# Hybrid ultrathin metasurface for broadband sound absorption

Marnix P. Abrahams  Université de Lorraine, CNRS, Institut Jean Lamour,
F-54000 Nancy, France  Mourad Oudich<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>Université de Lorraine, CNRS, Institut
Jean Lamour, F-54000 Nancy, France  Graduate Program in Acoustics, The
Pennsylvania State University, University Park, Pennsylvania 16802, USA  Yann
Revalor  Dassault Aviation, 92552 St-Cloud, France  Nicolas Vukadinovic
Dassault Aviation, 92552 St-Cloud, France  Badreddine Assouar
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Université de Lorraine, CNRS, Institut Jean Lamour, F-54000 Nancy, France

###### Abstract

To this day, achieving broadband low-frequency sound absorption remains a
challenge even with the possibilities promised by the advent of metamaterials
and metasurfaces, especially when size and structural restrictions exist.
Solving this engineering challenge relies on stringent impedance matching and
coupling of the multiple independent local resonators in metasurface
absorbers. In this letter, we present an innovative design approach to broaden
the sound absorption bandwidth at low-frequency regime. A hybrid metasurface
design is proposed where four coupled planar coiled resonators are also
coupled to a well designed thin planar cavity. This hybrid metasurface creates
a broad sound absorption band (130-200 Hz) that is twice as wide as that of
the traditional single layer metasurface utilizing four coiled cavities at a
deep sub-wavelength thickness ($<\ \nicefrac{{\lambda}}{{51}}$). This design
strategy open routes towards engineering a class of high performance thin
metasurfaces for ultra-broadband sound absorption while keeping the planar
size unchanged.

††preprint: AIP/123-QED

Figure 1:  (a) Hybrid meta-surface made of four coiled resonators and a planar back-cavity. A perforated plate with four holes is located between the coiled resonators and the back cavity where each hole connects each to the back cavity. The coiled (unit-cells) are numbered 1 through 4 and consist of an inlet hole and a coil, its backplate is perforated to allow flow through to the shared back-cavity. (b) Top-view of the metasurface indicating three different configurations for the positions of the holes connecting the coiled resonators to the back-cavity. These positions are labeled Pos.#1, Pos.#2, and Pos.#3. The values of the geometrical parameters in (a) and (b) are provided in Table 1. (c) The sound absorption coefficient for the three connecting holes configurations presented in solid line with symbol : Pos.#1 (blue circle), Pos.#2 (green diamond), and Pos.#3 (olive square), with the hole diameter $d_{c}$ being 0.85 mm. The dashed line corresponds to the four coiled metasurface without back cavity. (Symbols for illustration purposes only.)  Table 1: The design parameters depicted in Fig. 1 & 1 & Fig. 3 being static unless locally otherwise specified. $\text{d}_{\text{i,N}}$ [mm] |  $\text{L}_{\text{1}}$ [mm] |  $\text{L}_{\text{2}}$ [mm] |  $\text{L}_{\text{3}}$ [mm] |  $\text{L}_{\text{4}}$ [mm] |  W [mm] |  $\text{h}^{*}$ [mm] |  $\text{W}_{\text{c}}$ [mm] |  $\text{d}_{\text{c}}$ [mm] |  t [mm] |  $\text{t}_{\text{bc}}$ [mm]
---|---|---|---|---|---|---|---|---|---|---
7 | 460 | 430 | 410 | 380 | 165 | 40 | 12 | 1 | 2 | 4

The creation of thin rationally designed materials, namely metasurfaces[1],
capable of high sound absorption with broadband capability at a deep
subwavelength scale is highly desirable for a wide range of applications :
room acoustics, automobiles, aerospace etc. In traditional sound-absorbing
materials such as foam, fiber glass and mineral wool, the relationship between
the wavelength and the material thickness constrains the absorption spectrum.
This makes decreasing the material thickness difficult while maintaining the
desired sound absorption performance at low frequency[2]. Fortunately, the
advent of acoustic metamaterials has reshaped the entire field of acoustics
and particularly the sound absorption with the emergence of ultrathin
metasurfaces capable of deep-subwavelength high sound absorption[3, 4, 5, 6,
7, 8, 9, 10, 11].

