Abstract

In transformation optics, the space transformation is viewed as the deformation of a material. The permittivity and permeability tensors in the transformed space are found to correlate with the deformation field of the material. By solving the Laplace’s equation, which describes how the material will deform during a transformation, we can design electromagnetic cloaks with arbitrary shapes if the boundary conditions of the cloak are considered. As examples, the material parameters of the spherical and elliptical cylindrical cloaks are derived based on the analytical solutions of the Laplace’s equation. For cloaks with irregular shapes, the material parameters of the transformation medium are determined numerically by solving the Laplace’s equation. Full-wave simulations based on the Maxwell’s equations validate the designed cloaks. The proposed method can be easily extended to design other transformation materials for electromagnetic and acoustic wave phenomena.

© 2009 Optical Society of America

Full Article  |  PDF Article

Errata

Jin Hu, Xiaoming Zhou, and Gengkai Hu, "Design method for electromagnetic cloak with arbitrary shapes based on Laplace's equation: erratum," Opt. Express 17, 13070-13070 (2009)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-17-15-13070

References

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    [CrossRef] [PubMed]
  3. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  4. 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]
  5. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
    [CrossRef]
  6. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  7. U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006).
    [CrossRef]
  8. U. Leonhardt, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
    [CrossRef]
  9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2006).
    [CrossRef]
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    [CrossRef]
  23. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
    [CrossRef] [PubMed]
  24. M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008).
    [CrossRef] [PubMed]

2008

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
[CrossRef]

D. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008).
[CrossRef]

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008).
[CrossRef]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
[CrossRef] [PubMed]

Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, "Invisibility cloaks for irregular particles using coordinate transformations," Opt. Express 16, 6134-6145 (2008).
[CrossRef] [PubMed]

A. Nicolet, F. Zolla, and S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett. 33, 1584-1586 (2008).
[CrossRef] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008).
[CrossRef] [PubMed]

2007

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

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

2006

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

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (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]

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (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, 248 (2006).
[CrossRef]

U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006).
[CrossRef]

U. Leonhardt, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

2004

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

2003

A. Greenleaf, M. Lassas, and G. Uhlmann, "On non-uniqueness for Calderón’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

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, 248 (2006).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Chan, C. T.

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

Chen, B. W.

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Chen, H.

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

Cheng, Q.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008).
[CrossRef]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Chin, J. Y.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008).
[CrossRef]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

Cui, T. J.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
[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]

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, "On non-uniqueness for Calderón’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Guenneau, S.

Jiang, W. X.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (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]

Kattawar, G. W.

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Kwon, D.

D. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, "On non-uniqueness for Calderón’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Leonhardt, U.

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

U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006).
[CrossRef]

U. Leonhardt, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Li, Z.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Lin, X. Q.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

Liu, R.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Ma, H.

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008).
[CrossRef]

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
[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," New J. Phys. 8, 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," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Nicolet, A.

Pendry, J. B.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
[CrossRef] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008).
[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. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Qiu, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2006).
[CrossRef]

Qu, S.

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008).
[CrossRef]

Qu, S. B.

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Rahm, M.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
[CrossRef] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

Roberts, D. A.

M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

Ruan, Z.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2006).
[CrossRef]

Schurig, D.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 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," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

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

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
[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]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[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]

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, "On non-uniqueness for Calderón’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Wang, J.

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008).
[CrossRef]

Wang, J. F.

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Werner, D. H.

D. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (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," New J. Phys. 8, 248 (2006).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Xu, Z.

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008).
[CrossRef]

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[CrossRef]

Yan, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2006).
[CrossRef]

Yan, W.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2006).
[CrossRef]

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Yu, G. X.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
[CrossRef]

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Zhang, J. Q.

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
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J. Phys. D: Appl. Phys.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008).
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[CrossRef]

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[CrossRef]

Phys. Rev. A

H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Phys. Rev. E

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008).
[CrossRef]

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M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008)
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Figures (10)

Fig. 1.
Fig. 1.

The scheme of constructing an arbitrary cloak.

Fig. 2.
Fig. 2.

