Abstract

Coordinate-transformation cloaking is based on the design of a metamaterial shell made of an anisotropic, spatially inhomogeneous “transformation medium” that allows rerouting the impinging wave around a given region of space. In its original version, it is generally believed that, in the ideal limit, the radiation cannot penetrate the cloaking shell (from outside to inside, and viceversa). However, it was recently shown by Chen et al. that electromagnetic fields may actually penetrate the cloaked region, provided that this region contains double-negative transformation media which, via proper design, may be in principle used to (partially or totally) “undo” the cloaking transformation, thereby acting as an “anti-cloak.” In this paper, we further elaborate this concept, by considering a more general scenario of cloak/anti-cloak interactions. Our full-wave analytical study provides new insightful results and explores the effects of departure from ideality, suggesting also some novel scenarios for potential applications.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  4. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794.
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    [CrossRef]
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2008

A. Alù and N. Engheta, "Plasmonic and metamaterial cloaking: physical mechanisms and potentials," J. Opt. A 10, 093002 (2008).
[CrossRef]

B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

H. Chen, X. Luo, H. Ma, and C. T. Chan, "The anti-cloak," Opt. Express 16, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603.
[PubMed]

2007

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

M. Yan, Z. C. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

2006

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006), http://www.iop.org/EJ/article/1367-2630/8/10/247/njp6_10_247.html.
[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]

G. W. Milton and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462, 3027-3059 (2006).
[CrossRef]

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

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794.
[CrossRef] [PubMed]

2005

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, "Plasmonic and metamaterial cloaking: physical mechanisms and potentials," J. Opt. A 10, 093002 (2008).
[CrossRef]

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

Chan, C. T.

H. Chen, X. Luo, H. Ma, and C. T. Chan, "The anti-cloak," Opt. Express 16, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603.
[PubMed]

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

Chen, H.

H. Chen, X. Luo, H. Ma, and C. T. Chan, "The anti-cloak," Opt. Express 16, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603.
[PubMed]

B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Chen, H. S.

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Chen, H. Y.

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007).
[CrossRef]

Cummer, S. A.

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]

Engheta, N.

A. Alù and N. Engheta, "Plasmonic and metamaterial cloaking: physical mechanisms and potentials," J. Opt. A 10, 093002 (2008).
[CrossRef]

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

Jiang, X. Y.

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[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]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

Kong, J. A.

B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006), http://www.iop.org/EJ/article/1367-2630/8/10/247/njp6_10_247.html.
[CrossRef]

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Liang, Z. X.

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

Luo, X.

Luo, Y.

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Ma, H.

Ma, H. R.

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

Milton, G. W.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

G. W. Milton and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462, 3027-3059 (2006).
[CrossRef]

Mock, J. 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]

Neff, C. W.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Nicorovici, N. A. P.

G. W. Milton and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462, 3027-3059 (2006).
[CrossRef]

Pendry, J. B.

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794.
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, 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]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006), http://www.iop.org/EJ/article/1367-2630/8/10/247/njp6_10_247.html.
[CrossRef]

Qiu, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

M. Yan, Z. C. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007).
[CrossRef]

Ran, L. X.

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Ruan, Z.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Ruan, Z. C.

M. Yan, Z. C. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, 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. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794.
[CrossRef] [PubMed]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, 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. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794.
[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]

Wu, B. I.

B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Yan, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

M. Yan, Z. C. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007).
[CrossRef]

Yao, P. J.

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

Zhang, B.

B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Appl. Phys. Lett.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

J. Opt. A

A. Alù and N. Engheta, "Plasmonic and metamaterial cloaking: physical mechanisms and potentials," J. Opt. A 10, 093002 (2008).
[CrossRef]

Nat. Photonics

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007).
[CrossRef]

New J. Phys.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006), http://www.iop.org/EJ/article/1367-2630/8/10/247/njp6_10_247.html.
[CrossRef]

Opt. Express

Phys. Rev. B

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007).
[CrossRef]

Phys. Rev. E

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

Phys. Rev. Lett.

