# Transforming Room Acoustics with Causality-Driven Dual-Function Passive Metamaterials

Eric Ballestero, Yang Meng, Ping Sheng, Vincent Tournat, Vicent

Romero-García, Jean-philippe Groby

► To cite this version:

Eric Ballestero, Yang Meng, Ping Sheng, Vincent Tournat, Vicent Romero-García, et al. Transforming Room Acoustics with Causality-Driven Dual-Function Passive Metamaterials. Advanced Materials Technologies, 2025, 10 (11), pp.064074. 10.1002/admt.202402082. hal-05406057

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# Transforming room acoustics with causality-driven dual-function passive metamaterials

Eric Ballestero<fnref n="1" /> | Yang Meng<fnref n="1" /> | Ping Sheng<fnref n="2" /> | Vincent Tournat<fnref n="1" />

Vicent Romero-García<fnref n="3" /> | Jean-Philippe Groby<fnref n="1" />

**Correspondence**

Eric Ballestero, Av. Olivier Messiaen, 72085 Le Mans, France.

Email: eric.ballestero[a]univ-lemans.fr

**Funding information**

The authors gratefully acknowledge the ANR-RGC METARoom (No. ANR-18-CE08-0021) project. V. R.-G. acknowledges support from project CIAICO/2022/052 of the "Programa para la promoción de la investigación científica, el desarrollo tecnológico y la innovación en la Comunitat Valenciana".

Recent advances in acoustic metamaterials have overcome narrow-band limitations, allowing for optimal tuning of key functions such as sound absorption and diffusion. However, most existing designs focus on single functions, leaving a gap in multi-functional solutions that can address complex acoustic requirements. To fill this gap, we propose a Dual-Function Passive Acoustic Metamaterial (DFPAM) that effectively balances absorption and diffusion within the same structure by leveraging principles of causality and passivity. The DFPAM achieves high broadband sound absorption (⟨α⟩ ≈ 0.85) and diffusion (⟨δ⟩ ≈ 0.8), covering more than one octave in each configuration. With a compact thickness of only 12 cm, nearly half that of traditional treatments, this design uses just a fraction of the available space (1/3 for absorption, 2/3 for diffusion) while maximizing performance. This novel approach not only optimizes sound control but also provides a highly space-efficient solution. The DFPAM offers transformative possibilities for room acoustics and can be adapted to diverse environments, including building and aerospace applications, where multi-functional acoustic solutions are in increasing demand.

**KEYWORDS**

Acoustics | Metamaterial | Passivity | Asorption | Diffusion | Hybrid Materials

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<footnote n="1">¹Laboratoire d'Acoustique de l'Université du Mans (LAUM), Intitut d'Acoustique - Graduate School (IA-GS), UMR CNRS 6613, Le Mans, France.</footnote>

<footnote n="2">²Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China</footnote>

<footnote n="3">³Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universitat Politècnica de València, Spain.</footnote>
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# 1 | INTRODUCTION

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Over the past decades, acoustic metamaterials have garnered significant attention due to their ability to manipulate sound waves in unprecedented ways. Inspired by early advances in electromagnetism [1, 2, 3], acoustic metamaterials offer solutions that operate at deep-subwavelength scales, allowing substantial reductions in size and mass while maintaining high performance [4, 5, 6]. These innovations have been particularly impactful in traditional acoustic treatments, where the quarter-wavelength scaling issue presents challenges in low-frequency regimes [7]. Researchers have further optimized designs by exploring space folding strategies [8, 9], micro-perforated sheets [10, 11, 12], and slow-sound conditions [13, 14], leading to breakthroughs in both narrowband and broadband absorption [15, 16] and scattering [17, 18, 19].

Despite these advances, most existing acoustic metamaterial solutions remain configuration-dependent, addressing either sound absorption or diffusion, but rarely both [20, 21, 22, 23]. Hybrid solutions attempting to integrate these functions have often been constrained by physical and spatial trade-offs, leading to suboptimal performance. For instance, absorption requires dissipation of acoustic energy, while diffusion seeks to scatter sound uniformly with minimal dissipation, making it difficult to combine these functions within the same structure. Furthermore, recent reconfigurable approaches have shown promise but are still limited in the degree of performance variability achievable [24, 25, 26, 25, 27]. Continued advancements in material science and mechanical metamaterials are necessary to unlock the full potential of such designs [28, 29, 30, 31].

In this work, we adopt a causality-driven approach to design a Dual-Function Passive Acoustic Metamaterial (DFPAM) that optimizes both absorption and diffusion within a single structure. By leveraging the principles of causality and passivity, we determine the optimal volume distribution between these two contrasting

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>**FIGURE 1** Dual-functionality of the DFSPAM in a room. (a) Absorption configuration reducing reflected sound levels. (b) Diffusion configuration scattering back waves uniformly with almost no absorption.</figcaption></figure>

acoustic mechanisms. Sound absorption dissipates acoustic energy through visco-thermal effects at the mesoscopic scale [13, 32], while diffusion ensures spatial uniformity of the reflected sound field via phase gratings and multiple scatterers [33, 34]. For peak performance, these mechanisms must be isolated to avoid interference, making volume management critical [21]. Our dual-function metasurface design integrates these principles, resulting in a compact, highly efficient acoustic solution.

