Abstract

In this paper we perform a theoretical and numerical study of two-dimensional inside-out Eaton lenses under transverse-magnetic-polarized excitation. We present one example design and test its performance by utilizing full-wave Maxwell solvers. With the help of the WKB approximation, we further investigate the finite-wavelength effect analytically and demonstrate one necessary condition for perfect imaging at the level of wave optics, i.e. imaging with unlimited resolution, by the lens.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).
  2. J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. 8, 188 (1854).
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  4. J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).
  5. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
  6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).
  7. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006).
    [PubMed]
  8. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).
  9. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials 9, 129–132 (2010).
  10. V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lenses emulated by microdroplets,” Opt. Lett. 35, 3396–3398 (2010).
    [PubMed]
  11. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18, 21238–21251 (2010).
    [PubMed]
  12. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).
  13. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
  14. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011).
    [PubMed]
  15. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
    [PubMed]
  16. A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).
  17. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980–2990 (2008).
  18. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [PubMed]
  19. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).
  20. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [PubMed]
  21. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [PubMed]
  22. D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag. 52, 24–45 (2010).
  23. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).
  24. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [PubMed]
  25. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).
  26. J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).
  27. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).
  28. R. H. Good, “The generalization of the WKB method to radial wave equations,” Phys. Rev. 90, 131–137 (1953).
  29. B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986).
    [PubMed]
  30. R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. 51, 669–676 (1937).
  31. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010).
    [PubMed]
  32. COMSOL, www.comsol.com .
  33. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).
  34. Y. Zeng, J. Liu, and D. H. Werner, “General properties of two-dimensional conformal transformations in electrostatics,” Opt. Express 19, 20035–20047 (2011).
    [PubMed]

2011 (5)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[PubMed]

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011).
[PubMed]

Y. Zeng, J. Liu, and D. H. Werner, “General properties of two-dimensional conformal transformations in electrostatics,” Opt. Express 19, 20035–20047 (2011).
[PubMed]

2010 (6)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials 9, 129–132 (2010).

Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010).
[PubMed]

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18, 21238–21251 (2010).
[PubMed]

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lenses emulated by microdroplets,” Opt. Lett. 35, 3396–3398 (2010).
[PubMed]

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag. 52, 24–45 (2010).

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

2009 (3)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

2008 (1)

2006 (3)

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

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

J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006).
[PubMed]

2000 (2)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

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

1986 (1)

B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986).
[PubMed]

1953 (1)

R. H. Good, “The generalization of the WKB method to radial wave equations,” Phys. Rev. 90, 131–137 (1953).

1952 (1)

J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).

1937 (1)

R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. 51, 669–676 (1937).

1854 (1)

J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. 8, 188 (1854).

Bao, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

Danner, A. J.

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).

Di Falco, A.

Durand, B.

B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986).
[PubMed]

Durand, L.

B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986).
[PubMed]

Eaton, J. E.

J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[PubMed]

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Good, R. H.

R. H. Good, “The generalization of the WKB method to radial wave equations,” Phys. Rev. 90, 131–137 (1953).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).

Kehr, S. C.

Kildishev, A. V.

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials 9, 129–132 (2010).

Kundtz, N. B.

Kwon, D.-H.

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag. 52, 24–45 (2010).

Landy, N. I.

Langer, R. E.

R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. 51, 669–676 (1937).

Leonhardt, U.

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011).
[PubMed]

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

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

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

Liu, J.

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

Lock, J. A.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

Ma, Y. G.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

Maxwell, J. C.

J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. 8, 188 (1854).

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

Miñano, J. C.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

Ong, C. K.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

Pendry, J. B.

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

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

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

Schurig, D.

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

Shalaev, V. M.

Shamonina, E.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

Smith, D. R.

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18, 21238–21251 (2010).
[PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials 9, 129–132 (2010).

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

Smolyaninov, I. I.

Smolyaninova, V. N.

Solymar, L.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

Tyc, T.

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

Urzhumov, Y.

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[PubMed]

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

Werner, D. H.