B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

M. Yan, Z. C. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007).
[CrossRef]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Proc. R. Soc. London A

G. W. Milton and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462, 3027-3059 (2006).
[CrossRef]

Science

J. B. Pendry, D. Schurig, 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]

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Other

R. W. Ziolkowski and N. Engheta, "Introduction, history and fundamental theories of double-negative (DNG) metamaterials," in Metamaterials: Physics and Engineering Explorations, N. Engheta and R. W. Ziolkowski, eds., (Wiley-IEEE Press, 2006).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

COMSOL MULTIPHYSICS - User's Guide (COMSOL AB, 2005).

Supplementary Material (5)

» Media 1: MOV (2164 KB)     
» Media 2: MOV (2160 KB)     
» Media 3: MOV (2153 KB)     
» Media 4: MOV (2161 KB)     
» Media 5: MOV (2174 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Geometry of the problem. (a) Homogeneous circular cylinder in the auxiliary space. (b) Radial coordinate transformation, as in (2). Note that, in view of the vanishingly small character of the Δ2 and Δ3 parameters, the plotted case is hardly distinguishable from the ideal case. However, this case is strictly speaking different from the ideal case due to the presence of the Δ2 and Δ3 parameters. (c) Topological interpretation of the mapping, with

Fig. 2.
Fig. 2.

(a) Magnetic field (real part) map and time-domain animation (Media 1) for a configuration featuring a vacuum (ε 1 = ε 0, μ 1 = μ 0) inner cylinder and a DNG anti-cloak, with R 1 = 0.4λ 0, R 2 = 0.75λ 0, R 3 =1.7λ 0, R 4 = 2.5λ 0, Δ2 = R 2/200, Δ3 = R 3/200, and tanδ=10-4. (b) Magnified view with a superimposed map of the real part of the Poynting vector (normalized in the uncloaked regions, so that it is only indicative of the power flow direction).

Fig. 3.
Fig. 3.

As in Fig. 2, but with DNG (ε 1 =-(ε 0, μ 1= -μ 0) inner cylinder and DPS anti-cloak. See (Media 2) for a time-domain animation.

Fig. 4.
Fig. 4.

As in Fig.2 , but with ENG ((ε 1 = -(ε 0, μ 1 = μ 0) inner cylinder and MNG anti-cloak. See (Media 3) for a time-domain animation.

Fig. 5.
Fig. 5.

As in Fig. 2, but with MNG ((ε 1 = (ε 0, μ 1 = -μ 1) inner cylinder and ENG anti-cloak. See (Media 4) for a time-domain animation.

Fig. 6.
Fig. 6.

Geometry as in Fig. 2 (DPS inner cylinder, DNG anti-cloak). Exterior (a) and interior (b) parameters in (15) as a function of the parameter Δ3/R 3, for a fixed ratio Δ23 =R 2/R 3 ≈ 0.44 and different values of the loss-tangent. The blue dots indicate the values corresponding to the geometry of Fig. 2.

Fig. 7.
Fig. 7.

As in Fig. 2, but with a dielectric coaxial annular layer of radii Ra = λ 0 and Rb =1.5λ 0 (shown dashed) and permittivity εobj = 2ε 0 (tan δ = 10-4) inside the cloaked layer. See (Media 5) for a time-domain animation.

Fig. 8.
Fig. 8.

As in Fig. 6, but for a lossless cloak/anti-cloak configuration, and in the presence of a dielectric coaxial annular layer of radii Ra = λ 0 and Rb =1.5λ 0, inside the cloaked layer, for various values of the object permittivity εobj .

Fig. 9.
Fig. 9.

As in Fig. 8, but for a slightly lossy (tan δ = 10-4) cloak/anti-cloak configuration. The red dots indicate the values corresponding to the geometry of Fig. 7.