The proposed DFPAM is particularly suited to applications in room acoustics, where it can shift between highly absorbent and reverberant states. This flexibility is valuable in environments where acoustic conditions need to adapt dynamically, such as recording stu-

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<figure><img src="image_4.png" /><figcaption>**FIGURE 2** (a) 3D visualization of the Dual-Function Passive Acoustic Metamaterial (DFPAM) of side width *D* = 30 cm and depth *L* = 12 cm, with highlights on the micro unit cell (violet), UC<sub>μ</sub>, used for absorption measurement in a square impedance tube and the overall macro unit cell (orange), UC<sub>m</sub>, which is formed by two-fold mirror symmetry of the UC<sub>μ</sub>. (inset) geometrical parameters of the (x, y) ↦ (i, j) slit for the n<sup>th</sup> slit resonator section of height *h* and depth *a*<sub>z</sub>. (b) Mechanical switch for alternating between acoustic absorption (green) and diffusion (blue) states. Different colors for the grid illustrate different height levels allowing for superposition of grid panels.</figcaption></figure>

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dios, concert halls, and multipurpose buildings. Figure 1 illustrates the transformation enabled by the DFPAM, from sound-absorbing to diffusive configurations. Current reconfigurable acoustic treatments are bulky and expensive, often relying on large-scale installations such as quarter-wavelength structures. By contrast, the DFPAM offers dual-functionality in a significantly reduced volume, providing a compact and cost-effective solution for multi-purpose acoustic applications. This design holds potential not only for enhancing room acoustics but also for use in fields such as building engineering and aerospace environments, where space is at a premium, and multi-functional acoustic treatments are increasingly in demand.

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## 2 | PRINCIPLES & DESIGN

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Absorption and diffusion of acoustic waves are contrastive effects as the former needs to dissipate the acoustic energy and the latter needs to scatter the waves with minimal dissipation. Helmholtz resonators (HR) have been proven as very good candidates to tune both the dissipation and the phase of the reflections

coefficient in a controlled manner [35, 18]. In the present case, we intend to use a strategy based on rigidly-backed slits loaded by an array of HRs as shown in Fig. 2(a), which may or may not be identical depending on the acoustic function, i.e., absorption or diffusion. Having resonators in such an arrangement allows to either (i) affect sound absorption at multiple resonant frequencies (i.e., rainbow trapping [16, 36]) by introducing acoustic losses via visco-thermal boundary layers in the system with the aim to critically couple the entire resonant system, or (ii) affect the sound diffusion made by the overarching structure by introducing different dispersions in the slits. This creates slow sound conditions, thus reducing the effective wavelength propagating in the slits and effectively lowering the usual quarter wavelength resonant frequency. The latter results in structure cells that can alter the phase of the reflected wave without applying too much acoustic losses (i.e., phase-grating diffusers [37, 38]).

Acoustic absorption is expressed through a two-fold process, where the complex reflection coefficient $R(\omega)$ at radial frequency $\omega$ is first determined at the top of each slit by a Transfer Matrix Method (TMM). The lat-

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ter is then injected in a Bloch Wave Expansion covering the whole periodic surface by projecting (q, s) Bloch modes onto the absorption slits, which ultimately helps taking into account evanescent coupling between the slits. The final absorption coefficient of the periodic structure takes then the form

$$ \alpha(\omega) = 1 - \sum_{q,s \in Q} \frac{\operatorname{Re}(k_{z,qs})}{k_z^i} ||R_{qs}||^2, \quad (1) $$

where $R_{qs}$ is the (q, s) Bloch reflection coefficient.

The diffusion coefficient of a surface rates the uniformity of the scattered sound field, expressed as a spatial Fourier transform of the reflection coefficient of the locally-reacting diffusion slits $R(x, y, \omega)$. The directional diffusion coefficient, $\delta_{\psi}$, produced when a diffuser is illuminated by a plane wave at the incident angle $\psi = (\theta, \phi)$ can be estimated from a hemispherical auto-covariance function

$$ \delta_{\psi}(\omega) = \frac{\left[ \iint I_s(\theta, \varphi, \omega) dS \right]^2 - \iint I_s^2(\theta, \varphi, \omega) dS}{\iint I_s^2(\theta, \varphi, \omega) dS}, \quad (2) $$

where $I_s(\theta, \varphi, \omega) \propto |p_s(\theta, \varphi, \omega)|^2$ is proportional to the scattered intensity.

A more in-depth physical description of absorption and diffusion functions of the DFPAM are both expressed in their respective sections in the Supporting Information (SI). An additional element of the DFPAM overall design concerns the type of switch that allows to go from one state to the other. The herein proposed strategy introduces a mechanical panel switch made of thin rigid plates that sit on top of each slit, effectively closing it, and move along a drive shaft allowing translational as well as rotational movement of the covering plates (see Fig. 2(b)). By allowing some plates to superpose each other, this system helps toggle between absorption (green) and diffusion (blue) functions, where while one configuration is enabled (open) the other one is disabled (closed). For a better visualization and physical understanding, animations of the moving mechanism can be found as SI.

The proposed metasurface thus consists of a dis-

cretized $(x, y) \mapsto (i, j)$ 2D surface grid of square cells. This brings a few design rules to mind that will impact the subsequent work, i.e.,

1. **surface size.** The first design rule concerns the overall size of the structure as it is constrained by the experimental apparatus being used. The absorption of the metasurface is measured in a square impedance tube of $15 \times 15$ cm$^2$ section. Due to the Neumann boundary conditions at the edge of the square section imposed by the impedance tube, a two-fold mirror symmetry of the impedance tube sample (referred to as the micro unit cell, $UC_{\mu}$) must be performed in order to obtain the macro structure geometry, $UC_{m}$ (see Fig. 2), with dimensions now twice as big, i.e., $30 \times 30$ cm$^2$. These final dimensions, representing the overall structure, are the ones used to design the sound diffusion aspect of the metasurface;

2. **grid discretization.** This brings the second design rule which involves the degree of discretization of the surface, i.e., the total number of cells as well as their shape. For optimal sound diffusion the grating limit for the locally-reacting elements that cover the surface plane is provided as $\lambda_{min}/2 = 5$ cm, with $\lambda_{min}$ being the minimum wavelength taken into account. The latter is related to the maximum upper limit frequency of this study, $f_{max} = 3500$ Hz. Taking this high limit allows much more flexibility for the tuning of the sound diffusion performance over lower frequencies where local slits behave alike point-scatterers. This results in a $6 \times 6$ grid composed of $5 \times 5$ cm$^2$ square elements;

