Beyond phase grating diffusers
using locally-resonant metamaterials
Noé Jiménez(1) , Trevor J. Cox(2) , Jean-Philippe Groby(3) , Vicente Romero-García(3)(1) Instituto de Instrumentación para Imagen Molecular, CSIC-UPV, Valéncia, Spain, nojigon@upv.es(2) Acoustics Research Centre, University of Salford, Salford, UK(3) Laboratoire d’Acoustique de l’Université du Maine - CNRS UMR 6613, Le Mans, France

Abstract
We present sound diffusers based on locally resonant metamaterials that are one order magnitude thinner than theclassical designs based on phase gratings. Using a set of waveguides loaded by an array of optimized Helmholtz
resonators, we induce strong dispersive propagation inside the material. Doing this, waves in the low-frequency
regime are dramatically slowed down and, because of this, a deep-subwavelength resonance is produced. In
addition, this design allows an accurate tuning of the thermoviscous loses. In this way, the reflection coefficient
can be spatially tuned in both phase and magnitude, and we are able to mimic the classical QRD, PRD or
Binary/Ternary sound diffusers. Beyond these designs, we also propose an optimized panel showing efficientand broadband sound diffusion ranging from 250 Hz to 2 kHz using only 3 cm thickness, i.e., 46 times thinnerthan the wavelength at the lowest working frequency.

Keywords: Sound diffusers, Metamaterials, Metasurfaces, Metadiffusers.

1

INTRODUCTION

Sound diffusers are often locally-reacting reflecting surfaces with spatially dependent reflection coefficient usually designed to produce a uniform scattering pattern, i.e., the reflected waves by these surfaces are dispersed
in a broad range of directions [1]. To produce such a uniform far field scattering distribution, the spatiallydependent reflection coefficient of the panel should present a Fourier transform of uniform magnitude. This wastraditionally achieved by using phase grating diffusers, also known as Schröeder’s diffusers [2]. These surfaces
are rigid-backed slotted panels where each slit acts as a quarter wavelength resonator. However, this approach
results in very thick and heavy panels, limiting the use of phase grating diffusers at low-frequencies where thewavelength of sound in air is of the order of several meters [1].
To overcome this limitation, we present metadiffusers, sound diffusers based on metamaterials [3]. These structures are composed by optimized air cavities producing slow-sound conditions when waves propagate aroundtheir resonances. Metamaterials based on slow sound condition have been widely used to design acoustic absorbers [4, 5, 3, 6]. Using slow sound results in a decrease of the cavity resonance frequency and, hence, thestructure thickness can be drastically reduced to the deep-subwavelength regime [5]. Therefore, the thickness ofthe panel can be reduced to the deep subwavelength regime.
Here, we present deep-subwavelength thickness diffusers based on acoustic metamaterials to reduce the thicknessof Schröeder diffusers. The system works as follows: first, we consider a rigid panel of finite length with a setof N slits. Second, we modify the dispersion relations inside each slit by loading one of their walls with a setof Helmholtz resonators, as shown in Fig. 1 (b). The sound propagation becomes strongly dispersive in each slit
and the resulting sound speed, c p , is drastically reduced. Therefore, each slit behaves as a deep-subwavelength
resonator. As a consequence, the effective depth of the slits can be strongly reduced as L = c p /4 f holds. By
tuning the geometry of the Helmholtz resonators and the thickness of the slits, the dispersion relations insideeach slit can be modified. As a result, the phase of the reflection coefficients can be tailored to those of an
Schröeder phase grating diffuser, or can be optimized to obtain broadband sound diffusers.

2270

(a) Phase grating diffuser(b) Metadiffusere{jk x
0

pi(Á)

pi(Á)

ps(µ)

ps(µ)

(c) Helmholtz Resonator

{jknx

e

wc
{jknx

e{jk x enwn
ah ln lc

Ln
x2

n=N
x1

d

n=N
n=1

Figure 1. (a) Scheme of a QRD Schröeder diffuser composed by N = 7 slits or quarter wavelength resonators. (b) Metadiffusercomposed of N = 7 sub-wavelength slits, each of them loaded by M = 3 Helmholtz resonators, with slightly different geometry.

