Abstract

An approach to design an invisible cloak with controlled constitutive parameters and arbitrary shaped boundaries is presented. Helmholtz’s equation is adopted to establish a mapping between original and transformed coordinates inside the cloak. Then the constitutive parameters are obtained by the established mapping. The analytical solution of a regular cloak and the numerical solution of an irregular cloak both verify that that our method will guide electromagnetic wave efficiently and control the constitutive parameters of the cloak conveniently. It has great significance in realizing a cloak practically.

©2009 Optical Society of America

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780(2006).
    [Crossref] [PubMed]
  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,” Science 314, 977–980(2006).
    [Crossref] [PubMed]
  4. 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. Nanostruc. Fundam. Appl. 6, 87–95(2008).
    [Crossref]
  5. Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Arbitrary N-side Regular Polygonal Invisible Cloak,” in Proceedings of the 2008 International Workshop on Metamaterials, (Academic, Nanjing, China, 2008), 60–63.
  6. 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]
  7. D. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
    [Crossref]
  8. 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]
  9. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10, 043040 (2008).
    [Crossref]
  10. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16, 13414–13420 (2008).
    [Crossref] [PubMed]
  11. 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]
  12. H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
    [Crossref] [PubMed]
  13. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformal invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
    [Crossref]
  14. 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]
  15. J. Hu, X. Zhou, and G. Hu, “Design Method for Electromagnetic Cloak with Arbitrary Shapes Based on Laplace’s Equation,” Arxiv.0807.3837.
  16. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Design of optical cloak with nonlinear transformation,” Opt. Express 16, 5444–5452(2008).
    [Crossref] [PubMed]

2008 (11)

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. Nanostruc. 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. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10, 043040 (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]

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

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Design of optical cloak with nonlinear transformation,” Opt. Express 16, 5444–5452(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]

C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16, 13414–13420 (2008).
[Crossref] [PubMed]

H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
[Crossref] [PubMed]

2006 (3)

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]

Cai, W.

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]

Cheng, 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]

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

Chettiar, U. K.

Chin, J. Y.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformal 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]

Cui, T. J.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformal 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. Nanostruc. 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]

Guenneau, S.

Hu, G.

J. Hu, X. Zhou, and G. Hu, “Design Method for Electromagnetic Cloak with Arbitrary Shapes Based on Laplace’s Equation,” Arxiv.0807.3837.

Hu, J.

J. Hu, X. Zhou, and G. Hu, “Design Method for Electromagnetic Cloak with Arbitrary Shapes Based on Laplace’s Equation,” Arxiv.0807.3837.

Jiang, W. X.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformal 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]

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.

Kwon, D.

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

Leonhardt, U.

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

Li, C.

Li, F.

Li, L. W.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Arbitrary N-side Regular Polygonal Invisible Cloak,” in Proceedings of the 2008 International Workshop on Metamaterials, (Academic, Nanjing, China, 2008), 60–63.

Li, Z.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformal 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 conformal 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. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
[Crossref] [PubMed]

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]

Meng, F. Y.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Arbitrary N-side Regular Polygonal Invisible Cloak,” in Proceedings of the 2008 International Workshop on Metamaterials, (Academic, Nanjing, China, 2008), 60–63.

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.

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]

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 (2008).
[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]

H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
[Crossref] [PubMed]

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, 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. Nanostruc. Fundam. Appl. 6, 87–95(2008).
[Crossref]

Roberts, D. 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. Nanostruc. 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 (2008).
[Crossref]

Schurig, D.

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. Nanostruc. Fundam. Appl. 6, 87–95(2008).
[Crossref]

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]

Shalaev, V. M.

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. Nanostruc. 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]

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]

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]

H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
[Crossref] [PubMed]

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]

Wu, Q.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Arbitrary N-side Regular Polygonal Invisible Cloak,” in Proceedings of the 2008 International Workshop on Metamaterials, (Academic, Nanjing, China, 2008), 60–63.

Xu, Z.

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]

H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
[Crossref] [PubMed]

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]

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 (2008).
[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 (2008).
[Crossref]

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]

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).
[Crossref]

Zhang, K.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Arbitrary N-side Regular Polygonal Invisible Cloak,” in Proceedings of the 2008 International Workshop on Metamaterials, (Academic, Nanjing, China, 2008), 60–63.