3. **tessellation pattern.** The third design rule deals with the final tessellation pattern for each acoustical function, and depends on the double symmetries imposed by the acoustic measurement system. This aspect is discussed in more details in its own section later on;

4. **frequency range.** This work considers a frequency range $\Delta f_{\alpha} = [400 : 1100]$ Hz for absorption and $\Delta f_{\delta} = [1000 : 2500]$ Hz for diffusion. A broader unique frequency range that would cover 400 - 2500 Hz in both cases would negatively impact each func-

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tion performance due to wavelength considerations in each case. In the case of absorption, the frequency range is shifted to minimal ranges in order to use at best the deep-subwavelength properties of the resonating HRs and therefore maximise the lowest absorption frequency, which is mostly ruled by low frequency requirements from causality conditions. For diffusion, the frequency range is shifted to higher frequencies due to the size of the panel, where lateral edge diffraction still dominates around 500 Hz, resulting in high diffusion independently of the diffusive nature of the phase grating. This mostly depends on the physical measurement system as well as the contextual setup that would be intended. For the sake of exploring optimal wavelength-to-dimensions ratios, it would seem more interesting to push each frequency range close to the physical limits of each function; and

**5. surface area ratio.** The final design rule with a critical role in the geometrical design of the structure, and probably the most restrictive, deals with the proportion distribution of sound absorption to sound diffusion over the surface. This arises due to the fact that the more unit cells can be dedicated to one process, the better its performance. This however comes at the expense of the remaining acoustical state. Therefore, it is crucial to determine which ratio of sound absorption-to-diffusion over the surface would be the most suitable for broadband performances for each acoustical function.

## 3 | RESULTS

### Optimal absorption/diffusion ratio

By analyzing the sum rule [39] given by the passivity conditions of the acoustic system accounted for in this work, it is possible to evaluate the optimal value of the absorption-to-diffusion surface area ratio, $\phi_{\alpha}|_{\delta}$, with a given target absorption spectrum. The optimal value indicates the most efficient design of absorber that can achieve the target spectrum with the minimum fraction

of surface area. To derive the sum rule, we consider solely the structure of the absorber within the metal surface, i.e., the part for diffusion is currently assumed as rigid, as illustrated in Fig. 3(a). Then, it is easily validated that this acoustical system in the reflection problem satisfies continuity, causality, linearity, and time-translational invariance. Moreover, if we focus on passive acoustic treatments, the reflection energy is guaranteed to be less than (or equal to) the incident energy. This implies that $0 \le |R(\omega)| \le 1$, which serves as input for the absorption coefficient expressed as $\alpha(\omega) = 1 - |R(\omega)|^2$. It follows that a sum rule can be derived by either the Kramers-Kronig relation [40] or the method of Herglotz-Nevanlinna function [41] (the derivation using the latter method is given in SI), which provides

$$ \phi_{\alpha} \ge \frac{-c_0}{2\pi\gamma\phi_{\text{air}}L} \int_{0}^{\infty} \frac{\ln[1 - \alpha(\omega)]}{\omega^2} d\omega \equiv \phi_{\alpha,\text{min}}, \quad (3) $$

where $c_0$ and $\gamma$ are the adiabatic sound speed and the adiabatic index of air, respectively, $L$ is the total thickness of the DFPAM, and $\phi_{\text{air}} = V_{\text{air}}/V_{\text{tot}}$ denotes the filling fraction of the air volume over the overall absorber volume. The former refers to the total volume of air in the necks and cavities of the HRs as well as the slits below the resonators whilst the latter corresponds to $V_{\text{tot}} = LH$ as shown in Fig. 3(a). When a target absorption spectrum $\alpha(\omega)$ is given, Eq. (3) can be readily applied to evaluate the optimal (i.e., minimal) absorption surface ratio $\phi_{\alpha}$ defined as $\phi_{\alpha} = H/D$, i.e., the area ratio between the absorber and the whole surface as shown in Fig. 3(a). Then, the optimal absorption-to-diffusion ratio is obtained as $\phi_{\alpha}|_{\delta,\text{opt}} = \phi_{\alpha,\text{min}}/(1 - \phi_{\alpha,\text{min}})$.

A cross-parameter study is thus conducted to show the effects of $L$ and $\phi_{\text{air}}$ on the prediction of $\phi_{\alpha,\text{min}}$ to achieve the specific absorption spectrum shown in Fig. 3(b). The shape of this frequency spectrum, divided into two parts, is not quite arbitrary. Starting with the target frequency range defined as $\Delta f_a = [400 : 1100]$, the bottom end of the spectrum is going to impact substantially the parameters of length $L$ or surface absorption ratio $\phi_{\alpha}$. In order to aim for high absorption with reasonable parameter constraints $L$ and $\phi_{\alpha}$, an absorption coefficient lower than the average target
<figure><img src="image_5.png" /><figcaption>FIGURE 3 (a) Slab of a composite metamaterial divided into two domains, viz., a rigid domain of dimension $H \times L$ and an open domain of same dimensions composed of a slit of varying section, each loaded by a square Helmholtz resonator. (b) Target acoustic absorption spectrum for the composite metamaterial made of two polynomial functions. (c) Sum rule estimation of the minimal length $L_{min}$ in function of the surface absorption ratio $\phi_{\alpha}$ required to comply with the target absorption spectrum.</figcaption></figure>

coefficient needs to be set, in this case 0.8 at 400 Hz. In order to increase the average, a small linear increase of the absorption coefficient is applied. However, below 400 Hz, the absorption spectrum needs to be defined so as to provide a continuity in the lower part of the spectrum. This is enforced following the quadratic nature of the asymptotic expansion of the absorption coefficient at low frequencies which varies as ~$O(\omega)^2$ [42].

