Acoustic cloaking in three dimensions using acoustic metamaterialsHuanyang Chena and C. T. ChanDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay,Hong Kong, China

Received 18 September 2007; accepted 7 October 2007; published online 2 November 2007

A scheme to achieve two dimensional 2D acoustic cloaking is proposed by Cummer and SchurigNew J. Phys. 9, 45 2007 by mapping the acoustic equations to Maxwells equations of onepolarization in the 2D geometry. We find that the acoustic equation can be mapped to the directcurrent conductivity equation in three dimensions, which then allows the design ofthree-dimensional acoustic cloaking using the coordinate transformation scheme. The perfectcloaking effect is confirmed by solving for the scattering problem using the spherical-Besselfunction series expansion method. 2007 American Institute of Physics.DOI: 10.1063/1.2803315

The transformation media concept1,2 offers us methodsto manipulate electromagnetic EM waves to achieve invis-ibility cloaking.1,3 Similar concepts have in fact been dis-cussed for the conductivity equations by Greenleaf et al.,4,5and we will see in this article that we can achieve invisibilitycloaking in acoustic waves by mapping to the conductivityequations. Various wave manipulation strategies such as theconcentrator6 and rotation coating7 can be easily derived us-ing the transformation media concept. There are many de-signs to realize the invisibility cloak for em waves.8,9 Thecoordinate transformation method is based on the invarianceof Maxwells equations. This method can be extended toother classical waves if the invariance symmetry of the waveequations is kept under coordinate transformation. However,the wave equations in most other systems do not have thisinvariance symmetry. Milton et al.10 investigated how theconventional elastodynamic equations vary under curvilineartransformations and proposed modified equations of motionwhich is invariant under transformation. A scheme was thensuggested for realizing the cloaking for elasticity. Cummerand Schurig11 demonstrated acoustic cloaking in two dimen-sions by noting the equivalence of acoustics and electromag-netic in the two-dimensional 2D geometry. After comparingthe equations in acoustic case and em case, they obtained thegeneral transformation for the density and the bulk modulus.They also noted that the equivalence is specific to two di-mensions and such an approach cannot be generalized tothree dimensions.

In this paper, we will compare the acoustic field and dcconductivity equations in three dimensional 3D geometry.The invariance property of the acoustic equation will beidentified and, thus, a general transformation for the densityand the bulk modulus to achieve cloaking will be obtained.

The invariance of the conductivity equation is a corol-lary of the invariance of Maxwells equations.12 Supposethere is a potential Vx satisfying the conductivity equation,

x Vx = fx , 1where x is the conductivity tensor field and fx is thesource term.

We can write the acoustic equation as

1x

px = 2x

px . 2

Comparing the equations, we can identify the variable ex-changes,

Vx,x, fx px, 1x

,

2

xpx . 3

After mapping from each point x in one space to a corre-sponding point xx in another space, where Vxx=Vx. The conductivity equation in the new space is10

xVx = fx , 4with x=AxAT /det A and fx= fx /det A, where Ais the matrix with elements,

Aki =xk

xi. 5

In fact, we note that the concept of invisibility has beendiscussed in terms of such equations.4,5

Using the above procedure, we map each point x in onespace to a corresponding point xx in another space wherepxx= px, the acoustic equation becomes

1x

px = 2x

px , 6

where 1/ x=A1/ x AT /det A and 2 / x=2 / x /det A or x=xdet A.

Then we apply the method prescribed by Pendry et al.1and Greenleaf et al.4,5 to design a 3D acoustic cloaking sys-tem using the linear radial transformation r=a+rba /b.The relative mass density elements and the bulk modulus arethen

r =b a

br2

r a2,

= =b a

b,

aPermanent address: Institute of Theoretical Physics, Shanghai Jiao TongUniversity, Shanghai 200240, Peoples Republic of China. Electronic mail:kenyon@ust.hk

APPLIED PHYSICS LETTERS 91, 183518 2007

0003-6951/2007/9118/183518/3/$23.00 2007 American Institute of Physics91, 183518-1Downloaded 03 Jul 2008 to 148.228.150.163. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

=b a3

b3r2

r a2, 7

where a and b are the inner and outer radii of the cloak,respectively. It can also be applied to a 2D acoustic cloakingsystem and we can obtain the same results as Cummer andSchurig.11

The perfect cloaking effect will be confirmed by thespherical-Bessel function series expansion method in the fol-lowing section.

