from the distance between the resonators, the resonances can be
controlled by introducing porous materials into the volume of the resonators.
Such inserts may significantly decrease the quality factor of a standalone
Helmholtz resonator (see Supplementary Information), practically leading to
the destruction of the resonance. However, small intrinsic losses can be used
for the fine-tuning of quasi-BIC. For instance, the system supplemented by an
additional pair of the resonators can be considered, such that this additional
pair is supplemented by the absorbing inserts made of a porous material with
the flow resistivity $R_{f_{2}}$, while another pair has no such inserts [see
Fig. 4(a)]. Since there are four resonators in such a system, it is
characterized by four eigenmodes [see Figs. 4(b)], two of which are
characterized by the symmetric field distributions within the pairs and two –
by the antisymmetric [see Figs. 4(c)]. When the flow resistivity $R_{f_{2}}$
is zero, the antisymmetric eigenmodes are nearly degenerate, meaning that the
have equal real parts of the frequencies, but slightly different values for
the imaginary parts. Both the real and imaginary parts of the symmetric modes
are almost not affected by the change of $R_{f_{2}}$. At the same time, the
increase of $R_{f_{2}}$ results in the splitting of quasi-BIC [see Figs. 4(b)
and 4(c)], such that the frequency of one antisymmetric eigenmode remains the
same, while another antisymmetric eigenmode shifts towards the smaller real
frequencies, while its quality factor quickly decreases. Similar behaviour can
be also observed experimentally. In this case the spectral position of the
quasi-BIC measured in the first pair remains nearly unchanged [see Fig. 4(e)],
while the quasi-BIC measured in the second pair shifts towards the lower
frequencies with the increase of the absorbing insert size [see Fig. 4(d)].
These results indicate the spectral positions of BIC in arrays of Helmholtz
resonators can be fine-tuned via simple adjustment of the intrinsic losses
controlled by the size of the absorbing inserts.

The important remark is that in the numerical calculations, the insert
occupies the whole volume of the resonator, and the corresponding porous
material is characterized only by the flow resistivity, which is varied. For
the experimental verification, it is assumed that a small piece of porous
material with a large flow resistivity is roughly equivalent to a large piece
of a material with a low flow resistivity. In such an approximation, a porous
insert occupying the whole volume of the resonator can be represented by an
insert occupying only a part of the volume [see Fig. 4(d)], and it is much
easier to control the size of an insert contrary to its material parameters.

To push the concept further, the considered system of two pairs of coupled
resonators can be used to achieve an EP. For that, both pairs of the
resonators should be supplemented by porous inserts, such that in the first
pair, the inserts are characterized by the flow resistivity $R_{f_{1}}$ and in
the second pair – by $R_{f_{2}}$ [see Fig. 5(a)]. The tuning of the flow
resistivity allows to control both antisymmetric eigenmodes and achieve their
coalescence. As shown in Figs. 5(b) and 5(d), when
$R_{f_{1}}=$1\text{\,}\mathrm{Pa}\cdot\mathrm{s}\mathrm{/}\mathrm{m}^{2}$$ and
$R_{f_{2}}\approx$0.5\text{\,}\mathrm{Pa}\cdot\mathrm{s}\mathrm{/}\mathrm{m}^{2}$$
these eigenmodes have the same complex frequency, which can be considered as
the manifestation of an EP. To verify this result experimentally, the pressure
spectra is measured inside the first and the last resonator using the porous
inserts of different size, similarly to the idea developed in Ref. [51]. The
merging of the antisymmetric eigenmodes in this case should occur as the
spectral overlap of the quasi-BIC measured in both pairs. To provide such
measurements, the size of the porous inserts inside the first pair is fixed,
while the size of the inserts in the second pair is varied. Hence, the quasi-
BIC occurring in the first pair remains almost unchanged [see Fig. 5(c)],
while the spectral position of the quasi-BIC in the second pair shifts, such
that when $R_{f_{2}}$ is smaller than $R_{f_{1}}$, the quasi-BIC in both pairs
are excited at the same frequency [see Fig. 5(e)]. While the exact values of
the flow resistivity are not estimated in this work, it still can be concluded
that the provided measurements indirectly confirm the presence of the EP in
the considered system.