2

METHODS

2.1 Transfer matrix method
The system described before has been theoretically modelled by using the transfer matrix method. Under theassumption of plane waves travelling inside the metamaterial, either the transfer matrix or the scattering matrix
can be obtained, providing directly the reflection of the metamaterial, as well as its effective parameters.
The transfer matrix is used to relate the sound pressures (P) and normal acoustic particle velocities (V ) at the
beginning and at the end of each slit. The transfer matrix of the n-th slit, Tn , of length L, extending from y = 0
to y = L is written as
 n 
 n 
 n
 n 
n
P
P
T11 T12
P
n
.
(1)
=
T
=
n
n
T22
T21
V n y=LV n y=0V n y=L
For an identical set of M resonators, the transmission matrix Tn is written as nn
T11 T12= Mn∆lslit (Mns MnHR Mns )M .
Tn =
n
n
T22
T21
Here, the transmission matrix for each lattice step in the n-th slit, Mns , is written as a
 a 
nn
iZs sin ksn
 cos ks 2n2 ,
Ms = 
(2)
 a
 a 

i
n
n
sin
k
cos
k
s
s
Zsn
2
2
√
where thepslit characteristic impedance is written as Zsn = κsn ρsn /Ssn and Ssn = hn a, the complex wavenumber
is ksn = ω ρsn /κsn and κsn and ρsn are the complex and frequency dependent bulk modulus and density in eachslit, i.e., the effective acosutic parameters. The resonators are introduced as punctual scatters by a transmission
matrix MnHR as


1
0
,
(3)
MnHR =n1
1/ZHR
and the radiation correction of the n-th slit to the free space as


n
1 Z∆l
slit
Mn∆lslit =
,
0
1

(4)

2271

n
n ρ /φ n S , where S = d a, ρ the air
with the characteristic radiation impedance of the n-th slit Z∆l= −iω∆lslit
0 t 0
0
0
slit
n
density and ∆lslit the proper end correction that will be described later.
Finally, the reflection coefficient of the rigidly backed slit can be directly calculated from the elements of thematrix Tn as

Rn =n −Z Tn
T11
0 21
n
n .
T11 + Z0 T21(5)with Z0 = ρ0 c0 /S0 , and finally the absorption as α n = 1 − |Rn |2 . The effective parameters of each slit can beobtained from the transfer matrix elements as follows
s
 n
n 
n
T12
T
+
T
1
n
n
22
, Zeff=(6)
keff= cos−1 11
n .
L
2
T21
2.2 Visco-thermal losses model
The visco-thermal losses in the system are considered both in the Helmholtz resonators and in the slits by using
its effective complex and frequency dependent parameters. Considering only plane waves propagate inside themetamaterial, the effective parameters of the ducts that conform 2D resonators and the slits of width 2r are
given by [7]:
"
 #−1


tanh rGρtanh (rGκ ) −1
ρeff = ρ0 1 −, κeff = κ0 1 + (γ − 1)
,
(7)
rGρ
rGκ
p
p
with Gρ = iωρ0 /η and Gκ = iωPrρ0 /η, and where γ is the specific heat ratio of air, P0 is the atmosphericpressure, Pr is the Prandtl number, η the dynamic viscosity, ρ0 the air density and κ0 = γP0 the air bulk
modulus. The effective parameters of the n-th main slit, ρsn and κsn , are obtained by setting r = hn /2 in Eqs. (77). The visco-thermal losses inside the 2-dimensional resonator’s neck and cavity are modelled in the same way
by these effective parameters, ρnn,m , κnn,m and ρcn,m , κcn,m respectively, by setting r = wnn,m /2 and r = wcn,m /2 forthe m-th resonator located at the n-th slit.
2.3 Resonator impedance and end corrections
Using the effective parameters for the neck and cavity elements given by Eqs. (7-7), the impedance of a Helmholtz resonator, including a length correction due to the radiation can be written as [8]:
n,m
ZHR
= −i

