Abstract

The optical force distribution in the cylindrical cloak under arbitrary incident waves is presented. We show that on the inner surface of the cloak both the induced surface currents and polarization charges interact with the waves and give opposite radiation pressure onto the inner surface. The Lorentz force in the cloak can contribute to change the trajectory of the rays, while in some cases it may only reflect the rays having a tendency to decrease the total energy it carries. The force is symmetric and in balance. Therefore the total momentum transfer from the waves to the cylindrical cloak is zero.

© 2010 Optical Society of America

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  3. 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]
  4. W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
    [CrossRef]
  5. J. Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901 (2008).
    [CrossRef] [PubMed]
  6. D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Express 14, 9794 (2006).
    [CrossRef] [PubMed]
  7. B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
    [CrossRef]
  8. L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
    [CrossRef]
  9. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Phys. Rev. Lett. 97, 133902 (2006).
    [CrossRef] [PubMed]
  10. H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

2009 (1)

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

2008 (2)

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
[CrossRef]

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

2007 (1)

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

2006 (5)

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Express 14, 9794 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, 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]

Chen, H. S.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

Cheng, Q.

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
[CrossRef]

Chin, J.

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
[CrossRef]

Cui, T.

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Ding, K.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Grzegorczyk, T. M.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

Jiang, W. X.

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (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, Science 314, 977 (2006).
[CrossRef] [PubMed]

Kemp, B. A.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

Kong, J. A.

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Leonhardt, U.

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

Li, J.

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

Lin, X.

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
[CrossRef]

Luo, Y.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[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]

Pendry, J. B.

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

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

D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Express 14, 9794 (2006).
[CrossRef] [PubMed]

Ran, L.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, 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]

D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Express 14, 9794 (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, Science 314, 977 (2006).
[CrossRef] [PubMed]

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

D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Express 14, 9794 (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]

Tsang, L.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Wu, B. I.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

Yu, G.

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
[CrossRef]

Zhang, B.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

Zhang, J.

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

J. Phys. D (1)

W. X. Jiang, T. Cui, G. Yu, X. Lin, Q. Cheng, and J. Chin, J. Phys. D 41, 085504 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. A. (1)

H. S. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and B. I. Wu, Phys. Rev. A. 80, 011808(R) (2009).

Phys. Rev. B (1)

B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
[CrossRef]

Phys. Rev. Lett. (2)

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

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

Science (3)

J. B. Pendry, D. Schurig, 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)

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Lorentz force density (arrows) on the ideal linearly transformed cylindrical cloak under normal incidence ( α = 0 ) . The background pattern is the electric field with unit amplitude (volt/meter) propagating along the x direction. The axes are normalized by λ 0 .

Fig. 2
Fig. 2

The forces the ray feels when obliquely incident into an ideal linearly transformed cylindrical cloak with α = π / 6 : (a) in the x y plane, (b) viewed from the incident wave direction, (c) in the x z plane, and (d) in three-dimensional view. The blue arrows at the outer surface of the cloak are surface force density with unit of N / m 2 , while the red arrows distributed inside the cloak layer are bulk force density with unit of N / m 3 . The dashed circle represents the inner surface of the cloak.

Fig. 3
Fig. 3

Lorentz force density (arrows) on the cloak in the x y plane when the wave vector is along the z direction ( α = π / 2 ) . The background pattern represents the amplitude of the Poynting power.

Equations (9)

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E ρ c = f ( ρ ) sin   α   cos   ϕ e i k x f ( ρ ) cos   ϕ + i k z z ,
H ρ c = ϵ 0 μ 0 f ( ρ ) sin   ϕ e i k x f ( ρ ) cos   ϕ + i k z z ,
E ϕ c = f ( ρ ) ρ sin   α   sin   ϕ e i k x f ( ρ ) cos   ϕ + i k z z ,
H ϕ c = ϵ 0 μ 0 f ( ρ ) ρ cos   ϕ e i k x f ( ρ ) cos   ϕ + i k z z ,
E z c = cos   α e i k x f ( ρ ) cos   ϕ + i k z z ,     H z c = 0.
f ¯ = 1 2 Re { ( P ¯ ) E ¯ + ( μ 0 M ¯ ) H ¯ i ω P ¯ × μ 0 H ¯ + i ω μ 0 M ¯ × ϵ 0 E ¯ } .
f ¯ b = ϵ 0 4 [ f ( ρ ) ρ + f ( ρ ) f ( ρ ) ρ 2 ] [ z ̂   sin   2 α   cos   ϕ + ρ ̂ 2 f ( ρ ) ( 1 cos 2 α cos 2 ϕ ) + ϕ ̂ f ( ρ ) ρ cos 2 α   sin   2 ϕ ] .
f ¯ ρ = R 2 = ϵ 0 4 [ 1 f ( R 2 ) ] { z ̂   sin   2 α   cos   ϕ + ρ ̂ [ 1 + f ( R 2 ) ] ( 1 cos 2 α cos 2 ϕ ) + ϕ ̂ cos 2 α   sin   2 ϕ } .
f ¯ ρ = R 1 = ϵ 0 4 { [ ρ ̂ [ f 2 ( R 1 ) ( 1 cos 2 α cos 2 ϕ ) ] cos 2 α ] z ̂ f ( R 1 ) sin   2 α   cos   ϕ } .

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