Abstract

We demonstrate that a medium consisting of two adjoining distinct layers of transformation materials, corresponding respectively to two linear coordinate transformations, can behave effectively as that of the same region transformed by another linear transformation. The equivalence means that, irrespective of the direction of incident wave, the fields of the medium exterior to the transformed regions of the two configurations are exactly the same. This property can also apply to a domain that is transformed by a piecewise linear transformation function, and to a medium that is mapped by a general curved function. This proof is shown analytically based on a rigorous Fourier-Bessel analysis. The equivalence suggests that, for a given transformed domain, one can find an infinite number of complementary media that altogether can give a desired effective response of certain transformation path.

© 2009 OSA

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [CrossRef] [PubMed]
  3. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 (2006).
    [CrossRef]
  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(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  5. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007).
    [CrossRef]
  6. H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007).
    [CrossRef]
  7. G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009).
    [CrossRef]
  8. J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009).
    [CrossRef] [PubMed]
  9. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).
  10. H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 (2009).
    [CrossRef]
  11. N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
    [CrossRef] [PubMed]
  12. T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008).
    [CrossRef]
  13. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 (2009).
  14. V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
    [CrossRef] [PubMed]
  15. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
    [CrossRef]
  16. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009).
    [CrossRef]
  17. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
    [CrossRef]
  18. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006).
    [CrossRef]
  19. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10 509–14 (Engl. Transl.) (1968).
  20. H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008).
    [CrossRef]
  21. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
    [CrossRef] [PubMed]
  22. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transfromations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (2008).
    [CrossRef]
  23. 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(12), 125127 (2008).
    [CrossRef]
  24. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
    [CrossRef] [PubMed]
  25. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009).
    [CrossRef] [PubMed]
  26. Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
    [CrossRef] [PubMed]
  27. J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 (2009).
    [CrossRef]
  28. G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).
  29. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, (North-Holland, Amsterdam, 1962).
  30. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007).
    [CrossRef] [PubMed]
  31. V. G. Veselago, “About the wording of Fermat’s principle for light propagation in media with negative refraction index,” (2002) http://arxiv.org/abs/cond-mat/0203451 .
  32. M. Born, and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).
  33. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007).
    [CrossRef] [PubMed]
  34. S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997).
    [CrossRef]

2009 (8)

G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009).
[CrossRef]

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 (2009).

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009).
[CrossRef]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 (2009).
[CrossRef]

H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 (2009).
[CrossRef]

J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009).
[CrossRef] [PubMed]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009).
[CrossRef] [PubMed]

2008 (10)

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transfromations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (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(12), 125127 (2008).
[CrossRef]

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
[CrossRef] [PubMed]

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
[CrossRef] [PubMed]

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008).
[CrossRef]

2007 (5)

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007).
[CrossRef]

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007).
[CrossRef]

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

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007).
[CrossRef] [PubMed]

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007).
[CrossRef] [PubMed]

2006 (5)

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

1997 (1)

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997).
[CrossRef]

Allen, J.

Alù, A.

Briane, M.

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

Caorsi, S.

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997).
[CrossRef]

Castaldi, G.

Chan, C. T.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008).
[CrossRef]

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

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

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007).
[CrossRef]

Chen, H.

H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 (2009).
[CrossRef]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

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(12), 125127 (2008).
[CrossRef]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008).
[CrossRef]

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
[CrossRef] [PubMed]

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007).
[CrossRef]

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

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007).
[CrossRef] [PubMed]

Chen, J. S.

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008).
[CrossRef]

Chen, T.

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008).
[CrossRef]

Cherednichenko, K.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

Cui, T. J.

G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009).
[CrossRef]

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009).
[CrossRef]

Cummer, S.

Cummer, S. A.

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

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007).
[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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Engheta, N.

Galdi, V.

Gallina, I.

Guenneau, S.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).

Han, D.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

Hu, G.

Hu, J.

Jacob, Z.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

Jiang, W. X.

