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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 11 — May. 24, 2010
  • pp: 11537–11551
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Broadband cloaking and mirages with flying carpets

André Diatta, Guillaume Dupont, Sébastien Guenneau, and Stefan Enoch  »View Author Affiliations


Optics Express, Vol. 18, Issue 11, pp. 11537-11551 (2010)
http://dx.doi.org/10.1364/OE.18.011537


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Abstract

This paper extends the proposal of Li and Pendry [Phys. Rev. Lett. 101, 203901-4 (2008)] to invisibility carpets for infinite conducting planes and cylinders (or rigid planes and cylinders in the context of acoustic waves propagating in a compressible fluid). Carpets under consideration here do not touch the ground: they levitate in mid-air (or float in mid-water), which leads to approximate cloaking for an object hidden underneath, or touch either sides of a square cylinder on, or over, the ground. The tentlike carpets attached to the sides of a square cylinder illustrate how the notion of a carpet on a wall naturally generalizes to sides of other small compact objects. We then extend the concept of flying carpets to circular cylinders and show that one can hide any type of defects under such circular carpets, and yet they still scatter waves just like a smaller cylinder on its own. Interestingly, all these carpets are described by non-singular parameters. To exemplify this important aspect, we propose a multi-layered carpet consisting of isotropic homogeneous dielectrics rings (or fluids with constant bulk modulus and varying density) which works over a finite range of wavelengths.

© 2010 Optical Society of America

1. Introduction

There is currently a keen interest in electromagnetic metamaterials within which very unusual phenomena such as negative refraction and focussing effects involving the near field can occur [1–4

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Usp. Fiz. Nauk 92, 517 (1967).

]. A circular cylinder coated with a negative refractive index displays anomalous resonances [5

5. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994). [CrossRef]

] and can even cloak a set of dipoles located in its close neighborhood [6

6. G. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A 462, 3027 (2006). [CrossRef]

]. The dielectric cylinder itself can be made transparent with a plasmonic coating [7

7. A. Alu and N. Engheta, “Achieving Transparency with Plasmonic and Metamaterial Coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]

].

During the same decade, some theoretical and experimental progress has been made towards a better understanding of interactions between linear surface water waves and periodic structures, leading to stop bands and lensing effects [28–30

28. X. Hu, Y. Shen, X. Liu, R. Fu, J. Zi, X. Jiang, and S. Feng, “Band structures and band gaps of liquid surface waves propagating through an infinite array of cylinders,” Phys. Rev. E 68, 037301 (2003). [CrossRef]

] as well as metamaterials designed as fluid networks [31

31. S. Zhang, L. Yin, and N. Fang, “Focusing ultrasound with an acoustic metamaterial network,” Phys. Rev. Lett. 102, 194301 (2009). [CrossRef] [PubMed]

, 32

32. A. Sukhovich, L. Jing, and J. H. Page, “Negative refraction and focusing of ultrasound in two-dimensional phononic crystals,” Phys. Rev. B 77, 014301 (2008). [CrossRef]

]. However, a further control of surface water waves can be achieved with a structured cloak [33

33. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. 101, 134501 (2008). [CrossRef] [PubMed]

].

In this paper, we focus our analysis on cloaking of pressure waves with carpets [34

34. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

] making use of some correspondences with electromagnetics, as the latter have recently led to experiments in the electromagnetic context [35

35. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366 (2008). [CrossRef]

, 36

36. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009). [CrossRef]

]. Here, we would like to render e.g. pipelines lying at the bottom of the sea or floating in mid-water undetectable for a boat sonar. These pipelines are considered to be infinitely long straight cylinders with a cross-section which is of circular or square shape. A pressure wave incident from above (the surface of the sea) hits the pipeline, so that the reflected wave reveals its presence to the sonar boat. We would like to show that we can hide the pipeline under a cylindrical carpet (a metafluid) so that the sonar only detects the wave reflected by the bottom of the sea.

2. Governing equations for pressure and transverse electric waves

For an inviscid fluid with zero shear modulus, the linearized equations of state for small amplitude perturbations from conservation of momentum, conservation of mass, and linear relationship between pressure and density are

ρ0vt=p,pt=λ·v,

where p is the scalar pressure, v is the vector fluid velocity, ρ 0 is the unperturbed fluid mass density (a mass in kilograms per unit volume in meters cube), and λ is the fluid bulk modulus (i.e. it measures the substance’s resistance to uniform compression and is defined as the pressure increase needed to cause a given relative decrease in volume, with physical unit in Pascal). This set of equations admits the usual compressional wave solutions in which fluid motion is parallel to the wavevector.

In cylindrical coordinates with z invariance, the time harmonic acoustic equations of state simplify to (the exp(-jωt) convention is used throughout)

·(ρ01p)+ω2λ1p=0,
(1)

where ω is the angular pressure wave frequency (measured in radians per unit second). Importantly, this equation is supplied with Neumann boundary conditions on the boundary of rigid defects (no flow condition).

