Abstract

Elliptic cylindrical cloaks are investigated analytically in elliptic cylindrical coordinates. It is shown that both the linear and higher-order transformations produce an imperfect cloaking due to the poor symmetry of the coordinate system. The imperfection being in concordance with that obtained by numerical simulations, it cannot be eliminated by improving the computer techniques. The cloaking becomes almost perfect in the limit case of nearly circular cloaks with the advantage that none of the parameters is singular in the cloak shell. In circular cylindrical coordinates instead, a perfect cloaking is achieved with elliptic cloaks, as has been proven in other studies by numerical simulations. Analytic solutions to Maxwell’s equations are provided in these coordinates only in the limit case of nearly circular cloaks. Simple general expressions for material parameters are given also in circular cylindrical coordinates that can be then transformed in Cartesian coordinates.

© 2009 Optical Society of America

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References

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    [CrossRef]
  2. 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).
    [CrossRef] [PubMed]
  3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (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. H. 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]
  6. 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).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  11. Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93, 243502 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  24. E. Janke, F. Emde, and F. Losch, Tafeln Hoherer Functionen (Verlag, 1960).

2008 (12)

C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express 16, 19366-19374 (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]

Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16, 6134-6144 (2008).
[CrossRef] [PubMed]

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

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

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]

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

Y. Luo, J. Zhang, H. Chen, and B. I. Wu, “Full-wave analysis of prolate spheroidal and hyperboloidal cloaks,” J. Phys. D 41, 235101 (2008).
[CrossRef]

P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

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

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

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (2008).
[CrossRef]

2007 (2)

H. 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]

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]

2006 (4)

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

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).
[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]

1995 (1)

J. J. Stamnes, “Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips and slits,” Pure Appl. Opt. 4, 841-855 (1995).
[CrossRef]

Abramowitz, M.

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

Chen, B.

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

Chen, H.

Y. Luo, J. Zhang, H. Chen, and B. I. Wu, “Full-wave analysis of prolate spheroidal and hyperboloidal cloaks,” J. Phys. D 41, 235101 (2008).
[CrossRef]

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

H. 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]

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. Liu, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (2008).
[CrossRef]

Chin, J. Y.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Liu, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D 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]

Cojocaru, E.

E. Cojocaru, “Elliptic cylindrical invisibility cloak, a semianalytical approach using Mathieu functions,” eprint arXiv: 0808.1498 [physics. comp-ph] (2008).

E. Cojocaru, “Mathieu functions computational toolbox implemented in Matlab,” eprint arXiv: 0811.1970 [math-ph] (2008).

Cui, T. J.

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (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]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Liu, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D 41, 085504 (2008).
[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]

Emde, F.

E. Janke, F. Emde, and F. Losch, Tafeln Hoherer Functionen (Verlag, 1960).

Guenneau, S.

He, S.

P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

Janke, E.

E. Janke, F. Emde, and F. Losch, Tafeln Hoherer Functionen (Verlag, 1960).

Jiang, W. X.

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (2008).
[CrossRef]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Liu, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D 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]

Jin, Y.

P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93, 243502 (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.

Kong, J. A.

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

H. 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]

Kwon, D. H.

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

Leonhardt, U.

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

Li, C.

Li, F.

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]

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]

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (2008).
[CrossRef]

Liu, X. Q.

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

Losch, F.

E. Janke, F. Emde, and F. Losch, Tafeln Hoherer Functionen (Verlag, 1960).

Luo, Y.

Y. Luo, J. Zhang, H. Chen, and B. I. Wu, “Full-wave analysis of prolate spheroidal and hyperboloidal cloaks,” J. Phys. D 41, 235101 (2008).
[CrossRef]

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Ma, H.

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

Nicolet, A.

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).
[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).
[CrossRef]

Qiu, M.

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

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]

Qu, S.

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

Ran, L.

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Ruan, Z.

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

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]

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).
[CrossRef] [PubMed]

Smith, D. R.

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (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]

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).
[CrossRef] [PubMed]

Stamnes, J. J.

J. J. Stamnes, “Exact two-dimensional scattering by perfectly reflecting elliptical cylinders, strips and slits,” Pure Appl. Opt. 4, 841-855 (1995).
[CrossRef]

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]

Stegun, I.

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

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Wang, J.

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

Werner, D. H.

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

Wu, B. I.

Y. Luo, J. Zhang, H. Chen, and B. I. Wu, “Full-wave analysis of prolate spheroidal and hyperboloidal cloaks,” J. Phys. D 41, 235101 (2008).
[CrossRef]

H. 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]

Xu, Z.