The contour plot of the electric field Ex in the x-z plane for a plane electromagnetic wave of the electric field polarized in the x direction incident on the cloak along the z direction, the material parameters of the cloak are given by Eq. (11). (The inner and outer radii of the cloak are λ/4 and λ/2 respectively, where λ is the wavelength, gray arrows indicate directions of the power flow).

Fig. 3.
Fig. 3.

The simulation environment of a 2D arbitrary cloak with the inner boundary a′ and outer boundary b′ illuminated by a plane electromagnetic wave of frequency 20GHz.

Fig. 4.
Fig. 4.

The coordinate lines in the transformed space distorted by the arbitrary cloak shown in Fig. 3.

Fig. 5.
Fig. 5.

The contour plots of the electric fields Ez for the TE waves incident on the cloak in Fig. 3, (a) horizontally and (b) at an angle of 45° from the x direction. (The white lines indicate directions of the power flow).

Fig. 6.
Fig. 6.

The contour plots of the material parameters (a) εzz , (b) μxx , (c) μxy , and (d) μyy of the cloak.

Fig. 7.
Fig. 7.

The scheme of a 2D arbitrary cloak with separated embedded obstacles. (The simulation environment is the same as that used in Fig. 3).

Fig. 8.
Fig. 8.

The contour plots of the electric fields Ez for the TE waves incident on the cloak in Fig. 7 horizontally (a) and at an angle of 45° from the x direction (b). (The white lines indicate directions of the power flow).

Fig. 9.
Fig. 9.

The coordinate lines in the transformed space distorted by an arbitrary cloak.

Fig. 10.
Fig. 10.

The contour plots of the electric fields Ez for the TE waves of frequency 20 GHz incident on an arbitrary cloak (a) from left to right and (b) from bottom to top. (The size of square simulation region is 8cm×8cm and the white lines indicate directions of the power flow).

Equations (27)

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ε ' = A T / det A ,
μ ' = A T / det A ,
ε = ε 0 B / det A ,
μ = μ 0 B / det A .
B = diag [ λ 1 2 , λ 2 2 , λ 3 2 ] ,
ε = ε 0 diag [ λ 1 λ 2 λ 3 , λ 2 λ 3 λ 1 , λ 3 λ 1 λ 2 ] ,
μ = μ 0 diag [ λ 1 λ 2 λ 3 , λ 2 λ 3 λ 1 , λ 3 λ 1 λ 2 ] .
( 2 x 1 2 + 2 x 2 2 + 2 x 3 2 ) U i = 0 , i = 1,2,3 ,
{ 1 r 2 [ r ( r 2 r ) ] + 1 r sin θ [ θ ( sin θ 1 r θ ) ] + 1 r 2 sin 2 θ φ 2 } U i = 0 , i = 1,2,3 .
d d r ( r 2 d d r ) = 0 .
r = b 2 b a ( 1 a r ) .
r = a b 2 ( a b ) r + b 2 .
λ r = d r d r = a b 2 ( b a ) r ,
λ θ = λ φ = r r = ( b a ) r 2 ( r a ) b 2 ,
ε r = μ r = λ r λ θ λ φ = a b 6 ( r a ) 2 r 3 ( b a ) ,
ε θ = μ θ = ε φ = μ φ = λ θ λ φ r r = 1 a b 2 ( b a ) r .
{ x = c ξ η y = ± c ( ξ 2 1 ) ( 1 η 2 ) z = z } , 1 ξ < , 1 η 1 , < z < ,
h ξ = c ξ 2 1 ξ 2 η 2 ,
h η = c 1 η 2 ξ 2 η 2 ,
h z = 1 .
( 1 h ξ 2 2 ξ 2 + 1 h η 2 2 η 2 + 2 z 2 ) U i = 0 .
λ ξ = h ξ d ξ h ξ d ξ ,
λ η = h η h η ,
λ z = 1 .
ε ξ = μ ξ = λ ξ λ η λ z = ξ 2 1 ξ 2 1 b a b 1 ,
ε η = μ η = λ η λ z λ ξ = ξ 2 1 ξ 2 1 b 1 b a ,
ε z = μ z = λ z λ ξ λ η = ξ 2 η 2 ξ 2 η 2 ξ 2 1 ξ 2 1 b 1 b a .

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