Within the frequency range of interest, the spectrum in Fig. 3(b) provides the target absorption coefficient of the DFPAM design in this work. According to Fig. 3(c), $\phi_{\alpha,\min}$ is proportional to $1/L$. Hence, both $\phi_{\alpha,\min}$ and $\phi_{\alpha|\delta,\text{opt}}$ decrease with increasing depth $L$. There are diminishing returns in increasing porosity values over 0.75, which helps accounting for the finite wall thickness within the absorber. Conversely, a decrease of porosity provides great adverse effects, increasing significantly either the surface ratio $\phi_{\alpha,\min}$ or the total thickness $L$. For the design of the DFPAM structure in this work, the total thickness is fixed as $L = 12$ cm and a porosity $\phi_{\text{air}} = 0.8$ is ultimately found after the optimization procedure outlined thereafter. A depth of 12 cm was chosen in order to provide a solution that fits within dimensions similar to other metamaterial

designs reported in the literature [42, 43]. According to Fig. 3(c), it follows that $\phi_{\alpha,\min} \approx 0.33$. This implies that only about 1/3 of the surface is enough to comply with the target absorption spectrum, effectively leaving 2/3 of the remaining surface to be used for sound diffusion. This makes for a convenient allowance of 3 cells out of 9 for sound absorption in the present design.

In conjunction to the previous analysis, there is an additional implication behind the sum rule relation shown in (3). It shows that the use of sum rules for passive structures can only hint on a certain minimum depth of treatment, given some structural parameters, which can comply with a target absorption spectrum. Such conclusion is thus made regardless of the design strategy to be used, i.e., usage of porous/fibrous media or HR-based structures. However, reaching such theoretical minimum length ultimately depends on the physical design being implemented, and thus on its ability to reach minimal deep-subwavelength limits. This is where the main difference between porous/fibrous media and HR-based structures can play a major role. A way to highlight the differences between these two design strategies remains in the expression of the low frequency bulk modulus and
the viscous-thermal transition limit for porous media compared to HR structures (further details in SI).

The application of this analytical limit of passivity-induced sum rules to acoustics has already been sparsely discussed on the topics of porous materials [44] or quarter-wavelength structures [42] for finding the minimal length of a material required in function of a given absorption spectrum. Yet, its application to composite HR-based metamaterials, such as the ones displayed in Fig 3(a), in addition to the relation to a surface ratio and HR-induced volume filling fraction is a novel application case which allows to bring fruitful insights into the problem at hand.

## Surface tesselation

Starting from a 3 × 3 square grid for the $UC_μ$ with a combinatorial ratio of absorption cells to overall cells $\phi_α = 1/3$, there are a total of $B = 84$ possible combinations or spatial patterns which can be computed with a binomial formula "n chooses k". This is illustrated in Fig. 4(a) where the $η = 9$ cells are arranged in the 3 × 3 square grid and 3 example patterns are shown. The rigid boundary conditions at the edges of the surface (i.e., tube walls) imply that the resulting macro unit cell $UC_m$ possesses a two-fold mirror symmetry, as illustrated in Fig. 4(b). Due to such mirror symmetries, many patterns obtained in this way have rotational symmetries, hence reducing the number of different patterns. Further physical considerations regarding the

<figure><img src="image_6.png" /><figcaption>**FIGURE 4** (a) Illustration of the $B = 84$ possible configurations obtained in the binomial formula 9 choose 3. (b) Illustration of the two-fold mirror symmetries of the micro unit cell, $UC_μ$, making the macro unit cell, $UC_m$.</figcaption></figure>

choice of tesselation pattern, e.g., relation between the spatial distribution of the reflection coefficient and scattered sound pressure field, are detailed in the SI. Ultimately, out of the 42 unique patterns, the rhombus pattern $b = 63/29$ happens to be one of the best candidates available for sound diffusion.

## Geometry optimization

In the present work, we make use of non-linear constrained minimization algorithms [45] for maximising both absorption and diffusion as well as determining their respective geometries. In the case of absorption, the target cost function to minimise is that of $\epsilon_α = \int_{\Delta_α,\min}^{\Delta_α,\max} [1 - \alpha(f)] df / n_f$ for a frequency range $\Delta_α = [400 : 1100]$ Hz, whereas in the diffusion case the cost function is $\epsilon_δ = \int_{\Delta_δ,\min}^{\Delta_δ,\max} [1 - \delta(f)] df / n_f$ for $\Delta_δ = [1000 : 2500]$ Hz. The term $n_f = \Delta_{max} - \Delta_{min}$ is the number of discrete frequencies over which the cost function is averaged. These frequency ranges were chosen for (i) displaying the deep-subwavelength performance achievable for sound absorption, i.e., $L \approx \lambda_{α,max}/7$, and (ii)

<figure><img src="image_7.png" /><figcaption>**FIGURE 5** Final Pareto front obtained via cross-optimizations. (inset) Optimization cost function of absorption and diffusion for different surface ratios.</figcaption></figure>
362 show large broadband performance for sound diffusion outside the point scatterer regime of the metasurface, 363 $\lambda_\delta \le D$. Put in words, the two cost functions try to max- 364 imize as best as possible each coefficient over a certain 365 frequency range. The geometrical parameters which are 366 tuned during the optimization procedure are the width 367 of the slits, the heights of each slit section, as well as the 368 369 370 HR for each individual slit.

371 As previously explained, the performance of both 372 sound absorption and diffusion is greatly dependent on 373 the number of cells dedicated to each state. Figure 5 dis- 374 plays such behaviour through the Pareto front obtained 375 via the cross-optimizations, highlighting again a 1/x pro- 376 portionality trade-off between each acoustical aspect. 377 Figure 5(inset) illustrates such trend from an iteration 378 perspective, where the different optimization cost func- 379 tions of absorption and diffusion for different surface ra- 380 tios, viz., 1/9 (central abs. cell), 2/9 (diagonal abs. cells), 381 382 383 384

385 We can observe that both the average absorption 386 or diffusion coefficient over the target frequency range 387 decreases in function of the reduced space over which 388 each state is allowed to perform. In the absorption case, 389 this can be translated as a decrease of the number of 390 HRs that can work together in order to better dissipate 391 acoustic energy. For the diffusion case, the reduction of 392 the number of cells with fixed resonators per slit results 393 in less modularity in the distribution of the reflection co- 394 efficient at the surface.

