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Dynamics of elastic waves in two-dimensional phononic crystals with chaotic defect
Jie Chen, Jian-Chun Cheng, and Baowen Li
Citation: Applied Physics Letters 91, 121902 (2007); doi: 10.1063/1.2779967
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APPLIED PHYSICS LETTERS 91, 121902 共2007兲
Dynamics of elastic waves in two-dimensional phononic crystals
with chaotic defect
Jie Chen and Jian-Chun Cheng
Key Laboratory of Modern Acoustics, MOE, Nanjing University, Nanjing 210093, China
and Institute of Acoustics, Nanjing University, Nanjing 210093, China
Baowen Lia兲
Department of Physics and Centre for Computational Science and Engineering, National University
of Singapore, Singapore 117546, Singapore and NUS Graduate School of Intergrative Sciences
and Engineering, Singapore 117597, Singapore
共Received 28 June 2007; accepted 16 August 2007; published online 17 September 2007兲
The authors study dynamics of wave function in two-dimensional phononic crystals with different
billiard defects. It is found that the elastic wave is localized in the defect region with soft material.
The spatial statistical properties of wave function are studied. More strikingly, they find that given
the same area, the chaotic billiard contains more energy than the integrable billiards such as
rectangular and circular billiards. The dependence of this “chaotic effect” on wave number k is also
studied. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2779967兴
Two-dimensional 共2D兲 phononic crystal 共PC兲 is a periodic elastic structure. It has attracted increasing attention in
recent years1–3 due to its potential industrial applications. It
is well known that defects and/or impurities will cause wave
localization, thus affect the transport behavior of the system.4
Therefore, by embedding different billiards or impurities,
people expect to alter the properties of phononic
transmission.
In this letter, we study the wave dynamics in the PC with
defects of different geometries—stadium, rectangle, and
circle. In particular, we study the energy in the defect region
共Eb兲 and the whole PC system 共Epc兲 as a function of time by
the finite-difference time-domain method.5 The purpose is to
understand whether the underlying dynamics, instead of the
geometry, affect the wave dynamics in phononic crystals.
From the study of quantum chaos, it is well known that the
statistics of wave function and energy level spacing distribution in a quantum billiard is not determined by its geometry;
instead, it is determined by its underlying dynamics.6 For
example, the integrable systems have the same energy level
spacing distribution, Poisson distribution, no matter whether
it is a circular one or a rectangular one, while chaotic systems such as the Bunimovich stadium and Sinai billiard have
the Wigner distribution.6 The stadium billiard 关see Fig. 1共a兲兴
is a Hamiltonian system that was proved by Bunimovich7,8 to
be strongly chaotic.
The equations governing the motion of lattice displacements u共r , t兲 in the inhomogeneous system are given by
␳共r兲
⳵2
ui共r,t兲 = ⵜiJTJ共r,t兲
⳵t2
共i = 1,2,3;J = 1,2, . . . ,6兲,
共1兲
TI共r , t兲 = CIj共r兲S j共r , t兲, and SI共r , t兲 = ⵜTIju j共r , t兲共I = 1 , 2 , . . . ,
6 ; j = 1 , 2 , 3兲, where r = 共x , y , z兲共the z axis is taken to be parallel to the cylinder axis兲, ⵜ is
a兲
Author to whom correspondence should be addressed; electronic mail:
phylibw@nsu.edu.sg
冤
冥
0
0 ⳵/⳵z ⳵/⳵ y
⳵/⳵x 0
ⵜ = 0 ⳵/⳵ y 0 ⳵/⳵z 0 ⳵/⳵x ,
0 ⳵/⳵z ⳵/⳵ y ⳵/⳵x 0
0
共2兲
and ⵜT is the transposed matrix of ⵜ. ␳共r兲 and CIJ共r兲共I
= 1 , 2 , . . . , 6 ; J = 1 , 2 , . . . , 6兲 are the position-dependent mass
density and elastic stiffness tensor of the system, respectively, and TI共r , t兲 and SJ共r , t兲 are the Ith and Jth components
of the stress vector and strain vector, respectively. Note that
␳ and CIJ do not depend on z because of the homogeneity of
the system along the cylinder axis. The summation over repeated indices is assumed in the present letter. The total energy density ␧t is given by ␧t = ␧k + ␧ p, where ␧k
= 21 ␳共r兲v共r , t兲 · v共r , t兲 and ␧ p = 21 S共r , t兲C共r兲ST共r , t兲 represent
the kinetic and potential energy densities, respectively, with
v共r , t兲 is the velocity vector, S共r , t兲 is the strain vector 共S
= 关S1 , S2 , S3 , S4 , S5 , S6兴兲, ST is the transposed vector of S, and
C共r兲 is the elastic stiffness matrix of CIJ共r兲.