Cummer and Schurig have shown that this equation holds for anisotropic heterogeneous fluids in cylindrical geometries [17

17. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9, 45 (2007). [CrossRef]

], and they have drawn some comparisons with transverse electromagnetic waves. In the transverse electric polarization (longitudinal magnetic field parallel to the cylinder’s axis):

.(εr1Hz)+ω2ε0μ0Hz=0.
(2)

where Hz is the longitudinal (only non-zero) component of the magnetic field, εr is the dielectric relative permittivity, ε 0 μ 0 is the inverse of the square velocity of light in vacuum, and ω is the angular transverse electric wave frequency (measured in radians per unit second). Importantly, this equation is supplied with Neumann boundary conditions on the boundary of infinite conducting defects.

In this paper, we look at such ‘acoustic’ models, for the case of flying carpets which are associated with geometric transforms. While we report computations for a pressure field, results apply mutatis mutandis to transverse electric waves making the changes of variables

pHz,ρ0εr,λ1ε0μ0,
(3)

in Eq. (1). However, after geometric transform, Eq. (1) will involve an anisotropic (heterogeneous) density ρ̳ and a varying (scalar) bulk modulus λ, see Eq. (7), whereas Eq. (2) would involve an anisotropic (heterogeneous) permittivity ε̳ and a varying (scalar) permeability μ, see Eq. (6). It is nevertheless possible to work with a reduced set of parameters, to avoid a varying λ in acoustics (resp. μ in optics), as we shall see in the last section of the paper.

3. From transformation optics to transformation acoustics

Let us consider a map from a co-ordinate system {u,v,w} to the co-ordinate system {x,y,z} given by the transformation characterized by x(u,v,w),y(u,v,w) and z(u,v,w). This change of co-ordinates is characterized by the transformation of the differentials through the Jacobian:

(dxdydz)=Jxu(dudvdw),withJxu=(x,y,z)(u,v,w).

In electromagnetics, this change of coordinates amounts to replacing the different materials (often homogeneous and isotropic, which corresponds to the case of scalar piecewise constant permittivity and permeability) by equivalent inhomogeneous anisotropic materials described by a transformation matrix T (metric tensor) T=J T J/det(J). The idea underpinning acoustic invisibility [9

9. J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

] is that newly discovered metamaterials should enable control of the pressure waves by mimicking the heterogeneous anisotropic nature of T with e.g. an anisotropic density, in a way similar to what was recently achieved with the permeability and permeability tensors in the microwave regime in the context of electromagnetism [10

10. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

]. In transverse electric polarisation, the Maxwell operator in the transformed coordinates writes as

×(ε=1×Hl)μ0ε0ω2μ=Hl=0
(4)

where H l = Hz(x,y)e z, ε ̳ and μ ̳ are defined by:

ε'==εrT1andμ==T1.
(5)

We would like to deduce the expression of Eq. (2) in the transformed coordinates from Eq. (4), in order to draw analogies with acoustics via Eq. (1). For this, we establish that [37

37. F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003). [CrossRef]

]:

Property: Let M=(m1m0mm22000m33)=:(M̃00m33) be a real symmetric matrix. Then we have ∇×(M∇×(u(x,y)e z)) = -∇· (M̃-1 det(M̃)∇u(x,y)e z.

Indeed, we note that

×(M×(u(x,y)ez))=(x(m22uxmuy)+y(m11uymux))ez.

Furthermore, let M be defined as M'=(m11'm12'm21'm22')

We have ∇×(M∇×(u(x,y)e z) = -∇·(M u)e z, if and only if

m11'ux+m12'uy=m22uxmuy,m21'ux+m22'uy=m11uymux,

which is true if M = M̃-1det(M̃). Using the above property, from Eq. (4), we derive the transformed equation associated with Eq. (2):

·ε=T1Hz+ω2ε0μ0Tzz1Hz=0,
(6)

with ε ̳T -1 = εr -1 T̃/det(T̃). Here, T̃ denotes the upper diagonal part of the transformation matrix T and Tzz its third diagonal entry. Invoking the one-to-one correspondence Eq. (3), we infer that the transformed equation associated with Eq. (1) reads

·ρ=T1p+ω2λ1Tzz1p=0,
(7)

with ρ̳ T -1 = ρ 0 -1 T̃/det(T̃). In the sequel we will also consider a compound transformation. Let us consider three coordinate systems {u,v,w}, {X,Y,Z}, and {x,y,z} (possibly on different regions of spaces). The two successive changes of coordinates are given by the Jacobian matrices J xX and J Xu so that

Jxu=JxXJXu.
(8)

This rule naturally applies for an arbitrary number of coordinate systems.

4. Flying carpets over a flat ground plane

This section is dedicated to the study of carpets levitating above a ground plane, that conceal to certain extent any object placed anywhere underneath them from plane waves incident from above. The construction of the carpet is a generalization of those considered in [34

34. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

] to carpets flying over ground planes or located on either sides of rectangular objects, as shown in Fig. 1 with certain altitude y = y 0. In the context of pressure waves, this corresponds for instance to the physical situation of a carpet which flies in mid-air if y = 0 is the altitude of the ground, or a carpet which floats in mid-waters if y = 0 stands for the bottom of the sea. This formalism allows us to study carpets which are either flying/floating on their own, or which are touching a cylindrical object on the ground or in mid-air/water.

