Acoustic rainbow trapping
Jie Zhu1, Yongyao Chen2, Xuefeng Zhu1, 3, Francisco J. Garcia-Vidal4, Xiaobo Yin1, Weili
Zhang2* and Xiang Zhang1*
1
NSF Nano-scale Science and Engineering Center (NSEC), 3112 Etcheverry Hall, University
of California, Berkeley, California 94720, USA.
2
School of Electrical and Computer Engineering, Oklahoma State University, Stillwater,
Oklahoma 74078, USA.
3
Huazhong University of Science and Technology, Wuhan, Hubei 430074, China.
4
Departamento de Fisica Teorica de la Materia Condensada, Universidad Autonoma de
Madrid, E-28049 Madrid, Spain.
*
Corresponding Author: [email protected], [email protected]
Supplementary Information
Effective medium model for anisotropic acoustic metamaterials
Periodic groove array shown in Fig. 1a of main manuscript is equivalent to an anisotropic
homogeneous medium characterized by effective mass density tensor
modulus tensor
 eff
and bulk
 eff .
 eff
 x

 0
0

0
y
0
0

0
 z 
 eff
x 0

  0 y
0 0

0

0
 z 
(S1)
Here, we consider a two-dimensional (2D) case, where the structure extends infinitely
along the y direction, and the acoustic wave propagates in the x-z plane. The relevant
anisotropic medium parameters here are  x ,
z
and

. The corresponding governing
wave equation is given by [1]:
2
 1  2
2 
1 
  x  x 2    z  z 2    P  0


where

is angular frequency,
(S2)
P is pressure field. The effective medium properties shown
in manuscript can be obtained by using an analogy with electromagnetic waves. In a similar
corresponding electromagnetic system presented in [2], the anisotropic electromagnetic
medium is described by electric permittivity tensor
 eff
 eff
and magnetic permeability tensor
in Cartesian coordinates. For TM polarized electromagnetic waves propagating in the x-
z plane, the relevant effective medium parameters are  x ,
equations is thus given by
z
and
 . The governing wave
[( z 1 )
where

2
2
1 

(

)
  2  ]H y  0
x
2
2
x
z
is angular frequency,
(S3)
H y is the magnetic component parallel to the y direction.
From a comparison between equations (S2) and (S3), the following correspondence can be
obtained:  x
  z ,  z   x and    1 . It has been demonstrated that the effective
medium properties for similar electromagnetic waves system satisfy the relations:  x
 z   p / w air ,    y  w / p  air
 ,
[2,3]. Therefore, the acoustic metamaterial
presented in the main manuscript can be concluded having the following effective mass
densities and effective bulk modulus:
 x   p / w air ，  z   ，    p / w  air
(S4)
Generalized metamaterial dispersion equation using effective medium model
As illustrated Fig. 1b of main manuscript, the equivalent metamaterial waveguide is
comprised of two isotropic cladding media II, III and an anisotropic metamaterial core I. In
medium I, dispersion relation of the anisotropic metamaterial is given by
2 
Where

k x2
k2
 z
x /  z / 
is the angular frequency,
(S5)
cair is the speed of sound in air, k x
and
k z are the
wave-vector components along x and z directions, respectively. For guided modes, the
sound pressure of acoustic waves in medium I can be expressed with a transverse
oscillating field propagating along the z direction:
PI  [ A cos(k x x)  B sin(k x x)]e i  z
For
T / 2  x  T / 2
(S6)
where
PI
is the pressure field in metamaterial I, A and B are amplitude coefficients of the
transverse oscillating field.
and

x

kx  
is the wave-vector component along x direction,
is the propagation constant along z direction. Since the waveguide modes are
transversely confined in the metamaterial I, acoustic waves outside should be evanescent
along the x direction, which could maintain the confinement of guided waves propagating
along the z direction. The relevant pressure fields in media II and III can be expressed as
the following exponentially decay fields along the x direction:
PII  DII e  II ( x T /2) e i  z
For
PIII  DIII e III ( x T /2) e  i  z
where
Pi
is the pressure field,
Dj
For
T / 2 x  ;
   x  T / 2
denotes the amplitude coefficient.
transverse wave-vector component,
ki  
media II and III, satisfying the condition
i
(i  II, III)
i
ki  
(S7)
 i2   2  ki2
is the
is the wave-vector in cladding
for the evanescent fields. The
corresponding velocity fields in different regions could be obtained from
Vq  (i j )1 Pj  q ( j  I, II, III)
where the pressure field
, where
q  x, z . By matching the boundary conditions,
Pj and normal velocity component Vx
are continuous at the
interfaces between different regions, we can obtain the following general dispersion
equations for the guided modes in the heterostructure acoustic metamaterial waveguides:
k T 
k T
A sin  x   B cos  x
 x II
 2 
 2

k
T
 II k x


k T
A cos  x   B sin  x
 2 
 2






k T 
k T
A sin  x   B cos  x
 x III
 2 
 2

 III k x
k T 
k T
A cos  x   B sin  x
 2 
 2






(S8)
Derivation of Equation (1) in the manuscript
For the metamaterial waveguide shown in Fig. 1b of main manuscript, the media II and III
in our experiment are made of air and rigid body respectively having the mass densities of
cladding materials
II  air
and
III   , The general dispersion equations (S8) can be
reformulated into
1
2 2