Zhou, X.

J. Hu, X. Zhou, and G. Hu, “Design Method for Electromagnetic Cloak with Arbitrary Shapes Based on Laplace’s Equation,” Arxiv.0807.3837.

Zolla, F.

Appl. Phys. Lett. (1)

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

J. Phys. D: Appl. Phys. (1)

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]

New J. Phys. (1)

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10, 043040 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Photon. Nanostruc. Fundam. Appl. (1)

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. Nanostruc. Fundam. Appl. 6, 87–95(2008).
[Crossref]

Phys. Rev. A (1)

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 (2)

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformal 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]

Science (3)

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]

Other (2)

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Arbitrary N-side Regular Polygonal Invisible Cloak,” in Proceedings of the 2008 International Workshop on Metamaterials, (Academic, Nanjing, China, 2008), 60–63.

J. Hu, X. Zhou, and G. Hu, “Design Method for Electromagnetic Cloak with Arbitrary Shapes Based on Laplace’s Equation,” Arxiv.0807.3837.

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Figures (4)

Fig. 1.
Fig. 1. Geometrical graph of the section of irregular invisible cloak

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Fig. 2.
Fig. 2. Curves of constitute parameters with variable k inside the cloak

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Fig. 3.
Fig. 3. Distribution of electric-field near the cloak with different k. (a) k = 10π. (b) k = 30π. (c) k = 50π.

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Fig. 4.
Fig. 4. Distribution of εzz inside the cloak with different k. (a) k = 10π. (b) k = 30π. (c) k = 50π.

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Equations (12)

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

ε r ij = A j ′ i A j ′ i ε r i ′ j ′ [ det ( A ) ] − 1
μ r ij = A j ′ i A j ′ i μ r i ′ j ′ [ det ( A ) ] − 1
ε r ij = ( A i ′ i ) − 1 ( A j ′ i ) − 1 ε r ij [ det ( A ) ]
μ r i'j' = ( A i ′ i ) − 1 ( A j ′ i ) − 1 μ r ij [ det ( A ) ]
{ ( ∂ 2 ∂ x 1 ′ 2 + ∂ 2 x 2 ′ 2 + ∂ 2 x 3 ′ 2 ) x i + k 2 x i = 0 x i ∣ Γ outer = x ′ i , x i ∣ Γ inner = x i ∣ P
{ 1 r ′ 2 [ ∂ ∂ r ′ ( r ′ 2 ∂ r ∂ r ′ ) ] + k 2 r = 0 r ∣ r ′ = r 1 = 0 , r ∣ r ′ = r 2 = r 2
r = A · sin ⁡ kr ′ kr ′ − B · cos ⁡ kr ′ kr ′
A = k r 2 2 cos ⁡ k r 1 sin ⁡ [ k ( r 2 − r 1 ) ] , B = k r 2 2 sin ⁡ k r 1 sin ⁡ [ k ( r 2 − r 1 ) ]
A r ′ r = ∂ r ∂ r ′ = A ( cos ⁡ kr ′ r ′ − sin ⁡ kr ′ kr ′ 2 ) + B ( sin ⁡ kr ′ r ′ + cos ⁡ kr ′ kr ′ 2 )
A θ ′ θ = A φ ′ φ = r r ′ = A sin ⁡ kr ′ kr ′ 2 − B cos ⁡ kr ′ kr ′ 2
ε ′ r = μ ′ r = ( A sin ⁡ kr ′ kr ′ 2 − B cos ⁡ kr ′ kr ′ 2 ) 2 A ( cos ⁡ kr ′ r ′ − sin ⁡ kr ′ kr ′ 2 ) + B ( sin ⁡ kr ′ r ′ + cos ⁡ kr ′ kr ′ 2 )
ε ′ θ = ε ′ φ = μ ′ θ = μ ′ φ = A ( cos ⁡ kr ′ r ′ − sin ⁡ kr ′ kr ′ 2 ) + B ( sin ⁡ kr ′ r ′ + cos ⁡ kr ′ kr ′ 2 )

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