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

Yan, M.

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

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]

Yan, W.

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

Yang, P.

Yang, X. M.

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (2008).
[CrossRef]

Yao, K.

You, Y.

Yu, G. X.

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

Zhai, P. W.

Zhang, B.

H. 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]

Zhang, J.

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

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

Y. Luo, J. Zhang, H. Chen, and B. I. Wu, “Full-wave analysis of prolate spheroidal and hyperboloidal cloaks,” J. Phys. D 41, 235101 (2008).
[CrossRef]

Zhang, P.

P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

Zolla, F.

Appl. Phys. Lett. (3)

P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

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

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93, 194102 (2008).
[CrossRef]

J. Phys. D (2)

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

Y. Luo, J. Zhang, H. Chen, and B. I. Wu, “Full-wave analysis of prolate spheroidal and hyperboloidal cloaks,” J. Phys. D 41, 235101 (2008).
[CrossRef]

New J. Phys. (2)

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

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

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

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

Phys. Rev. B (1)

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (2)

H. 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]

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]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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E. Cojocaru, “Elliptic cylindrical invisibility cloak, a semianalytical approach using Mathieu functions,” eprint arXiv: 0808.1498 [physics. comp-ph] (2008).

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Figures (11)

Fig. 1
Fig. 1

Electric-field distributions obtained analytically for a linear transformation in the ( ξ , η , z ) elliptic cylindrical coordinates at different incident angle ϕ measured against the x-axis direction. Here four cases are considered: (a) ϕ = 0 , (b) π 2 , (c) π 6 , and (d) π 4 .

Fig. 2
Fig. 2

Electric-field distributions at ϕ = 0 when high-order transformations are considered as given by (a) Eq. (21) and (b) Eq. (23).

Fig. 3
Fig. 3

Electric-field distributions in the limit case of nearly circular cloaks that are obtained analytically for linear transformations in the ( ξ , η , z ) elliptic cylindrical coordinates when (a) c = 0.0001 m and (b) c = 0.008 m and in the ( ρ , θ , z ) circular cylindrical coordinates when (c) c = 0.0001 m and (d) c = 0.008 m .

Fig. 4
Fig. 4

Distributions of components ( μ ξ , μ η , ϵ z ) ( μ ρ , μ θ , ϵ z ) inside the cloak shell in the limit case of nearly circular cloaks when c = 0.0001 (○) and 0.01 m (+).

Fig. 5
Fig. 5

Electric-field distribution obtained analytically for a linear transformation in the ( u , v , z ) elliptic cylindrical coordinates at ϕ = 0 , all other parameters as in Fig. 1.

Fig. 6
Fig. 6

Electric-field distributions in the limit case of nearly circular cloaks that are obtained analytically for linear transformations in the ( u , v , z ) elliptic cylindrical coordinates when (a) c = 0.0001 m and (b) c = 0.008 m and in the ( ρ , θ , z ) circular cylindrical coordinates when (c) c = 0.0001 m and (d) c = 0.008 m .

Fig. 7
Fig. 7

Electric-field behavior in the limit case of nearly circular cloaks obtained with Eqs. (52, 53) when (a) c = 0.0001 m and (b) c = 0.008 m .

Fig. 8
Fig. 8

Distributions of components ( μ u , μ v , ϵ z ) ( μ ρ , μ θ , ϵ z ) inside the cloak shell in the limit case of nearly circular cloaks when c = 0.0001 (○), 0.005 (×), and 0.01 m (+).

Fig. 9
Fig. 9

Electric-field behavior at ϕ = 0 for a linear transformation in the ( ρ , θ , z ) circular cylindrical coordinates in case of an elliptic cylindrical cloak with confocal boundaries as in Fig. 1.

Fig. 10
Fig. 10

Electric-field behavior at ϕ = 0 in case of confocal elliptic boundaries as in Fig. 1 for high-order transformations in the ( ρ , θ , z ) circular cylindrical coordinates as given by (a) Eq. (59) and (b) Eq. (60).

Fig. 11
Fig. 11

(a) Cross section of an elliptic cylindrical cloak with concentric boundaries of minor semiaxes a and b in y direction and major semiaxes k a and k b in x direction [13] and (b) the electric-field behavior at ϕ = 0 for a linear transformation in the ( ρ , θ , z ) circular cylindrical coordinate system when a = 0.1 m , b = 0.2 m , and k = 2 .

