Abstract

We study an invisibility cloak with a twin cavity, simulated by a plane algebraic curve- hippopede. The cloaked region, which looks like eight for some sets of geometric parameters, is expanded from one single point. Using a geometric transformation approach, we demonstrate that the material parameters of cloaking layer can be exactly determined. Numerical simulations show that the incoming rays pass in and out the cloaking region twice, and return to their original trajectory outside the curved cloak. A notable feature is that the cloaking region has two hollow regions in which two objects can be hidden at one time and that they could not perceive each other.

© 2009 Optical Society of America

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  6. 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]
  7. A. Greenleaf, Y. Kurylev, M. Lassas, and G Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett.  99, 183901 (2007).
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  14. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.  90, 241105 (2007).
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    [CrossRef]
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    [CrossRef]
  18. J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.
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    [CrossRef] [PubMed]
  21. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.  101, 203901 (2008).
    [CrossRef] [PubMed]
  22. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (2009).
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  26. D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett.  92, 013505 (2008).
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    [CrossRef] [PubMed]
  31. J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17, 1308–1320 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
  33. J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (2008).
    [CrossRef]
  34. 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]
  35. 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]
  36. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16, 13414–13420 (2008).
    [CrossRef] [PubMed]
  37. H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
    [CrossRef] [PubMed]
  38. J. D. Lawrence,. A Catalog of Special Plane Curves, (New York: Dover, 1972).
  39. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (New York: McGraw-Hill, 1961).
  40. A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys.  28153–157 (2004).
    [CrossRef]
  41. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009).
    [CrossRef]
  42. 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]

2009 (7)

X. Chen, Y. Fu, and N. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express 17, 3581–3586 (2009).
[CrossRef] [PubMed]

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

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys.  42, 035408 (2009).
[CrossRef]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (2009).
[CrossRef] [PubMed]

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

Y. You, G. W. Kattawar, and P. Yang, “Invisibility cloaks for toroids,” Opt. Express 17, 6591–6599 (2009).
[CrossRef] [PubMed]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009).
[CrossRef]

2008 (15)

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]

Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16, 6134–6145 (2008).
[CrossRef] [PubMed]

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

D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett.  92, 013505 (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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (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, 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]

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

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

A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008).
[CrossRef]

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

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

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

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.  101, 203901 (2008).
[CrossRef] [PubMed]

2007 (5)

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

G. W. Milton and J. R. Willis, “On modifications of Newton’s second law and linear continuum elastodynamics,“ Proc. R. Soc. London, Ser. A 463, 855–880 (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]

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

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

2006 (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]

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

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

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

2004 (1)

A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys.  28153–157 (2004).
[CrossRef]

2003 (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.  24, 413–419 (2003).
[CrossRef] [PubMed]

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.  43, 773–793. (1996).
[CrossRef]

1994 (2)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

1984 (1)

R. V. Kohn and M. Vogelius, “Identification of an unknown conductivity by means of measurements at the boundary,” in Inverse problems, D. W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113–123.

Briane, M.

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

Chan, C. T.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (2009).
[CrossRef] [PubMed]

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

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

Chen, H.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (2009).
[CrossRef] [PubMed]

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

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

J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (2008).
[CrossRef]

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

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

Chen, J. S.

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

Chen, X.

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

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: Appl. Phys.  41, 085504 (2008).
[CrossRef]

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (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, 041904 (2009).
[CrossRef]

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

W. 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]

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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

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

Fu, Y.

Genon, A.

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.

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]

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.  24, 413–419 (2003).
[CrossRef] [PubMed]

Guenneau, S.

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]

A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys.  28153–157 (2004).
[CrossRef]

Hu, G.

Hu, J.

Huangfu, J.

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (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, 041904 (2009).
[CrossRef]

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

W. 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]

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.

Kohn, R. V.

R. V. Kohn and M. Vogelius, “Identification of an unknown conductivity by means of measurements at the boundary,” in Inverse problems, D. W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113–123.

Kong, J. A.

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (New York: McGraw-Hill, 1961).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (New York: McGraw-Hill, 1961).

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]

Kwon, D. H.

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

Lai, Y.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (2009).
[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]

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.  24, 413–419 (2003).
[CrossRef] [PubMed]

Lawrence, J. D.

J. D. Lawrence,. A Catalog of Special Plane Curves, (New York: Dover, 1972).

Legros, W.

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.

Leonhardt, U.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009).
[CrossRef]

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

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of Light,” arXiv: 0805.4778; Prog. Optics (to appear).

Li, C.

Li, F.

Li, J.

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.  101, 203901 (2008).
[CrossRef] [PubMed]

Li, L. W.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys.  42, 035408 (2009).
[CrossRef]

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]

Lin, X. Q.

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

Liu, R.

W. X. Jiang, 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]

Luo, X.

Luo, Y.

J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (2008).
[CrossRef]

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Ma, H.

McPhedran, R. C.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Meng, F. Y.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys.  42, 035408 (2009).
[CrossRef]

Meys, B.

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

Milton, G. W.

G. W. Milton and J. R. Willis, “On modifications of Newton’s second law and linear continuum elastodynamics,“ Proc. R. Soc. London, Ser. A 463, 855–880 (2007).
[CrossRef]

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

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[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]

Nicolet, A.

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]

A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys.  28153–157 (2004).
[CrossRef]

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Norris, A. N.