Fig. 1. Construction of a carpet above the x-axis. The transformation in Eq. (9) shrinks the region between the two curves y = y 0 (dotted red) and y = y 2(x) (dashed blue) and two vertical segments x = x 0 and x = x 1 into the region between the curves y = y 1(x) (solid black) and y = y 2(x) (dashed blue) and the vertical segments x = x 0 and x = x 1 (carpet). The curvilinear metric inside the carpet is described by the transformation matrix T, see Eq. (10), corresponding to permittivity and permeability given by Eq. (5); or to density and bulk modulus for a metafluid, see Eq. (7). The grey rectangle can be either filled with air/ambient fluid (flying/floating carpet) or be replaced by an infinite conducting/rigid cylinder.

4.1. The construction of carpets over horizontal planes

When the carpet is made on the y-axis (we call it a carpet flying above the x-axis), we consider a transformation mapping the region enclosed between two curves (x,y 0) and (x,y 2(x)) to the one comprised between (x,y 1(x)) and (x,y 2(x)) as in Fig. 1, where (x,y 0) is mapped on (x,y 1(x)) and (x,y 2(x)) is fixed point-wise. If we write α=y2y1y2y0,β=y1y0y2y0y2, then such a transformation is of the form

{x=xy=α(x)y+β(x)z=zwithinverse{x=xy=yβ(x)α(x')z=z}
(9)

In Eq. (9), the transformation (x ,y ,z ) ↦ (x,y,z) has the following Jacobian matrix

Jxx=(x,y,z)(x,y,z)=(100g1α0001)

in which we have set

g:=yx=1α2(αdx(yβ)dx)
=(y2y)(y2y0)(y2y1)2dy1dx+(y1y)(y1y0)(y2y1)2dy2dx.

Hence we get

T1=Jxx1JxxTdet(Jxx)=(1αg0g(1+g2)α0001α).
(10)

4.2. Analysis of the metamaterial properties

Let us now look at an interesting feature of the invisibility carpet. The eigenvalues of T -1 are given by i=1,2:λi=12α(1+α2+g2α2+(1)i4α2+(1+α2+g2α2)2),λ3=1α.

We note that λi, i = 1,2, and λ 3 are strictly positive functions as obviously 1+α2+g2α2>4α2+(1+α2+g2α2)2 and also α > 0. This establishes that T -1 is not a singular matrix for a two-dimensional carpet even in the case of curved ground planes, which is one of the main advantages of carpets over cloaks [34

34. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

]. The broadband nature of such carpets remains to be investigated when one tries to mimic their ideal material parameters with structured media.

The carpets in Fig. 2 are made of a semi-circle y = y 2(x) as an outer curve and a semi-ellipse y = y 1(x) as its inner curve, both centered at (a 0, b 1) with

y0=b0,y1(x)=b1+(1k0r0)r02(xa0)2,y2(x)=b1+r02(xa0)2.

Plugging the numerical values r 0 := 0.2,k 0 = 0.1,a 0 = 0 in, one gets g=(5x(50b1(b0b1)+125x2+50y(b1b0))(125x2)32, where b 0 is the y-coordinate of the ground (typically, the ground can be taken as y = 0, so that b 0 = 0) and b 1 is the hight, measured on the y-axis from the origin, at which the carpet is flying.

4.3. The construction of carpets over vertical planes

We note that for a carpet flying above the y-axis, if we consider the transformation mapping the region enclosed between two curves (x 0(y),y) and (x 2(y),y) to the one comprised between (x 1(y),y) and (x 2(y),y) as in Fig. 1, where again (x 0(y),y) is mapped on (x 1(y),y) and (x 2(y),y) is fixed point-wise, that is, if x = α(y)x+β(y) with α=x2x1x2x0 and β=x1x0x2x0x2,y=y,z=z, then the Jacobian of the inverse transformation now reads

Jxx=(x,y,z)(x,y,z)=(1αh0010001)

where h is defined as

h:=xy=1α2(αdβdy(xβ)dαdy)
=(x2x)(x2x0)(x2x1)2dx1dy+(x1x)(x1x0)(x2x1)2dx2dy.

and hence we get

T1=((1+h2)αh0h1α0001α).
(11)

The same analysis as in Section 4.2 literally applies here, as well.

4.4. Numerical results for a carpet over a plane

We first look at the case of a pressure plane wave incident upon a flat rigid ground plane (Neumann boundary conditions) and a carpet above it. The inner boundary of the carpet is rigid (Neumann boundary conditions). We report these results in Fig. 2 where we can see that the altitude y = 0.9 leads to less scattering than the other two flying carpets. Of course, the carpet attached to the ground plane leads to perfect invisibility.

We then look at the case of the rigid ground plane with a rigid circular obstacle on top of it. Some Neumann boundary conditions are set on the ground plane, the inner boundary of the carpet and the rigid obstacle. Once again, we can see in Fig. 2 that the altitude y = 0.9 for the flying carpet is the optimal one. Interestingly, we note that the flying carpet with an object underneath [Fig. 2(f)] scatters even less field than the carpet on its own [Fig. 2(c)].

