Supplementary material of “Implementation of acoustic demultiplexing with
membrane-type metasurface in low frequency range”
Xing Chen, Peng Liu, Zewei Hou, Yongmao Pei
State Key Lab for Turbulence and Complex Systems, College of Engineering, Peking University,
Beijing 100871, China
(1) The physical parameters used for designing the acoustic demultiplexer
The basic unit for designing the acoustic demultiplexer is composed of a massweighted elastic membrane, which is sealed by a back cavity with the depth of 10mm.
The radius R1 and thickness t1 of the membrane are 10mm and 0.3mm, respectively.
The mass, with a radius R2 of 3mm and thickness t2 of 0.8mm, is adhered to the center
of membrane. The elastic membrane is made of polyamide. Young's modulus E1,
density ρ1 and Poisson’s ratio ν1 for the membrane material is 200 MPa, 980 kg/m3 and
0.4, respectively. The central mass is selected as an iron disk, whose Young's modulus
E2, density ρ2 and Poisson’s ratio ν2 is 200 GPa, 7800 kg/m3 and 0.33, respectively.
(2) Reflected properties calculation with the surface Green's function method and
the finite element method validation
The mass-weighted elastic membrane structure can be divided into two parts, i.e.,
the annulus membrane and the central mass. The out-of-plane displacement of the
composite structure is denoted as i (i  1, 2) . The subscript is used to distinguish
different parts of the structure. For instance, the deflection of the annulus membrane is

Corresponding authors at: Department of mechanics, College of Engineering, Peking University,
Beijing, China. Tel.: +86-10-62757417;
E-mail address: [email protected] (Y.M. Pei).
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defined as 1 , while the motion of the mass is expressed by a translational displacement
 2 . Therefore, the free vibration equations of the annulus membrane and the central
mass can be written respectively as:
 1 21  T  21  0
 R22  2 22  F
F  2 R2T
1
r
(S1)
is the restoring force by the membrane. For a mass-weighted
r  R2
elastic membrane, the corresponding eigenmodes, as shown in Fig. S1, can expressed
as:
Wi (r )  Y0 (ki R1 ) J 0 (ki r )  J 0 (ki R1 )Y0 (ki r )
Wi (r )  Y0 (ki R1 ) J 0 (ki R2 )  J 0 (ki R1 )Y0 (ki R2 )
R2  r  R1
0  r  R2
(S2)
Combining Eq. (S2) with Eq. (S1), the eigenvalue equation is obtained:
Y0 (kR1 ) J 0 (kR2 )  J 0 (kR1 )Y0 (kR2 ) 
2 1
Y0 (kR1 ) J1 (kR2 )  J 0 (kR1 )Y1 (kR2 ) 
kR2  2
(S3)
A series of discrete wave numbers k i satisfying Eq. (S3) are related with the
axisymmetric eigenstates. Following the Eq. (1) in the manuscript, the surface-averaged
Green function
Gm
and related effective impedance Z m for the mass-weighted
membrane can be readily calculated. Considering the extra impedance Z c by the back
cavity, the impedance Z1 of single resonant unit satisfies the relationship: Z1 =Z m +Z c .
When a supercell is composed by three units in parallel, the total impedance can be
calculated in analogy to three parallel impedances in an electrical circuit:
1
Z tot
=Z11 +Z 21 +Z 31 . To verify the accuracy of the impedance theory, the reflected phase
of the supercell is studied by both the surface Green's function method and the finite
2
element method. The preliminary tensions are applied to the three units with the value
of 1MPa, 3.5MPa and 10MPa, respectively. The reflected phase with the surface
Green's function method and the finite element method is shown in Fig. S2, and good
agreement can be observed.
Fig. S1. (a) The first (b) The second axisymmetric eigenmodes of the mass-weighted
membrane structure.
Fig. S2. Theoretical and numerical solutions of the reflection phase for the supercell.
(3) Verifying the proposed design method with a preliminary experiment
Here，we conducted a preliminary experiment to verify our design method. A
supercell, composed of three resonant units, was fabricated as shown in Fig. S3(a).
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Three back cavities, whose depths were 10mm, were manufactured by 3D printing. The
elastic membrane, made of polyamide, had the thickness of 0.1mm, was glued on the
back cavity. The radius of each resonant unit was 20mm. The central masses, with the
radius R2=6mm for Unit 1, 5.5mm for Unit 2 and 3mm for Unit 3, were made by
aluminum. The measured reflected phase was plotted in Fig. S3(b), and has good
agreement with the impedance theory. Three phase jumps, corresponding to the hybrid
resonance of the three units, can be clearly observed. It is concluded that the desired
phase delay can be obtained simultaneously for several frequency components by
containing multiple resonant units in parallel. In our future work, we will optimize the
structure to obtain abundant phase delay in low frequency. Besides, the acoustic
demultiplexing phenomena by the membrane-type metasurface will be conducted.
FIG.S3 (a) The photograph of the supercell (b) The measured and theoretical reflected
phase of the supercell.
(4) Acoustic dispersive prism in free space
Here, we will demonstrate that the scheme for acoustic demultiplexing can steer
the wavefront at will. First, an acoustic demultiplexer is developed for realizing
dispersive prism in free space. As each discrete phase shift of /4 is covered by two
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identical supercells, the phase gradient of d/dx=/8p is obtained in 400Hz, 600Hz and
800Hz. Fig. S4 show the reflected pressure patterns for normal incidence. It is clearly
seen that the reflected angles are varied for different frequency components. Deduced
from Eq. (3) in the manuscript, the reflected angle can be calculated by
r  arc sin  c0 16 pf  . The mechanism of the dispersive direction is attributed to the
frequency-dependent wave path difference. Besides, the acoustic demultiplexer
possesses high selectivity. As shown in Fig. S4(d), the incident wave returns back in
500Hz. It is understood that the phase shifts are vanished out of the design frequencies
due to the sharp resonant properties. Thus, the metasurface behaves as a rigid surface.
Fig. S4. The reflected pressure patterns in (a) 400Hz (b) 600Hz (c) 800Hz (d) 500Hz.
The frequency-dependent reflected angle is observed in the design frequencies. Besides,
the acoustic demultiplexer possesses high selectivity, and acts as a rigid surface out of
the design frequencies.
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(5) Acoustic demultiplexer for planar focusing
Then, the idea of metasurface-based demultiplexer is expanded to realize novel
acoustic propagation mode. The second example is aiming at obtaining varied focal
spots in different frequencies. According to the geometrical relationship, the
compensated phase along the metasurface should satisfy the Eq. (5) in the manuscript.
In this case, the focal point is located at (0, 40) in 400Hz, where 0 is the reference
wavelength of 400Hz. As the frequency increases 200Hz, the focal length decreases
one reference wavelength in proportion. By combing the discrete states of the basic
units appropriately, the desired phase profiles are available for multiple frequencies. In
Fig. S5(a)~(c), the reflected pressure amplitudes are plotted in 400Hz, 600Hz and
800Hz. The focal spots are spatially separated. The normalized intensity of pressure
along the acoustic axis is shown in Fig. S5(d), which provides clear confirmation of the
acoustic focusing effect. The slight deviation between calculated focal length and
design value is observed due to the discrete phase profile.
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Fig. S5. The reflected pressure amplitude of acoustic focusing in (a) 400Hz (b) 600Hz
(c) 800Hz. (d) The reflected pressure intensity along the central axis is plotted for
different frequency.
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