Abstract

A dispersive full-wave finite-difference time-domain model is used to study the performance of bipolar cylindrical invisibility cloaking devices. We have considered two different cloaking structures generated by the mapping of the σ axis and the mapping of the τ axis of bipolar coordinates. The permittivity and permeability tensors for the cloaking devices are obtained from an effective medium approach in general relativity. The σ-axis mapped bipolar cylindrical cloak is found to be imperfect, and the cloaking performance is found to depend on the polarization of the incident waves, the direction of propagation of the waves, and the loss tangents of the metamaterial. Only the case of TM waves for the specific propagation direction shows good cloaking performance. On the other hand, the τ-mapped cloaking device shows good cloaking performance for all polarizations and directions of propagation. However, this structure has a singular boundary at the inner radius. Realistic cloaking materials with loss still show a cloak that is working, but attenuated backscattering waves exist.

© 2012 Optical Society of America

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    [CrossRef]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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  3. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef]
  4. J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, “Metamaterial electromagnetic concentrators with arbitrary geometries,” Opt. Express 17, 19656–19661 (2009).
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  5. H. Chen, C. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
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  6. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
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  7. D. Ahn, “Calculation of permittivity tensors for invisibility devices by effective medium approach in general relativity,” J. Mod. Opt. 58, 700–710 (2011).
    [CrossRef]
  8. Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
    [CrossRef]
  9. I. Tamm, “Electrodynamics of an anisotropic medium in the special theory of relativity,” J. Russ. Phys. Chem. Soc. 56, 248 (1924).
  10. J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118, 1396–1408 (1960).
    [CrossRef]
  11. A. Lakhtakia and T. G. Mackay, “Toward gravitationally assisted negative refraction of light by vacuum,” J. Phys. A 37, L505–L510 (2004).
    [CrossRef]
  12. T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Gravitation and electromagnetic wave propagation with negative phase velocity,” New J. Phys. 7, 75 (2005).
    [CrossRef]
  13. T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
    [CrossRef]
  14. M. W. Mccall, “On negative refraction in classical vacuum,” J. Mod. Opt. 54, 119–128 (2007).
    [CrossRef]
  15. M. W. McCall, “Classical gravity does not refract negatively,” Phys. Rev. Lett. 98, 091102 (2007).
    [CrossRef]
  16. H. Chen and C. Chan, “‘Cloaking at a distance’ from folded geometries in bipolar coordinates,” Opt. Lett. 34, 2649–2651 (2009).
    [CrossRef]
  17. U. Leonhardt and T. Tyc, “Broadband invisibility by nonEuclidean cloaking,” Science 323, 110–112 (2009).
    [CrossRef]
  18. J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
    [CrossRef]
  19. E. Cojocaru, “Exact analytical approaches for elliptic cylindrical invisibility cloaks,” J. Opt. Soc. Am. B 26, 1119–1128 (2009).
    [CrossRef]
  20. Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
    [CrossRef]
  21. D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
    [CrossRef]
  22. W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
    [CrossRef]
  23. H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
    [CrossRef]
  24. J. Jin, J. Jin, and J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).
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  29. J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
    [CrossRef]
  30. J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  31. Y. Zhao, C. Argyropoulos, and Y. Hao, “Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures,” Opt. Express 16, 6717–6730 (2008).
    [CrossRef]
  32. Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
    [CrossRef]
  33. C. Argyropoulos, E. Kallos, and Y. Hao, “Dispersive cylindrical cloaks under nonmonochromatic illumination,” Phys. Rev. E 81, 016611 (2010).
    [CrossRef]
  34. C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. 57, 1432–1441 (2009).
    [CrossRef]
  35. Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain (FDTD) analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
    [CrossRef]
  36. B. Kante, D. Germain, and A. Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
    [CrossRef]
  37. 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]
  38. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
    [CrossRef]
  39. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
    [CrossRef]

2012

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain (FDTD) analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

2011

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

D. Ahn, “Calculation of permittivity tensors for invisibility devices by effective medium approach in general relativity,” J. Mod. Opt. 58, 700–710 (2011).
[CrossRef]