A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008).
[CrossRef]

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.  101, 203901 (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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

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

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

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.  43, 773–793. (1996).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of Light,” arXiv: 0805.4778; Prog. Optics (to appear).

Post, E. J.

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

Qu, S.

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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

Remade, J. F.

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.

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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

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

Smith, D. R.

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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (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. 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]

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]

Tyc, T.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009).
[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]

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.  24, 413–419 (2003).
[CrossRef] [PubMed]

Vogelius, M.

R. V. Kohn and M. Vogelius, “Identification of an unknown conductivity by means of measurements at the boundary,” in Inverse problems, D. W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113–123.

Wang, J.

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.  43, 773–793. (1996).
[CrossRef]

Weng, C. N.

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

Willis, J. R.

G. W. Milton and J. R. Willis, “On modifications of Newton’s second law and linear continuum elastodynamics,“ Proc. R. Soc. London, Ser. A 463, 855–880 (2007).
[CrossRef]

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

Wu, B. I.

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (2008).
[CrossRef]

Wu, Q.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys.  42, 035408 (2009).
[CrossRef]

Xu, Z.

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]

You, Y.

Yu, G. X.

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

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

Yuan, N.

Zhai, P. W.

Zhang, J.

J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (2008).
[CrossRef]

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Zhang, K.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys.  42, 035408 (2009).
[CrossRef]

Zhang, Z. Q.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (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. Lett.  33, 1584–1586 (2008).
[CrossRef] [PubMed]

A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys.  28153–157 (2004).
[CrossRef]

Appl. Phys. Lett (6)

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

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

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

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

Eur. Phys. J. Appl. Phys (1)

A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys.  28153–157 (2004).
[CrossRef]

J. Appl. Phys (1)

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys.  75, 6036–6038 (1994).
[CrossRef]

J. Mod. Opt (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.  43, 773–793. (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys (2)

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

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys.  42, 035408 (2009).
[CrossRef]

New J. Phys (2)

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

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

Opt. Express (7)

Opt. Lett (1)

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]

Photonics 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,” Photonics Nanostruct. Fundam. Appl.  6, 87–95 (2008).
[CrossRef]

Phys. Rev. B (2)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (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 (3)

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.  101, 203901 (2008).
[CrossRef] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.  102, 093901 (2009).
[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]

Physiol. Meas (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.  24, 413–419 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (2)

A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008).
[CrossRef]

G. W. Milton and J. R. Willis, “On modifications of Newton’s second law and linear continuum elastodynamics,“ Proc. R. Soc. London, Ser. A 463, 855–880 (2007).
[CrossRef]

Science (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]

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

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

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009).
[CrossRef]

Other (6)

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

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of Light,” arXiv: 0805.4778; Prog. Optics (to appear).

R. V. Kohn and M. Vogelius, “Identification of an unknown conductivity by means of measurements at the boundary,” in Inverse problems, D. W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113–123.

J. D. Lawrence,. A Catalog of Special Plane Curves, (New York: Dover, 1972).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (New York: McGraw-Hill, 1961).

J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.

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Figures (5)

Fig. 1.
Fig. 1.

The hippopedal curves, described by Eq. (1), with different sets of parameters (a) a > b (dashed lines), (b) a = b (solid line), (c) a < b (dotted lines).

Fig. 2.
Fig. 2.

The magnitudes of material properties εxx ′, εyy ′, εzz ′, and εxy ′ inside the hippopedal cloaking layer (a = 1, b = 1) .

Fig. 3.
Fig. 3.

Snapshot of the electric field distribution. The plane waves are incident in the horizontal direction. (a) a = 0.12, b = 0.1, (b) a = 0.15, b = 0.15, (c) a = 0.11, b = 0.16.

Fig. 4.
Fig. 4.

Snapshot of the electric field distribution (a), and stream lines (b). The plane waves are incident at an oblique angle of 30° with a = 0.15, b = 0.15.

Fig. 5.
Fig. 5.

A schematic illustration of a few plane curves. (a) double folium, r = 4acosθsin2 θ (dashed line), (b) sinusoidal spirals, r = (acos3θ)1/3 (dotted line).

Equations (14)

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

r2=4b(absin2θ),
r(θ)=r(θ)=r(πθ)=r(π+θ).
(x2+y2)2+4b(bc)(x2+y2)4b2x2=0.
((xb)2+y2b2)((x+b)2+y2b2)=0.
Ωi:ri=2abb2sin2θ ,
Ωo:ro=2mabb2sin2θ ,
x'=rrx=(α+rir)x,y=rry=(α+rir)y,z=z,
ε'=ATdetA , μ=ATdetA,
A=(x',y',z')(x,y,z)=(α+4by2((a+b)r22by2)r5ri4bxy((a+b)r22by2)r5ri04bxy(ar22by2)r5riα+4bx2(ar22by2)r5ri0001).
εxx=μxx=rαr [(α+4by2((a+b)r22by2)r5ri)2+(4bxy((a+b)r22by2)r5ri)2] ,
εyy=μyy=rαr [α+(4bx2(ar22by2)r5ri)2+(4bxy(ar22by2)r5ri)2] ,
εzz=μzz=rαr ,
εzz=μxy=rαr [(4bxy(4br2(ar22by2)(ar2+b(x2y2))αr5ri((2a+b)r24by2))r10ri2)].
ε'=μ'=(r'rir'+4b2sin22θri2r'(r'ri)2b2sin2θri(r'ri)0r'r'ri0sym1α2r'rir').

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