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

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  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).
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  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]
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  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]

2009 (11)

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

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]

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

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

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]

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]

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]

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

2008 (14)

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]

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

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]

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

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]

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]

2007 (3)

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

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]

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

2006 (6)

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]

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]

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

2005 (2)

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

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

2003 (3)

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

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

2000 (2)

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]

1998 (1)

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]

1994 (1)

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]

1967 (1)

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

Alu, A.

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

Argyropoulos, C.

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

Bigoni, D.

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]

Briane, M.

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]

Brun, M.

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

Cardenas, J.

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]

Chan, C. T.

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]

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]

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

Chen, H.

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]

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

Chen, H. Y.

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

Chen, Y. B.

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]

Chen, Y. F.

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]

Chin, J. Y.

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]

Cui, T. J.

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]

Cummer, S. A.

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]

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

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]

Diatta, A.

Engheta, N.

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

Enoch, S.

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]

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, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. 101, 134501 (2008).
[CrossRef] [PubMed]

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]

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]

Fang, N.

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

Farhat, M.

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]

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. 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]

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]

Feng, L.

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]

Feng, S.

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]

Fu, R.

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]

Gabrielli, L. H.

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]

Greenleaf, A.

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

Guenneau, S.

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

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]

M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[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, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. 101, 134501 (2008).
[CrossRef] [PubMed]

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]

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]

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

Han, D. Z.

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]

Hao, Y.

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

Hu, X.

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]

Ji, C.

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]

Jiang, X.

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]

Jing, L.

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]

Justice, B. J.

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]

Kallos, E.

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

Kildishev, A. V.

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]

Kohn, R.V .

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]

Lai, Y.

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]

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]

Lassas, M.

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

Leonhardt, U.

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

Li, J.

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

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

Lipson, M.

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]

Liu, R.

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]

Liu, X.

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]

Liu, X. P.

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]

Lu, M. H.

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]

Mao, Y. W.

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]

McPhedran, R. C.

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]

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]

Milton, G.

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

Milton, G. W.

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]

Milton, G.W.

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]

Ming, N. B.

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]

Mock, J. J.

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]

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]

Movchan, A. B.

M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[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, 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, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. 101, 134501 (2008).
[CrossRef] [PubMed]

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]

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]

Movchan, N. V.

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]

Nemat-Nasser, S. C.

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]

Ng, J.

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]

Nicolet, A.

Nicorovici, N. A.

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

Nicorovici, N. A. P.

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]

Norris, A. N.

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

Padilla, W. J.

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]

Page, J. H.

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]

Pendry, J.

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]

Pendry, J. B.

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

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

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]

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]

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

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

Poitras, C. B.

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]

Popa, B. I.

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]

Rahm, M.

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]

Ramakrishna, S. A.

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

Sanchez-Dehesa, J.

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

Schultz, S.

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]

Schurig, D.

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]

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

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]

Serkov, S. K.

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]

Shalaev, V. M.

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]

Shen, H.

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]

Shen, Y.

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]

Shurig, D.

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

Smith, D. R.

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]

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]

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]

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]

Smolyaninov, I. I.

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]

Smolyaninova, V. N.

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]

Starr, A.

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]

Starr, A. F.

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]

Sukhovich, A.

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]

Tayeb, G.

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]

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]

Torrent, D.

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

Uhlmann, G.

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

Valentini, M.

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]

Veselago, V. G.

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

Vier, D. C.

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]

Vogelius, M. S.

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]

Weder, R.

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]

Weinstein, M. I.

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]

Willis, J. R.

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]

Xiao, J. J.

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]

Yin, L.

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

Zhang, S.

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

Zhang, Z. Q.

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]

Zhang, Z.-Q.

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]

Zhu, S. N.

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]

Zhu, Y. Y.

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]

Zi, J.

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]

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]

Zolla, F.

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

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]

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]

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

Appl. Phys. Lett. (2)

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

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

Int. J. Solids Structures (1)

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]

Inverse Probl. (1)

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]

J. Phys. A: Mathematical and Theoretical (2)

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]

Math. Res. Lett. (1)

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

N. J. Phys. (6)

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]

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]

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

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]

Nat. Photon. (1)

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]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

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

Phys. Rev. B (4)

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]

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]

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, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B 79, 033102 (2009).
[CrossRef]

Phys. Rev. E (4)

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]

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]

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

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

Phys. Rev. Lett. (10)

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. 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]

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

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]

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

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]

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]

Proc. Roy. Soc. Lond. A (2)

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

Rep. Prog. Phys. (1)

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

Science (4)

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]

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]

Usp. Fiz. Nauk (1)

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

Other (1)

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

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Figures (7)

Fig. 1.
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.

Fig. 2.
Fig. 2.

2D plot of the real part of the total magnetic field ℜ e (Hz ) (resp. pressure field ℜ e (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 y 1 ( x ) = b 1 + 1 / 2 0.04 x 2 and y 2 ( x ) = b 1 + 0.04 x 2 . (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.