cos(kn ln ) cos(kc lc ) − Zn kn ∆l cos(kn ln ) sin(kc lc )/Zc − Zn sin(kn ln ) sin(kc lc )/Zc,sin(kn ln ) cos(kc lc )/Zn − kn ∆l sin(kn ln ) sin(kc lc )/Zc + cos(kn ln ) sin(kc lc )/Zc(8)

where (note superscripts were omitted for the sake of simplicity) lnn,m and lcn,m are the neck and cavity lengths,
knn,m and kcn,m , are the effective wavenumbers and and Znn,m and Zcn,m effective characteristic impedance in theneck and cavities respectively, and ∆l n,m the correction length for the Helmholtz resonators. These correction
lengths are deduced from the addition of two correction lengths ∆l n,m = ∆l1n,m + ∆l2n,m as" n,m 3 #
wn,m
wn
n
n,m
wnn,m ,
(9)∆l1
= 0.41 1 − 1.35 n,m + 0.31
wc
wcn,m
"
 n,m 3
 n,m 4 #
 n,m 2
wn,m
wn
wn
wn
n
n,m
∆l2
= 0.41 1 − 0.235 n − 1.32
+ 1.54− 0.86
wnn,m .
(10)
ws
wns
wns
wns
The first length correction, ∆l1n,m , is due to pressure radiation at the discontinuity from the neck duct to thecavity of the Helmholtz resonator [9], while the second ∆l2n,m comes from the radiation at the discontinuity

2272

from the neck to the principal waveguide [10]. This correction only depends on the radius of the waveguides,
so it becomes important when the duct length is comparable to the radius, i.e., for small neck lengths and for
frequencies where knn,m wn,mn  1.
Another important end correction comes from the radiation from the slits to the free air. The radiation correction
for a periodic distribution of slits can be expressed as [11]:sin2 (nπσ n ).
n 3
n=1 (nπσ )∞n∆lslit= hn σ n ∑(11)√n ≈ − 2 ln [sin (πσ n /2)] /π.
with σ n = hn /d. Note for 0.1 ≤ σ n ≤ 0.7 this expression reduces to ∆lslit
2.4 Diffusion coefficient
The diffusion coefficient, dφ , is estimated from a polar response as [12]
2

π
R

Is (θ )dθ

−π

δφ =

−

Rπ

Is (θ )2 dθ

−π

Rπ

,

(12)

Is (θ )2 dθ−πwhere Is (θ ) is the polar scattering intensity for a wave with incident angle φ . This coefficient is normalized tothat of a plane reflector, δflat , to eliminate the effect of the finite size of the structure as δn = (δφ − δflat )/(1 −δflat ).

3

REDUCING THE THICKNESS OF A DIFFUSER USING METAMATERIALS

0.5

slit1
slit2
k0 a/π

f1

(b)

f2

0

(c)
600
c0

400
200

-π

0
0

1

2
f (kHz)34π

arg R (rad)1c˜p (m/s)(a)Re(k)a/π

Figure 2 (a) shows the dispersion relations inside two different slits, n = 1 and n = 2, obtained by using M = 2
HR with the geometrical parameters listed in Table 1 of Ref. [13]. First, above the resonance frequency of theHRs, fn , a band gap is generated. Below the resonance frequency of the HRs a dispersive band is observed
and the wavenumber is increased with respect to the wavenumber in air. In this regime, slow sound conditions
are produced, as shown in Fig. 2 (b), i.e., the phase speed inside the slits is strongly reduced. The phase of
the reflection coefficient produced by each slit is shown in Fig. 2 (c). We can see that for some frequencies thephase of the reflection coefficient of both slits (blue and red lines) is strongly modified when compared to thereflection phase of a slit without HRs (dashed line). At 2 kHz, the 1st slit (red curve) reacts inverting the phaseof the incoming wave, while for the 2nd slit (blue curve) this occurs at 3.2 kHz. In this way, by tuning thegeometry a specific phase profile can be tailored, while the total thickness of the panel can be greatly reduced
when compared with a quarter wavelength resonator of length L. By using these features, we show in this0

