Supplementary Information on “Three-dimensional Ultrathin
Planar Acoustic Lenses by Metamaterials” by
Yong Li, Gaokun Yu, Bin Liang, Xinye Zou, Guangyun Li, Su
Cheng, and Jianchun Cheng
I. Derivation of the complex transmission and reflection coefficient for
the hybrid labyrinthine structure
Figure S1 illustrates the considered hybrid labyrinthine structure, which consists
of three parts: Part I, the extra air channel inner a labyrinthine unit; Part II, the
labyrinthine unit with varied cross-section air channels labeled A-G, and Part III, an
extra air channel outside of the labyrinthine unit. Let ( pi ,ui ) , ( pr , ur ) and ( pt , ut )
denote the sound pressure and the axial components of the velocities, respectively, for
the
incident,
the
reflected
and
the
transmitted
fields,
and
( pL , uL ), ( L  A ~ G and I II III ) for the fields inside the channels and the inlets and
outlets of the three parts. The velocity components u are associated with the
corresponding sound pressure
p
via ik 0c0u  p / z with 0 c0 being the
characteristic acoustic impedance of air and k  2 /  being the wavenumber
respectively. Without loss of generality, a unit incident field, pi  eikz with i  1 is
assumed. It is noteworthy that the harmonic factor e  i t , with   kc0 , is omitted as
understood.
In the following derivations, first we derive the transfer matrix to connect the
fields between the inlets and outlets of the three parts. Then the Green function theory
is utilized to express fields of the incident, reflected and transmitted waves. Finally,
1
considering the continuity conditions of averaged pressure and volume velocity, the
complex transmission and reflection coefficient can be obtained from the transfer
matrix and the Green function theory.
Figure S1. The schematic diagram of the structure for the analytical derivations.
A: Transfer Matrix for the air channels
In the following derivation, we let r1 = (s - a) / 2 + w , r2 = (s + a) / 2 - w . Then
the field ( pL , uL ) of the regions A (viz. 0  z  w and r1  r  r1  d ), B (viz.
w  z  w  d and r1  r  r2 ), and C (viz. w  d  z  2w  d and r2  d  r  r2 ) can
be expressed as1
pA (r,z) = A+ eikz + A- e-ik( z-w) ,
u A (r , z ) 
1
 A eikz  A e  ik ( z  w )  ,
0 c0 
(S1)
(S2)
2

pB (r , z )   n (r )  Bn eikzn ( z  w)  Bn e ik zn ( z  w d ) ,
(S3)
n 0

uB (r , z )  n (r )
n 0
k zn
 Bn eikzn ( z  w)  Bneikzn ( z  wd ) ,
k 0c0
(S4)
pC (r , z )  C  eik ( z  wd )  C  e ik ( z  2 wd ) ,
uC (r , z ) 
(S5)
1
C  eik ( z  w d )  C  e  ik ( z  2 w d )  ,
0 c0
(S6)
where n (r ) is the transverse eigenmode
n (r )   kn r22  r12
J1 (kn r1 )Y0 (kn r )  Y1 (kn r1 )J 0 (kn r )
 2 kn2 r22  J1 (kn r1 )Y0 (kn r2 )  Y1 (kn r1 )J 0 (kn r2 )   4
2
(
,
)
with the wavenumber kn n = 0,1, 2, .... being the solution of
J1 (kn r1 )Y1 (kn r2 )  Y1 (kn r1 )J1 (kn r2 )=0 .
r2
The n (r ) satisfies the orthogonally relation 2  n (r )m (r )rdr   (r22  r12 ) mn , and
r1
the wavenumber along z direction can be expressed as kzn  k 2  kn2 . The symbols
A , A , C  and C  denote the coefficients in the thin channel, and the Bn  and Bn 
denote the coefficients of the n-th mode with the normal wavenumber kn . Note that
the higher modes are neglected in the region A because the breadth of region A is
quite small as compared with wavelengths.
Considering that the continuity of pressure should be satisfied on the boundary
between the region A and region B at z  w , we obtain


A eikw  A   Bn  Bneikzn d
n 0
 1S 
1
r1  d
r1
2n (r )rdr
(S7)
,
where S1   (r1  d ) 2   r12 . Also the pressure continuity on the boundary z  w  d
between region B and region C yields,
3