The density tensor can be written in r , , coordi-nates as follows,

1

x =1

bb a

r a2

r20 0

0b

b a0

0 0b

b a . 8

Together with the source term

2

x =

2

b3

b a3r a2

r2, 9

the acoustic wave equation in the cloak can be derived fromEq. 2,

p 2ar a2r2

pr

er = 2

b2

b a2r a2

r2p ,

10

where the primes are dropped for aesthetic reasons.After performing separation of variables, pr , ,

=Rr, the radial part Rr satisfies

ddrr a2dRdr + 2 b2b a2 r a2 ll + 1R

= 0. 11

The general form of the pressure field in the cloak can thusbe derived as

pr,, =

lm

plmjl2

bb a

r aYlm, , 12where and are the density and the bulk modulus of thebackground, Yl

m , is the spherical harmonics, and jlx isthe lth order spherical Bessel function hl

+x will be used asthe corresponding spherical Hankel function. Note that Eq.12 carries no Hankel function because of the singularity atr=a. The pressure in ra is zero, which can be confirmedeasily.13,14 For the incident waves of the form p0r , ,=lmplm

0 jl2 /rYlm ,, we assume the scatteringwaves for rb to be p+r , ,=lmplm

+ hl+2 /rYlm ,. Substitute them into the

boundary conditions,

p+r = b,, + p0r = b,,= pr = b,, ,

131 p+

rr = b,, +

p0

rr = b,,

=

1r

pr

r = b,, ,

the following equations can be obtained;

lmplm+ hl+2 b + plm0 jl2 b=

lmplmjl2

b ,

14

lm

plm+ hl

+2

b + plm0 jl2 b2 1=

lmplmjl2 b2 bb ab ab 1 .

Then plm+

=0 and plm= plm0

. It is obvious that there is no scat-tering in this case.

For example, if the incident wave is a plane wave in thez direction with the amplitude p0,

p0r,, = Rep0ei2/r cos . 15

Then the wave inside the cloak should be

pr,, = Rep0ei2/b/baracos 16

Figure 1 is the contour plot of the pressure in the x-z planey=0. The cloak is designed for the water background withb=1.5 wavelengths and a=0.75 wavelengths.

In conclusion, we have compared the conductivity equa-tion and the acoustic equation to obtain the general transfor-mation for the density and the bulk modulus in a 3D acousticcloaking. We also confirmed the perfect cloaking effect usingthe spherical-Bessel function series expansion method. The

FIG. 1. The contour plot of the acoustic pressure in the x-z plane y=0 withan incompressible spherical scatterer coated by a 3D acoustic cloaking. Theunit here is wavelength. The dotted lines outline the interior and exteriorboundaries of the cloak.

183518-2 H. Chen and C. T. Chan Appl. Phys. Lett. 91, 183518 2007

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realization of such cloak relies on the existence of acousticmetamaterials. Such materials can be designed using localresonances.

15,16 The above analysis is valid for any acousticwave source outside the cloak. We note that cloak makes theinside domain invisible. However, if there is a source terminside the cloaking region, the source is detectable from theoutside. For the case of EM waves, this has been noted anddiscussed.17

We thank Dr. Y. Lai and Professor Z. Q. Zhang for manyhelpful discussions. This work was supported by Hong KongRGC through Project 600305. Computation resources aresupported by Shun Hing Education and Charity Fund.

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183518-3 H. Chen and C. T. Chan Appl. Phys. Lett. 91, 183518 2007

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