4.5. Numerical results for a carpet over a square cylinder over a plane

Fig. 2. 2D plot of the real part of the total magnetic field ℜe(Hz) (resp. pressure field ℜe(p)): Scattering by a plane wave of wavelength 0.15 incident from the top on a flat ground plane with a carpet above it, here y1(x)=b1+1/20.04x2 and y2(x)=b1+0.04x2. (a) carpet touching the ground, (b) carpet flying at altitude b 1 = 0.7, (c) at b 1 = 0.9 and (d) at b 1 = 1. The optimal altitude (for a flying carpet) b 1 = 0.9 is noted; (e) same as before for an infinite conducting circular cylinder (resp. rigid cylinder) without carpet; (f) same as (e) with a flying carpet at altitude b 1 = 0.9.

4.6. Numerical results for a carpet over a square cylinder over a plane in grazing incidence

We now look at the case of grazing incidence, keeping otherwise the same configuration as in Fig. 3. We note that the total field for the square obstacle is exactly the same with and without the three carpets, which is a further evidence that carpets allow for multi-incidence cloaking. Of course, one can hide any object (e.g. a semi-disc) under the carpets in panel (b) and this will scatter as a square. However, in the case of a flying object, it is required that the lower boundary of the hidden object be flat (e.g. a semi-disc) in order to mimic the diffraction pattern associated with a square obstacle. Taking into account that it should be possible to design broadband carpets as their material parameters are non-singular, our numerics suggest that carpets are thus a very interesting alternative to invisibility cloaks. They can either reduce the scattering cross section of a rigid object or mimic that of another rigid object. Finally, we should emphasize that we numerically checked that our proposal for generalized carpets works fine for a pressure plane wave incident on the square at any angles ranging from normal to grazing incidence. This should not come as a surprise as the same holds true in the original proposal by Li and Pendry [34

34. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

].

Fig. 3. 2D plot of the real part of the total magnetic field ℜe(Hz) (resp. pressure field ℜe(p)) for a plane wave of wavelength 0.15 incident on an infinite conducting (resp. rigid) square cylinder of sidelength d = 0.4; (a) Plane wave incident from above on a cylinder lying on a flat ground plane on its own; (b) Plane wave incident from above on a cylinder flying over the ground plane on its own; (c) Same as (b) for a plane wave incident from the left; (d,e,f) same as (a,b,c) for a cylinder surrounded by three (tentlike) carpets on its sides.

5. Flying carpets surrounding a circular cylinder over a plane

However, whether this coated region is completely empty (e.g. [Fig. 5(b)]) or many objects are hidden inside it (see e.g. [Fig. 5(d)] where the small cylinder of radius R 0 has actually been put back inside the carpet, along with other objects), such a carpet produces a mirage effect that tricks an external observer into believing that this whole region is just the small rigid cylinder.

5.1. The construction of circular carpets

The carpet consists of a cylindrical region C(0,R 1) of radius R 1 to be coated and the coating itself which is the space between an inner cylinder of radius R 1 and an outer one of radius R 2 > R 1. As above, the material properties of this coating will be deduced by pullback, via a transformation that fixes angles just like in [9

9. J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

]. The transformation in [9

9. J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

] is indeed the particular case, of the one under consideration here, associated with the value R 0 = 0 of the radius of an imaginary small cylinder as explained below.

{r=R1+α(rR0)withα=R2R1R2R0θ=θz=zwithinverse{r=R0+1α(rR1)θ=θz=z.
(12)

The Jacobian J rr of the latter isJrr=(r,θ,z)(r,θ,z)=diag(1α,1,1).Let us denote by J xr the Jacobian of the change (r,θ,z) ↦ (x,y,z) from Cartesian to polar coordinates and J rx := J -1 xr1. The Jacobian J xx of the above transformation in Cartesian coordinates (x,y ,z ) ↦ (x,y,z) is obtained by applying the chain rule in Eq. (8) to get J xx = J xr J rr J rx, so that the tensor T -1 = J -1 xx J -T xx det(J xx) reads

T1=(1+(m21)cos2(θ)m(m21)sin(θ)cos(θ)m0(m21)sin(θ)cos(θ)mm2+(1m2)cos2(θ)m000mα2)=R(θ)diag(m,1m,mα2)R(θ),
(13)

where R(θ)=(cos(θ)sin(θ)0sin(θ)cos(θ)0001) is the matrix of the rotation with angle θ in the xy-plane and m=αrr=1R1R0(R2R0)R2r.

5.2. Analysis of the metamaterial properties

The tensor T -1 diagonalizes as diag(m,1m,mα2) where m=1R1R0(R2R0)R2r is bounded from below and above as it satisfies 0<αR0R1mα. This means that the material parameters are not singular, unlike the cloaking case as in [9

9. J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

].

Obviously, one realizes that, when R 0 tends to zero, one recovers the case in [9

9. J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

] where the material properties are no longer bounded, but one of them tends to zero whilst another one recedes to infinity, as we approach the inner boundary of the coated region, see also [14

14. R.V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24, 015016 (2008). [CrossRef]

].