The PC is a 2D square array with steel cylinders in silex
background, in which the lattice constant is 6 cm and the
radius of cylinders is 2.5 cm. In fact, other materials different from steel can be used as the background material. Three
kinds of defected PC are studied: PC with two void cylinders
关Fig. 1共c兲兴, PC with steel billiards 关Figs. 1共d兲–1共f兲兴, and PC
with epoxy resin billiards 关Figs. 1共g兲–1共i兲兴.
We first study the spatial distribution of wave function of
different configurations. Typical wave functions in different
configurations are shown in Fig. 2. Elastic wave in original
PC without defect is periodically distributed. In PC with
共soft material兲 defect, localization is clearly seen. This kind
of localization phenomenon has been also observed for electromagnetic wave in photonic crystals with defect.9 In PC
with steel 共harder兲 defect, elastic wave is localized in the
background material since steel is a hard material compared
with the background.
To compare the wave function in nonintegrable system
and integrable system more quantitatively, we calculate the
two point correlation function inside the epoxy resin billiard
at a given time. The results are shown in Fig. 3. The asterisks
denote circular billiard and solid circles denote stadium bil-
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91, 121902-1
© 2007 American Institute of Physics
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121902-2
Appl. Phys. Lett. 91, 121902 共2007兲
Chen, Cheng, and Li
FIG. 1. 共Color online兲 Schematic diagrams of different PC configurations.
The white region represents the silex background and the black region represents the steel rods. The blue region represents the epoxy resin. 共a兲 Geometry of stadium billiard: half length of the straight section a = 2.97 cm and
radius of the semicircle R = 2.5 cm. 共b兲 Original PC. 共c兲 PC with two void
cylinders. 共d兲 PC with steel stadium billiard. 共e兲 PC with steel circular billiard. 共f兲 PC with steel rectangular billiard. 共g兲 PC with epoxy resin stadium
billiard. 共h兲 PC with epoxy resin circular billiard. 共i兲 PC with epoxy resin
rectangular billiard. Circular billiard: radius R = 3.96 cm. Rectangular billiard: side length in the x direction Lx = 7.68 cm and side length in the y
direction Ly = 6.42 cm.
FIG. 2. 共Color online兲 Spatial wave function distribution in PC and PC with
different billiards at t = 100 ␮s with wave number k = 2.48⫻ 103. The amplitude has been normalized to unity. 关共a兲–共f兲兴 Corresponding to 共a兲–共f兲 in Fig.
1.
liard; solid 共red兲 and dashed 共blue兲 curves denote the best fit
function in integrable billiard. Second, the correlation for
of the discrete Corr共s兲 of stadium billiard and circular bilchaotic billiard does not change very much with time, while
liard, respectively, by the means of cubic spline interpolait does for integrable system, which means that the spatial
tion. The two point correlation function Corr共s兲 is defined
1 典
1
10
*共
distribution of function in integrable systems changes with
具
共
兲
兲
as Corr共s兲 = 共1 / ⌸共r兲兲 ␧t r + 2 s ␧t r − 2 s , where 兿共r兲 is the
time more significantly. Equally striking is the shape of the
2
local average intensity 兿共r兲 = 具兩␧t共r兲兩典 and 具典 denotes the enfitted curve. The fitted curve of stadium billiard shows a
semble average. The s is the distance between the two points
similar one for quantum wave function in chaotic billiard,10
r + s / 2 and r − s / 2. The ensemble average is taken by moving
namely, when ks ⬎ 0, Corr共s兲 is around zero with only a
r around a certain region which is about a wavelength. Acslight fluctuation.
10
cording to the quantum 共wave兲 chaos theory, in a closed
In order to study the time dependent energy variation in
chaotic billiard, the wave function at any point is a superpospace,
we launch a Gaussian pulse on the left side of the PC.
sition of infinite number of plane waves of the same wave
The
central
frequency of the Gaussian pulse is 500 kHz with
number with random phases, therefore, the two point correthe
highest
frequency of 1 MHz. Both the normalized total
lation function Corr共s兲 = J0共ks兲, where k is the wave number,
s is the distance of the two points, and J0 is the zeroth order
Bessel function. This has been numerically verified in chaotic billiards.10 The reason is that the waves that reflected
many times from boundary will eventually lose its memory
共coherence兲. However, in integrable billiard, the wave function does not follow J0共ks兲 because the wave inside an integrable billiard always keeps its coherence no matter how
many times it is reflected from the boundary.
