Abstract

Transformation optics is a recently appreciated methodology for the design of complex media that control the propagation of electromagnetic and other types of waves. The transformation optical technique involves the use of coordinate transformations applied to some region of space, providing a conceptual means to redirect the flow of waves. Successfully designed devices to date have made use of transformations acting on passive space only; however, the technique can also be applied when source distributions (e.g., current and charge) are included within the space being transformed. In this paper we present examples of source transformations that illustrate the potential of these expanded transformation optical methods. In particular, using finite-element full-wave simulations, we confirm the restoration of dipole radiation patterns from both a distorted ‘pin-wheel’ antenna and a bent dipole partially occluded by a cylindrical scatterer. We propose the technique of source transformations as a powerful approach for antenna design, especially in relation to conformal antennas.

© 2008 Optical Society of America

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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. V. M. Shalaev, "Transforming light," Science 322, 384-386 (2008).
    [CrossRef] [PubMed]
  3. J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396-1408 (1960).
    [CrossRef]
  4. E. J. Post, Formal Structure of Electromagnetics (Dover Publications, Inc., New York, 1962).
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    [CrossRef]
  6. A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwells equations," J. Mod. Opt. 43, 773-793.
  7. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refraction," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  8. 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]
  9. U. Leonhardt,"Optical Conformal Mapping," Science 312, 1777-1779 (2006).
    [CrossRef] [PubMed]
  10. Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (2008).
    [CrossRef]
  11. 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 41, 085504 (2008).
    [CrossRef]
  12. C. Li and F. Li, "Two-dimensional electromagnetic cloaks with arbitrary geometries," Opt. Express 16, 13414-13420 (2008).
    [CrossRef] [PubMed]
  13. W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16, 5444-5452 (2008).
    [CrossRef] [PubMed]
  14. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
    [CrossRef] [PubMed]
  15. D. Schurig, J. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007).
    [CrossRef] [PubMed]
  16. M. Tang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," Phys. Rev. B 77, 035122 (2008).
    [CrossRef]
  17. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, "Optical design of reflectionless complex media by finite embedded coordinate transformations," Phys. Rev. Lett. 100, 063903 (2008).
    [CrossRef] [PubMed]
  18. Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (2008).
    [CrossRef]
  19. H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
    [CrossRef]
  20. 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]
  21. 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]

2008 (9)

V. M. Shalaev, "Transforming light," Science 322, 384-386 (2008).
[CrossRef] [PubMed]

Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (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 41, 085504 (2008).
[CrossRef]

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

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16, 5444-5452 (2008).
[CrossRef] [PubMed]

M. Tang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," Phys. Rev. B 77, 035122 (2008).
[CrossRef]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, "Optical design of reflectionless complex media by finite embedded coordinate transformations," Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (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]

2007 (3)

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

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007).
[CrossRef] [PubMed]

2006 (5)

U. Leonhardt and T. 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]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

U. Leonhardt,"Optical Conformal Mapping," Science 312, 1777-1779 (2006).
[CrossRef] [PubMed]

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

2004 (1)

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

1960 (1)

J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396-1408 (1960).
[CrossRef]

Cai, W. S.

Chan, C. T.

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

Chen, H.

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (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]

Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (2008).
[CrossRef]

H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
[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 41, 085504 (2008).
[CrossRef]

Chettiar, U. K.

Chin, J. Y.

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 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 41, 085504 (2008).
[CrossRef]

Cummer, S. A.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and 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, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

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 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.

Kong, J. A.

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (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]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt,"Optical Conformal Mapping," Science 312, 1777-1779 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. Philbin, "General Relativity in Electrical Engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Li, C.

Li, F.

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 41, 085504 (2008).
[CrossRef]

Luo, Y.

Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (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]

Luo, Yu

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (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]

Pendry, J. B.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and 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. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007).
[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]

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 refraction," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwells equations," J. Mod. Opt. 43, 773-793.

Philbin, T.

U. Leonhardt and T. Philbin, "General Relativity in Electrical Engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Plebanski, J.

J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396-1408 (1960).
[CrossRef]

Psaltis, D.

M. Tang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," Phys. Rev. B 77, 035122 (2008).
[CrossRef]

Rahm, M.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, "Optical design of reflectionless complex media by finite embedded coordinate transformations," Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

Ran, L.

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (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]

Schurig, D.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and 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. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007).
[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]

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, S. A. Cummer, D. Schurig, J. B. Pendry, and 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. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007).
[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]

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 refraction," 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]

Tang, M.

M. Tang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," Phys. Rev. B 77, 035122 (2008).
[CrossRef]

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Ward, A. J.

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwells equations," J. Mod. Opt. 43, 773-793.

Wiltshire, M. C. K.

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

Wu, B.-I.

Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (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 41, 085504 (2008).
[CrossRef]

Zhang, J.

Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (2008).
[CrossRef]

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (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]

Appl. Phys. Lett. (1)

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

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwells equations," J. Mod. Opt. 43, 773-793.

J. Phys. D (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 41, 085504 (2008).
[CrossRef]

New J. Phys. (1)

U. Leonhardt and T. Philbin, "General Relativity in Electrical Engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Opt. Express (4)

Phys. Rev. (1)

J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396-1408 (1960).
[CrossRef]

Phys. Rev. B (3)

M. Tang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," Phys. Rev. B 77, 035122 (2008).
[CrossRef]

Y. Luo, J. Zhang, B.-I. Wu, and H. Chen, "Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator," Phys. Rev. B 78, 125108 (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]

Phys. Rev. Lett. (2)

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, "Optical design of reflectionless complex media by finite embedded coordinate transformations," Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

PIERS (1)

Yu Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, "Controlling the Emission of Electromagnetic Source," PIERS 4, 795-800 (2008).
[CrossRef]

Science (5)

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

V. M. Shalaev, "Transforming light," Science 322, 384-386 (2008).
[CrossRef] [PubMed]

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

U. Leonhardt,"Optical Conformal Mapping," Science 312, 1777-1779 (2006).
[CrossRef] [PubMed]

Other (1)

E. J. Post, Formal Structure of Electromagnetics (Dover Publications, Inc., New York, 1962).

Supplementary Material (5)

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Figures (4)

Fig. 1.
Fig. 1.

The process for the ‘pinwheel’ transformation is shown schematically. (a) The current is defined in Cartesian coordinates. (b) The current is transformed into cylindrical coordinates. (c) The current is transformed into ‘pinwheel’ coordinates. Note that to this point that when shown on a cartesian axis the current has not changed, it is simply expressed differently. (d) The expression for the current from (c) is used, but θ′→θ and r′→r. (e) The current distribution is expressed in a Cartesian basis. In (d) and (e) there is material introduced in the shaded region.

Fig. 2.
Fig. 2.

Vector field plots of transformed currents are shown. (a) Original current (b) Current after ‘pinwheel’ transformation with a rotation of Δθ=π (c)Current after a cloaking transform NOTE: The initial current used was shifted slightly to the left to avoid current in a region with diverging material parameters.

Fig. 3.
Fig. 3.

A line cut along y=0 of the current in the y direction is shown. Despite the stretching induced by the transformation, each region still carries a total current of exactly one Amp-m -1. This property will always be true under these transformations and allows the analytic current expression shown in Eq. 10 to be replaced by a discontinuity in the magnetic field in the limit δ→0.

Fig. 4.
Fig. 4.

The z-component of the magnetic field is plotted for (Media 1)(a) a pure dipole of lenth L=λ/2 (Media 2)(b) A dipole that has undergone a pinwheel rotation of π without material compensation. (Media 3)(c) A dipole that has undergone a pinwheel rotation of π with proper material compensation (Media 4)(d) A dipole that has been bent around a PEC scatterer (Media 5)(e) A dipole that has been bent around a PEC scatterer with proper ‘cloaking’ compensation. In each image there is a circle around the region in which the transformation has taken place. The current is carried on the wire inside of this region.

Equations (18)

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ε r m n = 1 A A m m A n n ε r mn
μ r m n = 1 A A m m A n n μ r mn
A m m = x m x m
j m = 1 A A m m j m
A 1 = ( cos ( θ ) sin ( θ ) 0 sin ( θ ) r cos ( θ ) r 0 0 0 1 )
r ' = r
θ ' = { θ r > R 1 θ + Δ θ ( 1 r R 1 ) r < R 1
A 2 = ( 1 0 0 Δ θ R 1 1 0 0 0 1 )
j = ( 0 1 δ * π e ( x ) 2 δ 0 )
j 1 = 1 δ * π exp ( ( r * cos ( θ ) ) 2 δ ) * ( r sin ( θ ) cos ( θ ) 0 )
j 2 = 1 R 1 δ * π exp ( ( r cos ( u ) ) 2 δ ) * ( R 1 r sin ( u ) R 1 cos ( u ) r Δ θ sin ( u ) 0 )
u = Δ θ ( 1 r R 1 ) θ
j 3 = 1 R 1 δ * π exp ( ( r * cos ( u ) ) 2 δ ) * ( R 1 sin ( u + θ ) + r Δ θ sin ( θ ) sin ( Δ θ ( r R 1 1 ) + θ ) R 1 cos ( u + θ ) + r Δ θ cos ( θ ) sin ( u ) 0 )
u = Δ θ ( 1 r R 1 ) θ
r = x 2 + y 2
θ = arctan ( y x )
r = { r r > R 2 ( r α ) ( R 2 R 1 ) R 2 α + R 1 α < r < R 2 R 1 α r r < α
θ = θ

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