Equations (63)

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x = c ξ η , y = c ξ 2 1 1 η 2 ,
ξ 2 1 ξ ( ξ 2 1 ψ ξ ) + 1 η 2 η ( 1 η 2 ψ η ) + c 2 ( ξ 2 η 2 ) ( 2 ψ z 2 ϵ 0 μ 0 2 ψ t 2 ) = 0 .
( ξ 2 1 ) d 2 F 1 d ξ 2 + ξ d F 1 d ξ + ( c 2 k 0 2 ξ 2 A ) F 1 = 0 ,
( 1 η 2 ) d 2 F 2 d η 2 η d F 2 d η + ( A c 2 k 0 2 η 2 ) F 2 = 0 ,
ξ = cosh u , η = cos v ,
x = c cosh u cos v , y = c sinh u sin v .
d 2 F 1 d u 2 ( A 2 q cosh 2 u ) F 1 = 0 ,
d 2 F 2 d v 2 + ( A 2 q cos 2 v ) F 2 = 0 ,
E z ( in ) = exp [ i k 0 ( x cos ϕ + y sin ϕ ) ] ,
E z ( in ) = 8 π n i n J p m n ( q , u ) S p m n ( q , v ) S p m n ( q , ϕ ) N p m n ( q ) ,
ξ = g ( ξ ) , η = η , z = z ,
ϵ ̿ ϵ 0 = T ̿ 1 , μ ̿ μ 0 = T ̿ 1 ,
T ̿ 1 = diag [ g 2 1 g ξ 2 1 , g ξ 2 1 g 2 1 , g ξ 2 1 ( g 2 η 2 ) g 2 1 ( ξ 2 η 2 ) ] ,
g 2 1 g ( g 2 1 E z g ) + 1 η 2 η ( 1 η 2 E z η ) + c 2 k 0 2 ( g 2 η 2 ) E z = 0 .
ξ > ξ 2 , E z = 8 π n i n [ α p m n ( in ) J p m n ( q , u ) + α p m n ( sc ) H p m n ( 1 ) ( q , u ) ] S p m n ( q , v ) S p m n ( q , ϕ ) ,
ξ 1 + δ < ξ < ξ 2 , E z = 8 π n i n [ α p m n ( 1 ) J p m n ( q , u g ) + α p m n ( 2 ) H p m n ( 1 ) ( q , u g ) ] S p m n ( q , v ) S p m n ( q , ϕ ) ,
α p m n ( 1 ) = 1 N p m n ( q ) , α p m n ( sc ) = α p m n ( 2 ) ,
α p m n ( 2 ) = α p m n ( 1 ) J p m n ( q , u g 1 ) H p m n ( 1 ) ( q , u g 1 ) ,
ξ = ξ 1 + ( ξ 1 ) ξ 2 ξ 1 ξ 2 1 ,
g ( ξ ) = ( ξ d ) D ,
g ( ξ ) = ( ξ 2 1 ) ( ξ ξ 1 ) 2 ( ξ 2 ξ 1 ) 2 + 1 ,
ξ = ξ 1 ξ 2 1 ( ξ 2 ξ 1 ) + ξ 1 ,
g ( ξ ) = ξ 2 ( ξ ξ 2 ) 2 ( ξ 2 1 ) ( ξ 2 ξ 1 ) 2 ,
ξ = ξ 2 ( ξ 2 ξ 1 ) ξ 2 ξ ξ 2 1 .
x ρ = x ρ , y ρ = y ρ .
ρ ξ large c ξ .
θ = arctan ( y x ) = arctan ( c ξ 2 1 sin v c ξ cos v ) ,
g ( ξ ) g ( ρ c ) = ( ρ c d ) ( c D ) ,
ϵ ̿ ϵ 0 = μ ̿ μ 0 diag ( g g ξ , g ξ g , g g ξ ) = diag ( ρ c d ρ , ρ ρ c d , ρ c d ρ D 2 ) ,
g = g ξ = g ρ ρ ξ = 1 D .
E z ( in ) = n α n ( in ) J n ( k 0 ρ ) exp ( i n θ ) ,
ρ > b , E z = n [ α n ( in ) J n ( k 0 ρ ) + α n ( sc ) H n ( k 0 ρ ) ] exp ( i n θ ) ,
a < ρ < b , E z = n [ α n ( 1 ) J n ( k 0 ( ρ c d ) D ) + α n ( 2 ) H n ( k 0 ( ρ c d ) D ) ] exp ( i n θ ) ,