G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009).
[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(5801), 977–980 (2006).
[CrossRef] [PubMed]

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(12), 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(6), 063903 (2007).
[CrossRef] [PubMed]

Kundtz, N.

Lai, Y.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 (2009).

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

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

Luo, X.

Luo, Y.

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(12), 125127 (2008).
[CrossRef]

Ma, H.

McPhedran, R. C.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

Mei, Z. L.

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009).
[CrossRef]

Milton, G. W.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

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

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 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(11), 113903 (2007).
[CrossRef] [PubMed]

Ng, J.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

Nicolet, A.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).

Nicorovici, N.-A. P.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

Pastorino, M.

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997).
[CrossRef]

Pendry, J.

J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 (2009).
[CrossRef]

Pendry, J. B.

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 (2009).

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

Qiu, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007).
[CrossRef] [PubMed]

Raffetto, M.

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997).
[CrossRef]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transfromations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (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(12), 125127 (2008).
[CrossRef]

Roberts, D. A.

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

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
[CrossRef] [PubMed]

Ruan, Z.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007).
[CrossRef] [PubMed]

Schurig, D.

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

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

Smith, D. R.

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
[CrossRef] [PubMed]

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 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(5801), 977–980 (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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Weng, C. N.

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (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,” N. J. Phys. 8(10), 248 (2006).
[CrossRef]

Wu, B. I.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007).
[CrossRef] [PubMed]

Xiao, J.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

Yan, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007).
[CrossRef] [PubMed]

Yang, T.

Yu, G. X.

G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009).
[CrossRef]

Zhang, B.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 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(12), 125127 (2008).
[CrossRef]

Zhang, X.

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008).
[CrossRef]

Zhang, Z. Q.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

Zhou, X.

Zolla, F.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).

Appl. Phys. Lett. (4)

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007).
[CrossRef]

G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009).
[CrossRef]

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008).
[CrossRef]

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

IEEE Trans. Antenn. Propag. (1)

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997).
[CrossRef]

J. Appl. Phys. (1)

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 (2009).
[CrossRef]

N. J. Phys. (5)

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008).
[CrossRef]

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

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

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007).
[CrossRef]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008).
[CrossRef]

Nature (1)

J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 (2009).
[CrossRef]

Opt. Express (6)

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

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transfromations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (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(12), 125127 (2008).
[CrossRef]

Phys. Rev. Lett. (3)

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[CrossRef] [PubMed]

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 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(11), 113903 (2007).
[CrossRef] [PubMed]

Prog. Opt. (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 (2009).

Science (4)

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (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(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Other (5)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10 509–14 (Engl. Transl.) (1968).

V. G. Veselago, “About the wording of Fermat’s principle for light propagation in media with negative refraction index,” (2002) http://arxiv.org/abs/cond-mat/0203451 .

M. Born, and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).

G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).

E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, (North-Holland, Amsterdam, 1962).

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Figures (4)

Fig. 1
Fig. 1

(a) An illustration of the coordinate transformation of Eqs. (1) and (2) in which ( r 0 , r 0 ) = ( 0.12 , 0.20 ) , a = 0.07 , a = 0.1 , b = 0.25 , b = 0.3. (b) the value of ε r / ε 0 versus r . (c) the value of ε θ / ε 0 versus r . (d) the value of ε z / ε 0 versus r .

Fig. 2
Fig. 2

A schematic illustration of equivalence of two configurations obtained from linear coordinate transformations. The equivalence means the electromagnetic fields in regions I and IV are identical for both configurations.

Fig. 3
Fig. 3

(left panel) A schematic illustration of equivalence of a multiple piecewise linear transformation to a single linear coordinate transformation (red line). (right panel) equivalence of a general curved transformation function to a single linear coordinate transformation.

Fig. 4
Fig. 4

Possible transformation paths that are equivalent to the linear one (red).