2010

H. Chen, C. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

C. Argyropoulos, E. Kallos, and Y. Hao, “Dispersive cylindrical cloaks under nonmonochromatic illumination,” Phys. Rev. E 81, 016611 (2010).
[CrossRef]

2009

C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. 57, 1432–1441 (2009).
[CrossRef]

B. Kante, D. Germain, and A. Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by nonEuclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

E. Cojocaru, “Exact analytical approaches for elliptic cylindrical invisibility cloaks,” J. Opt. Soc. Am. B 26, 1119–1128 (2009).
[CrossRef]

H. Chen and C. Chan, “‘Cloaking at a distance’ from folded geometries in bipolar coordinates,” Opt. Lett. 34, 2649–2651 (2009).
[CrossRef]

J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, “Metamaterial electromagnetic concentrators with arbitrary geometries,” Opt. Express 17, 19656–19661 (2009).
[CrossRef]

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[CrossRef]

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

2008

D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Y. Zhao, C. Argyropoulos, and Y. Hao, “Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures,” Opt. Express 16, 6717–6730 (2008).
[CrossRef]

Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

2007

M. W. Mccall, “On negative refraction in classical vacuum,” J. Mod. Opt. 54, 119–128 (2007).
[CrossRef]

M. W. McCall, “Classical gravity does not refract negatively,” Phys. Rev. Lett. 98, 091102 (2007).
[CrossRef]

2006

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (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]

2005

T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Gravitation and electromagnetic wave propagation with negative phase velocity,” New J. Phys. 7, 75 (2005).
[CrossRef]

2004

A. Lakhtakia and T. G. Mackay, “Toward gravitationally assisted negative refraction of light by vacuum,” J. Phys. A 37, L505–L510 (2004).
[CrossRef]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

1999

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1996

J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

1960

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118, 1396–1408 (1960).
[CrossRef]

1924

I. Tamm, “Electrodynamics of an anisotropic medium in the special theory of relativity,” J. Russ. Phys. Chem. Soc. 56, 248 (1924).

Ahn, D.

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain (FDTD) analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

D. Ahn, “Calculation of permittivity tensors for invisibility devices by effective medium approach in general relativity,” J. Mod. Opt. 58, 700–710 (2011).
[CrossRef]

Argyropoulos, C.

C. Argyropoulos, E. Kallos, and Y. Hao, “Dispersive cylindrical cloaks under nonmonochromatic illumination,” Phys. Rev. E 81, 016611 (2010).
[CrossRef]

C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. 57, 1432–1441 (2009).
[CrossRef]

Y. Zhao, C. Argyropoulos, and Y. Hao, “Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures,” Opt. Express 16, 6717–6730 (2008).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

Chan, C.

H. Chen, C. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

H. Chen and C. Chan, “‘Cloaking at a distance’ from folded geometries in bipolar coordinates,” Opt. Lett. 34, 2649–2651 (2009).
[CrossRef]

Chen, B.

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Chen, H.

H. Chen, C. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

H. Chen and C. Chan, “‘Cloaking at a distance’ from folded geometries in bipolar coordinates,” Opt. Lett. 34, 2649–2651 (2009).
[CrossRef]

Cheng, Q.

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

Cojocaru, E.

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

Cui, T. T.

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” 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, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Germain, D.

B. Kante, D. Germain, and A. Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 1995).

Hao, Y.

C. Argyropoulos, E. Kallos, and Y. Hao, “Dispersive cylindrical cloaks under nonmonochromatic illumination,” Phys. Rev. E 81, 016611 (2010).
[CrossRef]

C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. 57, 1432–1441 (2009).
[CrossRef]

Y. Zhao, C. Argyropoulos, and Y. Hao, “Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures,” Opt. Express 16, 6717–6730 (2008).
[CrossRef]

Y. Hao and R. Mittra , FDTD Modeling of Metamaterials: Theory and Applications (Artech House, 2009).

Holden, A.

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Huang, M.

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

Jiang, W. X.

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

Jiang, X.

Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Jin, J.

J. Jin, J. Jin, and J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

J. Jin, J. Jin, and J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

Jin, J. M.

J. Jin, J. Jin, and J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

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]

Kallos, E.

