Extraordinary acoustic transmission through a 1D grating with very
narrow apertures
Ming-Hui Lu, Xiao-Kang Liu, Liang Feng*, Jian Li, Cheng-Ping Huang, Yan-Feng Chen,†
Yong-Yuan Zhu, Shi-Ning Zhu and Nai-Ben Ming
National Laboratory of Solid State Microstructures and Department of Materials Science and
Engineering, Nanjing University, Nanjing, 210093, People’s Republic of China
Supplementary Information
Supplementary Texts
We analyze the EAT phenomena for the acoustic grating with very narrow apertures by the
rigorous analysis model based on rigorous coupling wave approximation (RCWA) method. This
model originates from interference between the diffracted wave and the incident wave at the
entrance of the aperture, which is coupling with the wave-guided mode. Fig. S1 indicates the
coupling process. It relies on two main features. Firstly, the pressure field inside the grating is
supposed as a superposition of two counter propagating waves, which being the zero-order
acoustic modes propagating forward and backward in the air aperture surrounded by steel walls.
Secondly, it is very important to take into account the composition of the diffracted waves above
and below the grating, then matching the boundary condition on each side of the grating. The
detailed process of the model is shown as followings.
*
Present address: Department of Electrical and Computer Engineering, University of California,
San Diego, CA 92093.
†
To whom correspondence should be addressed. E-mail: yfchen@nju.edu.cn.
The geometry of the square rods metallic grating transmission problem discussed in the paper is
shown in Fig. S2. An acoustic wave incident on the grating produces both forward-diffracted and
back-diffracted waves. We separate the whole area with three regions. Region I (the input region)
is a homogeneous air background, region III is the outgoing region with homogeneous air
background too, and region II is the grating region. So the representation of the acoustic pressure
field in three regions may be expressed as:




,
P   Pi  Pr  exp  j k x 0 x  k z 0 z    Rm exp  j k xm
 k zm


(1)
m


P    Tm exp  j k xm
x  k zm
( z  h) .
(2)
m
The pressure field in region II can be expressed as the rectangular wave guide mode:
P   cos( x)A exp  jqz   B exp  jqz ,
k x 0  k sin 
2
where 
, k
, then

k x 0  k sin 
(3)
2

 k xm  k x 0  m
d , and q and βare the

2 1/ 2
k zm  k 2  k xm


propagating constant and eigen wave vector of wave guide. From the pressure filed expressions.
We can obtain the acoustic particle velocity u with the equation:
u  
P
0
d t.
(4)
At the entrance of the aperture, considering the two boundary condition as following:
 Pz 0  Pz 0
 

u z z 0  u z z 0
 Pz h  Pz
h
 

u z z h  u z z h
,
(5)
The solutions of these equations can be determined as:
jqh

 A  2 g 0 1e F  

,
 jqh

F  
B  2 g 0 2 e

and F    1 e
2
jqh
 22 e  jqh
(6)

1
,
(7)
which is the enhancement factor of diffracted waves discussed in the paper.

 1 


Here,
2 


m

m
g m s m qf
 a 
 sin c

kzm
 2 
g m s m qf
 a  ,
 sin c

kzm
 2 
(8)
 g m  sin ck xm a / 2
a

1 2

where s m 
cosx e jkxm x dx .
a

a 
2


(9)
So substitute the Eqs. (1), (2), (3) and (6) into Eqs. (5), we can get the reflectivity and
transmissivity as:


k z 0  0 n qfs n
k 
qfs

 A  B   z 0 0n  n  2 g 0 1e jqh  2 e  jqh  F  

Rn  k
k zn
k zn
k zn
zn

 a 

4 g 0 sin c


qfs n
qfs n
1
 a 
 2 
Tn 
 4 g 0 sin c


 F   

2
jqh
k zn
k zn

1 e
1  2
 2 
.
(10)
Then the zero-order transmission may be expressed as:
t 0  Re(
k z
2
0
) T0 .
k z0
(11)
Now, how to determine the eigen wave vector and propagating constant of the aperture wave –
guide? Considering the real boundary, the β can be determined by the equation as
following:
 tana   jk g  jb  ,
(12)
where g-jb is the ratio of the conductance of the steel plate, which can be expressed as:
Z0
 0 c0
 g  jb 
,
jM st  Z 0
j st (d  a)   0 c 0
g  jb 
(13)
So the zero-order wave guide mode: β and propagating constant q can be obtained.
   jk g  jb  / a1 / 2
.

1/ 2
 q  k 2   2


(14)
Supplementary Figures and Legends
Supplementary Figure S1
Pi
Pdiffr
Pout
Supplementary Figure S1. Schematic of our theory model. The subwavelength narrow
apertures stimulate the diffracted waves Pdiffr, which interfere with the incident wave Pi and
coupled with the wave-guided mode at the entrance of the aperture. For every aperture, it is
affected by the diffracted waves excited by other apertures and the backscattering diffracted waves
excited by itself.
Pi
d
I
Ⅱh
III
a
Supplementary Figure S2. Geometry of the rectangular metallic acoustic grating
Acoustic grating
BK 3560C
BK free-field 1/8
inch Microphone
Output signal
Ultrasonic transducer
Computer
Input signal
Supplementary Figure S3. Schematic of the experimental setup. The experimental setup is
used to measure the transmission of ultrasonic wave in an acoustic grating, consisting of a
ultrasonic transducer, a flat rectangular acoustic grating of steel cylinders, an acoustic pulse
analyzer BK3560C, which can supply pulsed signal output and analyze the received signal by the
microphone. With the BK SSR software package, the zero-order transmission spectra could be
obtained