Abstract

The influence of a geometrical perturbation δ at the inner boundaries of both cylindrical and spherical invisibility cloaks on invisibility performance is presented. The analytic solutions for such influence in the case of the general coordinate transformation are given. We show that the cylindrical cloak is more sensitive than a spherical cloak to such a perturbation. The difference results from the different asymptotic properties of eigenfunctions for the cylindrical and spherical wave equations. In particular, the zeroth-order scattering coefficient for a cylindrical cloak determined by 1ln(δ) converges to zero very slowly. The noticeable scattering induced by the slow convergence speed can be decreased by choosing appropriate coordinate transformation functions. More interestingly, the slow convergence can be overcome dramatically by putting a PEC (PMC) layer at the interior boundary of the cloak shell for TM (TE) wave.

© 2008 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, 1780-1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  3. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
    [CrossRef]
  4. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069-1071 (2007).
    [CrossRef] [PubMed]
  5. H. S. Chen, B. I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
    [CrossRef] [PubMed]
  6. Z. C. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99, 113903 (2007).
    [CrossRef] [PubMed]
  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, 977-980 (2006).
    [CrossRef] [PubMed]
  8. B. L. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves” Phys. Rev. B 76, 121101 (2007).
    [CrossRef]
  9. W. Yan, M. Yan, Z. C. Ruan, and M. Qiu, “Perfect invisibility cloak constructed by arbitrary coordinate transformations,” http://www.arXiv:0712.1694 [physics. optics] (2007).
  10. A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, “Improvement of cylindrical cloaking with the SHS lining,” Opt. Express 15, 12717-12734 (2007).
    [CrossRef] [PubMed]

2007

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069-1071 (2007).
[CrossRef] [PubMed]

H. S. Chen, B. I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Z. C. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

B. L. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves” Phys. Rev. B 76, 121101 (2007).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, “Improvement of cylindrical cloaking with the SHS lining,” Opt. Express 15, 12717-12734 (2007).
[CrossRef] [PubMed]

2006

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]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[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]

Opt. Express

Opt. Lett.

Phys. Rev. B

B. L. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves” Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Phys. Rev. E

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Phys. Rev. Lett.

H. S. Chen, B. I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Z. C. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Science

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]

Other

W. Yan, M. Yan, Z. C. Ruan, and M. Qiu, “Perfect invisibility cloak constructed by arbitrary coordinate transformations,” http://www.arXiv:0712.1694 [physics. optics] (2007).

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Figures (6)

Fig. 1
Fig. 1

Schematic of an invisibility cloak, where a perturbation δ is introduced to make the actual inner boundary be at r = a + δ .

Fig. 2
Fig. 2

C 0 sc versus δ for a cylindrical cloak with f ( r ) = b ( r a ) ( b a ) , a = 1.2 π k 0 , b = 2 π k 0 , and the region inside the cloak shell being assumed to be air.

Fig. 3
Fig. 3

C 0 sc versus δ for p = 1 and f ( a ) = 2.5 , p = 1 and f ( a ) = 0.1 , p = 2 and f ( a ) = 0.1 . Other parameters are the same as those in Fig. 2.

Fig. 4
Fig. 4

C 0 sc versus δ for case (1) without a PEC layer at r = a + δ and case (2) with a PEC layer at r = a + δ . Other parameters are the same as those in Fig. 2.

Fig. 5
Fig. 5

(a) C 1 sc versus δ, (b) C 2 sc versus δ. Other parameters are the same as those in Fig. 2.

Fig. 6
Fig. 6

(a) C 1 sc versus δ, (b) C 2 sc versus δ. The spherical cloak is set at f ( r ) = b ( r a ) ( b a ) , a = π k 0 , b = 2 π k 0 , and the region inside the cloak shell is air.