5.3. Mirage effect for a cylinder surrounded by a carpet

We report the results of our simulations in Fig. 5 for a circular pipeline which is floating in mid-water, see panel (a). We then replace this pipeline by a circular carpet, see panel (b), which reflects a pressure plane wave from above at wavelength 0.5 in exactly the same way. When we add three small pipelines to the original one, see panel (c), the reflected field is obviously much different. However, when we surround the four pipelines by the circular carpet, see panel (d), the reflected wave is that of the original pipeline. Such a mirage effect, whereby a rigid obstacle hides other ones in its neighborhood, can thus be used for ‘sonar illusions’. For instance, an oil pipeline might reflect pressure waves like a coral barrier so that a sonar boat won’t catch its presence. Unlike for earlier proposals of approximate cloaks [13

13. A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

, 14

14. R.V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24, 015016 (2008). [CrossRef]

] scattering waves like a small highly conducting object, we emphasize here that we start the construction of the circular carpet by a finite size disc.

Fig. 4. Construction of a circular carpet of inner radius R 1 (solid dark) and outer radius R 2 (dashed blue) from cylinder 𝒮 (0,R 0) of smaller radius R 0 (dotted red). The transformation (12) shrinks the whole hollow cylindrical region 𝓐 (R 0,R 2) of inner radius R 0 and outer radius R 2 into its subset 𝓐 (R 1,R 2), the cylinder 𝒮 (0,R 0) being stretched to 𝒮 (0,R 1) whereas 𝒮 (0,R 2) is fixed point-wise. Such a carpet with permittivity and permeability given by Eq. (13) and Eq. (5), scatters waves as an infinite conducting cylinder 𝒮 (0,R 0).

6. Multilayered circular carpet for broadband mirage effect

6.1. Reduced material parameters for circular carpets

We now want to simplify the expression of the inverse of the transformation matrix T in order to avoid a varying (scalar) density (resp. permeability in optics). For this, we introduce the reduced matrix Tred1=diag(α,αm2,1α) which amounts to multiplying T -1 in Eq. (13) by α/m. We deduce from Eq. (1) the transformed governing equation for pressure waves in reduced coordinates:

r,θ·diag(1α,m2α)r,θP+ω2λ1αP=0,

and similarly for transverse electric waves using Eq. (2):

r,θ·diag(1α,m2α)r,θHz+ω2ε0μ0αHz=0,wherem=1R1R0(R2R0)R2randα=R2R1R2R0.

6.2. Homogenized governing equations for optics and acoustics

To illustrate our paper with a practical example, we finally choose to design a circular carpet using 40 layers of isotropic homogeneous fluids. These fluids have constant bulk modulus and varying density. We report these computations in Fig. 6. It is indeed well known that the homogenized acoustic equation for such a configuration takes the following form:

r,θ·(ρ01ρ=1r,θHz)+ω2<λ1>p=0,
Fig. 5. 2D plot of the real part of the total magnetic field ℜe(Hz): Scattering by a plane wave of wavelength 0.15 incident from above on a flat ground plane and (a) a circular object of radius R 0 = 0.2 flying at altitude on its own; (b) an empty cylindrical region of radius R 1 = 0.5, the “coated” region, surrounded by a carpet of inner radius R 1 and outer radius R 2 = 0.7. This hollow cylindrical carpet is designed to behave exactly like the object in (a) alone, irrespective of the form of any other additional object that may be enclosed inside; (c) same object as in (a) with now three additional small rigid cylinders touching it. The scattering is clearly different from that in (a); (d) now the three objects in as (c) have been hidden inside the carpet (b) and yet an outer observer will not be able to tell the scattering in (a) and in (d) apart.

where < λ -1 >= ∫0 1 λ -1(r)dr and with ρ̳ a homogenized rank 2 diagonal tensor (an anisotropic density) ρ̳ = Diag(ρr,ρθ) given by ρ̳ = Diag(< ρ -1 > ,< ρ >).

We note that if the cloak consists of an alternation of two homogeneous isotropic layers of fluids of thicknesses dA and dB, with bulk moduli λA and λB and densities ρA and ρB, we have

1ρr=11+η(1ρA+ηρB),ρθ=ρA+ηρB1+η,<λ1>=11+η(1λA+ηλB),

where η = dB/dA is the ratio of thicknesses for layers A and B and dA + dB = 1. Using the change of variables, we get ∇r,θ·(ε -1 r ε̳-1r,θp)+ε 0 μ 0 ω 2 Hz = 0 , where ε̳ is a homogenized rank 2 diagonal tensor (an anisotropic permittivity) ε̳ = Diag(εr,εθ) given by

ε==Diag(<ε1>1,<ε>).
(14)

6.3. Electromagnetic and acoustic paradigms: Reduced scattering with larger scatterer

The optical (resp. acoustic) parameters of the proposed layered circular carpet are therefore characterized by a spatially varying scalar permeability μ (resp. bulk modulus λ) and a spatially varying rank 2 permittivity (resp. density) tensor ε__ (resp. ρ̳) given by Eq. (14). We can further simplify the problem by choosing reduced optical (resp. acoustic) parameters, so that the permeability μ (resp. bulk modulus λ) is now constant, and all the variation is reported on the permittivity (resp. density), see Fig. 6. More precisely, εA (resp. ρA) varies in the range [0.1890;0.5493] and εB (resp. ρB) varies in the range [1.7987;2.1472]. We checked that this carpet is broadband as it works over the range of wavelengths λ ∈ [0.2,1.4286], see Fig. 6 and Fig. 7: a multilayered carpet of radius 1 surrounding a rigid obstacle of radius 0.32 scatters waves just like a rigid obstacle of radius R 0 =0.2. We note that the lower bound for the range of working wavelengths corresponds to the infinite conducting (resp. rigid) obstacle we want to mimic.