Acoustic absorption

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396 For the sake of simplicity, mathematical derivation of 397 the absorption coefficient can be found in the SI. Here, 398 a general overview of the physical mechanisms at play 399 will be given instead.

400 Formulating the absorption problem using the Trans- 401 fer Matrix Method (TMM), accounting for visco-thermal 402 boundary layer losses, allows to input the absorption

information of each slit into a larger analytical problem. 403 In the latter, the continuities of pressure and particle 404 velocities are formulated at the top of the surface, 405 i.e., at the interface of either closed or open slits, 406 accounting therefore for rigid (i.e., Neuman) or open 407 (i.e., Diriclet) boundary conditions over each cell and 408 projecting into the open slits by assuming plane wave 409 propagation inside. Finally, a Bloch wave expansion 410 (BWE) is performed over the larger surface (accounting 411 for the two-fold mirror symmetry implied by the mea- 412 surement system shown in Fig. 4(b)), as it is supposed 413 to be periodically repeated in space.

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The optimization of the absorption state cost func- 415 tion $\epsilon_\alpha$ defined in the previous section is performed 416 with $N = 8$ HRs into each slit over a total structure 417 depth $L = 12$ cm. It is worth noting that the more HRs 418 per slit, the more resonant frequencies can be created 419 and tuned to achieve high absorption. Yet, the more 420 HRs in the slit, the smaller the volume capacitance 421 of each HR, implying higher resonant frequencies. 422 The balance of 1 HR per 100 Hz in the frequency 423 range $\Delta_\alpha = [400 : 1100]$ Hz has been taken here as a 424 compromise between volume capacitance and number 425 of resonant frequencies to fit within the allocated 426 absorber volume. Optimizing the geometrical variables 427 of the slit sections and HRs in order to reach the highest 428 absorption coefficient over the target frequency range 429 results in the geometry illustrated in Figure 6, which 430 also provides numerical and experimental data for the 431 absorption state.

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Figures 6(a-h) show the resonant characteristics of 434 the final absorption geometry, highlighting the distri- 435 bution of the absolute normalized pressure field inside 436 the structure. It can be seen that for low frequencies, 437 c.f. Figs. 6(a-c), the larger HRs at the bottom are the 438 ones effectively trapping the acoustic energy, whereas 439 for higher frequencies, c.f. Figs. 6(d-h), it is the smaller 440 upper HRs. One can also observe that when reaching 441 high absorption many resonators start working in 442 conjunction around the same resonant frequency in 443 order to provide the required compliance for the sound 444 wave to be absorbed. Such figures prove that there 445

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<figure><img src="image_8.png" /><figcaption>**FIGURE 6** (a-h) Othorgonal view of the simulated pressure field inside the structure seen from the (x, z) plane for arbitrary frequencies spanning the target frequency range. (i) Simulated normalized sound pressure field at L/100 above the metasurface at 1100 Hz along with the velocity vector field **v** = **v**(x, y, z). (j) Analytical, numerical and experimental absorption coefficients overlayed with those of each individual slits. (k) Picture of the experimental setup of the absorption sample in the square impedance tube.</figcaption></figure>

is indeed a surface inter-slit coupling between the different slits, as detailed in the analytical modelling in the SI. The precision of the calculation of such coupling depends on the number of the Bloch waves effectively taken into account at the surface of the structure and comes at the expense of increased calculation time per optimization iteration.

Taking into account the surface coupling between slits allows to share the acoustic compliance load between the HRs in the slits. This enables a complementary performance between HRs of different slits that can be used for trapping acoustic energy and therefore makes the structure more efficient in its absorption capabilities compared to one that would use the same HR volume without inter-slit coupling. The fact that the HRs are arranged into a stairwell manner also allows for neighbouring HRs to share a part of the acoustic compliance load, as it can be seen through most of the acoustic pressure distributions shown in Fig. 6.

Compared to a similar 'rainbow-trapping absorber' geometry concept presented in Ref. [43] - which provides an almost perfect absorption spectrum over

the same frequency range by using a 5 × 3 rectangular cell grid with a dedicated surface ratio for absorption of 100%, and of same length *L* ≈ 12 cm - the design proposed herein not only enables a very good mean absorption spectrum of ⟨*α*⟩ ≈ 0.8 but does so within 1/3 of the dedicated surface/volume due to the account of evanescent coupling between slits.

Figure 6(i) shows the simulated normalized sound pressure level at *z* = −*L*/100 on top of the metasurface at 800 Hz. It illustrates the presence of evanescent modes at the surface of the structure by displaying the velocity vector field. From this, one can see the flux of acoustic energy going from the two lateral slits to the central one, and vice versa, as these are in anti-resonance. Phase-varying animations can be found in the SI video content. The presence of evanescent coupling at the top of the metasurface in the numerical model, which are also taken into account in the theory, provides a solid ground for explaining the excellent fit between the two absorption curves shown in Fig. 6(j), i.e., BWE with *q* = ±7 Bloch modes in continuous black
(-) and Finite Element Method (FEM) in orange circles (-). Some analytical-to-numerical discrepancies can be seen around 850-950 Hz which could be caused by evanescent coupling between HRs within a particular slit, thus slightly disrupting the plane wave propagation assumption inside it; hence this phenomenon not appearing at lower frequencies.

The case of $q = 0$ Bloch waves in dotted grey (- -) is also displayed as complementary information in order to illustrate the strong role of evanescent slit coupling in the build-up of the overall absorption coefficient spectrum. When no coupling is taken into account, one can see that the absorption spectrum is not only shifted to higher frequencies and wider in range but its spectrum also deteriorates of about 0.2 in average in the target frequency range compared to the other curves. Accounting for evanescent coupling thus helps making the best use of the resonant behaviour of the structure by shifting down its absorption spectrum and enhancing its average value.