Equations (45)

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r = α 1 r + β 1 = r 0 a r 0 a r + a r 0 a r 0 r 0 a , a r r 0 , θ = θ , z = z ,
r = α 2 r + β 2 = b r 0 b r 0 r + b r 0 b r 0 b r 0 , r 0 r b , θ = θ , z = z .
ε 1 r = r β 1 r ε 0 , ε 1 θ = r r β 1 ε 0 , ε 1 z = ( 1 α 1 ) 2 r β 1 r ε 0 , for   a r r 0 ,
ε 2 r = r β 2 r ε 0 , ε 2 θ = r r β 2 ε 0 , ε 2 z = ( 1 α 2 ) 2 r β 2 r ε 0 , for   r 0 r b .
1 ε z 1 r r ( r μ θ E z r ) + 1 ε z 1 r 2 ( 1 μ r 2 E z θ 2 ) + k 0 2 E z = 0 ,
E z I = n d n I J n ( k 0 r ) exp ( i n θ ) , for   r a
E z I I = n { d n I I J n ( k 1 ( r β 1 ) ) + f n I I H n ( 1 ) ( k 1 ( r β 1 ) ) } exp ( i n θ ) , for   a r r 0 ,
E z I I I = n { d n I I I J n ( k 2 ( r β 2 ) ) + f n I I I H n ( 1 ) ( k 2 ( r β 2 ) ) } exp ( i n θ ) , for   r 0 r b ,
E z I V = n { d n i n J n ( k 0 r ) + f n s c H n ( 1 ) ( k 0 r ) } exp ( i n θ ) , for   r b
d n i n J n ( k 0 b ) + f n s c H n ( 1 ) ( k 0 b ) = d n I I I J n ( k 2 ( b β 2 ) ) + f n I I I H n ( 1 ) ( k 2 ( b β 2 ) ) ,
k 0 d n i n J n ( k 0 b ) + k 0 f n s c H n ( 1 ) ( k 0 b ) = k 2 μ 2 θ ( b ) d n I I I J n ( k 2 ( b β 2 ) ) + k 2 μ 2 θ ( b ) f n I I I H n ( 1 ) ( k 2 ( b β 2 ) ) .
k 2 ( b β 2 ) = b r 0 b r 0 k 0 b b r 0 b b r 0 = k 0 b ,
k 2 μ 2 θ ( b ) = b r 0 b r 0 k 0 b β 2 b = b r 0 b r 0 k 0 b b b r 0 b b r 0 = k 0 b b ,
d n i n J n ( k 0 b ) + f n s c H n ( 1 ) ( k 0 b ) = d n I I I J n ( k 0 b ) + f n I I I H n ( 1 ) ( k 0 b ) ,
b d n i n J n ( k 0 b ) + b f n s c H n ( 1 ) ( k 0 b ) = b d n I I I J n ( k 0 b ) + b f n I I I H n ( 1 ) ( k 0 b ) .
d n I I I J n ( k 2 ( r 0 β 2 ) ) + f n I I I H n ( 1 ) ( k 2 ( r 0 β 2 ) ) = d n I I J n ( k 1 ( r 0 β 1 ) ) + f n I I H n ( 1 ) ( k 1 ( r 0 β 1 ) ) ,
k 2 μ 2 θ ( r 0 ) d n I I I J n ( k 2 ( r 0 β 2 ) ) + k 2 μ 2 θ ( r 0 ) f n I I I H n ( 1 ) ( k 2 ( r 0 β 2 ) ) = k 1 μ 1 θ ( r 0 ) d n I I J n ( k 1 ( r 0 β 1 ) ) + k 1 μ 1 θ ( r 0 ) f n I I H n ( 1 ) ( k 1 ( r 0 β 1 ) ) .
k 2 ( r 0 β 2 ) = b r 0 b r 0 k 0 b r 0 r 0 r 0 b r 0 = k 0 r 0 ,
k 1 ( r 0 β 1 ) = r 0 a r 0 a k 0 r 0 r 0 a r 0 r 0 a = k 0 r 0 ,
k 2 μ 2 θ ( r 0 ) = b r 0 b r 0 k 0 r 0 β 2 r 0 = b r 0 b r 0 k 0 r 0 b r 0 r 0 r 0 b r 0 = k 0 r 0 r 0 ,
k 1 μ 1 θ ( r 0 ) = r 0 a r 0 a k 0 r 0 β 1 r 0 = r 0 a r 0 a k 0 r 0 r 0 r 0 a r 0 r 0 a = k 0 r 0 r 0 ,
