Abstract

We propose a kind of folded geometry in bipolar coordinates. Transformation optics in such folded geometries has many intriguing applications, such as a kind of conceptual devices that can exactly cancel the scattering of a prespecified external object and thereby make it invisible. Numerical simulations are performed to confirm the properties.

© 2009 Optical Society of America

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    [PubMed]
  5. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).
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2009 (14)

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nature Mater. 8, 568 (2009).

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).

U. Leonhardt and T. Tyc, Science 323, 110 (2009).

Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 093901 (2009).
[PubMed]

T. Philbin, Physics (N.Y.) 2, 17 (2009).

H. Y. Chen, J. Opt. A, Pure Appl. Opt. 11, 075102 (2009).

A. Cho, Science 323, 701 (2009).
[PubMed]

X. Luo, T. Yang, Y. Gu, H. Y. Chen, and H. R. Ma, Appl. Phys. Lett. 94, 223513 (2009).

J. Zhang, Y. Luo, H. S. Chen, J. T. Huangfu, B.-I. Wu, L. Ran, and J. A. Kong, Opt. Express 17, 6203 (2009).
[PubMed]

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

H. Y. Chen, C. T. Chan, S. Y. Liu, and Z. F. Lin, New J. Phys. 11, 083012 (2009).

J. Ng, H. Y. Chen, and C. T. Chan, Opt. Lett. 34, 644 (2009).
[PubMed]

2008 (5)

H. Y. Chen, X. Luo, H. Ma, and C. T. Chan, Opt. Express 16, 14603 (2008).
[PubMed]

M. Yan, W. Yan, and M. Qiu, Phys. Rev. B 78, 125113 (2008).

T. Yang, H. Y. Chen, X. Luo, and H. Ma, Opt. Express 16, 18545 (2008).
[PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

J. Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901 (2008).
[PubMed]

2007 (4)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).

H. Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007).

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[PubMed]

A. V. Kildishev and E. E. Narimanov, Opt. Lett. 32, 3432 (2007).
[PubMed]

2006 (5)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[PubMed]

Leonhardt, Science 312, 1777 (2006).
[PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[PubMed]

U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006).

G. W. Milton and N.-A. P. Nicorovici, Proc. R. Soc. London, Ser. A 462, 3027 (2006).

2003 (2)

J. B. Pendry and S. A. Ramakrishna, J. Phys.: Condens. Matter 15, 6345 (2003).

A. Greenleaf, M. Lassas, and G. Uhlmann, Math. Res. Lett. 10, 685 (2003).

2000 (1)

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[PubMed]

1961 (1)

L. S. Dolin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 4, 964 (1961).

Ao, X.

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nature Mater. 8, 568 (2009).

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).

Chan, C. T.

Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 093901 (2009).
[PubMed]

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

H. Y. Chen, C. T. Chan, S. Y. Liu, and Z. F. Lin, New J. Phys. 11, 083012 (2009).

J. Ng, H. Y. Chen, and C. T. Chan, Opt. Lett. 34, 644 (2009).
[PubMed]

H. Y. Chen, X. Luo, H. Ma, and C. T. Chan, Opt. Express 16, 14603 (2008).
[PubMed]

H. Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007).

Chen, H. S.

J. Zhang, Y. Luo, H. S. Chen, J. T. Huangfu, B.-I. Wu, L. Ran, and J. A. Kong, Opt. Express 17, 6203 (2009).
[PubMed]

Y. Luo, J. Zhang, H. S. Chen, B.-I. Wu, and J. A. Kong, http://arxiv.org/abs/0808.0215.

Chen, H. Y.

X. Luo, T. Yang, Y. Gu, H. Y. Chen, and H. R. Ma, Appl. Phys. Lett. 94, 223513 (2009).

H. Y. Chen, C. T. Chan, S. Y. Liu, and Z. F. Lin, New J. Phys. 11, 083012 (2009).

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

J. Ng, H. Y. Chen, and C. T. Chan, Opt. Lett. 34, 644 (2009).
[PubMed]

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 093901 (2009).
[PubMed]

H. Y. Chen, J. Opt. A, Pure Appl. Opt. 11, 075102 (2009).

T. Yang, H. Y. Chen, X. Luo, and H. Ma, Opt. Express 16, 18545 (2008).
[PubMed]

H. Y. Chen, X. Luo, H. Ma, and C. T. Chan, Opt. Express 16, 14603 (2008).
[PubMed]

H. Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007).