α n ( 1 ) = α n ( in ) , α n ( sc ) = α n ( 2 ) ,
α n ( 2 ) = α n ( 1 ) J n [ k 0 ( a c d ) D ] H n [ k 0 ( a c d ) D ] .
u = h ( u ) , v = v , z = z ,
T ̿ 1 = diag ( 1 h , h , h cosh 2 h cos 2 v cosh 2 u cos 2 v ) ,
u ( 1 h E z u ) + v ( h E z v ) + c 2 k 0 2 h 2 ( cosh 2 h cos 2 v ) E z = 0 ,
2 E z h 2 + 2 E z v 2 + c 2 k 0 2 2 ( cosh 2 h cos 2 v ) E z = 0 .
u = u 2 u 1 u 2 u + u 1 ,
h ( u ) = u u 1 κ ,
ϵ ̿ ϵ 0 = μ ̿ μ 0 = diag [ κ , 1 κ , cosh 2 h cos 2 v κ ( cosh 2 u cos 2 v ) ] .
h ( u ) = ( u u 1 u 2 u 1 ) 2 u 2 , h ( u ) = u 2 [ 1 ( u 2 u u 2 u 1 ) 2 ] ;
ρ u large c e u 2 .
h ( u ) h [ ln ( 2 ρ c ) ] = ln ρ ln a κ = ln ( ρ a ) 1 κ ,
ϵ ̿ ϵ 0 = μ ̿ μ 0 diag ( κ , 1 κ , cosh 2 h κ cosh 2 u ) diag ( κ , 1 κ , c 2 4 κ ρ 2 ( ρ a ) 2 κ ) ,
1 ϵ z ρ ρ ( ρ μ θ E z ρ ) + 1 ϵ z ρ 2 θ ( 1 μ ρ E z θ ) + k 0 2 E z = 0 ,
ρ 2 2 Ψ ρ 2 + ρ Ψ ρ + [ k 0 2 c 2 4 κ 2 ( ρ a ) 2 κ l 2 κ 2 ] Ψ = 0 ,
2 Θ θ 2 + l 2 Θ = 0 .
F l ( k 0 c 2 ( ρ a ) 1 κ ) exp ( i l θ ) ,
E z = n [ α n ( 1 ) J n ( k 0 c 2 ( ρ a ) 1 κ ) + α n ( 2 ) H n ( k 0 c 2 ( ρ a ) 1 κ ) ] exp ( i n θ ) .
α n ( 2 ) = α n ( 1 ) J n ( k 0 c 2 ) H n ( k 0 c 2 ) .
ρ > b , E z = exp ( i k 0 ρ cos v ) ,
a < ρ < b , E z = exp [ i k 0 c 2 ( ρ a ) 1 κ cos v ] ,
R ( θ ) = 1 cos 2 θ a x 2 + sin 2 θ a y 2 = a x a y a y 2 cos 2 θ + a x 2 sin 2 θ ,
ρ = f ( ρ , θ ) , θ = θ , z = z ,
ϵ ̿ ϵ 0 = μ ̿ μ 0 = [ f [ 1 + ( 1 f f θ ) 2 ] ρ f ρ 1 f f θ 0 1 f f θ ρ f f ρ 0 0 0 f ρ f ρ ] .
ϵ ̿ x y z ϵ 0 = μ ̿ x y z μ 0 = [ t 11 x 2 + t 22 y 2 2 t 12 x y x 2 + y 2 ( t 11 t 22 ) x y + t 12 ( x 2 y 2 ) x 2 + y 2 0 ( t 11 t 22 ) x y + t 12 ( x 2 y 2 ) x 2 + y 2 t 11 y 2 + t 22 x 2 + 2 t 12 x y x 2 + y 2 0 0 0 t 33 ] ,
f ( ρ , θ ) = ρ R 1 ( θ ) R 2 ( θ ) R 1 ( θ ) R 2 ( θ ) .
f ( ρ , θ ) = ( ρ R 1 ) 2 ( R 2 R 1 ) 2 R 2 ,
f ( ρ , θ ) = R 2 [ 1 ( ρ R 2 ) 2 ( R 2 R 1 ) 2 ] .
f ( ρ , θ ) = [ ρ a χ ( θ ) ] b ( b a ) .
ϵ ̿ ϵ 0 = μ ̿ μ 0 = [ ρ a χ ρ [ 1 + ( a χ ρ a χ ) 2 ] a χ ρ a χ 0 a χ ρ a χ ρ ρ a χ 0 0 0 ( b b a ) 2 ρ a χ ρ ] ,

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