C. Argyropoulos, E. Kallos, and Y. Hao, “Dispersive cylindrical cloaks under nonmonochromatic illumination,” Phys. Rev. E 81, 016611 (2010).
[CrossRef]

Kante, B.

B. Kante, D. Germain, and A. Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Kong, J. A.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

Kwon, D. H.

D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[CrossRef]

T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Gravitation and electromagnetic wave propagation with negative phase velocity,” New J. Phys. 7, 75 (2005).
[CrossRef]

A. Lakhtakia and T. G. Mackay, “Toward gravitationally assisted negative refraction of light by vacuum,” J. Phys. A 37, L505–L510 (2004).
[CrossRef]

Landau, L.

E. Lifshitz, L. Pitaevskii, and L. Landau, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984), Vol. 8.

Lee, Y. Y.

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain (FDTD) analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

Leonhardt, U.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by nonEuclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

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

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

Liang, Z.

Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Lifshitz, E.

E. Lifshitz, L. Pitaevskii, and L. Landau, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984), Vol. 8.

Lin, X. Q.

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

Luo, Y.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

Lustrac, A.

B. Kante, D. Germain, and A. Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Ma, H.

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[CrossRef]

T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Gravitation and electromagnetic wave propagation with negative phase velocity,” New J. Phys. 7, 75 (2005).
[CrossRef]

A. Lakhtakia and T. G. Mackay, “Toward gravitationally assisted negative refraction of light by vacuum,” J. Phys. A 37, L505–L510 (2004).
[CrossRef]

Mccall, M. W.

M. W. Mccall, “On negative refraction in classical vacuum,” J. Mod. Opt. 54, 119–128 (2007).
[CrossRef]

M. W. McCall, “Classical gravity does not refract negatively,” Phys. Rev. Lett. 98, 091102 (2007).
[CrossRef]

Mittra, R.

Y. Hao and R. Mittra , FDTD Modeling of Metamaterials: Theory and Applications (Artech House, 2009).

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[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]

Pendry, J.

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Pendry, J. B.

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

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Peng, J.

Perczel, J.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

Philbin, T. G.

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

Pitaevskii, L.

E. Lifshitz, L. Pitaevskii, and L. Landau, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984), Vol. 8.

Plebanski, J.

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118, 1396–1408 (1960).
[CrossRef]

Qu, S.

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Ran, L.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

Robbins, D.

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Schurig, D.

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

Setiawan, S.

T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Gravitation and electromagnetic wave propagation with negative phase velocity,” New J. Phys. 7, 75 (2005).
[CrossRef]

Sheng, P.

H. Chen, C. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

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

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]

Stewart, W.

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Sullivan, D. M.

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).

Sun, X.

Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 1995).

Tamm, I.

I. Tamm, “Electrodynamics of an anisotropic medium in the special theory of relativity,” J. Russ. Phys. Chem. Soc. 56, 248 (1924).

Tyc, T.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by nonEuclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

Wang, J.

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Werner, D. H.

D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Wu, B. I.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

Xiao, Z.

Xu, Z.

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Yang, C.

Yang, J.

Yao, P.

Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Youngs, I.

J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Yu, G. X.

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

Zhang, J.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

Zhao, Y.

C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. 57, 1432–1441 (2009).
[CrossRef]

Y. Zhao, C. Argyropoulos, and Y. Hao, “Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures,” Opt. Express 16, 6717–6730 (2008).
[CrossRef]

Appl. Phys. Lett.

D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009).
[CrossRef]

C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. 57, 1432–1441 (2009).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Korean Phys. Soc.

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain (FDTD) analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

Y. Y. Lee and D. Ahn, “Dispersive finite-difference time-domain analysis of the elliptic cylindrical cloak,” J. Korean Phys. Soc. 60, 1349–1360 (2012).
[CrossRef]

J. Mod. Opt.

M. W. Mccall, “On negative refraction in classical vacuum,” J. Mod. Opt. 54, 119–128 (2007).
[CrossRef]

D. Ahn, “Calculation of permittivity tensors for invisibility devices by effective medium approach in general relativity,” J. Mod. Opt. 58, 700–710 (2011).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A

A. Lakhtakia and T. G. Mackay, “Toward gravitationally assisted negative refraction of light by vacuum,” J. Phys. A 37, L505–L510 (2004).
[CrossRef]

J. Phys. D

W. X. Jiang, T. T. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisibe cloaking,” J. Phys. D 41, 085504 (2008).
[CrossRef]

J. Russ. Phys. Chem. Soc.

I. Tamm, “Electrodynamics of an anisotropic medium in the special theory of relativity,” J. Russ. Phys. Chem. Soc. 56, 248 (1924).

