Scattering Theory Derivation of a 3D Acoustic Cloaking Shell
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PHYSICAL REVIEW LETTERS week ending PRL 100, 024301 (2008) 18 JANUARY 2008 Scattering Theory Derivation of a 3D Acoustic Cloaking Shell Steven A. Cummer,1,* Bogdan-Ioan Popa,1 David Schurig,2 David R. Smith,1 John Pendry,3 Marco Rahm,1 and Anthony Starr4 1Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA 2Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA 3Department of Physics, The Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom 4SensorMetrix, Inc., San Diego, California 92121, USA (Received 26 September 2007; revised manuscript received 16 November 2007; published 14 January 2008) Through acoustic scattering theory we derive the mass density and bulk modulus of a spherical shell that can eliminate scattering from an arbitrary object in the interior of the shell—in other words, a 3D acoustic cloaking shell. Calculations confirm that the pressure and velocity fields are smoothly bent and excluded from the central region as for previously reported electromagnetic cloaking shells. The shell requires an anisotropic mass density with principal axes in the spherical coordinate directions and a radially dependent bulk modulus. The existence of this 3D cloaking shell indicates that such reflectionless solutions may also exist for other wave systems that are not isomorphic with electromagnetics. DOI: 10.1103/PhysRevLett.100.024301 PACS numbers: 43.20.+g Pendry et al. [1] have shown that arbitrary coordinate outset. With a exp ÿi!t time dependence, these equa- transformations of Maxwell’s equations can be interpreted tions of motion are in terms of an electromagnetic material in the original rp i! r v; (1) coordinates with transformed permittivity and permeabil- 0 ity values. Consequently, the bending and stretching of i!p r rv; (2) electromagnetic fields specified by coordinate transforma- 0 tions can be implemented with electromagnetic materials, where p is scalar pressure, v is vector fluid velocity, is the enabling unexpected and interesting solutions such as elec- inhomogeneous fluid bulk modulus relative to 0, and is tromagnetic cloaking [1,2]. an inhomogeneous generalized fluid mass density tensor The degree to which this cloaking concept can be ex- relative to 0. These equations are the fluid version of the tended to other classes of waves is not known in general. more general elastodynamic equations considered in [5]. Miller [3] described an active approach for general wave Although anisotropic mass density is not a property com- cloaking based on sensing and nonlocal retransmission of monly encountered in natural materials, it naturally arises signals on the surface of an object. Leonhardt [4] has in the analysis of elastodynamics of strongly inhomoge- described how a 2D object can be cloaked in the short neous composite materials [8]. It is important to note that wavelength limit in a way that also applies to different the anisotropic mass density in (1) is an effective dynamic classes of waves. Milton et al. [5] showed that the coor- mass density that is not necessarily tied to the physical dinate transform approach cannot be extended in general to mass density of any of the individual elements of a com- elastic media. Cummer and Schurig [6] showed, however, posite material [9,10]. Milton et al. [5] describe a simple an exact analogy exists between 2D electromagnetics and conceptual model based on mass-loaded springs embedded acoustics for anisotropic materials and therefore that a 2D in a host matrix that is described by an anisotropic mass acoustic cloaking shell exists. This isomorphism does not density. An applied force near the internal resonant fre- extend to three dimensions, however, which means that a quency of such an inclusion will produce large motions 3D acoustic cloaking shell, if it exists, is different from its thereby giving the medium a small effective mass density. electromagnetic counterpart. If the internal springs vary with direction, the effective Chen et al. [7] performed a spherical harmonic scatter- mass density will also vary with direction and thus be ing analysis of the 3D spherical cloaking shell described in anisotropic. A composite material containing anisotropic [1] and confirmed that this shell renders any object in its mechanically resonant inclusions in a fluid host matrix will interior free of scattering in all directions. Here we use obey the scalar stress-strain relation because of the fluid scattering analysis as the starting point to derive a