Chen, S.

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

Cho, A.

A. Cho, Science 323, 701 (2009).
[PubMed]

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[PubMed]

Dolin, L. S.

L. S. Dolin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 4, 964 (1961).

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[PubMed]

A. Greenleaf, M. Lassas, and G. Uhlmann, Math. Res. Lett. 10, 685 (2003).

Gu, Y.

X. Luo, T. Yang, Y. Gu, H. Y. Chen, and H. R. Ma, Appl. Phys. Lett. 94, 223513 (2009).

Han, D. Z.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

Hou, B.

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

Huangfu, J. T.

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[PubMed]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).

A. V. Kildishev and E. E. Narimanov, Opt. Lett. 32, 3432 (2007).
[PubMed]

Kong, J. A.

J. Zhang, Y. Luo, H. S. Chen, J. T. Huangfu, B.-I. Wu, L. Ran, and J. A. Kong, Opt. Express 17, 6203 (2009).
[PubMed]

Y. Luo, J. Zhang, H. S. Chen, B.-I. Wu, and J. A. Kong, http://arxiv.org/abs/0808.0215.

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[PubMed]

Lai, Y.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 093901 (2009).
[PubMed]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[PubMed]

A. Greenleaf, M. Lassas, and G. Uhlmann, Math. Res. Lett. 10, 685 (2003).

Leonhardt,

Leonhardt, Science 312, 1777 (2006).
[PubMed]

Leonhardt, U.

U. Leonhardt and T. Tyc, Science 323, 110 (2009).

U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006).

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nature Mater. 8, 568 (2009).

J. Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901 (2008).
[PubMed]

Lin, Z. F.

H. Y. Chen, C. T. Chan, S. Y. Liu, and Z. F. Lin, New J. Phys. 11, 083012 (2009).

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

Liu, S. Y.

H. Y. Chen, C. T. Chan, S. Y. Liu, and Z. F. Lin, New J. Phys. 11, 083012 (2009).

Luo, X.

Luo, Y.

J. Zhang, Y. Luo, H. S. Chen, J. T. Huangfu, B.-I. Wu, L. Ran, and J. A. Kong, Opt. Express 17, 6203 (2009).
[PubMed]

Y. Luo, J. Zhang, H. S. Chen, B.-I. Wu, and J. A. Kong, http://arxiv.org/abs/0808.0215.

Ma, H.

Ma, H. R.

X. Luo, T. Yang, Y. Gu, H. Y. Chen, and H. R. Ma, Appl. Phys. Lett. 94, 223513 (2009).

Milton, G. W.

G. W. Milton and N.-A. P. Nicorovici, Proc. R. Soc. London, Ser. A 462, 3027 (2006).

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[PubMed]

Narimanov, E. E.

Ng, J.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

J. Ng, H. Y. Chen, and C. T. Chan, Opt. Lett. 34, 644 (2009).
[PubMed]

Nicorovici, N.-A. P.

G. W. Milton and N.-A. P. Nicorovici, Proc. R. Soc. London, Ser. A 462, 3027 (2006).

Pendry, J. B.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

J. Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901 (2008).
[PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[PubMed]

J. B. Pendry and S. A. Ramakrishna, J. Phys.: Condens. Matter 15, 6345 (2003).

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[PubMed]

Philbin, T.

T. Philbin, Physics (N.Y.) 2, 17 (2009).

Philbin, T. G.

U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006).

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).

Qiu, M.

M. Yan, W. Yan, and M. Qiu, Phys. Rev. B 78, 125113 (2008).

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, J. Phys.: Condens. Matter 15, 6345 (2003).

Ran, L.

