## Broadband cloaking and mirages with flying carpets

Optics Express, Vol. 18, Issue 11, pp. 11537-11551 (2010)

http://dx.doi.org/10.1364/OE.18.011537

Acrobat PDF (1386 KB)

### 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

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]

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]

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]

## 2. Governing equations for pressure and transverse electric waves

*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.

*z*invariance, the time harmonic acoustic equations of state simplify to (the exp(-

*jωt*) convention is used throughout)

*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).

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

*H*is the longitudinal (only non-zero) component of the magnetic field,

_{z}*ε*is the dielectric relative permittivity,

_{r}*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.

*ρ*̳ and a varying (scalar) bulk modulus

*λ*, see Eq. (7), whereas Eq. (2) would involve an anisotropic (heterogeneous) permittivity

*ε*̳ 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

*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:

**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]

**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]

**H**

_{l}=

*H*(

_{z}*x*,

*y*)

**e**

_{z},

*ε*

^{′}̳ and

*μ*

^{′}̳ are defined by:

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

**Property**:

*Let*

*be a real symmetric matrix. Then we have*∇×(

**M**∇×(

*u*(

*x*,

*y*)

**e**

_{z})) = -∇· (

**M**̃

^{-1}

*det*(

**M**̃)∇

*u*(

*x*,

*y*)

**e**

_{z}.

**M**

^{′}be defined as

**M**∇×(

*u*(

*x*,

*y*)

**e**

_{z}) = -∇·(

**M**

^{′}∇

*u*)

**e**

_{z}, if and only if

**M**

^{′}=

**M**̃

^{-1}det(

**M**̃). Using the above property, from Eq. (4), we derive the transformed equation associated with Eq. (2):

*ε*

^{′}̳

_{T}

^{-1}=

*ε*

_{r}^{-1}

**T**̃/det(

**T**̃). Here,

**T**̃ denotes the upper diagonal part of the transformation matrix

**T**and

*T*its third diagonal entry. Invoking the one-to-one correspondence Eq. (3), we infer that the transformed equation associated with Eq. (1) reads

_{zz}*ρ*̳

^{′}

_{T}

^{-1}=

*ρ*

_{0}

^{-1}

**T**̃/det(

**T**̃). 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

## 4. Flying carpets over a flat ground plane

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

*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.

### 4.1. The construction of carpets over horizontal planes

*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

### 4.2. Analysis of the metamaterial properties

**T**

^{-1}are given by

*λ*,

_{i}*i*= 1,2, and

*λ*

_{3}are strictly positive functions as obviously

*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]

*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

*r*

_{0}:= 0.2,

*k*

_{0}= 0.1,

*a*

_{0}= 0 in, one gets

*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

*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

*h*is defined as

### 4.4. Numerical results for a carpet over a plane

*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.

*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

*y*= -1.26 and is centered about

*x*= 0. In panel (d), it has three carpets shaped as tents attached to its sides. The construction of the carpets is as follows: The right-most tent is defined by

*x*

_{0}= 0.2,

*x*

_{1}(

*y*) = -

*y*-0.66,

*x*

_{2}(

*y*)= -

*y*- 0.56 (upper part) and

*x*

_{1}(

*y*)=

*y*+1.46,

*x*

_{2}(

*y*)=

*y*+1.56 (lower part). The left-most tent is defined by

*x*

_{0}= -0.2,

*x*

_{1}(

*y*) = -

*y*-1.46,

*x*

_{2}(

*y*) = -

*y*-1.56 (lower part), and

*x*

_{1}(

*y*) =

*y*+0.66,

*x*

_{2}(

*y*) =

*y*+0.56 (upper part). The pressure and bulk modulus (resp. permittivity and permeability) of these left-most and right-most tents is deduced from the expression of the transformation matrix for vertical walls, see Eq. (11). Finally, the uppermost tent is defined by

*y*

_{0}= -0.86,

*y*

_{1}(

*x*) = -

*x*-0.66,

*y*

_{2}(

*x*) = -

*x*-0.56 (right part) and

*y*

_{1}(

*x*) =

*x*-0.66,

*y*

_{2}(

*x*) =

*x*-0.56 (to the left). Here, we use the expression of the transformation matrix for horizontal walls, see Eq. (10). We now look at the case of a floating rigid square cylinder of sidelength 0.4 in mid-water (e.g. a pipeline). In panel (b), the obstacle is flying on its own. In panel (e), it has three tentlike carpets on its sides and yet still scatters the incoming wave as in (b). The construction of the carpets in panel (e) is as follows: The right-most tent is defined by

