Abstract

We extend the design of radially symmetric three-dimensional invisibility cloaks through transformation optics [1] to cloaks with a surface of revolution. We derive the expression of the transformation matrix and show that one of its eigenvalues vanishes on the inner boundary of the cloaks, while the other two remain strictly positive and bounded. The validity of our approach is confirmed by finite edge-elements computations for a non-convex cloak of varying thickness.

© 2009 Optical Society of America

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  1. J.B. Pendry, D. Shurig and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, "Optical conformal mapping," Science 312,1777 (2006).
    [CrossRef] [PubMed]
  3. A. Alu and N. Engheta, "Achieving Transparency with Plasmonic and Metamaterial Coatings," Phys. Rev. E 95,016623 (2005).
    [CrossRef]
  4. G. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462,3027 (2006).
    [CrossRef]
  5. F. Zolla, S. Guenneau, A. Nicolet, and J.B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069 (2007).
    [CrossRef] [PubMed]
  6. A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).
  7. 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 (2006).
    [CrossRef] [PubMed]
  8. P. Zhang, Y. Jin, and S. He, "Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak," Appl. Phys. Lett. 93, 243502 (2008).
    [CrossRef]
  9. U. Leonhardt and T. Tyc, "Broadband invisibility by non-euclidean cloaking," Science 323, 110 (2009).
    [CrossRef]
  10. W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007).
    [CrossRef]
  11. M. Farhat, S. Guenneau, A. B. Movchan, and S. Enoch, "Achieving invisibility over a finite range of frequencies," Opt. Express 16, 5656-5661 (2008).
    [CrossRef] [PubMed]
  12. C.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
    [CrossRef]
  13. Y. You, G. W. Kattawar, and P. Yang, "Invisibility cloaks for toroids," Opt. Express 17, 6591 (2009).
    [CrossRef] [PubMed]
  14. A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
    [CrossRef]
  15. R. V. Kohn, H. Shen,M. S. Vogelius, andM. I.Weinstein, "Cloaking via change of variables in electric impedance tomography," Inverse Probl. 24, 015016 (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 77, 066607 (2008).
    [CrossRef]
  17. 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]

2009

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

C.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
[CrossRef]

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

2008

M. Farhat, S. Guenneau, A. B. Movchan, and S. Enoch, "Achieving invisibility over a finite range of frequencies," Opt. Express 16, 5656-5661 (2008).
[CrossRef] [PubMed]

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]

R. V. Kohn, H. Shen,M. S. Vogelius, andM. I.Weinstein, "Cloaking via change of variables in electric impedance tomography," Inverse Probl. 24, 015016 (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]

P. Zhang, Y. Jin, and S. He, "Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak," Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

2007

W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007).
[CrossRef]

F. Zolla, S. Guenneau, A. Nicolet, and J.B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069 (2007).
[CrossRef] [PubMed]

2006

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 (2006).
[CrossRef] [PubMed]

J.B. Pendry, D. Shurig and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical conformal mapping," Science 312,1777 (2006).
[CrossRef] [PubMed]

G. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462,3027 (2006).
[CrossRef]

2005

A. Alu and N. Engheta, "Achieving Transparency with Plasmonic and Metamaterial Coatings," Phys. Rev. E 95,016623 (2005).
[CrossRef]

2003

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

1994

A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, "Achieving Transparency with Plasmonic and Metamaterial Coatings," Phys. Rev. E 95,016623 (2005).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007).
[CrossRef]

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]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (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 77, 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 77, 066607 (2008).
[CrossRef]

Cummer, S. A.

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 (2006).
[CrossRef] [PubMed]

Engheta, N.

A. Alu and N. Engheta, "Achieving Transparency with Plasmonic and Metamaterial Coatings," Phys. Rev. E 95,016623 (2005).
[CrossRef]

Enoch, S.

Farhat, M.

Feng, Y.

C.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
[CrossRef]

Genon, A.

A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
[CrossRef]

Greenleaf, A.

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Guenneau, S.

He, S.

P. Zhang, Y. Jin, and S. He, "Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak," Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

Hu, L.

C.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
[CrossRef]

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 77, 066607 (2008).
[CrossRef]

Jin, Y.

P. Zhang, Y. Jin, and S. He, "Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak," Appl. Phys. Lett. 93, 243502 (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 (2006).
[CrossRef] [PubMed]

Kattawar, G. W.

Kildiev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007).
[CrossRef]

Kohn, R. V.

R. V. Kohn, H. Shen,M. S. Vogelius, andM. I.Weinstein, "Cloaking via change of variables in electric impedance tomography," Inverse Probl. 24, 015016 (2008).
[CrossRef]

Lassas, M.

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Legros, W.

A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
[CrossRef]

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 (2006).
[CrossRef] [PubMed]

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]

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 77, 066607 (2008).
[CrossRef]

Meys, B.

A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
[CrossRef]

Milton, G.

G. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462,3027 (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, 977 (2006).
[CrossRef] [PubMed]

Movchan, A. B.

Nicolet, A.

Nicorovici, N. A.

G. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462,3027 (2006).
[CrossRef]

Pendry, J. B.

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 (2006).
[CrossRef] [PubMed]

Pendry, J.B.

