Abstract

Complex and interesting electromagnetic behavior can be found in spaces with non-flat topology. When considering the properties of an electromagnetic medium under an arbitrary coordinate transformation an alternative interpretation presents itself. The transformed material property tensors may be interpreted as a different set of material properties in a flat, Cartesian space. We describe the calculation of these material properties for coordinate transformations that describe spaces with spherical or cylindrical holes in them. The resulting material properties can then implement invisibility cloaks in flat space. We also describe a method for performing geometric ray tracing in these materials which are both inhomogeneous and anisotropic in their electric permittivity and magnetic permeability.

© 2006 Optical Society of America

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, "Optical conformal mapping," Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  3. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788 (2004).
    [CrossRef] [PubMed]
  4. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
    [CrossRef] [PubMed]
  5. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
    [CrossRef] [PubMed]
  6. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
    [CrossRef] [PubMed]
  7. D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).
  8. 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 (2006). In press.
    [CrossRef] [PubMed]
  9. A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
    [CrossRef]
  10. G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. London A 462, 1364 (2006).
  11. U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," (2006). http://xxx.arxiv.org/abs/cond-mat/0607418.
  12. T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
    [CrossRef]
  13. E.J. Post, Formal structure of electromagnetics (Wiley, New York, 1962).
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  15. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley-Interscience, New York, 1990).
  16. D. M. Shyroki, "Exact equivalent-profile formulation for bent optical waveguides," (2006). Unpublished.
  17. A. J. Ward and J. B. Pendry, "Refraction and geometry in maxwell’s equations," J. Mod. Opt. 43(4), 773 - 793 (1996).
    [CrossRef]
  18. Y. A. Kravtsov and Y. I. Orlov, Geometrical optics of inhomogeneous media (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  19. H. C. Chen, Theory of electromagnetic waves: A coordinate free approach, pp. 216-218 (McGraw-Hill, 1985).

2006

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

U. Leonhardt, "Optical conformal mapping," Science 312, 1777 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).

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 (2006). In press.
[CrossRef] [PubMed]

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. London A 462, 1364 (2006).

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

2005

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

2004

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788 (2004).
[CrossRef] [PubMed]

2003

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

1996

A. J. Ward and J. B. Pendry, "Refraction and geometry in maxwell’s equations," J. Mod. Opt. 43(4), 773 - 793 (1996).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

Basov, D. N.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

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 (2006). In press.
[CrossRef] [PubMed]

Driscoll, T.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

Engheta, N.

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

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 (2006). In press.
[CrossRef] [PubMed]

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, "Optical conformal mapping," Science 312, 1777 (2006).
[CrossRef] [PubMed]

Linden, S.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Milton, G. W.

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. London A 462, 1364 (2006).

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 (2006). In press.
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).

Nemat-Nasser, S.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

Nicorovici, N.-A. P.

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. London A 462, 1364 (2006).

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," 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, "Metamaterial electromagnetic cloak at microwave frequencies," Science (2006). In press.
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788 (2004).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, "Refraction and geometry in maxwell’s equations," J. Mod. Opt. 43(4), 773 - 793 (1996).
[CrossRef]

Rye, P. M.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

Schurig, D.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," 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, "Metamaterial electromagnetic cloak at microwave frequencies," Science (2006). In press.
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).

Smith, D. R.

D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," 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, "Metamaterial electromagnetic cloak at microwave frequencies," Science (2006). In press.
[CrossRef] [PubMed]

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

Starr, A. F.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (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 (2006). In press.
[CrossRef] [PubMed]

Vier, D. C.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Ward, A. J.

A. J. Ward and J. B. Pendry, "Refraction and geometry in maxwell’s equations," J. Mod. Opt. 43(4), 773 - 793 (1996).
[CrossRef]

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[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 (2004).
[CrossRef] [PubMed]

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett.

D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006).
[CrossRef]

J. Mod. Opt.

A. J. Ward and J. B. Pendry, "Refraction and geometry in maxwell’s equations," J. Mod. Opt. 43(4), 773 - 793 (1996).
[CrossRef]

Nature

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003).
[CrossRef] [PubMed]

Phys. Rev. E

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Proc. Roy. Soc. London A

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. London A 462, 1364 (2006).

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 (2006). In press.
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

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

U. Leonhardt, "Optical conformal mapping," Science 312, 1777 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788 (2004).
[CrossRef] [PubMed]

Other

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," (2006). http://xxx.arxiv.org/abs/cond-mat/0607418.

Y. A. Kravtsov and Y. I. Orlov, Geometrical optics of inhomogeneous media (Springer-Verlag, Berlin, 1990).
[CrossRef]

H. C. Chen, Theory of electromagnetic waves: A coordinate free approach, pp. 216-218 (McGraw-Hill, 1985).

E.J. Post, Formal structure of electromagnetics (Wiley, New York, 1962).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley-Interscience, New York, 1990).

D. M. Shyroki, "Exact equivalent-profile formulation for bent optical waveguides," (2006). Unpublished.

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Figures (3)

Fig. 1.
Fig. 1.