set of matrix yet will have an anisotropic dynamic response, and acoustic material parameters for a shell that renders a 3D therefore an anisotropic dynamic mass density, when object in the interior of the shell free of acoustic scattering. forces are applied. For an inviscid fluid with zero shear modulus, the small We consider these fields in a domain shown in Fig. 1,in perturbation dynamics are described by conservation of which a uniform fluid of isotropic density 0 and bulk momentum and the stress-strain relation. Because [6] modulus 0 is present for r>band r<a, while for a< showed that mass density anisotropy is required for 2D r<b the material is an inhomogeneous and anisotropic acoustic cloaking, we assume this anisotropy from the shell to be specified. A uniform plane wave is incident on 0031-9007=08=100(2)=024301(4) 024301-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 024301 (2008) 18 JANUARY 2008 2 @ r @f 2 2 k0 r ÿ n n 1 f 0; (5) @r r @r where we have used the wave number in the background 2 2 2 medium k0 ! =vp0. The resulting equations for g and h are in standard form (e.g., [11]) and their solu- tions are the associated Legendre functions g m K0Pn cos (the other Legendre function has been ex- cluded because of the domain) and the azimuthal harmon- ics h K1 cosm K2 sinm.However,(5) is not in general the spherical Bessel equation because of the r dependence of the acoustic properties of the shell. Chen et al. [7] showed that an important element in making the 3D electromagnetic cloaking shell scatter- free is a radial shift from r to r ÿ a in the equation for FIG. 1. The problem domain, in which a compressional uni- f r (which is the Riccati-Bessel equation in the electro- form plane wave propagating along the 0 direction is incident on a material shell with inner radius a and outer radius magnetic case) that is produced by the radially dependent b. parameters of the medium. Recognizing this as one poten- tial path for realizing a scatter-free shell, we now find the the shell from the 0 direction, without loss of conditions on r, , and that transform (5) into the generality. standard spherical Bessel equation in r ÿ a. These con- Symmetry dictates that if a cloaking shell exists, its ditions are scattering parameters must be independent of the incident r2 r ÿ a2 wave direction. Therefore, its material properties must ; (6) depend only on radius, and the principal axes of the mass r k1 density tensor must be aligned with spherical coordinate directions. The material properties to be determined are k1; (7) r r, r, r, and r. Moreover, symmetry implies that r r . These parameters are all assumed to be 2 2 2 2 k0r ksh r ÿ a ; (8) normalized with respect to the background density 0 and bulk modulus 0. It is straightforward to show that p satisfies the equation where k1 and ksh are constants to be determined later. Under these conditions (5) becomes !2 ÿ1 @ 2 @f 2 2 r rp p 0: (3) r ÿ a k r ÿ a ÿ n n 1f 0; (9) @r @r sh We first focus on the solution to (3) inside the anisotropic which has the solution f rbnksh r ÿ a, where bn x shell. In this region and in spherical coordinates, this is a spherical Bessel or Hankel function of order n. equation becomes The total pressure field in all regions can now be ex- pressed. For r>b, the spherical expansion of the incident 2 2 ! 2 @ r @p @ @p compressional plane wave gives 2 r p sin vp0 @r r @r sin @ @ X1 inc p K j k rP cos; (10) @2p n n 0 n 0; n0 2 @2 sin n where Kn i 2n 1 and Pn cos is the nth degree (4) Legendre polynomial. The scattered field is subject to the radiation condition and, because of the azimuthal invari- where we have used the compressionalp wave velocity in the ance of the source and geometry, can be written as background medium vp0 0=0 and r r. X1 The azimuthal mass density elements have been moved scat 1 p A h k rP cos; (11) outside their derivatives because of their purely radial n n 0 n n0 dependence. 1 Letting p r; ; f rg h through separation with hn k0r the spherical Hankel function of the first of variables and after multiplying all terms by kind, and An are constants to be determined by the bound- ÿ1 sin, the ordinary differential equation for f r ary conditions. For a<r<b, the azimuthal invariance becomes means that the pressure is given by 024301-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 024301 (2008) 18 JANUARY 2008 X1 X1 sh scat ÿk0 10 p Bnjnksh r ÿ aPn cos; (12) vr Anhn k0rPn cos; (16) n0 i!0 n0 X1 where the jn spherical Bessel function is used to ensure the sh ÿksh 0 v B j k r ÿ aP cos; (17) fields remain finite at r a.