 B e
 ik zn d
n
n 0
 Bn
 1S 
2
r2
r2  d
2n (r )rdr  C   C eikw
(S8)
,
where S2   r22   (r2  d ) 2 . Since the velocity continuity should also be satisfied at
these boundaries, we obtain
(
)
(
(
)
(
kzn +
DS
1 r1+d
ik d
Bn - Bn- e zn = 1 A+ eikw - A2pfn (r)r dr
k
DS
DS1 òr1
,
)
(S8)
and
kzn + kznd
DS
1 r2 -d
Bn e - Bn- = 2 C + - C - eikw
2pfn (r)r dr
k
DS
DS2 òr2
,
)
(S9)
2
2
where DS = p r2 - p r1 , and the orthogonality of n (r ) is utilized to obtain the Eqs.
(S9) and (S10). From Eqs. (S7~S10), a transfer matrix can be constructed to connect
the coefficients between region A and region C, which can be expressed as
C  
 A 
S1

M
 
1  
S2
C 
A 
(S10)
where
é b 2 - (g - 1)(a - 1)
ê
eikw
2b
ê
M1 = ê
b 2 - (g + 1)(a - 1)
ê
2b
êë
b 2 - (g - 1)(a + 1)
2b
b 2 - (g + 1)(a + 1) - ikw
e
2b
ù
ú
ú
ú,
ú
úû
(S11)
with
k S1
1  ei 2 k zn d
 n n
ik zn d
)(1  eik zn d )
n  0 k zn S (1  e

 
k S 2
1  ei 2 k zn d
 n n
ik zn d
)(1  eik zn d )
n  0 k zn S (1  e

 
(S12)
k S1S 2
2eik zn d
 
nn
k zn S (1  eik zn d )(1  eik zn d )
n 0

where  n 
1
S2

r2  d
r2
2n (r )rdr ,  n 
1 r1  d
2n ( r ) rdr .
S1 r1
4
Following the above process, the transfer matrix for the connection between the
coefficients of region C and region E can also be obtained
é E+
ê êë E
ù
é C+
DS2
=
M
ú
ê
DS1 2 êë C úû
ù
ú,
úû
(S13)
with
  2  (  1)(  1) ikw
e

2

M2  
  2  (  1)(  1)

2

 2  (  1)(  1)
2


.
2
  (  1)(  1) ikw 

e 
2


(S14)
According to Eqs. (S11) and (S14), we can establish the whole transfer matrix for the
labyrinthine units
G  
 A 
S1

M
M
M
 
1
2
1  .
S2
G 
A 
(S15)
At the inlet and outlet of the labyrinthine units (viz. Part II), one can readily arrive at
 A  1  1
     ikw
 A  2 e
pII |z 0
1 

1
 ikw  
e   0c0 (S1 ) U II |z 0 
(S16)
and
pII |z t

 eikw
  c (S )1U |    ikw
2
II z t 
 0 0
e
1  G  
 
1 G  
(S17)
where
pII |z 0 
2 r1  d
p(r , z ) |z 0 rdr
S1 r1
U II |z 0  2 
r1  d
r1
2
pII |z t 
S2

r2  d
r2
U II |z t  2 
r2  d
r2
uI (r , z ) |z 0 rdr
(S18)
p(r , z ) |z t rdr
uI (r , z ) |z t rdr
5
According to Eqs. (S16-S18), we can obtain the field relationship between inlet and
the outlet of the labyrinthine unit (Part II) as
S1
S1
m11 pII |z 0 
m12 0c0 (S1 ) 1U II |z 0
S 2
S 2
pII |z t 
S1
S1
0 c0 (S 2 ) U II |z t 
m21 pII |z 0 
m22 0 c0 (S1 ) 1U II |z 0
S 2
S 2
(S19)
1
with
 m11 m12  1 S1 eikw
 ikw
m