   
2
k T 
k T 
x   
 
A sin  x   B cos  x 
c

 air  
 2 
 2 

k
T
 air k x


k T 
A cos  x   B sin  x 
 2 
 2 
k T 
k T 
A sin  x   B cos  x   0
 2 
 2 
(S9)
After some derivations, one can obtain the following equations for the fundamental mode:
 air
x 
kx
   
 2  
 
c

 air  
2
1
2
 cot(k xT )
(S10)
where
k x    x   k0 ,  x  ( p w) air . T represents the width of metamaterial, and
k z (  )
represents the propagation momentum along z direction. The dispersion relation
( k z versus angular frequency  ) of the guided wave can be derived as:
k z  k0 1 
w2
tan 2  k0T 
p2
Here, it assumes that the thickness
T of linearly tapered metamaterial layer is a slow-
varying and smooth function of the distance
tapering angle  ,
(S11)
z
along the propagation axis, thus, for small
T ( z )    z . From equation (S11) and smooth function T ( z ) , one can
get the corresponding group velocity distribution (group velocity
axis
vg versus propagation
z ) of the fundamental guided mode
vg 
where
1

dk z / d
cair
2


 
1   air tan  k0 z    k0 z  air 
 x

 x 
2
(S12)
tan 2  k0 z   1
 
cot  k0 z    air 
 x 
2
2
air /  x  w / p . From equation (S11), one can find out an asymptote line at
f  cair / (4 z ) , where the acoustic waves are getting trapped. With that, the spatial
density of spectral components,
| df stop / dz | cair / (4 z 2 ) , is not uniform for the
linearly tapered metamaterial.
The analytical study at microscopic level
At microscopic level shown in Fig. 4a of the main manuscript, acoustic wave can propagate
inside and diffract out of the individual grooves (depth h). Evanescent fields associated with
diffracted waves at the openings of individual grooves are coherently coupled due to the
intermode interaction in the periodic aperture arrays, thus creating transversally
nonradiative acoustic waves propagating along the surfaces of the metamaterial. Here, we
consider the case of one-sided opening grooves. The pressure fields of diffracted acoustic
waves in medium II can be expressed as
PII 
where

Ce
n 
n   

n ( x  h /2)
n
e in  z
(h / 2  x  ,   z  )
(S13)
2
n and Cn are the wave vector components along the z direction of the
p
nth order diffracted wave and the corresponding coefficients.
transverse momenta along the x direction.

 n  n2  k02
are the
is the Bloch wave vector in the first Brillouin
zone. Since the propagating acoustic modes are tightly confined in metamaterials, the
acoustic waves outside the metamaterial region should be transversely evanescent (along
the X direction), satisfying the condition
k0  n
for the evanescent fields.
In region I, the perpendicularly oscillating waves are localized in individual grooves. The
pressure field is given by
PI  A cos(k0 x)  B sin(k0 x) (h / 2  x  h / 2,  w / 2  z  w / 2)
(S14)
where A and B are amplitude coefficients of the oscillating field along x direction. Since the
particle velocity at the rigid bottom of individual channel is zero, it requires that
(i air ) 1 PI  x |x  h /2  0  A sin(k0 h / 2)  B cos(k0 h / 2)  0
By matching the continuous condition for the volume velocity
 V dz
x
(S15)
at boundaries between
media I and II, one can obtain the following relation


 C  sinc  
n 
n n
n
p
 hw
  2 Ak0 sin  k0 
2
 2 p
(S16)
Then, a technique of average-field continuous boundary condition is applied for pressure
fields, which takes the integral on both sides of pressure at the opening region
( w 2  z  w 2)
of an aperture, thus making