Nat. Mater.

H. Chen, C. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef]

New J. Phys.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

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

T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Gravitation and electromagnetic wave propagation with negative phase velocity,” New J. Phys. 7, 75 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118, 1396–1408 (1960).
[CrossRef]

Phys. Rev. A

H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[CrossRef]

Phys. Rev. B

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[CrossRef]

B. Kante, D. Germain, and A. Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Phys. Rev. E

C. Argyropoulos, E. Kallos, and Y. Hao, “Dispersive cylindrical cloaks under nonmonochromatic illumination,” Phys. Rev. E 81, 016611 (2010).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

J. Pendry, A. Holden, W. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

M. W. McCall, “Classical gravity does not refract negatively,” Phys. Rev. Lett. 98, 091102 (2007).
[CrossRef]

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]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369(2009).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by nonEuclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

Other

J. Jin, J. Jin, and J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 1995).

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).

Y. Hao and R. Mittra , FDTD Modeling of Metamaterials: Theory and Applications (Artech House, 2009).

E. Lifshitz, L. Pitaevskii, and L. Landau, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984), Vol. 8.

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Figures (5)

Fig. 1.
Fig. 1.

Bipolar cylindrical coordinate system. The red, including dark red, and blue curves represent constant σ and τ contours, respectively. Line a is the half-distance between focal points. The region between σ1 and σ2, or between τ1 and τ2, is the cloaking device.

Fig. 2.
Fig. 2.

Constitutive parameter distribution: (a) the εzz (=μzz) distribution when σ1=0.75π, σ2=0.5π, and a=0.3m for the σ-axis mapped cloak, (b) the ε˜σσ (=μ˜σσ) distribution when τ1=2.0, τ2=1.3, and a=0.6m for the τ-axis mapped cloak, (c) the ε˜ττ (=μ˜ττ) distribution when τ1=2.0, τ2=1.3, and a=0.6m for the τ-axis mapped cloak, and (d) the ε˜zz (=μ˜zz) distribution when τ1=2.0, τ2=1.3, and a=0.6m for the τ-axis mapped cloak.

Fig. 3.
Fig. 3.

Ez(V/m) field distributions when an ideal σ-mapped bipolar cylindrical cloak is exposed to TE illumination with different incident axes: (a) x direction and (b) y direction. Hz(A/m) field distributions when an ideal σ-mapped bipolar cylindrical cloak is exposed to TM illumination with different incident axes: (c) x direction and (d) y direction. In all cases the structure is filled with a PEC at the cloaking region.

Fig. 4.
Fig. 4.

Ez(V/m) field distributions for an ideal τ-mapped bipolar cylindrical cloak filled with a PEC at the inner region (ττ1): (a) x direction and (b) y direction under TE illumination. Hz(A/m) field distributions for an ideal τ-mapped bipolar cylindrical cloak filled with a PEC at the inner region (ττ1): (c) x direction and (d) y direction under TM illumination.

Fig. 5.
Fig. 5.

Ez(V/m) field distributions for a lossy bipolar cylindrical cloak that is mapped along the σ-axis: (a) x direction and (b) y direction under TE illumination. Hz(A/m) field distributions for a lossy bipolar cylindrical cloak that is mapped along the σ-axis: (c) x direction and (d) y direction under TM illumination.