Still, the research within the field of acoustic metasurfaces shows that the
realization of a broadband high absorption rate at low-frequency regime
remains challenging, with the thickness being the leading limiting factor[12,
13]. One of the initial strategies adopted for the design of sound absorbing
metasurfaces is the utilization of thin membranes distributed in a rigid grid
panel[14, 15, 16, 17] . However, this type of thin resonating structure relies
on mastering the mechanical tension of the membranes which poses challenges in
fabrication. A different design methodology consists on combining a perforated
panel with a resonator coiled in a planar configuration such that the total
thickness of the whole structure can be reduced in the third dimension
($z$-direction)[18, 3, 5, 19, 10] . This design strategy enables perfect
absorption at deep subwavelength scale, but with a narrow frequency band
corresponding to the resonance frequency of the structure. Optimization
methods on the resonant cavity design were conducted to drive the high sound
absorption capability into low frequencies using internal patterning[7, 20],
or embedded and shaped neck[21, 22, 23, 24, 25].

Broadband sound absorption capability was later demonstrated with a super-cell
encompassing multiple unit-cells with coiled cavities. The resonance
frequencies of the latter were chosen judiciously to enable their mutual
coupling and achieve broadband high absorption rate [9, 6, 7, 17, 11, 26].
Recently, nonlocal coupling has been considered as a route for broadening the
absorption band [13, 27, 28]. In addition to coupling nearest-neighbor
resonators, nonlocality consists of coupling the cavities of second and higher
order neighboring cavities to broaden further the bandwidth. These strategies
for broadening the bandwidth of high absorption are mostly based on coupling
resonant units where a supercell has to be constructed with multiple units
which increases the size of the absorber in the planar direction. In other
words, the bandwidth interval strongly depends on the number of unit cells
which limits the metasurface practicability. In this letter, we aim on
overcoming this limitation for coiling-up space geometry cavity-based
metasurfaces.

Starting from a system of four coupled coiled cavities [5, 7], we present an
innovative design solution to broaden the sound absorption bandwidth beyond
what is allowed by the classical coupled cavities and without change in the
planar size of the metasurface. To do so, a hybrid concepts is introduced
where the four coupled coiled resonators are also coupled to a rationally
designed thin planar back cavity in the normal direction (Fig. 1). We will
demonstrate that judicious design of the planar back cavity enables an
enlargement of the bandwidth by a factor of two. We aim for a design that one
can integrate in the airplane fuselage wall to significantly mitigate the
motor-noise normally heard by the passengers, which ranges from 100 to 500 Hz.
Our proposed absorbing metasurface design uses non-local coupling for wide-
band absorption below 200 Hz with a sub-wavelength thickness of 50 mm
($\nicefrac{{\lambda}}{{51}}$).

We attain our design using the formulas (2 & 5) from Li and Assouar[5] from
which we create an optimized super-cell with equally spaced unitary absorption
peaks between 150 and 180 Hz. Making use of the commercial numerical
simulation software Comsol Multiphysics, we further optimize the design of the
metasurface. Figure 1(a) illustrates our hybrid metasurface that combines the
four coiled resonators with a perforated plate and a back-cavity that were
placed between the coils and back rigid wall. This added back cavity is shared
between the four unit-cells and connected with four holes of the same diameter
($d_{c}$). The geometrical parameters of the hybrid metasurface are indicated
in Fig. 1, and their values are provided in Table 1 which are adopted
throughout the entire letter unless otherwise specified.

