Supplementary Material
Broadband manipulation of acoustic wavefronts by pentamode metasurface
Ye Tian,1 Qi Wei,1,a) Ying Cheng,1 Zheng Xu,2 and Xiaojun Liu1,3,b)
1
Key Laboratory of Modern Acoustics, Department of Physics, Collaborative Innovation
Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
2
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
3
State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences,
Beijing 100190, China
1. Design procedure of the pentamode units
In order to match the background medium (with velocity c0 and density ρ0), the effective
velocity c′ and density ρ′ of the pentamode units should satisfy c   c0 0 . Then the
effective bulk modulus B′ and density ρ′ of the pentamode units can be obtained as
 B  cc0 0
.

    c0 0 c
S(1)
We design the eight pentamode units (denoted as U1 to U8) with the desired 1/c′ ranging from
5/6c0 to 20/3c0 with a step of 5/6c0. According to Eq. S(1), the desired B′ and ρ′ of the eight
pentamode units can be obtained, as listed in Tab. S1. Theoretical analyses have shown that
the effective bulk modulus of the pentamode metamaterial mainly depends on the thickness t
of the framework and the width w of the additional weights, while its effective density
approximates its bulk average density in the long wavelength limitation.1,2 Therefore, the
effective acoustic parameters can be tailored independently.3 A two-step procedure is
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considered to design the eight units based on the finite element method (FEM). Firstly, we
adjust t and w to make the effective bulk modulus equal to its desired value. Secondly, we
adjust h to ensure the effective density equaling to its desired value. Figure S1 gives the
schematic diagram of the eight designed pentamode units, and the geometric parameters are
listed in Tab. S2. The units U1 and U2 are made of aluminum while the other six units are
made of lead.
TAB. S1. Desired acoustic parameters of the eight pentamode units.
Unit
U1
U2
U3
U4
U5
U6
U7
U8
1/c′
5/6c0
5/3c0
5/2c0
10/3c0
25/6c0
5/c0
35/6c0
20/3c0
B′
6ρ0c02/5
3ρ0c02/5
2ρ0c02/5
3ρ0c02/10
6ρ0c02/25
ρ0c02/5
6ρ0c02/35
3ρ0c02/20
ρ′
5ρ0/6
5ρ0/3
5ρ0/2
10ρ0/3
25ρ0/6
5ρ0
35ρ0/6
20ρ0/3
FIG. S1. Schematic diagram of the eight pentamode units.
TAB. S2. Geometric parameters of the eight pentamode units.
Unit
U1
U2
U3
U4
U5
U6
U7
U8
t (mm)
0.540
0.183
0.613
0.441
0.330
0.195
0.137
0.105
w (mm)
1.5
3.0
1.0
1.5
2.0
3.0
3.5
3.8
2
h (mm)
2.95
6.08
2.58
4.18
4.89
5.08
5.47
5.85
2. Band structures of the pentamode units
Based on the FEM, we calculate the dispersion curves of the eight pentamode units, which
are shown in Figs. S2(a)-S2(h). The first and second branches emerging from the Γ point are
the dispersions of the shear and compression waves, respectively. The slopes of the branches
correspond to the phase velocities. The ratios of the effective shear modulus to bulk modulus
are about 0.028, 0.0078, 0.028, 0.018, 0.013, 0.0091, 0.0065 and 0.0050, resulting in the low
coupling between the shear and compression waves for all units. Thus, the units can actually
behave as pentamode materials. The slopes around the Γ point in the ΓJ and ΓX directions are
almost the same, indicating the isotropic property for the pentamode units. Besides, the
second branches are approximately straight, resulting in nearly constant phase velocities for
the compression waves over the entire investigated frequency range, which is desirable for
the construction of the proposed broadband metasurface.
FIG. S2. [(a)-(h)] The dispersion curves of the eight pentamode units in the first Brillouin
zone, along the ΓJX path.
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3. Construction procedure of the pentamode metasurface
As described by Eqs. (3) [ 1 c( x)  (1 lc0 )  sin t ( x)dx ] and (4) [  ( x)  0 c0 c( x) ], the
ideal metasurface has continuous profiles for velocity and density, which can be
approximated by a pentamode metasurface with discrete profiles for velocity and density. The
pentamode metasurface is constructed by using the eight pentamode units under the rule of
minimizing |1 c( x) 1 c | . For example, the section of the ideal metasurface where
5 6c0  5 12c0  1 c( x)  5 6c0  5 12c0 is constructed by the unit U1 since the reciprocal of
the effective velocity ( 1 c ) of U1 is 5/6c0. Figures S3(a)-S3(c) show the design schematics of
the pentamode metasurface for the cases of anomalous refraction, Bessel beam generation
and flat lens, respectively. According to Eq. (2) [ sin t ( x)  lc0 d [1 c( x)] dx ], the refraction
angle is proportional to the thickness l of the metasurface and the gradient of the velocity, but
the designed units provide only finite eight velocities. Hence we employ three layers of
pentamode units ( l  3a ) to enlarge the manipulation range of the refraction angle. The
vertexes of the hexagons of adjacent layers are not connected and the gaps between the
adjacent layers are filled with the background medium. It is noted that the thickness of the
metasurface ( l  3 20 ) is still sub-wavelength and the energy transmittance is greater than
95% over the entire investigated bandwidth. The major challenge for experimental
validations stems from the fabrication of the pentamode metamaterials. The proposed
pentamode units have small geometric size and their effective acoustic parameters are
sensitive to the machining precision. For example, the thickness of the metal framework is
only 0.105 mm for Unit 8, as shown in Fig. S1(a) and Tab. S2. The pentamode units may be
fabricated by the advanced 3D metal printing technology.