Tables (1)

Tables Icon

Table 1. Parameters for Unitary Transformation from Various Cylindrical Coordinate Systems to the Cartesian Coordinate System

Equations (37)

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

x=x(x1,x2,x3),y=y(x1,x2,x3),z=z(x1,x2,x3),
ds2=dx2+dy2+dz2=i,jγijdxidxj.
x1=U1+U2U1U2x1,x2=V1+V2V1V2x2,x3=W1+W2W1W2x3,
gij=xixkxjxlγkl,
g=g(x1,x2,x3)(x1,x2,x3)=γ(x1,x2,x3)(x1,x2,x3).
εij=±(g)1/2γ(g0jgi0g00gij),αij=(g)1/2γ[jkl]g0kgil,
βIJ=γg[jkl]g0kgil,(μ1)ij=±γg(gi0gj0g00gij).
εij=μij=gg00γgij,
{x=asinhτcoshτcosσy=asinσcoshτcosσz=z,
γij=(a2(cosσcoshτ)2000a2(cosσcoshτ)20001),
σ=σ2σ1σ2π(σπ)+σ1,σ[σ2,π],σ=σ2σ1σ2π(σπ)+2πσ1,σ(π,2πσ2],τ=τ,z=z,
σσ=σ2σ1σ2π,
gij=xixkxjxlγkl=diag((σ2σ1)2(σ2π)21a2(cosσcoshτ)2,1a2(cosσcoshτ)2,1).
εij=gγgij=1a2(cosσcoshτ)2×diag(σ2σ1σ2π,σ2πσ2σ1,σ2πσ2σ11(cosσcoshτ)2),
εji=εikγkj=diag(σ2σ1σ2π,σ2πσ2σ1,σ2πσ2σ1(cosσcoshτ)2(cosσcoshτ)2).
σ˜=σ˜,τ˜=τ2(τ2τ1)τ˜+τ1,z˜=z˜.
τ˜τ˜=τ2(τ1τ2)τ˜2,
g˜ij=diag(1a2(cosσ˜coshτ˜)2,τ22(τ1τ2)2a2τ˜4(cosσ˜coshτ˜)2,1).
ε˜ij=g˜γg˜ij=1a2(cosσ˜coshτ˜)2τ˜2τ2(τ1τ2)×diag(1,{τ2(τ1τ2)τ˜2}2,a2cosσ˜coshτ˜).
ε˜ji=diag(τ˜2τ2(τ1τ2),τ2(τ1τ2)τ˜2,τ˜2τ2(τ1τ2)(cosσ˜coshτ˜)2(cosσ˜coshτ˜)2).
εcartesian=UTεUandμcartesian=UTμU,
U=(Ω1Ω0Ω2Ω00Ω2Ω0Ω1Ω00001),
ε^i(ω)=εi+σ/jω,
ε^i(ω)=1ωp2ω2jωγ,
εxx(ω)=ε^1(ω)Ω12Ω0+ε^2Ω22Ω0,εyx(ω)=εxy(ω)=(ε^1ε^2)Ω1Ω2Ω0,εyy(ω)=ε^1Ω22Ω0+ε^2Ω12Ω0,εzz=ε^3,εyx=εzy=εxz=εyz=0.
(DxDy)=ε0(εxxεyxεxyεyy)(ExEy),
(ω2jωγ1)Ω0DX=ε0[(ω2jωγ1ωp12)Ω12+{ε2ω2jω(σ2+ε2γ1)σ2γ1}Ω22]Ex+ε0[(ω2jωγ1ωp12){ε2ω2jω(σ2+ε2γ1)σ2γ1}]Ω1Ω2Ey,(ω2jωγ1)Ω0Dy=ε0[(ω2jωγ1ωp12){ε2ω2jω(σ2+ε2γ1)σ2γ1}]Ω1Ω2Ex+ε0[(ω2jωγ1ωp12)Ω22+{ε2ω2jω(σ2+ε2γ1)σ2γ1}Ω12]Ey.