Figure 6: Experimental Setups. (a) Photo of the 2D anechoic chamber imitating
a free space. Close up photo of (b) the samples used in the 2D anechoic
chamber and (c) the samples and the box used for the transmission tube
measurements. (d) Photo of the transmission tube setup.

##  III Conclusion

In this work, coupling regimes of 2D Helmholtz resonators were investigated.
It was shown that quasi-BIC and EP can be easily achieved by tuning the
geometric parameters and intrinsic losses of the resonators. Experimental
demonstrations reveal the resonant enhancement of absorption associated with
quasi-BIC.The obtained results aim to extend the possibilities of physics-
based design of metamaterials and pave the route towards novel devices for the
efficient control of acoustic fields at the subwavelength scales.

##  IV Methods

###  IV.1 Experimental measurements

For the resonators placed in a free space, the measurements are performed in
the 2D anechoic chamber, representing a waveguide with walls covered by
absorbing mineral wood [see Figs. 6(a) and 6(b)]. In particular, the walls and
the bottom of the waveguide are made of $15\text{\,}\mathrm{mm}$ thick
aluminum plates, and the lid is made of $6\text{\,}\mathrm{mm}$ plexiglass.
Since the impedance contrast between air and most of the solid materials is
huge [52], the lid and the bottom can be considered as sound hard walls. The
height of the waveguide is $60\text{\,}\mathrm{mm}$, while the width and the
length are $650\text{\,}\mathrm{mm}$ and $1350\text{\,}\mathrm{mm}$,
respectively. The thickness of the mineral wood inserts is
$300\text{\,}\mathrm{mm}$ (from a wall to the tip of a pyramid). Generation
and recording of the sound signal is done using the analysis system Crysound
CRY6151B and 1/4” pressure field microphone Crysound CRY342. The resonators
themselves are fabricated by additive manufacturing technology on the Bambu
Lab X1-Carbon 3D printer (Shenzhen Tuozhu Technology Co., Ltd., Shenzhen,
China). To avoid gaps between the resonators and the lids, the top of the
resonators are supplemented by thin porous inserts.

The transmission loss measurements are conducted in the AED AcoustiTube
measurement system, the impedance tube. In this case, the resonators are
placed in a box with a rectangular cross-section [see Fig. 6(c) and 6(d)]. The
width of the box is $170\text{\,}\mathrm{mm}$, the height is
$160\text{\,}\mathrm{mm}$, and the thickness is $40\text{\,}\mathrm{mm}$. The
scattering coefficients are extracted from the measured T-matrix coefficients
using the conventional expressions [53] as:

$\displaystyle
t^{-}=\frac{2e^{ikL}}{T_{11}+T_{12}/Z_{L}+T_{21}Z_{0}+T_{2}2Z_{0}/Z_{L}},$ (1)
$\displaystyle
t^{+}=\frac{Z_{0}}{Z_{L}}\frac{2e^{ikL}(T_{11}T_{22}-T_{12}T_{21})}{T_{11}+T_{12}/Z_{L}+T_{21}Z_{0}+T_{2}2Z_{0}/Z_{L}},$
(2) $\displaystyle
r^{-}=\frac{T_{11}+T_{12}/Z_{L}-T_{21}Z_{0}-T_{2}2Z_{0}/Z_{L}}{T_{11}+T_{12}/Z_{L}+T_{21}Z_{0}+T_{2}2Z_{0}/Z_{L}},$
(3) $\displaystyle
r^{+}=\frac{-T_{11}+T_{12}/Z_{L}-T_{21}Z_{0}+T_{2}2Z_{0}/Z_{L}}{T_{11}+T_{12}/Z_{L}+T_{21}Z_{0}+T_{2}2Z_{0}/Z_{L}},$
(4)

