Abstract

We present a finite-element analysis of a diffraction problem involving a coated cylinder enabling the electromagnetic cloaking of a lossy object with sharp wedges located within its core. The coating consists of a heterogeneous anisotropic material deduced from a geometrical transformation as first proposed by Pendry et al. [Science 312, 1780 (2006)] . We analyze the electromagnetic response of the cloak in the presence of an electric line source in p polarization and a loop of magnetic current in s polarization. We find that the electromagnetic field radiated by such a source located a fraction of a wavelength from the cloak is perturbed by less than 1%. When the source lies in the coating, it seems to radiate from a shifted location.

© 2007 Optical Society of America

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References

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  1. J. B. Pendry, D. Shurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006).
    [CrossRef]
  4. A. Alu and N. Engheta, Phys. Rev. E 95, 016623 (2005).
    [CrossRef]
  5. G. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. A 462, 3027 (2006).
    [CrossRef]
  6. A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 73 (1996).
    [CrossRef]
  7. A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
    [CrossRef]
  8. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
    [CrossRef]
  9. 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).
    [CrossRef] [PubMed]
  10. J. P. Bérenger, J. Comput. Phys. 114, 185 (1994).
    [CrossRef]

2006 (5)

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

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

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

G. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. A 462, 3027 (2006).
[CrossRef]

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).
[CrossRef] [PubMed]

2005 (1)

A. Alu and N. Engheta, Phys. Rev. E 95, 016623 (2005).
[CrossRef]

1996 (1)

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 73 (1996).
[CrossRef]

1994 (2)

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

J. P. Bérenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, Phys. Rev. E 95, 016623 (2005).
[CrossRef]

Bérenger, J. P.

J. P. Bérenger, J. Comput. Phys. 114, 185 (1994).
[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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Engheta, N.

A. Alu and N. Engheta, Phys. Rev. E 95, 016623 (2005).
[CrossRef]

Felbacq, D.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

Genon, A.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

Guenneau, S.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Kuhlmey, B.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

Legros, W.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

Leonhardt, U.

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

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

Meys, B.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

Milton, G.

G. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. 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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Nicolet, A.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

Nicorovici, N. A.

G. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. 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, Science 314, 977 (2006).
[CrossRef] [PubMed]

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

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 73 (1996).
[CrossRef]

Philbin, T. G.

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

Remacle, J. F.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

Renversez, G.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

Schurig, D.

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).
[CrossRef] [PubMed]

Shurig, D.

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

Smith, D. R.

J. B. Pendry, D. Shurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [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).
[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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Ward, A. J.

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 73 (1996).
[CrossRef]

Zolla, F.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

J. Appl. Phys. (1)

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, J. Appl. Phys. 75, 6036 (1994).
[CrossRef]

J. Comput. Phys. (1)

J. P. Bérenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 73 (1996).
[CrossRef]

New J. Phys. (1)

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

Phys. Rev. E (1)

A. Alu and N. Engheta, Phys. Rev. E 95, 016623 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

G. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. A 462, 3027 (2006).
[CrossRef]

Science (3)

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

U. Leonhardt, 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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Other (1)

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Real part of the longitudinal electric field of a line source located at point r s = ( 0.84 , 0 ) that radiates in a vacuum.

Fig. 2
Fig. 2

Real part of the longitudinal electric field of the same line source that radiates in the presence of an F-shaped metallic obstacle.

Fig. 3
Fig. 3

Real part of the longitudinal electric field for the same line source that radiates in the presence of an F-shaped metallic obstacle surrounded by an invisibility cloak. Note that the cloak is of the same order as the wavelength: R 2 R 1 = 0.8 0.4 = λ .

Fig. 4
Fig. 4

Real part of E z along the x-axis. The vertical dashed bold lines represent the boundary of the cloak.

Fig. 5
Fig. 5

Real part of H z along the x-axis. The vertical dashed bold lines represent the boundary of the cloak. Note that R e { H z } is theoretically infinite at the source place.

Fig. 6
Fig. 6

Real part of the longitudinal electric field for a line source located at point r s = ( 0.96666 , 0 ) (엯) that radiates within a broad cloak characterized by R 1 = 0.4 and R 2 = 1.2 . For an exterior observer, the field seems to be emitted by a shifted source located at a point r s = ( 0.84 , 0 ) (⋆): this is the mirage effect. Note that the shift, which is 0.12666, is nearly one third of the wavelength, λ = 0.4 . Compare the field outside the invisibility cloak with Fig. 1.

Equations (8)

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( d x d y d z ) = J x u ( d u d v d w ) , J x u = ( x , y , z ) ( u , v , w ) .
ε ͇ = ε T 1 , μ ͇ = μ T 1 .
T 1 = ( ( T 1 ) 11 ( T 1 ) 12 0 ( T 1 ) 21 ( T 1 ) 22 0 0 0 r R 1 α 2 r ) ,
( T 1 ) 11 = 1 R 1 sin 2 ( θ ) r + R 1 cos 2 ( θ ) R 1 + r ,
( T 1 ) 22 = 1 R 1 cos 2 ( θ ) r + R 1 sin 2 ( θ ) R 1 + r ,
( T 1 ) 12 = ( T 1 ) 21 = R 1 cos ( θ ) sin ( θ ) ( R 1 2 r ) ( R 1 r ) r .
× ( μ ͇ 1 × E l ) μ 0 ε 0 ω 2 ε ͇ E l = i ω I s μ 0 δ r s e z ,
× ( ε ͇ 1 × H l ) μ 0 ε 0 ω 2 μ ͇ H l = × ( ε ͇ 1 j T ) ,

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