1

2
f (kHz)

3

4

0

1

2
f (kHz)3

4

Figure 2. (a) Dispersion relation inside the (blue) first and (red) second slits of a metadiffuser for the losslesscase (continuous lines) and accounting for the thermo-viscous losses (dashed lines), and wavenumber in air(dashed-dotted). The resonance frequencies of the HR are shown as f1 and f2 . (b) Corresponding phase speed.
(c) Phase of the reflection coefficient for each individual slit.

2273

Meta-QRD(a1)

Meta-PRD
(b1)

2 cm

3.5 cm

35 cm

42 cm

(a2)

(b2)

π/2

π/2

arg R

π

arg R

π

0

0
-π/2

-π/2

-π

-π
-1

-0.5

0
x (m)

0.5

-0.2

1

(a3)

-0.1

0
x (m)

0.1

0.2

(b3)
|p |

2

1

1
1.2

0.5

1.5

0.5

y (m)

1

0

0

0.8

y (m)

1
0.5

0.6
0.4

metadiffuser

0.2

0
-2

-1

0

1

0

2

metadiffuser

x (m)

-1

(a4)

|p|sp||

2

0

0.5

1

1.2

1

1

0.5

0.8

0

0.6

1
0.5

y (m)

y (m)

3

-0.5

x (m)

(b4)

1

0

0.4
0.2

QRD

0

PRD

0
-0.2

-3

-2

-1

0

1

2

3

x (m)

(a5)
-20
-30

-10

0

10

-1

0

30
40

-50

-20
-30

-10

0

10

Meta (FEM)
Meta (TMM)

20
30
40

-50

50

-60

60

-70

70

-70

-80

80

-80
-90

-30 -24 -18 -12

1

-40

50

-60

0.5

x (m)

Flat panel
ScröederMeta (FEM)
Meta (TMM)

20

-40

-90

-0.5

(b5)

-6 0 dBps(µ)

60-30 -24 -18 -127080-6 0 dBps(µ)

Figure 3. (a) Phase and (b) magnitude of the spatially-dependent reflection coefficient of a QRD (black line) and the QRmetadiffuser (red doted). (c) Scaled scheme of the QR-metadiffuser with N = 5 and M = 2. (d) Near field pressure distribution at
2 kHz of QR-metadiffuser with thickness L = 2 cm (e) phase grating QRD of thickness L = 27.4 cm and (f) flat plane reflector.
(g) Far-field polar distribution of the QR-metadiffuser obtained by TMM (continuous blue) and FEM (dotted black), the reference
QRD (dashed-grey), and a plane reflector with same width of the diffusers (continuous red).

2274

article that the phase profile of Schroeder and ternary sequence diffusers can be mimicked by a sub-wavelength
metadiffuser in a given frequency band. Therefore by tuning the geometry of a metadiffuser we can maximize
sound diffusion in a broad frequency band for room acoustics applications using a deep sub-wavelength panel.