Fig. 3.
Fig. 3.

2D plot of the real part of the total magnetic field ℜ e (Hz ) (resp. pressure field ℜ e (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.

Fig. 4.
Fig. 4.

Construction of a circular carpet of inner radius R 1 (solid dark) and outer radius R 2 (dashed blue) from cylinder �� (0,R 0) of smaller radius R 0 (dotted red). The transformation (12) shrinks the whole hollow cylindrical region �� (R 0,R 2) of inner radius R 0 and outer radius R 2 into its subset �� (R 1,R 2), the cylinder �� (0,R 0) being stretched to �� (0,R 1) whereas �� (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 �� (0,R 0).

Fig. 5.
Fig. 5.

2D plot of the real part of the total magnetic field ℜ e (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.

Fig. 6.
Fig. 6.

2D plot of the real part of the total magnetic field ℜ e (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).

Fig. 7.
Fig. 7.

2D plot of the real part of the total magnetic field ℜ e (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.

Equations (29)

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ρ 0 v t = p , p t = λ · v ,
· ( ρ 0 1 p ) + ω 2 λ 1 p = 0 ,
. ( ε r 1 H z ) + ω 2 ε 0 μ 0 H z = 0 .
p H z , ρ 0 ε r , λ 1 ε 0 μ 0 ,
( dx dy dz ) = J xu ( du dv dw ) , with J xu = ( x , y , z ) ( u , v , w ) .
× ( ε = 1 × H l ) μ 0 ε 0 ω 2 μ = H l = 0
ε ' = = ε r T 1 and μ = = T 1 .
× ( M × ( u ( x , y ) e z ) ) = ( x ( m 22 u x m u y ) + y ( m 11 u y m u x ) ) e z .
m 11 ' u x + m 12 ' u y = m 22 u x m u y , m 21 ' u x + m 22 ' u y = m 11 u y m u x ,
· ε = T 1 H z + ω 2 ε 0 μ 0 T zz 1 H z = 0 ,
· ρ = T 1 p + ω 2 λ 1 T zz 1 p = 0 ,
J xu = J xX J Xu .
{ x = x y = α ( x ) y + β ( x ) z = z with inverse { x = x y = y β ( x ) α ( x ' ) z = z }
J xx = ( x , y , z ) ( x , y , z ) = ( 1 0 0 g 1 α 0 0 0 1 )
g : = y x = 1 α 2 ( α dx ( y β ) d x )
= ( y 2 y ) ( y 2 y 0 ) ( y 2 y 1 ) 2 d y 1 dx + ( y 1 y ) ( y 1 y 0 ) ( y 2 y 1 ) 2 d y 2 dx .
T 1 = J xx 1 J xx T det ( J xx ) = ( 1 α g 0 g ( 1 + g 2 ) α 0 0 0 1 α ) .
y 0 = b 0 , y 1 ( x ) = b 1 + ( 1 k 0 r 0 ) r 0 2 ( x a 0 ) 2 , y 2 ( x ) = b 1 + r 0 2 ( x a 0 ) 2 .
J xx = ( x , y , z ) ( x , y , z ) = ( 1 α h 0 0 1 0 0 0 1 )
h : = x y = 1 α 2 ( α d β d y ( x β ) d α d y )
= ( x 2 x ) ( x 2 x 0 ) ( x 2 x 1 ) 2 d x 1 d y + ( x 1 x ) ( x 1 x 0 ) ( x 2 x 1 ) 2 d x 2 d y .
T 1 = ( ( 1 + h 2 ) α h 0 h 1 α 0 0 0 1 α ) .
{ r = R 1 + α ( r R 0 ) with α = R 2 R 1 R 2 R 0 θ = θ z = z with inverse { r = R 0 + 1 α ( r R 1 ) θ = θ z = z .
T 1 = ( 1 + ( m 2 1 ) cos 2 ( θ ) m ( m 2 1 ) sin ( θ ) cos ( θ ) m 0 ( m 2 1 ) sin ( θ ) cos ( θ ) m m 2 + ( 1 m 2 ) cos 2 ( θ ) m 0 0 0 m α 2 ) = R ( θ ) diag ( m , 1 m , m α 2 ) R ( θ ) ,
r , θ · diag ( 1 α , m 2 α ) r , θ P + ω 2 λ 1 α P = 0 ,
r , θ · diag ( 1 α , m 2 α ) r , θ H z + ω 2 ε 0 μ 0 α H z = 0 , where m = 1 R 1 R 0 ( R 2 R 0 ) R 2 r and α = R 2 R 1 R 2 R 0 .
r , θ · ( ρ 0 1 ρ = 1 r , θ H z ) + ω 2 < λ 1 > p = 0 ,
1 ρ r = 1 1 + η ( 1 ρ A + η ρ B ) , ρ θ = ρ A + η ρ B 1 + η , < λ 1 > = 1 1 + η ( 1 λ A + η λ B ) ,
ε = = Diag ( < ε 1 > 1 , < ε > ) .

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