Figure 2:  (a) The sound absorption coefficient dependence on the hole
diameter ($d_{c}$) of the connecting holes in the design displayed in Fig. 1
for the holes location ”Pos. $\\#1$”. (b) The absorption dependence on back-
cavity thickness ($t_{bc}$) when the connecting hole diameter $d_{c}$ is equal
to 1 mm and the total thickness is kept at 50 mm while altering the thickness
$\text{h}^{\text{*}}$ through the formula
$\text{h}^{\text{*}}=50\text{[mm]}-3\text{t}-\text{t}_{\text{bc}}$. (Symbols
for illustration purposes only.)

In our design optimization, it is important to realize that the holes
connecting the coiled cavities to the back-cavity (connecting holes) can be
located at any position along the length of each coiling, which divides the
channels of the coiled resonator into different resonating domains thereby
greatly affecting the absorption behavior. In order to find the optimal
location, an optimization process was conducted, and three of the
possibilities for the connecting holes localization considered are shown: at
the corners (Position 1), at the end of the coiled cavity (Position 2), and at
the same position as the inlet holes (first end of the cavity) (Position 3).
These positions are illustrated in Fig. 1 and labeled Pos.#1, Pos.#2 and
Pos.#3.

Figure 1 presents the sound absorption rates for different configurations of
the metasurface. The dashed line corresponds to the classical metasurface with
the four coiled cavities without a back cavity nor its adjoining perforated
plate. The three solid lines with markers correspond to the hybrid metasurface
which includes the back cavity with its perforated plate. These three curves
are associated with the three considered positions of the connecting holes as
described previously: circular marker for Pos.#1, diamond for Pos.#2, and
square for Pos.#3. The case of the classical metasurface (dashed line) (w/o
back cavity) exhibits near unitary absorption rate at four frequencies equally
spaced between 150-180 Hz and the coupled coiled resonator super-cell acquire
a 32 Hz bandwidth at 50% absorption (155-187 Hz). When adding the back cavity
with the perforated panel, two observations can be made. First, the absorption
bandwidth increases no matter the location of the connecting holes (solid line
curves in comparison with the dashed line curve) by between 5 to 16 Hz.
Secondly, the absorption coefficient is reduced when considering positions #2
and #3 of the connecting holes while the bandwidth increases. The enlargement
of the bandwidth of high absorption is made at the cost of not reaching
unitary absorption rate. Hence, there is a trade-off between the absorption
bandwidth and coefficient: the increase of the absorption band causes a
reduction of the absorption coefficient. However, even if perfect absorption
could not be attained when widening the bandwidth, the hybrid metasurface
still provides high absorption coefficient of at least 70% (Fig. 1c). Taking
this trade-off into account alongside our objective, we argue that the
location of the connecting holes at ‘Pos. #1’ is comparatively the best
scenario.

Considering ‘Pos. #1’ for the connecting holes, we have also conducted further
optimization on their diameter and the back-cavity thickness, and the results
are illustrated in Fig. 2. In figure 2, the increase of the diameter increases
the bandwidth significantly while the absorption coefficient is reduced. Also
from 2, it is clear that the thickness of the back cavity has a significant
impact on the absorption bandwidth and rate. Considering the trade-off evident
throughout these parametric optimizations, we state the optimal diameter for
the connecting holes and the back cavity thickness as $d_{c}=1mm$ and
$t_{bc}=4mm$ respectively, leading to an absorption coefficient above 50% for
a bandwidth of 70 Hz (130-200 Hz).

Figure 3:  (a) The hybrid metasurface similar to the one in Fig. 1 but with
individual back-cavities acquired by splitting the shared back-cavity in equal
dimensions. (b) The sound absorption dependence on the hole locations shown in
Fig. 1, for the split back-cavity design in (a). $1_{a,b}$ and $2_{a,b}$
indicate two frequencies 164 Hz and 212 Hz, respectively, for both curves to
compare the different designs by their pressure field and intensity which are
presented in Fig. 4. (c) Absorption spectra for both the designs shown in Fig.
1 and Fig. 3 (solid lines with markers) along with the case where no back-
cavity is included in the design (dashed line). (Symbols for illustration
purposes only.)