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[PubMed]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, Science 323, 366 (2009).
[PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics Nanostruct. Fundam. Appl. 6, 87 (2008).

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[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, Science 314, 977 (2006).
[PubMed]

Tyc, T.

U. Leonhardt and T. Tyc, Science 323, 110 (2009).

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[PubMed]

A. Greenleaf, M. Lassas, and G. Uhlmann, Math. Res. Lett. 10, 685 (2003).

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nature Mater. 8, 568 (2009).

Wen, W.

H. Y. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[PubMed]

Wu, B.-I.

J. Zhang, Y. Luo, H. S. Chen, J. T. Huangfu, B.-I. Wu, L. Ran, and J. A. Kong, Opt. Express 17, 6203 (2009).
[PubMed]

Y. Luo, J. Zhang, H. S. Chen, B.-I. Wu, and J. A. Kong, http://arxiv.org/abs/0808.0215.

Xiao, J. J.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

Yan, M.

M. Yan, W. Yan, and M. Qiu, Phys. Rev. B 78, 125113 (2008).

Yan, W.

M. Yan, W. Yan, and M. Qiu, Phys. Rev. B 78, 125113 (2008).

Yang, T.

X. Luo, T. Yang, Y. Gu, H. Y. Chen, and H. R. Ma, Appl. Phys. Lett. 94, 223513 (2009).

T. Yang, H. Y. Chen, X. Luo, and H. Ma, Opt. Express 16, 18545 (2008).
[PubMed]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nature Mater. 8, 568 (2009).

Zhang, J.

J. Zhang, Y. Luo, H. S. Chen, J. T. Huangfu, B.-I. Wu, L. Ran, and J. A. Kong, Opt. Express 17, 6203 (2009).
[PubMed]

Y. Luo, J. Zhang, H. S. Chen, B.-I. Wu, and J. A. Kong, http://arxiv.org/abs/0808.0215.

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nature Mater. 8, 568 (2009).

Zhang, Z. Q.

Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 093901 (2009).
[PubMed]

Zhang, Z.-Q.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 253902 (2009).
[PubMed]

Appl. Phys. Lett. (2)

H. Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007).

X. Luo, T. Yang, Y. Gu, H. Y. Chen, and H. R. Ma, Appl. Phys. Lett. 94, 223513 (2009).

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

L. S. Dolin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 4, 964 (1961).

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

H. Y. Chen, J. Opt. A, Pure Appl. Opt. 11, 075102 (2009).

J. Phys.: Condens. Matter (1)

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Figures (3)

Fig. 1
Fig. 1

(a) Bipolar coordinates σ and τ isosurfaces with the foci located at ( a , 0 ) and ( + a , 0 ) . (b) Geometric interpretation of σ, which is the angle from the directed line PF 2 to F 1 P , where F 1 and F 2 are the two foci and P is the point described in bipolar coordinates. When y 0 , σ [ 0 , π ] . (c) Same as (b), when y 0 , σ [ π , 0 ] .

Fig. 2
Fig. 2

(a) Physical space, (b) illusion space, and (c) coordinate transformation in bipolar coordinates.

Fig. 3
Fig. 3

(a) Obliquely incident TE plane wave in free space. (b) The structure with the material described in Eq. (3) ( ε s = μ s = 1 ) and a DNM is invisible to any incoming waves. (c) The external cloak (invisibility umbrella) can cloak a “1” shape scatterer at a distance. (d) Without the external cloak, the “1” shape scatterer causes some scattering.

Equations (6)

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x = a sinh τ cosh τ + cos σ ,
y = a sin σ cosh τ + cos σ ,
σ = f ( σ ) = { σ , 0 σ π σ , σ 1 σ < 0 σ 2 + σ 1 σ 2 σ 1 ( σ + σ 2 ) σ 2 , σ 2 σ < σ 1 σ , π σ < σ 2 } ,
ε σ = 1 d f ( σ ) d σ ,
ε τ = d f ( σ ) d σ ,
ε z = d f ( σ ) d σ ( cos τ + cos σ cos τ + cos f ( σ ) ) 2 ,

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