 21 m22  2 S2 e
1
 1
 M1M 2 M1  ikw
1
e
1 
.
eikw 
(S20)
For the air channels in Part I and Part III, it is readily to obtain the similar
equation to connect the fields of inlet and outlet, which can expressed as
pI |z=t = n11 pI |z=0 +n12 r0c0 S I-1U I |z=0
r0c0 S I-1U I |z=t = n21 pI |z=0 +n22 r0c0 S I-1U I |z=0
pIII |z=t = n11 pIII |z=0 +n12 r0c0 S III-1U III |z=0 ,
r0c0 S III-1U III |z=t = n21 pIII |z=0 +n22 r0c0 S III-1U III |z=0
(S21)
(S22)
where
2p
pI |z=0 =
SI
s-a
2
ò
0
U I |z=0 = 2p ò
s-a
2
0
pIII |z=0 =
2p
S III
ò
2p
pI (r, z) |z=0 r dr ; pI |z=t =
SI
ò
s-a
2
0
uI (r, z) |z=0 r dr ; U I |z=t = 2p ò
s-a
2
0
s
s+a
2
pI (r, z) |z=0 r dr ; pIII |z=t =
s
2p
S III
ò
pI (r, z) |z=t r dr ,
uI (r, z) |z=t r dr ,
s
s+a
2
pI (r, z) |z=t r dr ,
s
U III |z=0 = 2p òs+a uI (r, z) |z=0 r dr ; U I |z=t = 2p ò s+a uI (r, z) |z=t r dr ,
2
(S23)
2
2
é
æ s - aö
æ s + aö ù
2
ú,
SI = p ç
, S III = p ê s - ç
è 2 ÷ø
è 2 ÷ø ú
êë
û
é n
n ù é cos(kt) sin(kt) ù
ê 11 12 ú = ê
ú.
ê n21 n22 ú êë sin(kt) cos(kt) úû
ë
û
2
B. Green function theory
6
The duct Green’s function is govern by 2
 2 G ( r , z )  k 2G ( r , z )  
 (r  r0 ) ( z  z0 )
2 r
,
(S24)
and the solution of Eq. (25) is written as

n (r0 )n (r )
n 0
2i R 2 k 2  kn2
G(r , z, r0 , z0 )  
e
i k 2  kn2 z  z0
e
i k 2  kn2 z  z0

(S25)
where the transverse eigenmode j n (r) = J 0 (kn r) / J 0 (kn s) , with the wavenumber
kn , (n  0,1, 2, ...) , the solution of J1 (kn s) = 0 , satisfies the orthogonally relation
s
2p ò j n (r)j m (r)r dr = p s 2d mn .
0
Using the Green function theory, the pressure field for z  0 is
p(r, z) = pi + pr = e ikz + e -ikz + 2p ò
r1 -w
0
G(r, z,r0 , z0 )
¶p(r0 , z0 )
r dr
¶z0 z =0 0 0
0
+2p ò
r1 +d
r1
s
¶p(r0 , z0 )
¶p(r0 , z0 )
G(r, z,r0 , z0 )
r0 dr0 + 2p ò G(r, z,r0 , z0 )
r0 dr0
r
+w
2
¶z0 z =0
¶z0
z =0
0
(S26)
0
By the equation of motion and the assumption the velocity distribution are uniform at
the inlet, the Eq. (S26) can be simplified as
p(r, z) |z=0 = 2 + ik
r -w
r0c0
U I |z=0 2p ò G(r,0,r0 ,0)r0 dr0
0
SI
1
r +d
s
rc
rc
+ik 0 0 U II |z=0 2p ò G(r,0,r0 ,0)r0 dr0 + ik 0 0 U III |z=0 2p ò G(r,0,r0 ,0)r0 dr0
r
r +w
DS1
S III
(S27)
1
1
2
Then the averaged pressure field in the I, II and III region can be expressed as
pI | z  0  2 
0 c0
pII |z 0  2 
pIII |z 0  2 
SI
U I |z 0 g11 
0c0
S I S1
0 c0
S I S III
0 c0
S I S1
U I |z 0 g12 
U I |z 0 g13 
U II |z 0 g12 
0 c0
S1
U II |z 0 g 22 
0 c0
S1S III
0 c0
S I S III
U III |z 0 g13
0 c0
S1S III
U II |z 0 g 23 
0 c0
S III
U III |z 0 g 23
(S28)
U III |z 0 g33
7
where
r1 -w r1 -w
(2p )2
g11 =
ik ò ò G(r,0,r0 ,0)r0 dr0 r dr,
0
0
SI
g12 =
g13 =
(2p )2
ik ò
S I DS1
(2p ) 2
S I S III
r1 -w
r1 +d
r1
0
ik ò
ò
r1 -w
ò
s
r2 +w
0
G(r,0,r0 ,0)r0 dr0 r dr
G(r,0,r0 ,0)r0 dr0 r dr ,
(S29)
r1 +d r1 +d
(2p )2
g 22 =
ik ò ò G(r,0,r0 ,0)r0 dr0 r dr,
r1
r1
DS1
(2p ) 2
g 23 =
ik ò
DS1S III
r1 +d
ò
s
r2 +w
r1
G(r,0,r0 ,0)r0 dr0 r dr ,
s
s
(2p )2
g33 =
ik ò ò G(r,0,r0 ,0)r0 dr0 r dr
r2 +w r2 +w
S III
For z  t , the Green function can be written as
n (r0 )n (r )