Figure 4:  Acoustic pressure amplitude fields (in Pa) along with the intensity
flow fields (indicated by the arrows) at the middle plane through the coiled
resonators (upper panel) and back-cavities (bottom panel) of the design
depicted in Fig. 3. The green dots indicate the holes location for the
configurations Pos. $\\#1$ and Pos. $\\#2$ (Fig.1b). The fields are presented
at frequencies 164 Hz (a) and 212 Hz (b) indicated by ($1_{a,b}$) and
($2_{a,b}$) respectively in Fig. 3.

Further improvements of the absorption quality and bandwidth are acquired by
amending the back-cavity design through splitting it in four equal parts, one
per coil as depicted by Fig. 3. The new super-cell consists of four paralleled
unit-cells, each one consists of a series of two connected resonators. The
first layer remains the same, the prior individualized four coiled resonators
while, the second layer now consisting of four identical resonators each of
which individually connects to a unit-cell in the first layer without any of
the latter sharing any of the former. The new design uses the same parametric
values from Table 1 as well as the previously optimized connecting hole
locations and dimensions. Considering the fact that the back-cavities are no
longer shared among the unit-cells, the connecting hole location previously
optimized might no longer be in its optimal position. To check the validity
hereof, two of the resulting graphs are depicted in Fig. 3, from which it can
be deduced that the optimal connecting hole location remains to be ‘Pos #1’.
In fact, while the configuration of Pos.#2 tend to offer a larger bandwidth,
it displays a low absorption coefficient at around 170Hz (below 50%), while
the configuration of Pos.#1 maintains a relatively high absorption coefficient
of 70% at the same frequency.

A comparison between the shared back-cavity design (Fig. 1) and the split back
cavity design (Fig. 3) is made for the sound absorption performance. The
results are presented in Fig. 3. The comparison shows that splitting the back
cavity has the effect of smoothing the absorption curve while maintaining the
desired bandwidth and the overall absorption rate. The smoothing effect
improves the overall absorption quality by getting rid of the local minima.
Besides, to compare our hybrid metasurface capability to that of the classical
coiled resonator based metasurfaces, we have added in Fig. 3 the curve of the
absorption spectrum for the classical metasurface (w/o back-cavity). The
addition of the optimized shared back cavity doubles the absorption bandwidth
from 155-187 Hz (18.7%) to 130-200 Hz (42.4%) at absorption coefficient of
50%, without passing the minimum local absorption coefficient at 180 Hz in the
design w/o back cavity, leading to a doubling of total amount absorbed sound
energy even though the maximum absorption is decreased from unity to 80%. In
addition, the smoothing effect caused by the split in the back-cavity creates
a further overall increase of the absorption by smoothing away the local
minima and so, improving the minimal absorption within the absorption band by
roughly 15% without having any effect on the bandwidth. The split also smooths
the highest absorption peak whereby reducing the maximal absorption by 5% (25%
comparatively to the case without back-cavity).

Figure 3 contains four arrows labeled 1${}_{\text{a,b}}$ and
2${}_{\text{a,b}}$ at 164 Hz and 212 Hz, respectively, indicating the largest
difference in the absorption coefficients for the two different connecting
holes locations. These arrows are chosen to display the pressure amplitude
distribution within the four coiled cavities and back cavities, displayed in
Fig. 4. The objective is to understand the coupling mechanism between the
resonators that lead to these absorption coefficient differences. We also
indicate in the Fig. 4 the intensity flows (black arrow heads) at the mid-
planes of the coils and cavities. At 164 Hz (1${}_{\text{a,b}}$ in Fig. 3), we
observe in Fig. 4 that for the case of connecting holes located at in Pos.#1
(1${}_{\text{a}}$ ), the four coiled cavities display high acoustic pressure
amplitudes along with the back cavities, comparatively to the low amplitude in
the coils in case of the situation for Pos.#2 (fields corresponding to
1${}_{\text{b}}$ ). This lead to a higher absorption coefficient in the
spectrum at 164 Hz. In addition, all four coiled cavities are “active” in
$1_{\text{a}}$ meaning that nonlocal coupling is acquired, though the largest
coiled cavity (coil 1 $\sim$ the bottom right coil) clearly absorbs the
largest percentage of the acoustic energy. However, in the case of
1${}_{\text{b}}$ corresponding to the back holes at Pos.#2, the coiled
cavities contribution to the sound absorption capability is almost negligible
giving the low pressure amplitude while, the back-cavities display a
relatively high pressure amplitude hence being essential to the absorption.
For all the observation made from Fig. 4(a), we can conclude that better
absorption is attainable when all coiled and back cavities couple and
contribute to the overall absorption.

