Abstract

The design of electromagnetic cloaks based on the coordinate transformation requires a suitable geometrical definition of both the internal and the external surfaces of the cloak itself. We describe a straightforward method to design the electromagnetic cloak of a 3d-generic star domain whose surface is defined just by a set of points distributed over it. We also present numerical simulation for the ray tracing of a light beam inside the material calculated for an asymmetric three-dimensional example.

© 2009 Optical Society of America

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312(5781), 1780-1782 (2006).
    [CrossRef]
  2. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14(21), 9794-9804 (2006).
    [CrossRef]
  3. J. Sun, J. Zhou, and L. Kang, "Homogeneous isotropic invisible cloak based on geometrical optics," Opt. Express 16(22), 17768-17773 (2008).
    [CrossRef]
  4. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16(8), 5444-5452 (2008).
    [CrossRef]
  5. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
    [CrossRef]
  6. M. Farhat, S. Guenneau, A. B. Movchan, and S. Enoch, "Achieving invisibility over a finite range of frequencies," Opt. Express 16(8), 5656-5661 (2008).
    [CrossRef]
  7. L. Ulf and T. Tomas, "Broadband Invisibility by Non-Euclidean Cloaking," Science 323(5910), 110-112 (2009).
  8. P. Alitalo and S. Tretyakov, "Electromagnetic cloaking with metamaterials," Materials Today 12(3), 22-29 (2009).
    [CrossRef]
  9. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Two-dimensional metamaterial structure exhibiting reduced visibility at 500 nm," Opt. Lett. 33(12), 1342-1344 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  11. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (2009).
    [CrossRef]
  12. 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]
  13. H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 78(3), 036608 (2008).
    [CrossRef]
  14. H. Ma, S. Qu, Z. Xu, and J. Wang. "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16(20), 15449-15454 (2008).
    [CrossRef]
  15. A. Nicolet, F. Zolla, and S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett. 33(14), 1584-1586 (2008).
    [CrossRef]
  16. 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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
    [CrossRef]
  17. J. Zhang, Y. Luo, H. Chen, and B. I. Wu. "Cloak of arbitrary shape," J. Opt. Soc. Am. B 25(11), 1776-1779 (2008).
    [CrossRef]
  18. W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformation makes perfect invisibility cloak with arbitrary shape," New J. Phys. 10 (2008).
    [CrossRef]
  19. CGAL, Computational Geometry Algorithms Library.http://www.cgal.org.

2009 (4)

L. Ulf and T. Tomas, "Broadband Invisibility by Non-Euclidean Cloaking," Science 323(5910), 110-112 (2009).

P. Alitalo and S. Tretyakov, "Electromagnetic cloaking with metamaterials," Materials Today 12(3), 22-29 (2009).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (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]

2008 (9)

2007 (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312(5781), 1780-1782 (2006).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14(21), 9794-9804 (2006).
[CrossRef]

Alitalo, P.

P. Alitalo and S. Tretyakov, "Electromagnetic cloaking with metamaterials," Materials Today 12(3), 22-29 (2009).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (2009).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16(8), 5444-5452 (2008).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

Chen, H.

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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16(8), 5444-5452 (2008).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

Chin, J. Y.

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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

Cui, T. J.

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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

Davis, C. C.

Enoch, S.

Farhat, M.

Guenneau, S.

Hu, G.

Hu, J.

Hung, Y. J.

Jiang, W. X.

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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

Kang, L.

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16(8), 5444-5452 (2008).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

Liu, R.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

Luo, Y.

Ma, H.

H. Ma, S. Qu, Z. Xu, and J. Wang. "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16(20), 15449-15454 (2008).
[CrossRef]

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 78(3), 036608 (2008).
[CrossRef]

Milton, G. W.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

Movchan, A. B.

Nicolet, A.

Pendry, J. B.

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14(21), 9794-9804 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312(5781), 1780-1782 (2006).
[CrossRef]

Qiu, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformation makes perfect invisibility cloak with arbitrary shape," New J. Phys. 10 (2008).
[CrossRef]

Qu, S.

H. Ma, S. Qu, Z. Xu, and J. Wang. "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16(20), 15449-15454 (2008).
[CrossRef]

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 78(3), 036608 (2008).
[CrossRef]

Ruan, Z.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformation makes perfect invisibility cloak with arbitrary shape," New J. Phys. 10 (2008).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312(5781), 1780-1782 (2006).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14(21), 9794-9804 (2006).
[CrossRef]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16(8), 5444-5452 (2008).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

Smith, D. R.

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14(21), 9794-9804 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312(5781), 1780-1782 (2006).
[CrossRef]

Smolyaninov, I. I.

Sun, J.

Tomas, T.

L. Ulf and T. Tomas, "Broadband Invisibility by Non-Euclidean Cloaking," Science 323(5910), 110-112 (2009).

Tretyakov, S.