The thick blue line shows the path of the same ray in (A) the original Cartesian space, and under two different interpretations of the electromagnetic equations, (B) the topological interpretation and (C) the materials interpretation. The position vector x is shown in both the original and transformed spaces, and the length of the vector where the transformed components are interpreted as Cartesian components is shown in (C).

Fig. 2.
Fig. 2.

Rays traversing a spherical cloak. The transformation media that comprises the cloak lies between the two spheres.

Fig. 3.
Fig. 3.

Rays traversing a cylindrical cloak at an oblique angle. The transformation media that comprises the cloak lies in an annular region between the cylinders.

Equations (58)

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

F α β , μ + F β μ , α + F μ α , β = 0
G , α α β = J β
( F α β ) = ( 0 E 1 E 2 E 3 E 1 0 c B 3 c B 2 E 2 c B 3 0 c B 1 E 3 c B 2 c B 1 0 )
( G α β ) = ( 0 c D 1 c D 2 c D 3 c D 1 0 H 3 H 2 c D 2 H 3 0 H 1 c D 3 H 2 H 1 0 )
( J β ) = ( J 1 J 2 J 3 )
G α β = 1 2 C α β μ ν F μ ν
C α β μ ν = det ( Λ α α ) 1 Λ α α Λ β β Λ μ μ Λ ν ν C α β μ ν
Λ α α = x α x α
ε i j = det ( Λ i i ) 1 Λ i i Λ j j ε ij
μ i j = det ( Λ i i ) 1 Λ i i Λ j j ε ij
ε i j = det ( g i j ) 1 2 g i j ε
μ i j = det ( g i j ) 1 2 g i j μ
g i j = Λ k i Λ l j δ kl
r = ( x i x j δ ij ) 1 2 = ( x i x j g i j ) 1 2
r = ( x i x j δ i j ) 1 2
r = b a b r + a
x i r = x i r δ i i
x i = b a b x i δ i i + a x i r δ i i
x j x i r = x i x k δ kj r 3 + 1 r δ j i
Λ j i = r r δ j i a x i x k δ j i δ kj r 3
( Λ j i ) = ( r r a x 2 r 3 axy r 3 axz r 3 ayx r 3 r r ay 2 r 3 ayz r 3 azx r 3 azy r 3 r r az 2 r 3 )
( x i ) = ( r , 0 , 0 )
det ( Λ j i ) = r a r ( r r ) 2
ε i j = μ i j = b b a [ δ i j 2 ar a 2 r 4 x i x j ]
ε = μ = b b a ( I 2 ar a 2 r 4 r r )
det ( ε ) = det ( μ ) = ( b b a ) 3 ( r a r ) 2
Z ij = δ 3 i δ 3 j
T ij = δ 1 i δ 1 j + δ 2 i δ 2 j
ρ i = T j i x j
( Λ j i ) = ( ρ ρ ax 2 ρ 3 axy ρ 3 0 ayx ρ 3 ρ ρ ay 2 ρ 3 0 0 0 1 )
Λ j i = ρ ρ T j i a ρ i ρ k δ i i δ kj ρ 3 + Z j i
( x i ) = ( ρ i ) = ( ρ , 0 , 0 )
det ( Λ j i ) = ρ a ρ ρ ρ
ε = μ = ρ ρ a T 2 a 2 ρ 3 ( ρ a ) ρ ρ + ( b b a ) 2 ρ a ρ Z
det ( ε ) = det ( μ ) = ( b b a ) 2 ρ a ρ
× E = B t × H = D t
E = E 0 e i ( k 0 k · x ωt ) H = 1 η 0 H 0 e i ( k 0 k · x ω t )
D = ε 0 ε E B = μ 0 μ H
k × E 0 μ H 0 = 0 k × H 0 + ε E 0 = 0
k × ( μ 1 ( k × E 0 ) ) + ε E 0 = 0
K ik ε ijk k j
( K μ 1 K + ε ) E 0 = 0
det ( K μ 1 K + ε ) = 0
det ( K n 1 K + n ) = 1 det ( n ) ( knk det ( n ) ) 2
H = f ( x ) ( knk det ( n ) )
d x d τ = H k
d k d τ = H x
( k 1 k 2 ) × n = 0
H ( k 2 ) = 0
H k · n > 0
H = 1 2 b a b ( knk det ( n ) )
H k = k 2 ar a 2 r 4 ( x · k ) x
H x = 2 ar a 2 r 4 ( x · k ) k + 3 ar 2 a 2 r 6 ( x · k ) 2 x ( b b a ) 2 ( ar a 2 r 4 ) x
H = 1 2 ρ a ρ ( knk det ( n ) )
H = 1 2 kTk 1 2 2 a 2 ρ 4 ( ρ · k ) 2 + 1 2 [ b ( ρ a ) ρ ( b a ) ] 2 ( kZk 1 )
ρ x = T
H k = Tk 2 a 2 ρ 4 ( ρ · k ) ρ + [ b ( ρ a ) ρ ( b a ) ] 2 Zk
H x = 3 2 a 2 ρ 6 ( ρ · k ) 2 ρ 2 a 2 ρ 4 ( ρ · k ) Tk + ( b b a ) 2 a 2 ρ 4 ( kZk 1 ) ρ

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