Equations (21)

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ϵ r = μ r = f ( r ) r f ( r ) , ϵ θ = μ θ = r f ( r ) f ( r ) , ϵ z = μ z = f ( r ) f ( r ) r ,
H z = n = + a n in J n ( k 0 r ) e in θ + a n sc H n ( k 0 r ) e in θ ( b < r ) ,
H z = n = + a n 1 J n ( k 0 f ( r ) ) e in θ + a n 2 H n ( k 0 f ( r ) ) e in θ ( a + δ < r < b ) .
H z = n a n 3 S n ( r ) e in θ ( r < a + δ ) ,
E θ = n k 0 a n 3 T n ( r ) e in θ ( i ω ϵ 0 ) ,
a n in J n ( k 0 b ) + a n sc H n ( k 0 b ) = a n 1 J n ( k 0 b ) + a n 2 H n ( k 0 b ) ,
k 0 a n in J n ( k 0 b ) + k 0 a n sc H n ( k 0 b ) = f ( b ) k 0 ϵ θ ( b ) a n 1 J n ( k 0 b ) + f ( b ) k 0 ϵ θ ( b ) a n 2 H n ( k 0 b ) ,
a n 1 J n ( k 0 f ( a + δ ) ) + a n 2 H n ( k 0 f ( a + δ ) ) = a n 3 S n ( a + δ ) ,
f ( a + δ ) k 0 ϵ θ ( a + δ ) a n 1 J n ( k 0 f ( a + δ ) ) + f ( a + δ ) k 0 ϵ θ ( a + δ ) a n 2 H n ( k 0 f ( a + δ ) ) = k 0 a n 3 T n ( a + δ ) .
C n sc a J n ( k 0 δ f ) T n ( a + δ ) δ f J n ( k 0 δ f ) S n ( a + δ ) δ f H n ( k 0 δ f ) S n ( a + δ ) a H n ( k 0 δ f ) T n ( a + δ ) ,
C 0 sc a J 0 ( k 0 δ f ) T 0 ( a + δ ) + δ f J 1 ( k 0 δ f ) S 0 ( a + δ ) δ f H 1 ( k 0 δ f ) S 0 ( a + δ ) a H 0 ( k 0 δ f ) T 0 ( a + δ ) .
C 0 sc a J 0 ( k 0 δ f ) T 0 ( a + δ ) i 2 S 0 ( a + δ ) ( π k 0 ) a H 0 ( k 0 δ f ) T 0 ( a + δ ) ,
C 0 sc = J 1 ( k 0 δ f ) H 1 ( k 0 δ f ) .
C n sc A n B n ,
A n = k 0 a T n ( a + δ ) n S n ( a + δ ) k 0 a T n ( a + δ ) + n S n ( a + δ ) , B n = J n ( k 0 δ f ) H n ( k 0 δ f ) .
ϵ r = μ r = f ( r ) 2 r 2 f ( r ) , ϵ θ = μ θ = ϵ ϕ = μ ϕ = f ( r ) .
π e = n = 1 + m = n n a ( n , m ) in j n ( k 0 r ) P n m ( cos θ ) e i m ϕ + a ( n , m ) sc h n ( k 0 r ) P n m ( cos θ ) e i m ϕ ( b < r ) ,
π e = n = 1 + m = n n f ( r ) f ( r ) r [ a ( n , m ) 1 j n ( k 0 f ( r ) ) + a ( n , m ) 2 h n ( k 0 f ( r ) ) ] P n m ( cos θ ) e i m ϕ ( a + δ < r < b ) .
H θ = n = 1 + m = n n a ( n , m ) 3 m S n ( r ) P n m ( cos θ ) e i m ϕ ( sin θ μ 0 )
E θ = n = 1 + m = n n a ( n , m ) 3 T n ( r ) P n m ( cos θ ) e i m ϕ ( i ω ϵ 0 ) ,
C ( n , m ) sc δ f j n ( k 0 δ f ) T n ( a + δ ) ( k 0 δ f j n ( k 0 δ f ) + j n ( k 0 δ f ) ) S n ( a + δ ) ( k 0 δ f h n ( k 0 δ f ) + h n ( k 0 δ f ) ) S n ( a + δ ) δ f h n ( k 0 δ f ) T n ( a + δ ) ,

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