4

QUADRATIC RESIDUE METADIFFUSERS

The quadratic residue sequence is given by sn = n2 modN, where mod is the least non-negative remainder ofthe prime number N. If the phase grating diffuser is based on quarter wavelength resonators, the depth of
the slits is given by Ln = sn λ0 /2N, where λ0 is the design wavelength. Here, we use optimization methods
to tune the geometry of the metamaterial for its spatially-dependent reflection coefficient to match the one ofregular QRD at single frequency. The resulting geometry is shown in Fig. 3 (a1). Figure 3 (a2) shows thephase of the reflection coefficient along the surface for a N = 5 QRD with a design frequency of 500 Hz
and a total thickness of L = 27.4 cm, and a QR-metadiffuser of L = 2 cm thickness and M = 2 Helmholtz
resonators of same dimensions, calculated using the transfer matrix method (TMM). Perfect agreement is foundbetween the reflection coefficients of the QR-metadiffuser and the targeted phase grating QRD. The near fieldpressure distributions are shown in Figs. 3 (a3-a4) for the QR-metadiffuser and a QRD. Excellent agreement
is observed between both diffusers, where it is clear how the field is scattered in other directions rather than
specular one. Figure 3 (a5) shows the far-field calculation for both structures considering 6 repetitions of theunit cell in order to clearly generate the characteristic set of diffraction grating lobes of the QRD in the far-field.
Excellent agreement is obtained with the polar response using the TMM and a full-wave numerical solution
using the finite element method (FEM) accounting for the thermo-viscous losses. Notice that the presented
QR-metadiffuser is 17.1 times thinner than the usual QRD (34 times smaller than the QRD design wavelength).

5

PRIMITIVE ROOT METADIFFUSERS

The second numerical sequence presented here is the primitive root sequence, given by sn = rn modP, where P
is a prime number and r is the primitive root of P. The primitive root sequence have N = P − 1 different values.
A primitive root diffuser (PRD) is constructed using a set of N wells with depths Ln = sn λ0 /2N. The scatteredfield by these diffusers presents a notch at specular directions at multiples of the design frequency [14]. The
obtained PR-metadiffuser design is shown in Figure 3 (b1) and the corresponding geometrical parameters of the
PR-metadiffuser are listed in Table 2 of Ref.[13]. Figure 3 (b2) show both the phase of the spatially-dependent
reflection coefficient of a P = 7 phase grating PRD of thickness L = 17.1 cm with d = 7 cm, and for a PRmetadiffuser of L = 3.5 cm and M = 1 HR with the same lateral dimensions. Excellent agreement is foundbetween both responses. Figures 3 (b3-b4) show the near field corresponding to the PR-metadiffuser and areference PRD. The characteristic notch is observed at normal reflection angle, i.e., θ = 0. Note, because thestructures are not symmetric, neither is the field. The far-field is presented in Fig. 3 (b5), where good agreement
is found between the theory and the full-wave numerical solutions. Both panels produce the same scattering, but
the thickness of the PR-metadiffuser is around 10 times thinner than the phase grating PRD (20 times smaller
than the design wavelength).

6

BROADBAND OPTIMAL METADIFFUSERS

To obtain a metadiffuser useful for room acoustic applications, its diffusion must be broadened in frequency.
Thus, we extend the bandwidth of the optimization procedure. In particular, we look for deep-subwavelength
thickness metadiffusers presenting maximum normalized diffusion coefficient in the frequency range from 250
Hz to 2 kHz. A set of N = 11 slits separated by d = 12 cm was used, and the thickness of the panel wasconstrained to L = 3 cm. Figure 4 (a) shows the frequency dependent diffusion coefficient calculated according
to ISO 17497-2:2012 normalized to a flat reflector of same dimensions for a thick QRD with a design frequencyof 250 Hz (LQRD = 56 cm), a thin QRD with the same thickness of the metadiffuser LQRD,thin = 3 cm, and the2275optimized metadiffuser. Over the optimized frequency range, the diffusion coefficient of the metadiffuser takes
a mean value of δn ≈ 0.65, with peaks of δn = 0.91. When compared to the thick QRD, its frequency band isextended to one octave below while the metadiffuser thickness is 46 times smaller than the wavelength.
(a)
(b)