A1Exn+1=c1Dxn+1B1Eyn+1c2DXn+A2Exn+B2Eyn+c3Dxn1A3Exn1B3Eyn1,F1Eyn+1=c1Dyn+1B1Exn+1c2Dyn+F2Eyn+B2Exn+c3Dyn1F3Eyn1B3Exn1,
A1=(1Δt2+γ12Δt+ωp124)Ω12+{ε2Δt2+(σ2+ε2γ1)2Δt+σ2γ14}Ω22,A2=(2Δt2ωp122)Ω12+(2ε2Δt2σ2γ12)Ω22,A3=(1Δt2γ12Δt+ωp124)Ω12+{ε2Δt2(σ2+ε2γ1)2Δt+σ2γ14}Ω22,B1={(1ε2)Δt2+(γ1σ2ε2γ1)2Δt+(ωp12σ2γ1)4}Ω1Ω2,B2={2(1ε2)Δt2(ωp12σ2γ1)2}Ω1Ω2,B3={(1ε2)Δt2(γ1σ2ε2γ1)2Δt+(ωp12σ2γ1)4}Ω1Ω2,c1=(1ε0Δt2+γ12ε0Δt)Ω0,c2=2ε0Δt2Ω0,c3=(1ε0Δt2γ12ε0Δt)Ω0,F1=(1Δt2+γ12Δt+ωp124)Ω22+{ε2Δt2+(σ2+ε2γ1)2Δt+σ2γ14}Ω12,F2=(2Δt2ωp122)Ω22+(2ε2Δt2σ2γ12)Ω12,F3=(1Δt2γ12Δt+ωp124)Ω22+{ε2Δt2(σ2+ε2γ1)2Δt+σ2γ14}Ω12.
b1Exn+1=c1Dxn+1a1Dyn+1c2Dxn+a2Dyn+b2Exn+d1Eyn+c3Dxn1a3Dyn1b3Exn1d2Eyn1,f1Eyn+1=c1Dyn+1e1Dxn+1c2Dyn+e2Dxn+f2Eyn+g1Exn+c3Dyn1e3Dxn1f3Eyn1g2Exn1,
a1=B1c1F1,a2=B1c2F1,a3=B1c3F1,b1=A1(B1)2F1,b2=A2B1B2F1,b3=A3B1B3F1,d1=B2B1F2F1,d2=B3B1F3F1,e1=B1c1A1,e2=B1c2A1,e3=B1c3A1,f1=F1(B1)2A1,f2=F2B1B2A1f3=F3B1B3A1,g1=B2A2B1A1,g2=B3A3B1A1.
jωΩ0Dx=ε0{jω(ε1Ω12+ε2Ω22)+σ1Ω12+σ2Ω22}Ex+ε0{jω(ε1ε2)Ω1Ω2+(σ1σ2)Ω1Ω2}Ey,jωΩ0Dy=ε0{jω(ε1ε2)Ω1Ω2+(σ1σ2)Ω1Ω2}Ex+ε0{jω(ε1Ω22+ε2Ω12)+σ1Ω22+σ2Ω12}Ey.
A1both lossy=ε1Ω12+ε2Ω222Δt+σ1Ω12+σ2Ω224,A2both lossy=σ1Ω12+σ2Ω222,A3both lossy=ε1Ω12+ε2Ω222Δt+σ1Ω12+σ2Ω224,B1both lossy=(ε1ε22Δt+σ1σ24)Ω1Ω2,B2both lossy=σ1σ22Ω1Ω2,B3both lossy=(ε1ε22Δt+σ1σ24)Ω1Ω2,c1both lossy=12ε0ΔtΩ0,c2both lossy=0,c3both lossy=12ε0ΔtΩ0,F1both lossy=ε1Ω22+ε2Ω122Δt+σ1Ω22+σ2Ω124,F2both lossy=σ1Ω22+σ2Ω122,F3both lossy=ε1Ω22+ε2Ω122Δt+σ1Ω22+σ2Ω124.
(ω2jωγ3)Bz(ω)=μ0(ω2jωγ3ωp32)Hz(ω).
(1Δt2+γ32Δt+ωp324)Hzn+1=(1μ0Δt2+γ32μ0Δt)Bzn+12μ0Δt2Bzn+(1μ0Δt2γ32μ0Δt)Bzn1+(2Δt2ωp322)Hzn(1Δt2γ32Δt+ωp324)Hzn1.
jωBz(ω)=μ0(μ3jω+σ3)Hz(ω),
(μ32Δt+σ34)Hzn+1=12μ0ΔtBzn+112μ0ΔtBzn1σ32Hzn+(μ32Δtσ34)Hzn1.

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