Abstract

The design of electromagnetic invisibility cloaks is based on singular mappings prescribing zero or infinite values for material parameters on the inner surface of the cloak. Since this is only approximately feasible, an asymptotic analysis is necessary for a sound description of cloaks. We adopt a simple and effective approach for analyzing electromagnetic cloaks - instead of the originally proposed singular mapping, nonsingular mappings asymptotically approaching the ideal one are considered. Scattering and radiation from this type of imperfect cylindrical cloaks is solved analytically and the results are confirmed by full-wave finite element simulations. Our analysis sheds more light on the influence of this kind of imperfection on the cloaking performance and further explores the physics of cloaking devices.

© 2008 Optical Society of America

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References

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  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004).
    [Crossref] [PubMed]
  2. C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47–49 (2007).
    [Crossref] [PubMed]
  3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  4. 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]
  5. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
    [Crossref] [PubMed]
  6. Z. 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. B. Zhang, H. Chen, B. -I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
    [Crossref]
  8. H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
    [Crossref] [PubMed]
  9. 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]
  10. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Electromagnetic cloaking in the visible frequency range,” arXiv:0709.2862v1, (2007).
  11. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
    [Crossref]
  12. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
    [Crossref]
  13. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
    [Crossref]
  14. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007).
    [Crossref]
  15. L. Tsang, J. A. Kong, and K. -H. Ding, “Scattering of Electromagnetic Waves: Theories and Applications,” John Wiley & Sons Inc., New York, 41 (2000).
  16. G. Tyras, “Radiation and propagation of electromagnetic waves,” Academic Press, New York, 274 (1969)

2007 (7)

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]

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

H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47–49 (2007).
[Crossref] [PubMed]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

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

2006 (4)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (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]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (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]

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

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

Chen, H.

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

H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

Chettiar, U. K.

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

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[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, 977–980 (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]

Davis, C. C.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Electromagnetic cloaking in the visible frequency range,” arXiv:0709.2862v1, (2007).

Ding, K. -H.

L. Tsang, J. A. Kong, and K. -H. Ding, “Scattering of Electromagnetic Waves: Theories and Applications,” John Wiley & Sons Inc., New York, 41 (2000).

Guenneau, S.

Hung, Y. J.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Electromagnetic cloaking in the visible frequency range,” arXiv:0709.2862v1, (2007).

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, 977–980 (2006).
[Crossref] [PubMed]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

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

Kong, J. A.

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

H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

L. Tsang, J. A. Kong, and K. -H. Ding, “Scattering of Electromagnetic Waves: Theories and Applications,” John Wiley & Sons Inc., New York, 41 (2000).

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47–49 (2007).
[Crossref] [PubMed]

Luo, Y.

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

Milton, G.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (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, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Neff, C. W.

Z. 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]

Nicolet, A.

Pendry, J. B.

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (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]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

Popa, B. -I.

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]

Qiu, M.

Z. 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]

Ran, L.

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

Ruan, Z.

Z. 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]

Schurig, D.

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (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, 977–980 (2006).
[Crossref] [PubMed]

Shalaev, V. M.

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

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

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, 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (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. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Smolyaninov, I. I.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Electromagnetic cloaking in the visible frequency range,” arXiv:0709.2862v1, (2007).

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47–49 (2007).
[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, 977–980 (2006).
[Crossref] [PubMed]

Tsang, L.

L. Tsang, J. A. Kong, and K. -H. Ding, “Scattering of Electromagnetic Waves: Theories and Applications,” John Wiley & Sons Inc., New York, 41 (2000).

Tyras, G.

G. Tyras, “Radiation and propagation of electromagnetic waves,” Academic Press, New York, 274 (1969)

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Wegener, M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47–49 (2007).
[Crossref] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Wu, B. -I.

H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

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

Yan, M.

Z. 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]

Zhang, B.

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

H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

Zolla, F.