G(r , z, r0 , z0 )  
n 0
2i R
k k
2
2
2
n
e
i k 2  kn2 z  z0
e
i k 2  kn2 z  z0  2t

(S30)
,
and the pressure filed is written as
p(r, z) = pt = 2p ò
r1 -w
0
G(r, z,r0 , z0 )
¶p(r0 , z0 )
r dr
¶z0 z =t 0 0
0
+2p ò
r1 +d
r1
s
¶p(r0 , z0 )
¶p(r0 , z0 )
G(r, z,r0 , z0 )
r0 dr0 + 2p ò G(r, z,r0 , z0 )
r dr
r2 +w
¶z0 z =t
¶z0 z =t 0 0
0
.(S30)
0
Following the same process, the averaged pressure field in the Part I, II and III region
at z  t can be expressed as
pI |z=t = -
r0c0
pII |z=t = pIII |z=t = -
SI
U I |z=t g11 -
r0c0
S I DS2
r0c0
S I S III
r0c0
S I DS2
U I |z=t g12¢ U I |z=t g13 -
U II |z=t g12¢ -
r0 c0
DS2
U II |z=t g22
¢ -
r 0c0
DS2 S III
r 0c0
S I S III
U III |z=t g13
r0c0
DS2 S III
U II |z=t g23
¢ -
r 0c0
S III
U III |z=t g23
¢
(S31)
U III |z=t g33
where
8
g12¢ = ik
g 22
¢ = ik
g 23
¢ = ik
(2p ) 2
S I DS2
r1 -w
r2
0
r2 -d
ò ò
G(r,t,r0 ,t)r0 dr0 r dr
(2p ) 2 r2 r2
GG(r,t,r0 ,t)r0 dr0 r dr
DS2 òr2 -d òr2 -d
( )
(2p )2
DS2 S III
r2
s
r2 -d
r2 +w
ò ò
.
(S32)
GG(r,t,r0 ,t)r0 dr0 r dr
Subscribing Eqs. (S20-S23) and (S29) into Eqs. (S33), we can obtain three equations,
which can be expressed as
é
ê
ê
ê
A ê
M
ê
ê
ê
ê
ë
ù
U I |z=0 ú
SI
ú
ú
r 0c0
U II |z=0 ú = MB ,
DS1
ú
ú
r0c0
ú
U |
S III III z=0 ú
û
r 0c0
(S32)
with
é MA
11
ê
A
A
M = ê M 21
ê
A
êë M 31
é
ê
ê
ê
ê
B
M = 2ê
ê
ê
ê
ê
ë
A
M12
A
M 22
A
M 32
A ù
M13
ú
A
M 23 ú
ú
A
M 33
úû
ù
ú
ú,
ú
SI
DS1
DS1
S III
ú
n21 g12¢ +
m11 +
m21 g 22
n21 g23
¢ +
¢ ú
DS2
DS2
DS2
DS2
ú
ú
SI
DS1
ú
n21g13 +
m21g 23
¢ + ( n11 + n21 g33 )
S III
S III
ú
û
DS1
S III
n11 + n21 g11 +
m21g12¢ +
n g
SI
S I 21 13
(S32)
where
9
A
M11
= ( n11 + n21 g11 ) g11 + m21g12¢ g12 + n21g13 g13 + ( n12 + n22 g11 )
A
M12
= éë( n11 + n21 g11 ) g12 + m21 g12¢ g22 + n21 g13 g23 + m22 g12¢ ùû
A
M13
= éë( n11 + n21 g11 ) g13 + m21 g12¢ g23 + n21g13 g33 + n22 g13 ùû
A
M 21
= éë n21g12¢ g11 + ( m11 + m21g 22
¢ ) g12 + n21 g23
¢ g13 + n22 g12¢ ùû
DS1
SI
S III
SI
SI
DS2
A
M 22
= éë n21g12¢ g12 + ( m11 + m21g22
¢ ) g 22 + n21 g23
¢ g23 + ( m12 + m22 g 22
¢ ) ùû
A
M 23
= éë n21g12¢ g13 + ( m11 + m21 g22
¢ ) g23 + n21 g 23
¢ g33 + n22 g23
¢ ùû
A
M 31
= éë n21g13 g11 + m21g23
¢ g12 + ( n11 + n21g33 ) g13 + n22 g13 ùû
A
M 32
= éë n21g13 g12 + m21g 23
¢ g 22 + ( n11 + n21g33 ) g 23 + m22 g23
¢ ùû
DS1
. (S32)
DS2
S III
DS2
SI
S III
DS1
S III
A
M 33
= éë n21g13 g13 + m21 g23
¢ g 23 + ( n11 + n21 g33 ) g33 + ( n12 + n22 g33 ) ùû
According to Eqs. (S35), we can obtain the volume velocity at the three inlets
0c0
SI
U I |z  0 ,
0c0
S1
U II |z 0 ,
0c0
S III
U III |z 0 . Then according to the Eqs. (S20-S23), (S29)
and the derived volume velocity at the inlets, the volume velocity at three outlets
0c0
SI
U I |z t ,
0c0
S2
U II |z t ,
0c0
S III
U III |z t can also be obtained. As a results, according to
the Green function theory, the reflection and transmission coefficient for the plane
wave component can be expressed as
R=
pr (r, z)
rc
» 1- 0 20 (U I |z=0 +U II |z=0 +U III |z=0 )
pi (r, z)
pp
p (r, z) r 0c0
T= t
»
(U | +U II |z=t +U III |z=t ) eik ( z-t )
pi (r, z) p p 2 I z=t
(S33)
C. Validation
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To demonstrate the validity of the analytical results, the simulated and analytical
complex transmission and reflection coefficient is calculated and illustrated in Fig. S2.
It can be obviously observed that both the amplitude and the phase part of the
transmission and reflection coefficients agree well with the analytical ones, indicating
the solid proof of the validity of the analytical formulas.
Figure S2. (a) The simulated (lines) and analytical (dots) power transmission and
reflection coefficient. (b) The simulated (lines) and analytical (dots) phase
information of the transmission and reflection coefficient.