where $Z_{0}$, $Z_{L}$ are acoustic impedances of the media at the sides of
the structure, such that $L$ is the thickness of the structure, and the
superscripts indicate the positive ($+$) and ($-$) negative direction of the
incident wave propagation. The amplitude coefficients then can be defined as
$T^{+,-}=|t^{+,-}|^{2}$ and $R^{+,-}=|r^{+,-}|^{2}$. Correspondingly, the
absorption coefficients can be defined as
$A^{+,-}=1-|r^{+,-}|^{2}-|t^{+,-}|^{2}$. When the system is symmetric,
$A=A^{+}=A^{-}$. Note that all measured spectra are slightly smoothed using
the Savitzky-Golay filter of order $3$ and with a window width of $11$.

###  IV.2 Numerical calculations

All numerical calculations are performed in COMSOL Multiphysics using the
“Pressure Acoustics” physics. The incident plane wave is introduced as the
background pressure field with the amplitude $p_{0}=$1\text{\,}\mathrm{Pa}$$.
The pressure inside the resonators is defined as a root mean square of
pressure calculated at several sampling points located inside the resonators.
In order to account for the thermoviscous dissipation in the boundary layer of
the resonators’ walls, the “Thermoviscous Boundary Layer Impedance” condition
is applied to the corresponding boundaries. Porous inserts are included via
the “Poroacoustics” feature utilizing the Delany-Bazley-Miki model.

###### Acknowledgements.

The authors thank Badreddine Assouar for the fruitful discussions. M.Kr., S.K.
and A.B. acknowledges the BASIS foundation.

##  V Author contributions

M.Kr., M.K., F.K., M.M., and T.Y. performed the experimental measurements,
M.Kr., S.K. and F.K. provided numerical calculations. S.K. and M.Kr. suggested
the idea and developed the concept of the work. A.M., M.M., S.M. and A.B.
supervised the project. All authors contributed to writing and editing of the
manuscript.

##  VI Competing interests

The authors declare no competing interests.

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Supplementary Information for

Acoustic Bound States in the Continuum and Exceptional Points in Coupled
Helmholtz Resonators

##  S1 Comparison of the Experimental and Numerical Results

The experimental results presented in the main text are in a good agreement
with the numerical calculations. However, the resonant peaks in the pressure
and transmission loss spectra are noticeably shifted with respect to each
other. Such a difference can be explained by the limited accuracy of the
samples manufacturing. For instance, even slight inaccuracies of order
$0.5\text{\,}\mathrm{mm}$ may lead to the significant shift of the
transmission loss resonances, as shown in Fig. S1. The corresponding measured
spectrum lies within the range of the numerical curves, and hence it can be
concluded that the experimental results reproduce the numerical ones.

Figure S1: The influence of the geometric parameters of the resonators on
numerical calculations. Transmission loss spectra for the case of two coupled
resonators place in the rectangular box. The distance between the resonators
is $d_{x}=$100\text{\,}\mathrm{mm}$$ and the outer radius is
$R=$30\text{\,}\mathrm{mm}$$, while the inner radius $r$ and the slit width
$w$ are varied. The vertical dashed lines indicate the position of peaks in
the experimentally measured spectra.

It is also should be mentioned that when a pair of the resonators is
considered the pressure spectra inside them are slightly different. In
particular, the pressure spectra measured and calculated inside one of the
resonators are characterized by Fano-shaped resonances corresponding to the
antisymmetric mode [see Figs. S2(b) and S2(d)], while for the second resonator
the corresponding peaks have Lorentzian-like shape [see Figs. S2(c) and
S2(e)].

Figure S2: Pressure spectra in the open system of coupled resonators. (a)
Schematic illustration of the system consisting of two resonators excited by a
plane wave. The pressure spectra (b) calculated and (d) measured inside the
resonator I and the corresponding spectra (c) calculated and (e) measured
inside the resonator II.