Fig. 6. 2D plot of the real part of the total magnetic field ℜe(Hz): Scattering by a plane wave of wavelength 0.2 incident from above on a flat ground plane and (a) a circular object of radius R 0 = 0.2 flying at altitude on its own; (b) an empty cylindrical region of radius R 1 = 0.32, the “coated” region consisting of 40 layers of isotropic homogeneous dielectric, see closer view in (c), with permittivity ε, given in (d), surrounded by a carpet of inner radius R 1 and outer radius R 2 = 1. The red curves represents the variation of εθ = m 2/α with respect to r ∈ [0.32; 1]. The piecewise constant blue curve is a staircase approximation of the red curve, considering an alternation of 40 layers of density εA ∈ [0.1890; 0.5493] and εB ∈ [1.7987; 2.1472] using the homogenized formula in Eq. (14).

7. Conclusion

In this paper, we have proposed some models of flying carpets which levitate (or float) in mid-air (or mid-water). Such cloaks can be built from optical metamaterials or acoustic metafluids: as explained by Pendry and Li in a recent work, one can for instance emulate required anisotropic density and heterogeneous bulk modulus with arrays of rigid plates with a hemispherical sack of gaz attached to them [38

38. J. B. Pendry and J. Li, “An acoustic metafluid: realizing a broadband acoustic cloak,” N. J. Phys. 10, 115032 (2008). [CrossRef]

]. But other designs proposed by Torrent and Sanchez-Dehesa would work equally well [18

18. D. Torrent and J. Sanchez-Dehesa, “Anisotropic mass density by two-dimensional acoustic metamaterials,” N. J. Phys. 10, 023004 (2008). [CrossRef]

]. However, such flying carpets lead to some approximate cloaking as they do not touch the ground (the inner boundary of the carpet in the original design of Pendry and Li is attached to the ground [34

34. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

]. Interestingly, other authors also looked at quasi-cloaking with simplified carpets [39

39. E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79, 063825 (2009). [CrossRef]

]. We have also explained how one can hide an object located in the close neighborhood of a rigid circular cylinder, which in some sense can be classified as an external cloaking whereby a large scatterer hides smaller ones located nearby. Such an ostrich effect (which buries its head in the sand) has already been observed in the context of dipoles and even finite size obstacles located closeby a cylindrical perfect lens which displays anomalous resonances [6

6. G. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A 462, 3027 (2006). [CrossRef]

, 40

40. N. A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” N. J. Phys. 10, 115020 (2008). [CrossRef]

]. However, here the coating does not contain any negatively refracting material, and this is an anisotropy-led rather than plasmonic-type cloaking mechanism. Actually, it is possible to use complementary media to cloak finite size objects (rather than only dipoles) at a finite distance [41

41. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102, 093901 (2009). [CrossRef] [PubMed]

].

Fig. 7. 2D plot of the real part of the total magnetic field ℜe(Hz) for the same optogeometric parameters as in Fig. 6; Scattering by a plane wave of wavelength 0.5 (a–b) and 1.4286 (c–d), incident from above.

We have discussed some applications, with the sonar boats or radars cases as typical examples. Another possible application would be protecting parabolic antennas from the negative impact of their ‘supporting cable’. The feasibility of such carpets is demonstrated using a ho-mogenization approach enabling us to design a multi-layered acoustic metafluid leading to a mirage effect over a finite range of wavelengths. We note that the route towards anamorphism discussed in this paper is very different from the proposal of Nicolet et al. [42

42. A. Nicolet, F. Zolla, and C. Geuzaine, “Generalized Cloaking and Optical Polyjuice,” ArXiv:0909.0848v1.

] whereby an anisotropic heterogeneous object is placed within the coating of a singular cloak to mimic the scattering of another object.

Moreover, all computations hold for electromagnetic carpets built with heterogeneous anisotropic permittivity and scalar permittivity, which could be emulated using tapered waveguides as in [43

43. I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic Metamaterials Emulated by Tapered Waveguides: Application to Optical Cloaking,” Phys. Rev. Lett. 102, 213901 (2009). [CrossRef] [PubMed]

]. We therefore believe that the designs we proposed in this paper might foster experimental efforts in approximate cloaking for both acoustic and electromagnetic waves. Last, but not least, we note that the group of C.T. Chan has proposed an alternative approach to optical illusion based upon a negatively refracting medium [44

44. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “llusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102, 253902 (2009). [CrossRef] [PubMed]

, 45

45. J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34, 644 (2009). [CrossRef] [PubMed]

].

Acknowledgments

A. Diatta and S. Guenneau acknowledge funding from EPSRC grant EP/F027125/1.

References and links

1.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Usp. Fiz. Nauk 92, 517 (1967).