Appl. Phys. Lett. (1)

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

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Nat. Photonics (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

New J. Phys. (1)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

Opt. Lett. (1)

Phys. Rev. B (1)

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

Phys. Rev. E (1)

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. (2)

H. Chen, B. -I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

Z. 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 (4)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47–49 (2007).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (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, 977–980 (2006).
[Crossref] [PubMed]

Other (3)

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Electromagnetic cloaking in the visible frequency range,” arXiv:0709.2862v1, (2007).

L. Tsang, J. A. Kong, and K. -H. Ding, “Scattering of Electromagnetic Waves: Theories and Applications,” John Wiley & Sons Inc., New York, 41 (2000).

G. Tyras, “Radiation and propagation of electromagnetic waves,” Academic Press, New York, 274 (1969)

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Figures (4)

Fig. 1.
Fig. 1.

Mapping between the ‘transformation’ and ‘physical’ domain and the difference between a nonsingular (one-to-one) mapping corresponding to a imperfect cloak and the singular mapping of a ideal cloak. R 2 and R 1 are the outer and inner radius of the cloak, respectively, while ρ 1 is the effective size of the concealed object in the ‘transformation’ domain.

Fig. 2.
Fig. 2.

FE simulation results for the scattering of a TE plane wave with E 0 = 1 V m and ϕi =0 on a imperfect cloak with ρ 1=0.01m, R 2=2R 1=0.5m, λ 0=1.2R 1=0.3m, εc =3.5 and μc =1. a) shows the distribution of Ez (i.e. it’s real part). b) is the distribution of the amplitude of the electric field, |Ez |. This is a near-resonance case with particularly chosen values of εc and μc aiming to support the conclusions from the text. Outside the cloak, the field is perturbed only slightly since ρ 1 is small compared to λ 0. Notice how the wavefronts in a) are kept straight and parallel. However, the field in r<R 1 is very high because of the resonance. Also, notice that this is clearly not the zeroth angular mode which usually dominates if the cloaking is effective Ref. [6].

Fig. 3.
Fig. 3.

FE simulation results for the case R 2=2R 1=0.5m, λ 0=1.2R 1=0.3m, εc =μc =1, for the case of a very thin electric current source line located at Rs =0.1m, ϕs =0 and carrying a total amount of current I=1A. Electric field is given in V/m and magnetic field in A/m.

Fig. 4.
Fig. 4.

a) Radiated power, per unit length along z, for a thin electric current line located at Rs =0.1m, ϕs =0 carrying a total amount of current I=1A. b) Scattering width. λ 0=0.3m, R 1=0.25m. ρ 1=R 1 means that there is no cloak and ρ 1=0 represents the case of an ideal cloak. The radiation peak around ρ 1=0.05m for εc =1, μc =4, is due to the structure resonance mentioned in the text - here it’s for the m=4 angular mode.

Equations (29)