##  S2 Tuning of Intrinsic Losses

While the main text is dedicated to the systems of coupled resonators, it is
still reasonable to analyse a standalone resonator. In particular, two
configurations are considered, such that the resonator is oriented “away” from
and “towards” the incident field [see Fig. S3(a) and S3(b)]. In both cases the
resonance occurs near $900$ Hz, though for the configuration A the pressure is
slightly than for the B [see Fig. S3(c)]. When the resonator is supplemented
by a porous insert, the spectral width of the resonance increases with the
increase of the flow resistivity, while the pressure decreases [see Fig. S3(d)
– S3(f)]. Large values of the flow resistivity practically result in the
disappearance of the resonance as the intrinsic losses become too large and
the pressure spectra flattens. Hence, even for the case of a standalone
resonator it is reasonable to consider small values of flow resistivity, which
can be achieved via considerations of the partial inserts, occupying only a
small part of the resonator’s volume, as discussed in the main text.

Figure S3: Standalone resonator. Schematic geometry of the open system with a
standalone resonator oriented (a) “away” from (configuration A) and (b)
“towards” (configuration B) the incident field. In both cases the resonators
are supplemented by a porous inserts shown by yellow color. Pressure spectra
calculated inside the resonators (c) without porous inserts and with inserts
characterized by the flow resistivity $R_{f}=1000$ Pa$\cdot$s/m2. The vertical
red lines indicate the pressure peak corresponding to the configuration A
without the insert. Pressure spectra calculated inside the resonators as
functions of the flow resistivity of the porous inserts for (d) configuration
A and (f) configuration B.

When a second resonator is introduced into the system, their coupling results
in the splitting of the resonance. As discussed in the main text, one part of
the split resonance corresponds to the symmetric mode and shifts towards the
lower frequencies with the increase of the coupling strength. The second
resonance corresponds to the antisymmetric mode having larger frequency and
larger quality factor than the symmetric one. When a porous insert is
introduced into one of the resonators within the pair [see Figs. S4(a) and
S4(b)], the amplitude of the quasi-BIC in the resonator with the insert
quickly decreases with the increase of the flow resistivity [see Figs. S4(c)
and S4(f)]. At the same time, the change of the corresponding amplitude in the
resonator without the insert does not change so drastically [see Figs. S4(d)
and S4(e)]. In other words, the increase of intrinsic losses in one of the
resonators results in the asymmetric excitation of the resonators, as it
should be expected.

Figure S4: Tuning of losses in one of the resonators within a pair. Schematic
illustration of the considered open system consisting of two resonators, such
that the porous insert is placed inside (a) the resonator I and (b) the
resonator II. Pressure spectra for the case when the insert is placed in the
resonator I, calculated inside (c) the resonator I and (d) the resonator II.
Pressure spectra for the case when the insert is placed in the resonator II,
calculated inside (e) the resonator I and (f) the resonator II.

In the case when the inserts are introduced into the both resonators [see Fig.
S5(a)], the resonance amplitude also decreases with the increase of the flow
resistivity, such that at $R_{f}=200$ Pa$\cdot$s/m2 the antisymmetric mode
becomes indistinguishable. Therefore, tuning of the quasi-BIC via intrinsic
losses should be done with materials characterized by the flow resistivity
below $10$ Pa$\cdot$s/m2 or even below $1$ Pa$\cdot$s/m2. Such values are much
smaller than the corresponding values of conventional materials, like rock
wool or melamine foam. Hence, the experimental verification should be done
with the inserts occupying only a small part of the resonators’ volume, as
discussed in the main text.

Figure S5: Tuning of losses in both resonators within a pair. (a) Schematic
illustration of the considered open system consisting of the two resonators
supplemented by porous inserts with the flow resistivity $R_{f}$. Pressure
spectra calculated inside (b) the resonator I and (c) the resonator II.