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FIG. S3. Design schematics of the pentamode metasurface for the cases of (a) anomalous
refraction, (b) Bessel beam generation and (c) flat lens. The total numbers of the pentamode
units for the three cases are 267, 1053 and 459, respectively.
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4. Contrast between the Bessel beam generation and the flat lens
FIG. S4. Pressure field distributions for [(a), (b)] Bessel beam generation and [(c), (d)] flat
lens under normal incidence at [(a), (c)] ka = π/16 and [(b), (d)] ka = π/10.
Figures S4 and S5 show the pressure field distributions and the cross-section intensity
profiles for the Bessel beam generation and flat lens. For the case of Bessel beam generation,
it is found in Figs. S4(a) and S4(b) that the transmitted waves are focused in the area which is
similar to the theoretical Bessel formation zone (drawn in red lines), and the transmitted
waves resist diffraction in a relatively long distance at both frequencies. Figure S5(a)
indicates a strong intensity for the transmitted beams along the y-axis. Figures S5(b) and S5(c)
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show that the transverse profiles of the intensity fields are all Bessel-like, which almost stay
at different distances. The results demonstrate the well Bessel-beam generation. For the case
of flat lens, it is found in Figs. S4(c) and S4(d) that the transmitted waves are focused on the
focal point. Figures S5(d)-S5(f) show that the intensities are strong at the focal point.
However, the intensities decrease significantly with the locations away from the focal point.
The results confirm the excellent broadband focusing effect. Overall, the transmitted energy
is focused in a diamond area and the transmitted beams resist diffraction for the case of
Bessel beam generation, while the transmitted energy is focused on a point for the case of flat
lens. Therefore, we can clearly distinguish the case of flat lens from that of Bessel beam
generation based on the results in Figs. S4 and S5.
FIG. S5. (a) Intensity profiles along the axial direction (x = 0) for the case of Bessel beam
generation at ka = π/16 (black dashed line) and ka = π/10 (red solid line), respectively.
Intensity profiles along the transverse direction at y = 400a (black solid line), y = 500a (red
dashed line) and y = 600a (blue dot-dashed line) for the case of Bessel beam generation at (b)
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ka = π/16 and (c) ka = π/10, respectively. (d) Intensity profiles along the axial direction (x = 0)
for the case of flat lens at ka = π/16 (black dashed line) and ka = π/10 (red solid line),
respectively. Intensity profiles along the transverse direction at y = 100a (black solid line), y
= 200a (red dashed line) and y = 300a (blue dot-dashed line) for the case of flat lens at (e) ka
= π/16 and (f) ka = π/10, respectively.
5. Effects induced by the losses
FIG. S6. Pressure field distributions of anomalous refraction for (a) loss-free case, (b)
low-loss case and (c) high-loss case at ka ~ π/10, respectively.
Take the anomalous refraction as an example. We use the thermally conducting and
viscous model instead of the ideal loss-free model in the FEM simulation, and the resulting
pressure field distribution is plotted in Fig. S6(b). There is no remarkable difference in
pressure field distributions between Figs. S6(b) and S6(a) (the ideal loss-free case). Since the
working frequency (f = 5505.89 Hz) is low, the thermally conducting and viscous losses
should be very low. We calculate the total transmitted energies and find out that the loss is
smaller than 0.1%. Considering a possible high loss in experiment induced by other factors,
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we also investigate the high-loss case for the metasurface by the FEM simulation. The
resulting pressure field distribution is plotted in Fig. S6(c). It is found that the functionality of
the metasurface remains the same although the transmittance is lower than that in the
loss-free case, since the loss only weakens the amplitude but does not change the phase of the
transmitted waves.
References
1
A. N. Norris and A. J. Nagy, “Metal water: A metamaterial for acoustic cloaking,” in
Proceedings of Phononics, Santa Fe, New Mexico, USA, May 29-June 2 2011, pp. 112–113.
2
A.-C. Hladky-Hennion, J. O. Vasseur, G. Haw, C. Croënne, L. Haumesser, and A. N. Norris,
Appl. Phys. Lett. 102, 144103 (2013).
3
M. Kadic, T. Bückmann, R. Schittny, P. Gumbsch, and M. Wegener, Phys. Rev. Appl. 2,
054007 (2014).
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