-30
-40
-50
-60
-70

3 cm
1.32 moptimized
optimizeopti
mized
d range
ra

-30
-40
-50
-60
-70

40
50
60
70

-80

80

-80

80

-30 -24 -18 -12 -6 0 dB

-90

-30 -24 -18 -12 -6 0 dB(e) QRD [L = 3 cm ]

|ps |
0

f (Hz)

δn
0.2

1

|ps |

1000
500
250

500

1000

2000

4000

f (Hz)-50(f)

QRD [L = 3 cm]
QRD [L = 56 cm]
Meta [L = 3 cm]

(i)

0.4

1000
500

500

1000
f (Hz)

2000

4000

1

-3
0
1

)m ( y

250

2
3

-2

metadiffuser

500

-50

(g)

|ps |

250

0.2

1000

50

0

2000

f (Hz)α0.60θ (deg)QRD [L = 56 cm ]

1

1

250

1

0
θ (deg)50|ps |Metadiffuser
02000

f (Hz)

250

0

2000

f (Hz)

2000

0.4

0
125

(c)

Flat plane

0.6

0.8

50
60
70

r=1m
r=5m
r→∞
flat planef = 2000 Hz
0 10-20 -10

-90

(h)

0
125

40

(d)

1
0.8

r=1m
r=5m
r→∞
flat planef = 300 Hz
0 10-20 -10

1

1000
500
250

-50

0
θ (deg)50-50

0θ (deg)50

Figure 4. (a) Normalized diffusion coefficient of a 3 cm QRD (dashed black), 56 cm QRD (dashed-dotted red) and optimized
metadiffuser using TMM (blue) integrated in third of octaves. The third octave integration is shown in thick lines according to ISO
17497-2:2012.

7

CONCLUSIONS

We have presented a novel concept of sound diffuser called metadiffusers. These new structures are based onmetamaterials comprising slotted panel, with slits loaded by a set of Helmholtz resonators. The propagation
inside each slit presents strong dispersion and the sound speed can be significantly reduced so that each slit
effectively behaves as a deep-subwavelength resonator. Different designs were presented based on numbertheoretical sequences as quadratic residue and primary root sequences. It was shown that the structures can be
optimized to work in a broadband frequency range covering 3 octaves. In particular, we presented a 3 cm thickdiffuser efficient from 250 to 2000 Hz, demonstrating the potential of the metadiffusers to be used in criticallistening environments due to their deep-subwavelength nature: the thickness of the panel was 1/46 times the
wavelength corresponding to the lowest working frequency, i.e., about a twenty times thinner than the traditional
designs.
Finally, it should be remarked that the performance of the panel drops off at high frequency, which is notdesirable for an acoustic diffuser high frequency diffusion prevents perceptual aberrations. This will be achievedin future studies by modifying the metamaterial structure.

2276

ACKNOWLEDGEMENTS
This work was supported by Generalitat Valenciana through grant APOSTD/2017/042. The authors acknowledge
financial support from COST Action DENORMS - CA15125, supported by COST (European Cooperation in
Science and Technology).

REFERENCES[1] Trevor J Cox and Peter D’Antonio. Acoustic absorbers and diffusers: theory, design and application. CRC
Press, 3rd edition, 2016.
[2] Manfred R Schröder. Diffuse sound reflection by maximum- length sequences. The Journal of the AcousticalSociety of America, 57(1):149–150, 1975.
[3] Noé Jiménez, Vicent Romero-García, Vincent Pagneux, and Jean-Philippe Groby. Quasiperfect absorption
by subwavelength acoustic panels in transmission using accumulation of resonances due to slow sound.
Phys. Rev. B, 95:014205, 2017.
[4] J-P Groby, R Pommier, and Y Aurégan. Use of slow sound to design perfect and broadband passive soundabsorbing materials. J. Acoust. Soc. Am., 139(4):1660–1671, 2016.
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Geneva, Switzerland, 2012.
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Acoustics, 60(2):167–186, 2000.

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