2.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 86, 3966–3969 (2000). [CrossRef]

3.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]

4.

S. A. Ramakrishna, “Physics of negative refraction,” Rep. Prog. Phys. 68, 449 (2005). [CrossRef]

5.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994). [CrossRef]

6.

G. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A 462, 3027 (2006). [CrossRef]

7.

A. Alu and N. Engheta, “Achieving Transparency with Plasmonic and Metamaterial Coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]

8.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

9.

J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

10.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

11.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]

12.

A. Diatta, S. Guenneau, A. Nicolet, and F. Zolla, “Tessellated and stellated invisibility,” Opt. Express 17, 13389–13394 (2009). [CrossRef] [PubMed]

13.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

14.

R.V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24, 015016 (2008). [CrossRef]

15.

R. Weder, “A Rigorous Analysis of High-Order Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical 41, 065207 (2008). [CrossRef]

16.

R. Weder, “The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical 41, 415401 (2008). [CrossRef]

17.

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9, 45 (2007). [CrossRef]

18.

D. Torrent and J. Sanchez-Dehesa, “Anisotropic mass density by two-dimensional acoustic metamaterials,” N. J. Phys. 10, 023004 (2008). [CrossRef]

19.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, J. Pendry, J. Pendry, M. Rahm, and A. Starr, “Scattering Theory Derivation of a 3D Acoustic Cloaking Shell,” Phys. Rev. Lett. 100, 024301 (2008). [CrossRef] [PubMed]

20.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007). [CrossRef]

21.

M. Farhat, S. Guenneau, S. Enoch, A. B. Movchan, F. Zolla, and A. Nicolet, “A homogenization route towards square cylindrical acoustic cloaks,” N. J. Phys. 10, 115030 (2008). [CrossRef]

22.

M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B 79, 033102 (2009). [CrossRef]

23.

M. Farhat, S. Guenneau, and S. Enoch, “Ultrabroadband Elastic Cloaking in Thin Plates,” Phys. Rev. Lett. 103, 024301 (2009). [CrossRef] [PubMed]

24.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8, 248 (2006). [CrossRef]

25.

M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]

26.

A. N. Norris, “Acoustic cloaking theory,” Proc. Roy. Soc. Lond. A 464, 2411 (2008). [CrossRef]

27.

D. Bigoni, S. K. Serkov, M. Valentini, and A. B. Movchan, “Asymptotic models of dilute composites with imperfectly bonded inclusions,” Int. J. Solids Structures 35, 3239 (1998). [CrossRef]

28.

X. Hu, Y. Shen, X. Liu, R. Fu, J. Zi, X. Jiang, and S. Feng, “Band structures and band gaps of liquid surface waves propagating through an infinite array of cylinders,” Phys. Rev. E 68, 037301 (2003). [CrossRef]

29.

L. Feng, X. P. Liu, M. H. Lu, Y. B. Chen, Y. F. Chen, Y. W. Mao, J. Zi, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, “Refraction control of acoustic waves in a square-rod-constructed tunable sonic crystal,” Phys. Rev. B 73, 193101 (2006). [CrossRef]

30.

M. Farhat, S. Guenneau, S. Enoch, G. Tayeb, A. B. Movchan, and N. V. Movchan, “Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of nearly touching rigid square cylinders,” Phys. Rev. E 77, 046308 (2008). [CrossRef]

31.

S. Zhang, L. Yin, and N. Fang, “Focusing ultrasound with an acoustic metamaterial network,” Phys. Rev. Lett. 102, 194301 (2009). [CrossRef] [PubMed]

32.

A. Sukhovich, L. Jing, and J. H. Page, “Negative refraction and focusing of ultrasound in two-dimensional phononic crystals,” Phys. Rev. B 77, 014301 (2008). [CrossRef]

33.

M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. 101, 134501 (2008). [CrossRef] [PubMed]

34.

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

35.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366 (2008). [CrossRef]

36.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009). [CrossRef]

37.

F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003). [CrossRef]

38.

J. B. Pendry and J. Li, “An acoustic metafluid: realizing a broadband acoustic cloak,” N. J. Phys. 10, 115032 (2008). [CrossRef]

39.

E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79, 063825 (2009). [CrossRef]

40.

N. A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” N. J. Phys. 10, 115020 (2008). [CrossRef]

41.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102, 093901 (2009). [CrossRef] [PubMed]

42.

A. Nicolet, F. Zolla, and C. Geuzaine, “Generalized Cloaking and Optical Polyjuice,” ArXiv:0909.0848v1.

43.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic Metamaterials Emulated by Tapered Waveguides: Application to Optical Cloaking,” Phys. Rev. Lett. 102, 213901 (2009). [CrossRef] [PubMed]

44.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “llusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102, 253902 (2009). [CrossRef] [PubMed]

45.

