[3D Printed quadratic residue metadiffuser - design and measurements of an
optimized deep-subwavelength sound diffuser]
Eric Ballestero1 , Vicent Romero-García2 , Noé Jiménez3 , Jean-Philippe Groby2 , Haydar Aygun1 , Stephen Dance1(1) The Acoustics Group, London South Bank University, United Kingdom.
(2) Laboratoire d’Acoustique de l’Université du Mans, LAUM - UMR CNRS 6613, Le Mans Université, Avenue Olivier

Messiaen, 72085 LE MANS CEDEX 9, France.
(3) Instituto de Instrumentación para Imagen Molecular, Consejo Superior de Investigaciones Científicas (CSIC), Universitat

Politècnica de València, València, Spain.

Abstract
It has been two years since the first publication of the novel concept of metadiffusers. Heavily based on critical
coupling theory introduced in the case of perfect absorption in deep-subwavelength metamaterials, metadiffusers
are presented in a similar fashion, i.e. slotted panels with thin slits, each one backed with a set of critically coupled Helmholtz resonators (HRs). Instead of focusing on absorptive and transmitting properties, metadiffusers
emphasize on perfect or pseudo-perfect reflection of sound. Here a particular interest is given to the scattering
patterns obtained by the customization of the metamaterial’s geometry. The deep-subwavelength nature of thestructure leads to dimensions 20 to 46 times smaller than the design wavelength, i.e. about 1/20th to 1/10th of
the thickness of traditional designs. This presentation aims at introducing the physical aspects of sound propagation in and out of the metadiffuser, emphasizing on deep-subwavelength critical coupling, on the consequent
strong dispersion relations and on the custom scattered fields. This theoretical overview will then be pairedwith numerical and experimental data of a 3D-printed quadratic residue metadiffuser (QRM). Finally, potential
applications of such structures will be discussed in critical industry environments.
Keywords: metamaterials, sound diffusion, slow sound1

INTRODUCTION

Sound diffusers are locally-reacting reflecting surfaces with a spatially dependent reflection coefficient designed
to produce uniform scattering, i.e. the reflected waves by these surfaces are scattered in many different directions [1]. This means that diffusers tend to present a uniform Fourier transform magnitude of its spatiallydependent reflection coefficient distribution. The generation of spatially dependent reflecting surfaces is commonly achieved by using phase grating diffusers, also known as Schröeder’s diffusers [2], that are rigidly-backed
slotted panels where each slit acts as a quarter wavelength resonator. The common approach for designingphase-grating diffusers is to implement an effective well depth sequence in function of the range of frequencies
being targeted for optimal scattering. In traditonal means, this is done by using rigidly-backed slotted panels
where each slit acts as a quarter-wavelength resonator (QWR). Therefore, Schröeder diffusers can easily become
thick and heavy structures when designing for mid- to low frequencies of audio sound, e.g. the height or maximum well depth of a traditional Quadratic Residue Diffuser (QRD) is of the order of half the wavelength ofthe low cut-off design frequency; often being in the order of one or more meters.
In this work, we present an experimental validation of the concept of metadiffusers, i.e. deep-subwavelength
acoustic diffusers based on slow sound acoustic metamaterials [3, 4, 5] dramatically reducing the effective
thickness otherwise required by Schröeder diffusers. By tuning the geometry of the HRs and the thicknessof the slits, the phase of the metamaterial’s reflection coefficient can be tailored to that of usual diffusers with,however, extreme deep-subwavelength scales. We present herein the validation of a metadiffuser based on one
presented in a previous work of some of the authors of the current abstract [1].

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Figure 1. (a) Phase and (b) magnitude of the spatially-dependent reflection coefficient generated by six repetitions of a QRD (black line) and the QRM (red doted). (c) Picture the QRM (one period).