As shown in Fig. 3, the difference is quite obvious. First,
in the chaotic billiard, the correlation function decays very
fast with the increase of the distance between the two points,
FIG. 3. 共Color online兲 Two point spatial correlation function Corr共s兲 vs ks in
whereas in the integrable case, it shows strong oscillation
stadium
circle and curve兲 and circular billiard 共asterisk
and to IP:
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to thebilliard
terms 共solid
at: http://scitation.aip.org/termsconditions.
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meaning that there exists coherent structure in the wave
dotted curve兲. 共a兲 t = 180 ␮s; and 共b兲 t = 360 ␮s.
222.66.175.227 On: Thu, 23 Apr 2015 14:18:15
121902-3
Appl. Phys. Lett. 91, 121902 共2007兲
Chen, Cheng, and Li
FIG. 5. 共Color online兲 Evolution of normalized total energy in the billiard
Eb for different wave numbers k. 共a兲 ka = 0.82 and 共b兲 ka = 40.82.
FIG. 4. 共Color online兲 Energy variation with time. 共a兲 Eb of PC with steel
billiards. 共b兲 Eb of PC with epoxy resin billiards. 关共c兲 and 共d兲兴 Variation of
ratio ␩共␥兲 with ␥ under the condition that the area of the billiard remains
constant.
Lyapunov ␭ is nonzero, the underlying dynamics is completely chaotic.
Another important factor is the wave number k. In order
to make the wave “sense” the underlying classical dynamics
of the billiard,12,13 one needs to go to the so-called semiclassical limit, namely, ka Ⰷ 1. This means that the wavelength
must be much smaller than the typical length scale of the
system, a and R. We have calculated Eb for stadium billiard
and circular billiard for different ka, and the results are
shown in Fig. 5. Solid 共red兲 and dotted 共blue兲 curves refer to
stadium billiard and circular billiard, respectively. It is
clearly seen that in the case of large wavelength such as ka
= 0.82, the wave function does not sense the chaotic dynamics of the billiard; thus, there is very little difference between
the stadium billiard and the circular billiard. Thus, Eb of
these two billiards are almost the same. As ka is increased,
Eb of the stadium billiard becomes larger than that of the
circular billiard and the difference in Eb between these two
billiards enhances correspondingly, as shown in Fig. 5共b兲.
In summary, we have presented a comparative study of
the wave function in phononic crystals embedded with nonintegrable and integrable billiards. The energy both in the
billiards shows that in the short wavelength limit, the chaotic
billiard can trap more energy than the integrable counterpart
does. Spatial distribution of wave function shows that elastic
wave is localized in defect region with soft material. The two
point correlation function demonstrates the difference of the
two kinds of billiards.
energies in the billiard 共Eb兲 are shown in Fig. 4. Solid 共red兲,
dashed 共blue兲, and thin 共black兲 curves refer to stadium, rectangular, and circular billiards, respectively. The area of the
embedded billiards remains the same for different shapes.
As shown in Fig. 4共a兲, Eb of steel stadium billiard is
almost the same as that of steel rectangular billiard and is
only a little larger than that of steel circular billiard given the
same billiard area. This is because steel is a hard material
compared with silex; therefore, most of the energy is reflected by the hard billiard and localized in the soft background, which is shown in Figs. 2共c兲 and 2共d兲. It is the reflection of the steel billiard that plays the significant role in
this case.
As to the epoxy resin billiard, it is quite different. There
are two impressive phenomena shown in Fig. 4共b兲. One is
the similarity in energy variation displayed by rectangular
and circular billiards. This indicates that it is the underlying
physical property of the system that determines the dynamic
behavior, not the geometry. Rectangular and circular billiards
The work was supported by the Ministry of Education of
contain almost the same energy in spite of the different
China under Grant Nos. 705017 and 20060284035 and by
boundary shapes. The other is that the stadium billiard conthe National Natural Science Foundation of China for Exceltains more energy than any of the other two. We call this
lent Youth under Grant No. 10125417. One of the authors
characteristics “chaotic effect” in this letter. This distinction
共B.L.兲 was supported in part by a FRG grant of NUS. Anreflects the difference of nonintegrable system and integrable
other author 共J.C.兲 is currently a postgraduate student at
system.
NUS.
To further investigate this chaotic effect of stadium bil1
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