P. Alitalo and S. Tretyakov, "Electromagnetic cloaking with metamaterials," Materials Today 12(3), 22-29 (2009).
[CrossRef]

Ulf, L.

L. Ulf and T. Tomas, "Broadband Invisibility by Non-Euclidean Cloaking," Science 323(5910), 110-112 (2009).

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (2009).
[CrossRef]

Wang, J.

H. Ma, S. Qu, Z. Xu, and J. Wang. "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16(20), 15449-15454 (2008).
[CrossRef]

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 78(3), 036608 (2008).
[CrossRef]

Wu, B. I.

Xu, Z.

H. Ma, S. Qu, Z. Xu, and J. Wang. "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16(20), 15449-15454 (2008).
[CrossRef]

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 78(3), 036608 (2008).
[CrossRef]

Yan, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformation makes perfect invisibility cloak with arbitrary shape," New J. Phys. 10 (2008).
[CrossRef]

Yan, W.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformation makes perfect invisibility cloak with arbitrary shape," New J. Phys. 10 (2008).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (2009).
[CrossRef]

Zhang, J.

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (2009).
[CrossRef]

Zhou, J.

Zhou, X.

Zolla, F.

Appl. Phys. Lett. (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91(11), 111105 (2007).
[CrossRef]

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

Materials Today (1)

P. Alitalo and S. Tretyakov, "Electromagnetic cloaking with metamaterials," Materials Today 12(3), 22-29 (2009).
[CrossRef]

Nat Mater. (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat Mater. 8(7), 568-571 (2009).
[CrossRef]

New J. Phys. (1)

W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformation makes perfect invisibility cloak with arbitrary shape," New J. Phys. 10 (2008).
[CrossRef]

Opt. Express (6)

Opt. Lett. (3)

Science (2)

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312(5781), 1780-1782 (2006).
[CrossRef]

L. Ulf and T. Tomas, "Broadband Invisibility by Non-Euclidean Cloaking," Science 323(5910), 110-112 (2009).

Other (3)

CGAL, Computational Geometry Algorithms Library.http://www.cgal.org.

H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 78(3), 036608 (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 (Statistical, Nonlinear, and Soft Matter Physics) 77(6) 066607 (2008).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Cross section of the cloak for an arbitrary shaped object. The cloak is the hollow shell defined by its internal (a) and external (b) surfaces.

Fig. 2.
Fig. 2.

(a) a set of geometric points uniformly distributed on the object to cloak; (b) the discrete polar function ai (φi ,θi ) corresponding to the set of points sketched in (a); (c) the continuous function a(φ,θ) obtained via interpolation from the discrete function ai (φi ,θi ) presented in (b); (d) The object is smoothly reconstructed by reverting the points of function a(φ,θ) of sketch (c) in the Cartesian space.

Fig. 3.
Fig. 3.

Matrix of images representing a color map of the dielectric tensor εij in cut at y=0. Some elements shows negative values and ε 11 is very high near the sharper borders.

Fig. 4.
Fig. 4.

Ray tracing simulation for the object described in Fig. 2 cloaked using S=1.5 and put on a reflecting surface (ground). The object to cloak is represented in gray-scale, the external surface of the cloak is sketched as a green grid while the light beams are plotted in red: (a) top-right view of the ray tracing in presence of the cloaked object (b) top-right view of the ray tracing on a empty surface (c) side view of the ray tracing in presence of the cloaked object (d) side view of the ray tracing on a empty surface. The light beams plotted in (a) and (c) are perfectly superimposed to the ones plotted in (b) and (d) in the section of space outside the cloak, this means the object will be invisible for any external detector.

Equations (18)

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

r = αr + a (φ,θ)
x1=αx1+a1(φ,θ)x2=αx2+a2(φ,θ)x3=αx3+a3(φ,θ)
Λji = xixj = (α+a1x1a1x2a1za2x1α+a2x2a2za3x1a3x2α+a3z)
gij = Λki gkl Λlj = Λli δkl Λlj = Λli Λjj = Λ2
εij = det(Λji)1 gij ε0
xi = (α+ar)xi
xi = 1α (1ar) xi
xir = xir
Λji = αδji + ar (δjixixjr2) + (aφϕxj+ϕθθxj) xir
c4ω4 1det(ε) [kiεijkjω2c2det(ε)]2 = 0
ω = c2 kiεijkjdet(ε) .
dkedt = ωxe = c22ω ki xe (εijdet(ε)) kj
dxedt = ωke = c2ω εeikidet(ε) (i,j,e=1,,3)
dk̂rdτ = 12 k̂i k̂j ξr (εijdet(ε))
dξr = εrik̂idet(ε)
k̂i(2) = k̂i(1) = (δijninj)kj(1)
k̂i(2) εij k̂j(2) det (ε)=0
k̂i(2) = [k̂(2)(k̂(1)·n)]ni+k̂i(1)

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