Equations (18)

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ε ( r ) = { ε 1 , 0 < r < R 2 ε 0 , r > R 2 , μ ( r ) = { μ 1 , 0 < r < R 2 μ 0 , r > R 2 ,
r = f ( r ) = { r , r < R 1 , r > R 4 R 1 ( R 2 + Δ 2 r R 2 + Δ 2 R 1 ) , R 4 ( r R 3 + Δ 3 R 4 R 3 + Δ 3 ) , R 2 < r < R 4 . R 1 < r < R 2 ,
ε r ( r ) = ε ( r ) ( r r ) [ df ( r ) dr ] 1 , ε ϕ ( r ) = ε ( r ) ( r r ) df ( r ) dr , μ z ( r ) = μ ( r ) ( r r ) df ( r ) dr .
H z ( inc ) ( r , ϕ ) = exp ( i k 0 x ) = n = i n J n ( k 0 r ) exp ( inϕ ) ,
H z ( r , ϕ ) = n = { ( a n ( v ) + δ v 5 i n ) J n [ g ( r ) ] + b n ( v ) Y n [ g ( r ) ] } exp ( inϕ ) ,
R v 1 < r < R v
v = 1 , . . . , 5 ,
E ϕ ( r , ϕ ) = 1 ε ϕ ( r ) H z ( r , ϕ ) r , E r ( r , ϕ ) = 1 ε r ( r ) H z ( r , ϕ ) r ϕ .
a n ( 5 ) = i b n ( 4 ) , a n ( 4 ) = i n i b n ( 4 ) , b n ( 2 ) = 0 , a n ( 1 ) = a n ( 2 ) .
a 0 ( 1 ) = a 0 ( 2 ) ~ 2 π k 0 2 R 2 R 3 Λ log ( k 0 R 4 Δ 3 R 4 R 3 ) , a 0 ( 3 ) ~ Y 1 ( k 0 R 2 ) k 0 R 3 Λ log ( k 0 R 4 Δ 3 R 4 R 3 ) g ,
b 0 ( 3 ) ~ J 1 ( k 0 R 2 ) k 0 R 3 Λ log ( k 0 R 4 Δ 3 R 4 R 3 ) , a 0 ( 5 ) = i b 0 ( 4 ) ~ 2 log ( k 0 R 4 Δ 3 R 4 R 3 ) ,
a n ( 1 ) = a n ( 2 ) ~ 4 n i n ε 1 π k 0 R 3 Ω n [ R 4 ( R 2 R 1 ) Δ n ε 1 μ 1 R 1 ( R 4 R 3 ) Δ 2 ] n ,
a n ( 3 ) ~ 2 1 n i n Ψ n ( 2 ) ( k 0 R 2 ) ( n 1 ) ! k 0 R 3 Ω n ( k 0 R 4 Δ 3 R 4 R 3 ) n , b n ( 3 ) ~ 2 1 n i n Ψ n ( 1 ) ( k 0 R 2 ) ( n 1 ) ! k 0 R 3 Ω n ( k 0 R 4 Δ 3 R 4 R 3 ) n ,
a n ( 5 ) = i b n ( 4,5 ) ~ π i n + 1 2 2 n [ J n + 1 ( k 0 R 3 ) Ψ n ( 2 ) ( k 0 R 2 ) + Y n + 1 ( k 0 R 3 ) Ψ n ( 1 ) ( k 0 R 2 ) n ! ( n 1 ) ! Ω n ]
× ( k 0 R 4 Δ 3 R 4 R 3 ) 2 n ,
Ω n = Ψ n ( 1 ) ( k 0 R 2 ) Y n 1 ( k 0 R 3 ) + Ψ n ( 2 ) ( k 0 R 2 ) J n 1 ( k 0 R 3 ) ,
Ψ n ( 1 ) ( x ) = n ( 1 ε 1 ) J n ( x ) + ε 1 x J n + 1 ( x ) , Ψ n ( 2 ) ( x ) = n ( ε 1 1 ) Y n ( x ) ε 1 x Y n + 1 ( x ) .
Q e = 2 π m = a n ( 5 ) 2 , Q i = ( R 1 2 R 2 3 R 2 2 ) R 2 R 3 0 2 π H z ( r , ϕ ) 2 rdrϕ 0 R 1 0 2 π H z ( r , ϕ ) 2 rdrϕ .

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