Furthermore, is the orderly nature of the intensity flow in the coils
(1${}_{\text{a}}$ & 1${}_{\text{b}}$), proof that the design has good
impedance matching. In the coils of 1${}_{\text{a}}$ the orderly flow becomes
disorderly and dies down after the connecting hole, suggesting that the
acoustic intensity flow of the acoustic wave after the connecting hole becomes
low and eventually negligible likely meaning: almost perfect destructive
interference exists in this part of the coil after the hole.

At 212 Hz (2${}_{\text{a,b}}$ in Fig. 3, the absorption is highest for
connecting hole location at Pos. #2 (2${}_{\text{b}}$) and nonlocal coupling
is once more acquired among the paralleled unit-cells. Figure 4 shows high
pressure amplitude in the shortest coiled cavities (top ones) which is logical
as these operates at higher frequency in comparison to the case of
1${}_{\text{a,b}}$. Furthermore, the top back cavities display a high pressure
amplitude for the case of connecting holes in Pos.#2 (bottom right panel of
Fig.4(b)) in comparison to Pos.#1 (bottom left panel of Fig.4(b)). The
combination of resonating back cavities and coiled cavities (highlighted by
the high pressure amplitude) lead to an enhancement of the absorption
coefficients. Proper impedance matching still exists as shown by the intensity
flow in Fig. 4 as per the same reasoning previously mentioned.

So, the aforementioned comparisons make two things clear: first, the back-
cavities are the stronger absorbers, and second, the highest rate of
absorption is acquired when both the coils and the back-cavities work in
tandem.

In summary, a new sub-wavelength (50 mm$\sim\nicefrac{{\lambda}}{{51}}$) thick
design has been introduced, consisting of four coiled resonators in parallel
and each independent resonator being in series with another resonator,
bringing about low-frequency broadband absorption from 130-200 Hz at 50%
absorption. The design doubles the bandwidth of the classical super-cell
design of only four paralleled HR, even though it’s peak absorption is lowered
by 25%. The improvements are attributed to the combination of the proper
impedance matching and the nonlocal coupling seen from the intensity flow and
pressure fields respectively, and the back-cavities are responsible for the
majority of the absorption within the design. The latter being easily
introduced to other designs, makes it a good candidate to improve future
designs.

###### Acknowledgements.

The authors acknowledge the joint research laboratory MOLIERE and la Région
Grand Est for supporting this work.

## Conflict of interest

The authors have no conflicts of interest to disclose.

## Data Availability Statement

Restrictions apply to the availability of these data, which were used under
license for this study. Data are available from the authors upon reasonable
request.