Qiu, C.W.

C.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
[CrossRef]

Remacle, J.F.

A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
[CrossRef]

Schurig, D.

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 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007).
[CrossRef]

Shen, H.

R. V. Kohn, H. Shen,M. S. Vogelius, andM. I.Weinstein, "Cloaking via change of variables in electric impedance tomography," Inverse Probl. 24, 015016 (2008).
[CrossRef]

Shurig, D.

J.B. Pendry, D. Shurig and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

Smith, D. R.

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 (2006).
[CrossRef] [PubMed]

Smith, D.R.

J.B. Pendry, D. Shurig and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (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 (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, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Vogelius, M. S.

R. V. Kohn, H. Shen,M. S. Vogelius, andM. I.Weinstein, "Cloaking via change of variables in electric impedance tomography," Inverse Probl. 24, 015016 (2008).
[CrossRef]

Xu, X.

C.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
[CrossRef]

Yang, P.

You, Y.

Zhang, P.

P. Zhang, Y. Jin, and S. He, "Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak," Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

Zolla, F.

Appl. Phys. Lett.

P. Zhang, Y. Jin, and S. He, "Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak," Appl. Phys. Lett. 93, 243502 (2008).
[CrossRef]

Inverse Probl.

R. V. Kohn, H. Shen,M. S. Vogelius, andM. I.Weinstein, "Cloaking via change of variables in electric impedance tomography," Inverse Probl. 24, 015016 (2008).
[CrossRef]

J. Appl. Phys.

A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 756036 (1994).
[CrossRef]

Math. Res. Lett.

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Nat. Photon.

W. Cai, U. K. Chettiar, A. V. Kildiev, and V. M. Shalaev, "Optical Cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

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.W. Qiu, L. Hu, X. Xu, and Y. Feng, "Spherical cloaking with homogeneous isotropic multilayered structures," Phys. Rev. E 79, 047602 (2009).
[CrossRef]

A. Alu and N. Engheta, "Achieving Transparency with Plasmonic and Metamaterial Coatings," Phys. Rev. E 95,016623 (2005).
[CrossRef]

Proc. R. Soc. London A

G. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462,3027 (2006).
[CrossRef]

Science

J.B. Pendry, D. Shurig and D.R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical conformal mapping," Science 312,1777 (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 (2006).
[CrossRef] [PubMed]

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

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Figures (4)

Fig. 1.
Fig. 1.

3D plot of (T -1)11, as given by Eq. 5, Eq. 6 and Eq. 7 within the cloak with boundaries given by Eq. 10. The symmetries are noted.

Fig. 2.
Fig. 2.

3D plot of the magnitude ε = of the total electric field for a plane wave of wavenumber k=2π/0.3 incident from above on an non-convex invisibility cloak.

Fig. 3.
Fig. 3.

2D plot of E 1 2 + E 2 2 + E 3 2 generated by a slice of Fig. 2 in the xz-plane for y=0.

Fig. 4.
Fig. 4.

2D plot of E 1 2 + E 2 2 + E 3 2 generated by a slice of Fig. 2 in the yz-plane for x=0.

Equations (10)

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

ρ ( θ , ϕ ) = a 0 , 0 + Σ ( m , n ) 2 { ( 0 , 0 ) } { a m , n cos ( m θ + n ϕ ) + b m , n sin ( m θ + n ϕ ) } ,
{ ρ ( ρ , ϕ ) = R 1 ( ϕ ) + ρ R 2 ( ϕ ) R 1 ( ϕ ) R 2 ( ϕ ) , θ = θ , 0 < θ 2 π , ϕ = ϕ , π 2 π 2
J ( ρ , ϕ ) = ( ρ ( ρ , ϕ ) , θ , ϕ ) ( ρ , θ , ϕ ) .
ε = = ε T 1 , and μ = = μ T 1 ,
T 1 = ( c 13 2 + ρ ( ρ , ϕ ) 2 c 11 ρ 2 0 c 13 ρ 0 c 11 0 c 13 ρ 0 c 11 ) ,
c 11 ( ϕ ) = R 2 ( ϕ ) R 2 ( ϕ ) R 1 ( ϕ ) ,
c 13 ( ϕ ) = R 2 ( ϕ ) ρ R 2 ( ϕ ) ( R 2 ( ϕ ) R 1 ( ϕ ) ) 2 d R 1 ( ϕ ) d ϕ + R 1 ( ϕ ' ) R 1 ( ϕ ) ρ ( R 2 ( ϕ ) R 1 ( ϕ ) ) 2 d R 2 ( ϕ ) d ϕ ,
λ j = c 13 2 + ρ 2 + c 11 2 ρ 2 2 c 11 ρ 2 + ( 1 ) j 2 ( c 13 2 + ρ 2 + c 11 2 ρ 2 c 11 ρ 2 ) 2 4 ρ 2 ρ 2 , j = 1 , 2 , and λ 3 = c 11 .
× ( μ = 1 × E ) k 2 ε = E = 0
R 1 ( ϕ ) = 0.2 + 0.02 cos ( 4 ϕ ) , R 2 ( ϕ ) = 0.4 + 0.02 cos ( 8 ϕ ) .

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