2DESIGN OF METADIFFUSERS

A quadratic residue diffuser (QRD) is made from a numerical sequence given by sn = n2 modN, where mod is
the least non-negative remainder of the prime number N. Whereas phase grating diffusers are based on quarter
wavelength resonators (wells) – with well depths given by Ln = sn λ0 /2N, where λ0 is the design wavelength– the metadiffusers’ ability to create impedance changes across their surface is achieved by coupled deepsubwavelength resonance between the slits and the HRs, which takes into account the thermo-visous losses
happening when sound propagates in very narrow regions. By making a parametric design of such strategy,
allowing the geometry of the slits and HRs to change and using genetic algorithms, it is then possible for a
metadiffuser to mimick the spatially-dependent reflection coefficient R(x) of a given surface whilst conserving
a minimal thickness. Thus, let be a classical QRD with N = 5 and total thickness of L = 27.4 cm with side
Nd = 35 cm to be designed for a low cut-off frequency of 500 Hz. The resulting reflection coefficient of theQRD could then be mimicked by a metadiffuser structure, as shown in Figs. 1 (a,b).
As illustrated, perfect agreement is found between the reflection coefficients of the QRM and the target QRD
phase. The picture of the designed metadifuser structure is shown in Fig. 1 (c), with a deep sub-wavelength
thickness of L = 2 cm.

3EXPERIMENTAL SET-UPThe QRM was experimentally tested in the anechoic chamber of the Laboratoire d’Acoustique de l’Universitédu Mans (LAUM). The metasurface was excited by a maximum length sequence (MLS) radiated by a source
placed at 2 m from it at 0◦ incidence (normal). The structure was placed on a turning table covering all
azimutal angles. The scattering pressure field, ps , was evaluated in the polar angles, along a semi-circumferenceof 1 m radius centered in the center of the metadiffuser. Therefore all the scattering directions can be capturedby the experimental set-up. The scattering field was then obtained by Fourier transforming the impulse responseof the system after windowing it in the corresponding time period in order to remove the direct sound from thesource.

4RESULTS

We evaluated a single repetition of the QRM and an equivalent rigid panel of the same dimensions. Figure 2shows the experimental results for the evaluated QRM (red lines) as well as for the rigid panel (blue lines).
In both cases the experimental scattered field, ps (θ ) across the x-axis is plotted for two different frequencies.
The first one at 700 Hz, where the QRM behaves as a rigid panel would, i.e. even spreading of sound dueto the diffraction of the finite surface sample. The second frequency, at 1500 Hz, shows however the evenspreading of the acoustic energy in all the directions produced by the QRM whereas the flat surface acts in amore directive fashion.

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(a)

(b)

Figure 2. Experimental polar plots of the scattered acoustic field, ps (θ ), for (a) 700 Hz and (b) 1500 Hz.
Experimental evaluation of the far field of a flat square rigid panel (blue line) and the QRM (red line) ofdimensions 35 × 35 × 2 cm3 .

5CONCLUSION

The concept of metadiffuser – as a deep subwavelength locally reacting surface with tailored acoustic scattering– was experimentally validated, thus demonstrating the potential of metadiffusers to be used in critical listening
environments. Due to their deep-subwavelength nature, the thickness of the panels can be strongly reduced; by
11 times in the case herein presented. In the context of smart building design and sustainability, metadiffusers
can be used to save space and to produce lightweight materials, improving the performance of acoustic solutions
using less resources. Moreover, the proposed designs have the potential to meet the aesthetic requirements thatare mandatory for modern auditoria design.

ACKNOWLEDGEMENTS
This article is based upon work from COST Action DENORMS CA15125, supported by COST (EuropeanCooperation in Science and Technology). This study was financed by London South Bank University, the Royal
Opera House, Covent Garden, and the United Kingdom Acoustics Network (UKAN). The authors gratefully
acknowledge the ANR-RGC METARoom (ANR-18-CE08-0021) project and the project HYPERMETA funded
under the program Étoiles Montantes of the Région Pays de la Loire. N.J. acknowledges financial support fromGeneralitat Valenciana through grant APOSTD/2017/042.

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Schröder, M. R. Diffuse sound reflection by maximum length sequences. Jour. Acous. Soc. Am., 57, 149-150(1975).

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Groby, J.-P., Huang, W., Lardeau, A. and Aurégan, Y. The use of slow waves to design simple soundabsorbing materials, J. Appl. Phys., 117, 124903 (2015).

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[5]

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