## References

## References

  * Assouar  _et al._ [2018] B. Assouar, B. Liang, Y. Wu, Y. Li, J.-C. Cheng, and Y. Jing, Nature Reviews Materials 3, 460–472 (2018).
  * Yang and Sheng [2017] M. Yang and P. Sheng, Annual Review of Materials Research 47, 83–114 (2017).
  * Cai  _et al._ [2014] X. Cai, Q. Guo, G. Hu, and J. Yang, Applied Physics Letters 105 (2014).
  * Jiménez  _et al._ [2016] N. Jiménez, W. Huang, V. Romero-García, V. Pagneux, and J.-P. Groby, Applied Physics Letters 109 (2016).
  * Li and Assouar [2016] Y. Li and B. M. Assouar, Applied Physics Letters 108 (2016).
  * Peng, Ji, and Jing [2018] X. Peng, J. Ji, and Y. Jing, The Journal of the Acoustical Society of America 144, EL255–EL261 (2018).
  * Donda  _et al._ [2019] K. Donda, Y. Zhu, S.-W. Fan, L. Cao, Y. Li, and B. Assouar, Applied Physics Letters 115 (2019).
  * Zhu  _et al._ [2019] Y. Zhu, K. Donda, S. Fan, L. Cao, and B. Assouar, Applied Physics Express 12, 114002 (2019).
  * Ji  _et al._ [2020] J. Ji, D. Li, Y. Li, and Y. Jing, Frontiers in Mechanical Engineering 6 (2020).
  * Ma  _et al._ [2023] F. Ma, H. Zhang, X. Wang, C. Liu, and J. H. Wu, Small Methods 7 (2023).
  * Wu  _et al._ [2022] Y. Wu, Q. Liang, J. He, J. Feng, and T. Chen, Applied Acoustics 195, 108846 (2022).
  * Yang  _et al._ [2017] M. Yang, S. Chen, C. Fu, and P. Sheng, Materials Horizons 4, 673–680 (2017).
  * Zhou  _et al._ [2021] Z. Zhou, S. Huang, D. Li, J. Zhu, and Y. Li, National Science Review 9 (2021).
  * Mei  _et al._ [2012] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, and P. Sheng, Nature Communications 3 (2012).
  * Yang  _et al._ [2015] M. Yang, C. Meng, C. Fu, Y. Li, Z. Yang, and P. Sheng, Applied Physics Letters 107 (2015).
  * Ma  _et al._ [2014] G. Ma, M. Yang, S. Xiao, Z. Yang, and P. Sheng, Nature Materials 13, 873–878 (2014).
  * Guo  _et al._ [2022] J. Guo, X. Zhang, Y. Fang, and R. Qu, International Journal of Mechanical Sciences 213, 106872 (2022).
  * Liang and Li [2012] Z. Liang and J. Li, Physical Review Letters 108 (2012).
  * Huang  _et al._ [2018] S. Huang, X. Fang, X. Wang, B. Assouar, Q. Cheng, and Y. Li, Applied Physics Letters 113 (2018).
  * Donda  _et al._ [2022] K. Donda, Y. Zhu, A. Merkel, S. Wan, and B. Assouar, Extreme Mechanics Letters 56, 101879 (2022).
  * Huang  _et al._ [2019] S. Huang, X. Fang, X. Wang, B. Assouar, Q. Cheng, and Y. Li, The Journal of the Acoustical Society of America 145, 254–262 (2019).
  * Guo  _et al._ [2020] J. Guo, X. Zhang, Y. Fang, and Z. Jiang, Applied Physics Letters 117 (2020).
  * Bi  _et al._ [2023] S. Bi, F. Yang, S. Tang, X. Shen, X. Zhang, J. Zhu, X. Yang, W. Peng, and F. Yuan, Materials 16, 1597 (2023).
  * Zhang and Xin [2022] L. Zhang and F. Xin, The Journal of the Acoustical Society of America 151, 1191–1199 (2022).
  * Duan  _et al._ [2020] M. Duan, C. Yu, Z. Xu, F. Xin, and T. J. Lu, Applied Physics Letters 117 (2020).
  * Huang  _et al._ [2020] S. Huang, Z. Zhou, D. Li, T. Liu, X. Wang, J. Zhu, and Y. Li, Science Bulletin 65, 373–379 (2020).
  * Zhu  _et al._ [2021] Y. Zhu, A. Merkel, K. Donda, S. Fan, L. Cao, and B. Assouar, Physical Review B 103 (2021).
  * Wang  _et al._ [2023] X. Wang, R. Dong, Y. Li, and Y. Jing, Reports on Progress in Physics 86, 116501 (2023).