Experimental results (□) are also displayed in Fig. 6(j), with a picture of the stereolithography (SLA) 3D-printed sample inserted in the square impedance tube presented in Fig. 6(k). Further details on the measurement method and data processing used can be found in the Methods section. One can see that the experimental absorption spectrum is in good agreement with both analytical and numerical curves but displays some variations nonetheless related to manufacturing and experimental uncertainty conditions about visco-thermal losses [46]. Nonetheless, a good agreement between the different curves can still be observed, where the different absorption spectra cover the same frequency range of $\Delta\alpha = [400 : 1100]$ Hz and have the same approximate average absorption coefficient $\langle\alpha\rangle \sim 0.8$.

## Acoustic diffusion

As for the acoustic absorption section, mathematical derivation of the diffusion coefficient is shown in SI for simplicity. Instead, a general overview of the physical mechanisms at play will be briefly given.

The diffusion coefficient defines the uniformity of

the spatial distribution of a given scalar function, here the scattered sound pressure level. In a way, it is a measure of the isotropy of the scattered field around the diffuser. As such, the diffusion coefficient is defined as a spatial auto-covariance of a given scalar function. A diffusion coefficient of unity hence means that said distribution is entirely uniform. In a polar coordinates context, this means that the distribution is akin to that of a sphere, i.e., the energy is uniformly spread in all directions. Inversely, a diffusion coefficient of zero means that the distribution is exactly non-uniform, implying thus a perfect Dirac function with one positive point and the rest at zero.

The computational time required in order to compute the diffusion function can easily lead to extensively long optimization runs when considering many HRs per slit. Fortunately, a single HR already allows to manipulate efficiently the phase of the reflected acoustic wave. The diffusion configuration thus consists of $N = 1$ HR for all 24 slits, aiming for a perfect diffusion over the range $\Delta\delta = [1000 : 2500]$ Hz.

Figure 7(a) shows the experimental measurement setup used for measuring sound diffusion. The latter is performed in an anechoic chamber in order to minimize any acoustic reflections of the environment when measuring those made by the metasurface. A speaker is placed at the ceiling of the room and will generate an Maximum Length Sequence (MLS) signal which will be recorded in order to gather impulse responses (IRs) at multiple microphone positions. These cover the upper hemispherical space on top of the surface with a radius of 1 m, i.e., just outside the near-to-far field limit (Rayleigh distance) $d_F = 2D^2/\lambda_{max} = 0.7$ m. IRs are then processed in order to isolate the sound being scattered back from the surface, thus yielding the scattered pressure field distribution, allowing ultimately the expression of the sound diffusion coefficient. A similar setup is implemented numerically in COMSOL Multiphysics. For more details concerning the experimental and/or numerical setups, see the Methods section.

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<figure><img src="image_9.png" /><figcaption>FIGURE 7 (a) Picture of the experimental setup for sound diffusion in an anechoic chamber. (b) Analytical, numerical and experimental frequency-dependent diffusion coefficient $\delta(\omega)$. (c-e) Analytical, numerical and experimental sound scattered pressure fields obtained at 1500 Hz. (f) Numerical cross-section of the scattered pressure field at 1500 Hz illustrating the expected dip obtained in the optimized sound scattered pressure field.</figcaption></figure>

Figure 7(b) displays the analytical, numerical and experimental sound diffusion coefficients obtained throughout this study. It also displays the diffusion coefficient given by a flat reference surface of same dimensions as the sample metasurface. This allows to normalize the diffusion coefficient of the latter with respect to that of a flat reflector, so that the new normalized diffusion coefficient then represents how much more acoustic energy is being scattered back from said sample. By observing the diffusion curves, one can see that indeed the normalized diffusion coefficient of the structure possesses maximal values between 1000 Hz and 2500 Hz. These are far from perfect diffusion (i.e., $\delta = 1$) but still achieve a good average diffusion coefficient of $\langle\delta\rangle \approx 0.75$, or $\langle\delta_n\rangle \approx 0.58$ if normalized.

At this point, it is important to note that increasing the diffusion coefficient towards unity becomes increasingly more difficult to achieve for broadband performance due to (i) the cell discritization of the metasurface, (ii) the physical mechanisms at play in each cell, and most importantly (iii) the fact that perfect diffusion in a 3D space implies that all points in the $(\theta, \varphi)$ referential must be perfectly uniform. Slight deviations

in the latter will quickly decrease the diffusion coefficient, even more so than in a 2D case. A previous study [19] discussed about the tenets of going from perfect diffusion around a single frequency to broadband performance. A decrease of the diffusion coefficient was demonstrated when progressively increasing the frequency range being considered. Overall, in accordance to other sources [47, 48], it was concluded that although not perfect, a diffusion coefficient of around 0.6 to 0.7 could already be considered as a very good number for displaying the good sound diffusion characteristic of a surface. One can note that a spherical Lambertian distribution possesses a diffusion coefficient $\delta_L \approx 0.7$.

When comparing the analytical, numerical and experimental diffusion coefficients, apparent variations – although discrete – can be seen. These can be explained through the differences between analytical assumptions and numerical/experimental conditions. On the one hand, the analytical formulation considers an almost infinite far-field radiation from a baffled surface by performing a spatial Fourier transform over the surface reflection coefficient. When optimizing for perfectly uni-

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form spherical distributions, this leads to distribution
fields with great acoustic energy radiated everywhere,
specially at grazing angles. This appears easily when
applying a Sommerfeld radiation condition on the sys-
tem, where no radiation from external elements is added
to the one originating from the considered sample. On
the other hand, numerical and experimental conditions
have, ideally, impedance matching boundary conditions
around the diffusion sample. Due to the finite size of the
surface radiating sound, it implies that less energy is be-
ing radiated at grazing angles. To this end, a correction
to the diffusion coefficient has been applied, where the
spatial Fourier transform being performed ignores the
first 15° in the elevation axis. Doing so allows to find a
better fit between the different curves, even for that of
the flat reflector.