*x*

_{0}= 0.2,

*x*

_{1}(

*y*)=-

*y*+0.4,

*x*

_{2}(

*y*)=-

*y*+0.5 (upper part) and

*x*

_{1}(

*y*)=

*y*+0.4,

*x*

_{2}(

*y*)=

*y*+0.5 (lower part). Similarly, the left-most tent is defined by

*x*

_{0}= -0.2,

*x*

_{1}(

*y*) = -

*y*-0.4,

*x*

_{2}(

*y*) = -

*y*-0.5 (lower part) and

*x*

_{1}(

*y*) =

*y*-0.4,

*x*

_{2}(

*y*) =

*y*-0.5 (upper part). Finally, the uppermost tent is defined by

*y*

_{0}= 0.2,

*y*

_{1}(

*x*) = -

*x*+0.4,

*y*

_{2}(

*x*) = -

*x*+0.5 (right) and

*y*

_{1}(

*x*) =

*x*+0.4,

*y*

_{2}(

*x*) =

*x*+0.5 (left). We note that while forward and backward scattering in panels (d) and (e) are not negligible (as would be the case for acoustic/electromagnetic cloaks), these are instantly recognizable as that of a square obstacle on the ground (panel a) or floating in mid-air/water (panel b). The next question to address is whether such a generalized cloaking works at other incidences.

## 5. Flying carpets surrounding a circular cylinder over a plane

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

*R*

_{0}. The radius of such an effective rigid cylinder can be of any size smaller than that of the coated region. Such a carpet acts in a way similar to the invisibility cloaks proposed by Greenleaf et al. and Kohn et al. which are based upon the blow up of a small ball (of radius

*η*≪ 1) rather than a point, thereby leading to approximate invisibility. However, in our case, the radius of this ball (a disc in 2D) is finite, and our claim is that the carpet mimics the electromagnetic response of the rigid circular cylinder, just like previous carpets did for planes. The circular carpet actually plays the opposite role to the super-scatterer of Nicorovici, McPhedran and Milton whereby a cylinder surrounded by a coating of negative refractive index material scatters as a cylinder of diameter larger than the coating itself [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]

### 5.1. The construction of circular carpets

*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]

**312**, 1780–1782 (2006). [CrossRef] [PubMed]

*R*

_{0}= 0 of the radius of an imaginary small cylinder as explained below.

^{3}\

*C*(0,

*R*

_{0}) of a solid cylinder

*C*(0,

*R*

_{0}) of radius

*R*

_{0}, where

*R*

_{0}<

*R*

_{1}. It coincides with the identity map outside the solid cylinder

*C*(0,

*R*

_{2}), fixing its boundary point-wise, but now maps the region 𝓐 (

*R*

_{0},

*R*

_{2}) between the two coaxial cylinders of respective radii

*R*

_{0}and

*R*

_{2}into the space 𝓐 (

*R*

_{1},

*R*

_{2}) between the cylinders of radii

*R*

_{1}and

*R*

_{2}, as in Fig. 4. More precisely, in 𝓐 (

*R*

_{0},

*R*

_{2}) this geometric transformation can be expressed as

**J**

_{rr′}of the latter is

**J**

_{xr}the Jacobian of the change (

*r*,

*θ*,

*z*) ↦ (

*x*,

*y*,

*z*) from Cartesian to polar coordinates and

**J**

_{rx}:=

**J**

^{-1}

_{xr}1. 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**

_{r′x′}, so that the tensor

**T**

^{-1}=

**J**

^{-1}

_{xx′}

**J**

^{-T}

_{xx′}det(

**J**

_{xx′}) reads

*θ*in the

*xy*-plane and

### 5.2. Analysis of the metamaterial properties

**T**

^{-1}diagonalizes as diag

**312**, 1780–1782 (2006). [CrossRef] [PubMed]

*R*

_{0}tends to zero, one recovers the case in [9

**312**, 1780–1782 (2006). [CrossRef] [PubMed]

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

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]

## 6. Multilayered circular carpet for broadband mirage effect

### 6.1. Reduced material parameters for circular carpets

**T**in order to avoid a varying (scalar) density (resp. permeability in optics). For this, we introduce the reduced matrix

**T**

^{-1}in Eq. (13) by

*α*/

*m*. We deduce from Eq. (1) the transformed governing equation for pressure waves in reduced coordinates:

### 6.2. Homogenized governing equations for optics and acoustics

*λ*

^{-1}>= ∫

_{0}

^{1}

*λ*

^{-1}(

*r*)

*dr*and with

*ρ*̳ a homogenized rank 2 diagonal tensor (an anisotropic density)

*ρ*̳ = Diag(

*ρ*,

_{r}*ρ*) given by

_{θ}*ρ*̳ = Diag(<

*ρ*

^{-1}> ,<

*ρ*>).

*d*and

_{A}*d*, with bulk moduli

_{B}*λ*and

_{A}*λ*and densities

_{B}*ρ*and

_{A}*ρ*, we have

_{B}*η*=

*d*/

_{B}*d*is the ratio of thicknesses for layers

_{A}*A*and

*B*and

*d*+

_{A}*d*= 1. Using the change of variables, we get ∇

_{B}_{r,θ}·(

*ε*

^{-1}

_{r}

*ε*̳

^{-1}∇

_{r,θp})+

*ε*

_{0}

*μ*

_{0}

*ω*

^{2}

*H*= 0 , where

_{z}*ε*̳ is a homogenized rank 2 diagonal tensor (an anisotropic permittivity)

*ε*̳ = Diag(

*ε*,

_{r}*ε*) given by

_{θ}### 6.3. Electromagnetic and acoustic paradigms: Reduced scattering with larger scatterer

*μ*(resp. bulk modulus

*λ*) and a spatially varying rank 2 permittivity (resp. density) tensor

*(resp.*ε _ _

*ρ*̳) 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,

*ε*(resp.