II. Discussion on the viscosity
In the numerical simulation, the acoustic viscosity of the air in channels
contained in the metamaterial is completely neglected. In practice however, acoustic
wave that propagates within narrow channels is inevitably subject to the dissipation
effect stemming from the shear viscosity in air. If the metamaterial unit composing
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the GRIN lens contains air channels with extremely small sizes, as in the previous
work 3, such an effect will become pronounced and may notably affect the focusing
performance of the resulting lens. In physics, the additional loss should stem from the
viscous friction on the wall of the channels. The thickness of the viscous layer can be
expressed as dv µ 2m / rw , where w = 2p c / l ,  and  refer to the angular
frequency, mass density and the coefficient of viscosity, respectively 1. For acoustic
propagation in air under room temperature,  /   15.6 m2/s and c  343 m/s, thus
the thickness of the viscous layer is calculated as dv  0.12 103  . It means that in
our experiments, the value of d v should be approximately 0.078 mm. As a
comparison, the width of the channels in labyrinthine units and the maximum width
of the channels between adjacent units are fixed to be 13 mm and 15 mm, respectively.
It is therefore reasonable to conclude that that the viscous loss effect is negligible for
the proposed lens due to the fact that d v is more than two order of magnitude smaller
than the minimum width of air channels. This is also verified by the fact that the
numerical results agree quite well with the measured data although no viscosity effect
has been taken into consideration in the simulation, and both of them indicate a
considerably high transmission efficacy of the acoustic focusing.
III. The working bandwidth of the lens
We have further investigated the bandwidth of the focusing effect of the designed lens
and found that although the lens is designed to work at 815Hz, the focusing effect can
be achieved within a frequency ranging from 500Hz to 1100Hz. The corresponding
results for these two particular frequencies that can be regarded as the upper and the
lower limits of the working band are shown in Fig. S3. The underlying mechanism
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accounting for such functionality lies in the fact that the coiling-up-space structure is
able to mimic gradient index materials even when the frequency deviates from the
center frequency, except that the effective parameters are not defined perfectly.
2
Figure S3. (a) The simulated spatial distributions of the intensity field p of the
GRIN lens at 500Hz. (b) The simulated acoustic intensity distribution along the axis
(red line) and along the transverse cross-section in the radial plane through the focal
point (blue line) for the GRIN lens at 500Hz.. (c) The simulated spatial distributions
of the intensity field of the GRIN lens at 1100Hz. (d) The simulated acoustic intensity
distribution along the axis (red line) and along the transverse cross-section in the
radial plane through the focal point (blue line) for the GRIN lens at 1100Hz.
1. Morse, P. M. & Ingard, K. U. Theoretical acoustics. 1st edn, (Princeton University
Press, 1987).
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2. Feng, Q., Huang, Z., Yu, G. & Meng, X. Acoustic attenuation performance through
a constricted duct improved by an annular resonator. J Acoust Soc Am 134, EL345351 (2013).
3. Li, Y. et al. Acoustic focusing by coiling up space. Appl. Phys. Lett. 101, 233508
(2012).
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