Grazing angle variations can be better seen in Fig-
ures 7(c-e) which show the polar scattered pressure
distributions obtained at the frequency f = 1500 Hz
through the three different methods presented thus far.
Despite these variations, the resulting scattered fields
behind each diffusion coefficient value are in a relatively
good agreement with theoretical target values. These
distribution presents a typical notch around θ ≈ 30° and
φ ≈ 0°, and is well reproduced throughout all three po-
lar distributions. This notch is made more visible in the
numerical cross-section view presented in Fig.7(f). The
rest of the target distribution is also well represented,
i.e., normal peak and lateral notches, although slightly
more smoothed out due to numerical and experimental
scattering values being less dynamic in their range than
with perfectly analytical conditions.

Numerical room simulations

A numerical room acoustic simulation is performed in
order to demonstrate the transforming acoustics of the
room while in either state. In order to obtain room im-
pulse responses (RIRs) with meaningful time-domain in-
formation, a publicly available and academically-verified
GPU-accelerated finite-difference time-domain (FDTD)
numerical solver is used for the simulations [49]. More
details concerning the FDTD simulations can be found

in the Methods section.

The room of dimensions (x, y, z) 5 × 4 × 3 m is dec-
orated with 8 metamaterial blocks of 0.9 m side length
on the walls; two per wall and each composed of 3 × 3
repetitions of the UC<sub>m</sub>. An illustrative setup is shown
in Fig. 8(a). The frequency-dependent impedances of
each slit of the metamaterial are integrated in the FDTD
scheme via an RLC circuit fitting method [50]. The re-
cceiver array is composed of (x, y, z) 3×3×3 microphone
positions. This allows to perform a Spatial Decomposi-
tion Method (SDM) [51] in order to obtain the directions
of arrival (DOA) and amplitudes of the many wavefronts
crossing the receiver array. By forward-integrating the
RIRs through different time windows, this allows to ob-
tain spatio-temporal information describing the state of
sound field isotropy at the receiver position.

Figures 8(b-e) show such spatio-temporal informa-
tion for the diffusion and absorption states along with
the same in-plane cross-sectional cut of the normalized
pressure p – p<sub>ref</sub> at a time t = 10 ms, where p<sub>ref</sub> is the
pressure in a room of same geometry with rigid bound-
aries only. Normalizing the computed pressure in this
way allows to remove the original spherical wavefront
and further put in evidence the acoustical contributions
of each state compared to a room with no particular
treatments. As such, it can be observed in Figs. 8(b,c)
that the diffusion model effectively scatters the imping-
ing wavefronts when interacting with the metamaterial
blocks. Spatio-temporal insets clearly show a rapid evo-
lution of the sound field isotropy at the receiver, going
from a very localized sound level peak due to the inci-
dent wave and first few early reflections arriving within
5 ms to a much more even sound field distribution at
just 20 ms. This growth of the sound field isotropy in-
dicates a rapid build up of a statistically diffuse sound
field in the room. In contrast, the acoustic wave propa-
gation in the absorption state presents very little scat-
tering outside the metamaterial edge diffraction, and
its spatio-temporal distributions are more uneven and
evolve much slowly, indicating a longer time for the
room to reach diffuse conditions with lower sound pres-
sure levels.

Figure 8(f) displays the cumulative energy decay

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<figure><img src="image_10.png" /><figcaption>**FIGURE 8** (a) Simulated 3D room model with dimensions (x, y, z) 5 × 4 × 3 m with a source and a receiver array. Room impulse responses are forward-integrated via increasing time windows. (b,c,d,e) Cut planes showing the (b,c) diffusion and (d,e) absorption FDTD normalized pressure fields (with respect to an empty model) at *t* = 10 ms along with the in-plane spatio-temporal sound pressure level distributions for different forward-time integrations. (f) Cumulative Energy Decay Curves (EDC) for the absorption and diffusion states (and empty room in gray) and their respective Early Decay Times (EDTs). (g) Room frequency responses for both absorption and diffusion states.</figcaption></figure>

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curves between the absorption and diffusion room states. One can observe a significant level difference between the two decay rates, where the diffuse energy curve decays linearly with time; a sign of a homoge-

neous sound field decay; while the absorption energy curve decays more rapidly in the first half of the impulse response before decaying linearly in a similar way than the diffuse state. By calculating the -10 dB slope and ex-

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trapolating it to a 60 dB decay, Early Decay Times (EDTs) between absorption and diffusion states vary by a factor of 3, i.e., EDT<sub>α</sub> ≈ 0.54 s and EDT<sub>δ</sub> ≈ 1.72 s. EDT is more closely associated with the perceived clarity and intimacy of a space due to the early changes in reverberation time [52, 53, 54]; a quality explained by how the human ear and brain memory integrate varying acoustic level changes [55]. This significant variation in EDTs clearly indicates a strong change in the perceived acoustics by switching between either state of the BSPAM. This is also backed by a clear frequency spectrum alteration between the two room configurations, as shown by the room frequency responses in Fig. 8(g). It clearly shows a great gap in sound levels between the two spectra, coinciding with the frequency range of the absorption state, i.e., 400-1100 Hz. There is a 10 dB maximum difference between the two states, with an average of ~ 8 dB difference over the absorption frequency range. This results in a ~85% sound pressure level reduction, as predicted with the designed average absorption coefficient ⟨α⟩ = 0.85 of the BSPAM, and consequently achieves a perceived noise reduction of ~45%; almost a factor two in perceived sound reduction between the two states. Simulated impulse responses convoluted with anechoic audio samples can be found in the SI. In these, one can note the persistence of flutter echoes in the room while in the absorption state due to the lack of diffusion.