_{A}*ρ*) varies in the range [0.1890;0.5493] and

_{A}*ε*(resp.

_{B}*ρ*) varies in the range [1.7987;2.1472]. We checked that this carpet is broadband as it works over the range of wavelengths

_{B}*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.

## 7. Conclusion

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]

## Acknowledgments

## References and links

1. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Usp. Fiz. Nauk |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

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. |

4. | S. A. Ramakrishna, “Physics of negative refraction,” Rep. Prog. Phys. |

5. | N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B |

6. | G. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A |

7. | A. Alu and N. Engheta, “Achieving Transparency with Plasmonic and Metamaterial Coatings,” Phys. Rev. E |

8. | U. Leonhardt, “Optical conformal mapping,” Science |

9. | J.B. Pendry, D. Shurig, and D.R. Smith, “Controlling electromagnetic fields,” Science |

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 |

11. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. |

12. | A. Diatta, S. Guenneau, A. Nicolet, and F. Zolla, “Tessellated and stellated invisibility,” Opt. Express |

13. | A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. |

14. | R.V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. |

15. | R. Weder, “A Rigorous Analysis of High-Order Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical |

16. | R. Weder, “The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical |

17. | S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. |

18. | D. Torrent and J. Sanchez-Dehesa, “Anisotropic mass density by two-dimensional acoustic metamaterials,” N. J. Phys. |

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. |

20. | H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. |

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. |

22. | M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B |

23. | M. Farhat, S. Guenneau, and S. Enoch, “Ultrabroadband Elastic Cloaking in Thin Plates,” Phys. Rev. Lett. |

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. |

25. | M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. |

26. | A. N. Norris, “Acoustic cloaking theory,” Proc. Roy. Soc. Lond. A |

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 |

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 |

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 |

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 |

31. | S. Zhang, L. Yin, and N. Fang, “Focusing ultrasound with an acoustic metamaterial network,” Phys. Rev. Lett. |

32. | A. Sukhovich, L. Jing, and J. H. Page, “Negative refraction and focusing of ultrasound in two-dimensional phononic crystals,” Phys. Rev. B |

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. |

34. | J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. |

35. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science |

36. | L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. |

37. | F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E |

38. | J. B. Pendry and J. Li, “An acoustic metafluid: realizing a broadband acoustic cloak,” N. J. Phys. |

39. | E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A |

40. | N. A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” N. J. Phys. |

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. |

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. |

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. |

45. | J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. |

**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

- V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Usp. Fiz. Nauk 92, 517 (1967).
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 86, 3966–3969 (2000). [CrossRef]
- 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]
- S. A. Ramakrishna, “Physics of negative refraction,” Rep. Prog. Phys. 68, 449 (2005). [CrossRef]
- 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]
- G. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. Lond. A 462, 3027 (2006). [CrossRef]
- A. Alu and N. Engheta, “Achieving Transparency with Plasmonic and Metamaterial Coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]
- U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- A. Diatta, S. Guenneau, A. Nicolet, and F. Zolla, “Tessellated and stellated invisibility,” Opt. Express 17, 13389–13394 (2009). [CrossRef] [PubMed]
- A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
- 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]
- R. Weder, “A Rigorous Analysis of High-Order Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical 41, 065207 (2008). [CrossRef]
- R. Weder, “The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks,” J. Phys. A: Mathematical and Theoretical 41, 415401 (2008). [CrossRef]
- S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9, 45 (2007). [CrossRef]
- D. Torrent and J. Sanchez-Dehesa, “Anisotropic mass density by two-dimensional acoustic metamaterials,” N. J. Phys. 10, 023004 (2008). [CrossRef]
- 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]
- H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007). [CrossRef]
- 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]
- 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]
- M. Farhat, S. Guenneau and S. Enoch, “Ultrabroadband Elastic Cloaking in Thin Plates,” Phys. Rev. Lett. 103, 024301 (2009). [CrossRef] [PubMed]
- 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]
- M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]
- A. N. Norris, “Acoustic cloaking theory,” Proc. Roy. Soc. Lond. A 464, 2411 (2008). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- S. Zhang, L. Yin, and N. Fang, “Focusing ultrasound with an acoustic metamaterial network,” Phys. Rev. Lett. 102, 194301 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003). [CrossRef]
- J. B. Pendry and J. Li, “An acoustic metafluid: realizing a broadband acoustic cloak,” N. J. Phys. 10, 115032 (2008). [CrossRef]
- E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79, 063825 (2009). [CrossRef]
- 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]
- 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]
- A. Nicolet, F. Zolla, and C. Geuzaine, “Generalized Cloaking and Optical Polyjuice,” ArXiv:0909.0848v1.
- 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]
- 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]
- J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34, 644 (2009). [CrossRef] [PubMed]

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