 C sinc  
n 
n
n
cos(k0 h)
w
 A
2
k h
cos  0 
 2 
(S17)
From Equation (S11), one can obtain that at the trapping frequency, the momentum along
the propagating direction
   . In reality, due to the finite-sized unit cell in the
metamaterial (lattice constant
p  0 ), the propagating momentum has a maximum at
max   / p , where the phase velocity ( v phase  stop /  max ) is very small but not zero and
 d / d 
the group velocity ( vgroup
   max
) equals to zero, corresponding to the lower
boundary of a band-gap caused by the locally resonance in grooves. For the wave vector
 n , the terms n  (0, 1)
are dominant diffraction orders and higher order modes can be
neglected as an approximation. One can obtain
 0   1  ( / p)2  k02
0   / p ,  1   / p , and
. The equations (S16) and (S17) can be truncated into

p  p
 p

 h
C0 0sinc  0 2   C1 1sinc   1 2   w  2 Ak0 sin  k0 2 







cos(k0 h)
w
 w

C0sinc  0   C1sinc   1   A
2
k h
 2

cos  0 
 2 
where
k0   air  air
(S18)
. Equations (S18) represent the coupling between the localized
sound oscillation in individual unit-cells of metamaterial and mutual coupling among
the cells, which gives the “sound trapping equation” in the microscopic scale
 w
k0 sin 
p 2 

cot(k0 h) 
2
 
2
 p   k0
 
It needs to mention that from Equation (S13),
C0
should equal to
(S19)
C1
so as to make
diffracted evanescent waves form a standing wave along the propagation direction of z.
Therefore, from Equation (S18) one can get
C0 =C1  A
cos(k0 h)
 w
k h
2 cos  0  sinc 

 2 
 p 2
Balance breakage through geometrical modifications for sound releasing.
(S20)
Figure S1: As revealed in the microscopic model, sound trapping is a result of balanced
interplay between the localized sound oscillation in individual unit-cells of metamaterial and
mutual coupling among the cells, thus the key for sound releasing is to disrupt the balanced
interaction through external stimuli, and sound releasing can be achieved by modifying the
geometry of such metamaterials waveguides in the following two ways. (a) Schematic of
achieving acoustic wave releasing through mechanically-driven geometrical modification to
the metamaterial waveguide. Such effect can be accomplished either through lattice
modification (green folded lines), or through the separation and distance adjustment
between the substrate and suspended metamaterial waveguide layer (blue fold lines).
Width, depth and period of the grooves are w, h and p, respectively. The distance between
metamaterial wedge and substrate is S. A uniform waveguide is added as guidance and
exhibition of released wave. Operating frequency is set at 5KHz. (b) Sound trapping without
external stimuli. c and d present the sound releasing through external stimuli. (c) The
mutual coupling among grooves is modified through the lattice change by compressing the
period of grooves from p=6.35 mm to p=3.18 mm. (d) The localized sound oscillation in
individual grooves is perturbed by the cavity modulation. The distance between the
metamaterial waveguide layer and rigid substrate changes from S=0 to S=0.48mm. b, c,
and d are drew from the Finite-difference-time-domain (FDTD) simulation results.
FDTD simulations
In the above 2D FDTD simulation, the units for space and time domain are millimeter (mm)
and millisecond, respectively. The simulated region is surrounded by perfectly matched
layers to absorb the scattering waves. The meshed grid in space is chosen as x  0.05
mm and z  0.1 mm, respectively. The sampling in time is selected to achieve the
numerical stability of the simulation, and one time step satisfied the condition:
t  c 1
( x ) 2  ( z ) 2 , where cair  343 (m/s) is the speed of sound in air. The total
simulation has 20000 time steps, which are long enough to get steady state. The materials
of slats arrays are chosen from rigid solids impenetrable for sound waves. The material
filled in the gaps among the slats and the background medium is chosen as air with mass
density air
 1.21 (kg m3 ) and bulk modulus  air  1.01105 pa .
Methods used for calculating the Figure 2b.
FEM simulations on acoustic pressure field distribution were carried by COMSOL
MultiphysicsTM 4.1 with the partial differential equation module. The materials applied in
simulations were Metamaterial (Medium I), air (Medium II), and Brass (Medium III) as
shown by the schematic diagram in Figure 1b of the original manuscript, where the material
Brass is treated as a fluid-like material by ignoring the shear modulus for simplicity, and
the material parameters of the Metamaterial are
tensor), and
   p / w  air
 x   p / w air ，  z  
(density
(bulk modulus), respectively. Mur absorbing boundary
conditions were given to the outer boundaries of simulation domain so there will be no
interference from reflected acoustic wave. The largest mesh size was set lower than 1/10 of
the lowest wavelength.
3D full wave FEM simulation on actual geometric design
Figure S2: Normalized acoustic field intensities from the simulation result versus the
positions (distance from the start of metamaterial in z direction) also shows clear rainbow
trapping effect. The frequencies of incident waves applied in the simulation changes from
4.5 kHz to 9.5 kHz by step of 25 Hz. The measuring positions inside each groove are same
as the ones applied in the experiments.
Video S1
Demonstration video of a pulse signal travelling along the metamaterials. The wave clearly
slows down during the propagation and eventually stops at the trapping location. The
simulation was done with COMSOL.
Snapshot from Video S1:
Figure S3: Normalized acoustic field intensities from FEM simulation versus the positions
(distance from the start point in z direction) at 5 equally separated times. The wave packet
show clear slowing down and compression effect when traveling along the metamaterial
waveguide.
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