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

E H , ε µ , j ( e ) j ( m ) ,
r = g ( r ) ,
ε r ε r = µ r µ r = r d g ( r ) d r r , ε ϕ ε ϕ = µ ϕ µ ϕ = r r d g ( r ) d r , ε z ε z = µ z µ z = j z j z = r r d g ( r ) d r ,
E z ( r ) = E z ( r ) , H r ( r ) = 1 g ( r ) H r ( r ) , H ϕ ( r ) = r r H ϕ ( r ) .
ε r = µ r = g 1 ( r ) r d g 1 ( r ) d r , ε ϕ = µ ϕ = r d g 1 ( r ) d r g 1 ( r ) , ε z = µ z = g 1 ( r ) d g 1 ( r ) d r r ,
ε r = ε ϕ = ε c , μ r = μ ϕ = μ c , ε z = α 2 ε c , μ z = α 2 μ c , j z = α 2 j z , α = R 1 ρ 1 .
1 r r ( r E z r ) + 1 r 2 2 E z r 2 + k 0 2 μ ε E z = i ω μ 0 μ j z ,
E int , z = E 0 m = i m exp ( i m ( ϕ ϕ i ) ) a m J m ( α k c r ) , k c = ε c μ c k 0 , for r < ρ 1 ,
E scatt , z = E 0 m = i m exp ( i m ( ϕ ϕ i ) ) b m H m ( k 0 r ) , for , r < ρ 1 ,
a m = 1 π R 1 i 2 μ c α μ c k 0 H m ( k 0 R 1 α ) J m ( k c R 1 ) α k c H m ( k 0 R 1 α ) J m ( k c R 1 ) ,
b m = α k c J m ( k 0 R 1 α ) J m ( k c R 1 ) μ c k 0 J m ( k 0 R 1 α ) J m ( k c R 1 ) μ c k 0 H m ( k 0 R 1 α ) J m ( k c R 1 ) α k c H m ( k 0 R 1 α ) J m ( k c R 1 ) ,
a m 1 2 m 1 ( m 1 ) ! γ J m ( k c R 1 ) + m γ J m ( k c R 1 ) ( k 0 ρ 1 ) m , γ = μ c k c R 1 , m 1 .
a 0 γ J 0 ( k c R 1 ) ln ( 1 k 0 ρ 1 ) + γ J 0 ( k c R 1 )
b m i π 1 2 2 m m ! ( m 1 ) ! J m ( k c R 1 ) m γ J m ( k c R 1 ) J m ( k c R 1 ) + m γ J m ( k c R 1 ) ( k 0 ρ 1 ) 2 m , m 1 .
b 0 i π 2 J 0 ( k c R 1 ) + 1 2 γ J 0 ( k c R 1 ) ( k 0 ρ 1 ) 2 J 0 ( k c R 1 ) ln ( 1 k 0 ρ 1 ) + γ J 0 ( k c R 1 ) ,
J m ( k c R 1 ) + m γ J m ( k c R 1 ) = 0 , m 0 .
j z = I δ ( r ρ s ) δ ( ϕ ϕ s ) r , ρ s = R s α .
E < , z = m = exp ( i m ( ϕ ϕ s ) ) A m J m ( α k c r ) , r < ρ s ,
E > , z = m = exp ( i m ( ϕ ϕ s ) ) ( B m J m ( α k c r ) + C m H m ( α k c r ) ) , ρ s < r < ρ 1 ,
E out , z = m = exp ( i m ( ϕ ϕ s ) ) D m H m ( k 0 r ) , ρ 1 < r .
A m = μ 0 μ c ω I 4 H m ( k c R s ) + B m ,
B m = μ c k 0 H m ( k 0 ρ 1 ) H m ( k c R 1 ) α k c H m ( k 0 ρ 1 ) H m ( k c R 1 ) α k c H m ( k 0 ρ 1 ) J m ( k c R 1 ) μ c k 0 H m ( k 0 ρ 1 ) J m ( k c R 1 ) C m ,
C m = μ 0 μ c ω I 4 J m ( k c R s ) ,
D m = 1 π R 1 i 2 α α k c H m ( k 0 ρ 1 ) J m ( k c R 1 ) μ c k 0 H m ( k 0 ρ 1 ) J m ( k c R 1 ) C m .
A m μ 0 μ c ω I 4 ( J m ( k c R s ) H m ( k c R 1 ) + m γ H m ( k c R 1 ) J m ( k c R 1 ) + m γ J m ( k c R 1 ) H m ( k c R s ) ) , γ = μ c k c R 1 ,
B m μ 0 μ c ω I 4 J m ( k c R s ) H m ( k c R 1 ) + m γ H m ( k c R 1 ) J m ( k c R 1 ) + m γ J m ( k c R 1 ) ,
D m μ 0 ω I 2 m + 1 ( m 1 ) ! γ J m ( k c R s ) J m ( k c R 1 ) + m γ J m ( k c R 1 ) ( k 0 ρ 1 ) m , for m 1 , and
B 0 μ 0 μ c ω I 4 J 0 ( k c R s ) H 0 ( k c R 1 ) ln ( 1 k 0 ρ 1 ) + γ H 0 ( k c R 1 ) J 0 ( k c R 1 ) ln ( 1 k 0 ρ 1 ) + γ J 0 ( k c R 1 ) ,
D 0 μ 0 ω I 4 γ J 0 ( k c R s ) J 0 ( k c R 1 ) ln ( 1 k 0 ρ 1 ) + γ J 0 ( k c R 1 ) ,

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