740 | DISCUSSION

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In this work, we present a dual-function metasurface capable of achieving high values of absorption, ⟨α⟩ ≈ 0.85, and diffusion, ⟨δ⟩ ≈ 0.8, for different broadband frequency ranges covering more than one octave in each configuration, i.e., Δ<sub>α</sub> = [400 : 1100] Hz and Δ<sub>δ</sub> = [1000 : 2500] Hz, while still maintaining compact dimensions.

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The design of the passive metamaterial relies heavily on causality-induced sum rules in order to find the best volume distribution between the different functions, resulting in a ratio of 1/3 for sound absorption and 2/3 for

sound diffusion. This was done by proposing a methodology that can evaluate the absorption-to-diffusion ratio $\phi_{\alpha|\delta}$ with solely the total thickness of the metasurface as well as the target absorption spectrum required. This is an important parameter in the design process as it can save a great amount of time that could be otherwise spent on empirically finding the most suitable configuration. The resulting numerical and experimental measurements show that the acoustic absorption spectrum found through geometry optimizations of the structure conversely obeys to the laws of causality.

Additionally, a study on surface tessellations was conducted in order to choose a good surface tiling out of the 84 apparent possibilities for alternating between acoustic absorption and diffusion states. In complement, a cross-optimization study was then performed between different tessellations with more or less absorption/diffusion surface ratios in order to illustrate the inverse proportionality between the two states.

Lastly, a time-domain room acoustic numerical simulation was performed in order to better demonstrate the transforming room acoustics between either state of the BSPAM. Combined with a spatio-temporal analysis, results of Early Decay Time (EDT) and room frequency responses show a strong and noticeable difference in acoustical parameters in the room between the two states. This resulted in high isotropy variations in room sound fields, a factor three difference between EDTs and almost a factor two in perceived sound level reduction in the central octave.

Thus, the work presented in this study pushes further on the combinatorial aspect behind the design of multi-purpose passive acoustic metamaterials by juxtaposing two previously known design concepts, with very different design limitations, and integrating both of them within the same geometry volume. This multi-state approach hence proposes an alternative route to the theme of reconfigurability. By combining both acoustic absorption and diffusion, the final metasurface design could clearly help enhance the acoustic experience in rooms, going from very absorbent to very reverberant, consequently changing the notion of room volume and room impression. The present metamaterial also offers

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deep-subwavelength dimensions almost twice as small than traditional alternatives using only a third and two thirds of the available space for absorption and diffusion, respectively. This allows many prospects for re-configurable acoustic applications [50], including control studio rooms, musical auditoria, conference rooms, but could also be extrapolated to building and aerospace environments [56] such as cabin insulation, engine noise control, or noise barrier design.

## Methods

### Acoustic absorption measurement

The waveguide has a square cross-sectional area of $A = 15 \times 15 \text{ cm}^2$ and is made of 3 cm thick acrylic. The cut-off frequency of the first transverse mode is thus $f_c = c_0/A \approx 1133 \text{ Hz}$. The pressure field is evaluated at $\mathcal{P} = 7$ non-equidistant points located along the z axis referenced in Fig. 2(a). A GRAS 46BD 1/4" CCP Pressure Standard Microphone Set was used, together with the conditioner GRAS Power Module Type 12AQ. The microphone is moved along the z axis, driven by the stepping motor Nanotec TYPE 4H8618M1408, and controlled using CHARLYROBOT F4350 CERNEX. The frequency range is spanned linearly with the emission of $N_f = 200$ monochromatic plane waves. An analyzer Stanford SRS MODEL SR785 was used for the input electrical signal. The data then recorded is processed for frequencies below the cut-off frequency by considering the acoustic field in the tube at any point as a sum of forward and backward plane waves, where the complex amplitudes for each direction of wave propagation can be retrieved after a system inversion. The complex reflection coefficient is then obtained from the ratio of the two complex amplitudes, from which the absorption coefficient can be determined.

### Acoustic scattered field measurement

An experimental procedure based on ISO 17497-2:201218 was developed here to determine the sound scattering properties of the metamaterial. The physical sample was placed at the center of a concentric arc of evenly spaced microphone positions, all within an

anechoic environment for keeping unwanted acoustical contributions from the measurement system as minimal as possible, as shown in Fig. 7(a). Microphone positions ranged from $\varphi = 0^\circ$ to $\varphi = 90^\circ$ around the surface with a spacing of $6^\circ$ between each and along an arc radius of 1 m. Measurements were performed for normal incidence by locating the source 2.5 m away from the surface. Each IR was subtracted from the one obtained from the anechoic background and windowed to extract the scattered sound field [33].

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### Numerical FDTD room simulations

A publicly available time-domain solver was used for the simulations, and can be found at https://github.com/bsxfun/pffdtd. PFFDTD [49] (pretty-fast FDTD) is an implementation of finite-difference time-domain (FDTD) simulation for 3D room acoustics. Written in Python, C and CUDA, it includes a set of tools for simulation setup and processing of input/output signals. Room acoustic simulations performed in this study used a 7-point Cartesian scheme with 10.5 points per wavelength, allowing less than 1% of numerical dispersion error up to a high-cut frequency of 3500 Hz.

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## Acknowledgements

The authors gratefully acknowledge the ANR-RGC METARoom (No. ANR-18-CE08-0021) project. V. R.-G. acknowledges support from project CIAICO/2022/052 of the "Programa para la promoción de la investigación científica, el desarrollo tecnológico y la innovación en la Comunitat Valenciana". E.B. acknowledges N. Pajusco and B. Hamilton for their great help in the numerical room simulations.

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## Author contributions

E. B., V. T., P.S. V. R-G. and J-P. G. designed the research. E.B. conducted the research. E. B., V. T., V. R-G. and J-P. G. wrote the article.

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## Competing Interests

The authors declare no conflict of interest.

## Supporting Information

Supporting information covering theoretical derivations and numerical details can be found along this manuscript.

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