Abstract

In this paper, we derive the material parameter formulae for designing an electromagnetic invisibility anti-cloak of two-dimensional arbitrary geometry, which is conformal with the cloaked object. Different shapes of electromagnetic invisibility anti-cloaks are proposed to verify the correctness and effectiveness of the proposed formulae. The simulation results show that the invisibility anti-cloak can break cloak shielding and make the external electromagnetic waves into the cloak. This is not only to realize the transfer of information, but will not affect the role of cloak of stealth.

© 2013 OSA

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  1. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
    [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006).
    [CrossRef]
  4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006).
    [CrossRef] [PubMed]
  5. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys.8(10), 248 (2006).
    [CrossRef]
  6. 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,” Science314(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
    [CrossRef]
  8. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90(24), 241105 (2007).
    [CrossRef]
  9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008).
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  14. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express16(19), 14603–14608 (2008).
    [CrossRef] [PubMed]
  15. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009).
    [CrossRef] [PubMed]
  16. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
    [CrossRef]
  17. I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]

2011 (2)

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011).
[CrossRef] [PubMed]

2010 (2)

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010).
[CrossRef]

2009 (3)

2008 (4)

2007 (2)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90(24), 241105 (2007).
[CrossRef]

2006 (7)

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006).
[CrossRef]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys.8(10), 248 (2006).
[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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

Alù, A.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009).
[CrossRef] [PubMed]

Briane, M.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys.8(10), 248 (2006).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Castaldi, G.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009).
[CrossRef] [PubMed]

Chan, C. T.

Chen, H.

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90(24), 241105 (2007).
[CrossRef]

Chen, H. Y.

Chen, X.

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[CrossRef] [PubMed]

Engheta, N.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009).
[CrossRef] [PubMed]

Fu, Y.

Galdi, V.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009).
[CrossRef] [PubMed]

Gallina, I.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009).
[CrossRef] [PubMed]

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011).
[CrossRef] [PubMed]

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,” Science314(5801), 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. Photonics1(4), 224–227 (2007).
[CrossRef]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011).
[CrossRef] [PubMed]

Kwon, D. H.

D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010).
[CrossRef]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006).
[CrossRef]

Li, C.

Li, F.

Luo, X.

Luo, X. D.

Ma, H.

Ma, H. R.

Milton, G. W.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys.8(10), 248 (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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Ng, J.

Pendry, J.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 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. Express14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006).
[CrossRef]

Popa, B.-I.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[CrossRef] [PubMed]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008).
[CrossRef]

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008).
[CrossRef]

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 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. Express14(21), 9794–9804 (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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 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. Express14(21), 9794–9804 (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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011).
[CrossRef] [PubMed]

Werner, D. H.

D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010).
[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,” New J. Phys.8(10), 248 (2006).
[CrossRef]

Yang, T.

Yuan, N.

Appl. Phys. Lett. (1)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90(24), 241105 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010).
[CrossRef]

Nat. Photonics (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

New J. Phys. (2)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006).
[CrossRef]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys.8(10), 248 (2006).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Photon. Nanostruct.: Fundam. Appl. (1)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008).
[CrossRef]

Phys. Rev. B (1)

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006).
[CrossRef] [PubMed]

Science (3)

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Wave Motion (1)

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011).
[CrossRef]

Other (1)

Comsol Multiphysics” (Comsol AB), < http://www.comsol.com >.

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic diagram of stealth cloak with anti-cloak. (b) Coordinate transformation of the anti-cloak.

Fig. 2
Fig. 2

Total electric field distribution near the anti-cloak under an incident TE plane-wave from left to right for (a) a dieletic cylindrical object ( ε r =2, μ r =1 ) with radius 100mm in a cloak; (b) a cylindrical object ( ε r =2, μ r =1 ) surrounded by an ideal anti-cloak embedded in the cloak; (c) a cylindrical object ( ε r =2, μ r =1 ) surrounded by a simplified anti-cloak embedded in the cloak.

Fig. 3
Fig. 3

Total electric fields distribution near the anti-cloak filled with dielectric ( ε r =2, μ r =1 ) under an incident TE plane-wave from left to right for (a) an elliptical anti-cloak; (b) a -contour anti-cloak; (c) an arbitrary geometry anti-cloak.

Fig. 4
Fig. 4

Total electric field distribution near the anti-cloak filled with dielectric ( ε r =2, μ r =1 ) under an incident TE plane-wave from left to right for (a) a hexagon anti-cloak; (b) a discretization anti-cloak for unknown boundary object.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

{ u'=u'(u,v,w) v'=v'(u,v,w) w'=w'(u,v,w).
ε ¯ ¯ '= A ε ¯ ¯ A T detA , μ ¯ ¯ '= A μ ¯ ¯ A T detA ,
A=[ u' / u u' / v u' / w v' / u v' / v v' / w w' / u w' / v w' / w ].
r'= R 3 ( θ ) R 2 ( θ ) R 3 ( θ ) r+ R 2 ( θ ), θ'=θ, z'=z, 0r R 3 ( θ ).
r'= R 1 ( θ ) R 2 ( θ ) R 1 ( θ ) r+ R 2 ( θ ), θ'=θ, z'=z, 0r R 1 ( θ ).
r'= t i r+ R 2 (θ), i=1,2.
{ ε rr ={ r' R 2 (θ') r' + 1 r'[r' R 2 (θ')] [ R 2 (θ') θ ] 2 } ε r ε rθ = ε θr = 1 r' R 2 (θ') R 2 (θ') θ ε r ε θθ = r' r' R 2 (θ') ε r ε zz = r' R 2 (θ') t i 2 r' ε r .

Metrics