J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34, 644 (2009). [CrossRef] [PubMed]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(160.1050) Materials : Acousto-optical materials
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials
(230.3205) Optical devices : Invisibility cloaks
(290.5839) Scattering : Scattering, invisibility

ToC Category:
Physical Optics

History
Original Manuscript: April 1, 2010
Revised Manuscript: May 13, 2010
Manuscript Accepted: May 13, 2010
Published: May 17, 2010

Citation
André Diatta, Guillaume Dupont, Sébastien Guenneau, and Stefan Enoch, "Broadband cloaking and mirages with flying carpets," Opt. Express 18, 11537-11551 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11537


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References

  1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Usp. Fiz. Nauk 92, 517 (1967).
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 86, 3966–3969 (2000). [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]
  4. S. A. Ramakrishna, “Physics of negative refraction,” Rep. Prog. Phys. 68, 449 (2005). [CrossRef]
  5. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994). [CrossRef]
  6. G. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A 462, 3027 (2006). [CrossRef]
  7. A. Alu and N. Engheta, “Achieving Transparency with Plasmonic and Metamaterial Coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]
  8. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
  9. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
  10. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]
  11. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]
  12. A. Diatta, S. Guenneau, A. Nicolet, and F. Zolla, “Tessellated and stellated invisibility,” Opt. Express 17, 13389–13394 (2009). [CrossRef] [PubMed]
  13. A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
  14. R.V . Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24, 015016 (2008). [CrossRef]
  15. R. Weder, “A Rigorous Analysis of High-Order Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical 41, 065207 (2008). [CrossRef]
  16. R. Weder, “The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical 41, 415401 (2008). [CrossRef]
  17. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9, 45 (2007). [CrossRef]
  18. D. Torrent and J. Sanchez-Dehesa, “Anisotropic mass density by two-dimensional acoustic metamaterials,” N. J. Phys. 10, 023004 (2008). [CrossRef]
  19. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering Theory Derivation of a 3D Acoustic Cloaking Shell,” Phys. Rev. Lett. 100, 024301 (2008). [CrossRef] [PubMed]
  20. H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007). [CrossRef]
  21. M. Farhat, S. Guenneau, S. Enoch, A. B. Movchan, F. Zolla, and A. Nicolet, “A homogenization route towards square cylindrical acoustic cloaks,” N. J. Phys. 10, 115030 (2008). [CrossRef]
  22. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B 79, 033102 (2009). [CrossRef]
  23. M. Farhat, S. Guenneau and S. Enoch, “Ultrabroadband Elastic Cloaking in Thin Plates,” Phys. Rev. Lett. 103, 024301 (2009). [CrossRef] [PubMed]
  24. G.W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8, 248 (2006). [CrossRef]
  25. M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]
  26. A. N. Norris, “Acoustic cloaking theory,” Proc. Roy. Soc. Lond. A 464, 2411 (2008). [CrossRef]
  27. D. Bigoni, S. K. Serkov, M. Valentini, and A. B. Movchan, “Asymptotic models of dilute composites with imperfectly bonded inclusions,” Int. J. Solids Structures 35, 3239 (1998). [CrossRef]
  28. X. Hu, Y. Shen, X. Liu, R. Fu, J. Zi, X. Jiang, and S. Feng, “Band structures and band gaps of liquid surface waves propagating through an infinite array of cylinders,” Phys. Rev. E 68, 037301 (2003). [CrossRef]
  29. L. Feng, X. P. Liu, M. H. Lu, Y. B. Chen, Y. F. Chen, Y. W. Mao, J. Zi, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, “Refraction control of acoustic waves in a square-rod-constructed tunable sonic crystal,” Phys. Rev. B 73, 193101 (2006). [CrossRef]
  30. M. Farhat, S. Guenneau, S. Enoch, G. Tayeb, A. B. Movchan, and N. V. Movchan, “Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of nearly touching rigid square cylinders,” Phys. Rev. E 77, 046308 (2008). [CrossRef]
  31. S. Zhang, L. Yin, and N. Fang, “Focusing ultrasound with an acoustic metamaterial network,” Phys. Rev. Lett. 102, 194301 (2009). [CrossRef] [PubMed]
  32. A. Sukhovich, L. Jing, and J. H. Page, “Negative refraction and focusing of ultrasound in two-dimensional phononic crystals,” Phys. Rev. B 77, 014301 (2008). [CrossRef]
  33. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. 101, 134501 (2008). [CrossRef] [PubMed]
  34. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]
  35. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366 (2008). [CrossRef]
  36. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461-463 (2009). [CrossRef]
  37. F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003). [CrossRef]
  38. J. B. Pendry and J. Li, “An acoustic metafluid: realizing a broadband acoustic cloak,” N. J. Phys. 10, 115032 (2008). [CrossRef]
  39. E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79, 063825 (2009). [CrossRef]
  40. N. A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” N. J. Phys. 10, 115020 (2008). [CrossRef]
  41. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102, 093901 (2009). [CrossRef] [PubMed]
  42. A. Nicolet, F. Zolla, and C. Geuzaine, “Generalized Cloaking and Optical Polyjuice,” ArXiv:0909.0848v1.
  43. I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev and V. M. Shalaev, “Anisotropic Metamaterials Emulated by Tapered Waveguides: Application to Optical Cloaking,” Phys. Rev. Lett. 102, 213901 (2009). [CrossRef] [PubMed]
  44. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “llusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102, 253902 (2009). [CrossRef] [PubMed]
  45. J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34, 644 (2009). [CrossRef] [PubMed]

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