( d n I I I d n I I ) J n ( k 0 r 0 ) + ( f n I I I f n I I ) H n ( 1 ) ( k 0 r 0 ) = 0 ,
( d n I I I d n I I ) J n ( k 0 r 0 ) + ( f n I I I f n I I ) H n ( 1 ) ( k 0 r 0 ) = 0 ,
d n I I = d n I I I , f n I I = f n I I I .
d n I I J n ( k 0 a ) + f n I I H n ( 1 ) ( k 0 a ) = d n I J n ( k 0 a ) ,
a d n I I J n ( k 0 a ) + a f n I I H n ( 1 ) ( k 0 a ) = a d n I J n ( k 0 a ) .
r = α 3 r + β 3 = b a b a r + a b b a b a , θ = θ , z = z ,
ε 3 r = r β 3 r ε 0 , ε 3 θ = r r β 3 ε 0 , ε 3 z = ( 1 α 3 ) 2 r β 3 r ε 0 .
E ^ z I = n d ^ n I J n ( k 0 r ) exp ( i n θ ) , for   r a ,
E ^ z I I + I I I = n { d ^ n I I + I I I J n ( k 3 ( r β 3 ) ) + f ^ n I I + I I I H n ( 1 ) ( k 3 ( r β 3 ) ) } exp ( i n θ ) , for   a r b ,
E ^ z I V = n { d n i n J n ( k 0 r ) + f ^ n s c H n ( 1 ) ( k 0 r ) } exp ( i n θ ) , for   r b ,
d n i n J n ( k 0 b ) + f ^ n s c H n ( 1 ) ( k 0 b ) = d ^ n I I + I I I J n ( k 0 b ) + f ^ n I I + I I I H n ( 1 ) ( k 0 b ) ,
b d n i n J n ( k 0 b ) + b f ^ n s c H n ( 1 ) ( k 0 b ) = b d ^ n I I + I I I J n ( k 0 b ) + b f ^ n I I + I I I H n ( 1 ) ( k 0 b ) ,
d ^ n I I + I I I J n ( k 0 a ) + f ^ n I I + I I I H n ( 1 ) ( k 0 a ) = d ^ n I J n ( k 0 a ) ,
a d ^ n I I + I I I J n ( k 0 a ) + a f ^ n I I + I I I I H n ( 1 ) ( k 0 a ) = a d ^ n I J n ( k 0 a ) .
f n s c = f ^ n s c , d n I = d ^ n I , d n I I = d n I I I = d ^ n I I + I I I , f n I I = f n I I I = f ^ n I I + I I I .
E z I = E ^ z I , E z I V = E ^ z I V , H θ I = H ^ θ I , H θ I V = H ^ θ I V .
d n I I J n ( k 0 a ) + f n I I H n ( 1 ) ( k 0 a ) = 0.
d ^ n I I + I I I J n ( k 0 a ) + f ^ n I I + I I I H n ( 1 ) ( k 0 a ) = 0.
f n s c = f ^ n s c , d n I I = d n I I I = d ^ n I I + I I I , f n I I = f n I I I = f ^ n I I + I I I .
( d n i n d n I I I ) J n ( k 0 b ) + ( f n s c f n I I I ) H n ( 1 ) ( k 0 b ) = 0 ,
( d n i n d n I I I ) J n ( k 0 b ) + ( f n s c d n I I I ) H n ( 1 ) ( k 0 b ) = 0 ,
( d n I I d n I ) J n ( k 0 a ) + ( f n I I ) H n ( 1 ) ( k 0 a ) = 0 ,
( d n I I d n I ) J n ( k 0 a ) + ( f n I I ) H n ( 1 ) ( k 0 a ) = 0 ,
d n i n = d n I I = d n I I I , f n s c = f n I I = f n I I I = 0.

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