Wu, Q.

Zeng, Y.

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

Camb. Dublin Math. J. (1)

J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. 8, 188 (1854).

IEEE Antennas Prop. Mag. (1)

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag. 52, 24–45 (2010).

J. Opt. Soc. Am. A (1)

Nat. Materials (3)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials 9, 129–132 (2010).

Nat. Nanotechnology (1)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

Nat. Photonics (1)

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).

New J. Phys. (1)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. (2)

R. H. Good, “The generalization of the WKB method to radial wave equations,” Phys. Rev. 90, 131–137 (1953).

R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. 51, 669–676 (1937).

Phys. Rev. A (1)

B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986).
[PubMed]

Phys. Rev. Lett. (2)

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[PubMed]

Prog. Opt. (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).

Science (3)

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

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

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[PubMed]

Trans. IRE Antennas Propag. (1)

J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).

Other (9)

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).

COMSOL, www.comsol.com .

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Inside-out Eaton lens (1 ≤ r ≤ 2). The source is located at r0 = 0.5. Light rays (blue) are described by Hamilton’s Eq. (A.2).

Fig. 2
Fig. 2

(a) Dependence of the function s(r) on the mode order n. Here the wavelength is 0.3. (b) Effect of the wavelength λ on the phase factor γn.

Fig. 3
Fig. 3

(a) A schematic of the design. (b) The radius of the metallic wire as a function of its position. Here the permittivity of the metal is ɛm = −0.6.

Fig. 4
Fig. 4

Full-wave simulations of the design, with the excitation source placed at different locations. r0 equals (a) 0.2, (b) 0.3, (c) 0.4, and (d) 0.5. The wavelength is fixed at 0.3, and the permittivity of metal is ɛm = −0.6 + 0.01i. The amplitude of the magnetic field |H| is plotted.

Fig. 5
Fig. 5

The effect of metallic loss δ on the lens performance. The source is located at r0 = 0.2. The wavelength λ is set to be 0.3 and Re(ɛm) = −0.6.

Equations (46)

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

n ( r ) = 2 a r 1 .
1 ɛ ( r ) 2 H + 1 ɛ ( r ) H = ω 2 c 2 H ,
2 H r 2 + ( 1 r 1 ɛ d ɛ d r ) H r + 1 r 2 2 H θ 2 + k 0 2 ɛ H = 0 ,
f n + ( k 0 2 ɛ n 2 r 2 + ɛ 2 3 r 2 ɛ 2 + 2 r ɛ ɛ + 2 r 2 ɛ ɛ 4 r 2 ɛ 2 ) f n = 0 .
f n + ( k 0 2 n 2 1 / 4 r 2 ) f n = 0 ,
f n ( r ) = r [ a n H n ( 1 ) ( k 0 r ) + b n H n ( 2 ) ( k 0 r ) ] ,
f n + s ( r ) k 0 2 f n = 0 ,
s ( r ) = 2 r r n 2 1 / 4 r 2 k 0 2 r + 1 r 2 ( 2 r ) 2 k 0 2 .
τ ( r ) = τ 0 ( r ) + 1 k 0 τ 1 ( r ) + 1 k 0 2 τ 2 ( r ) + .
( d τ 0 d r ) 2 = s ( r ) , d τ 1 d r = i τ 0 2 τ 0 .
f n ( r ) A n + s ( r ) 1 / 4 exp [ i k 0 r n r s ( r ) d r ] + A n s ( r ) 1 / 4 exp [ i k 0 r n r s ( r ) d r ] ,
f n ( r ) B n + | s ( r ) | 1 / 4 exp [ k 0 r n r | s ( r ) | d r ] .
f n ( r ) A n s ( r ) 1 / 4 exp [ i k 0 r n r s ( r ) d r ] + A n s ( r ) 1 / 4 exp [ i π 2 i k 0 r n r s ( r ) d r ] ,
k 0 r n r s ( r ) d r = k 0 r n r s 0 d r + k 0 r n 1 [ s ( r ) s 0 ] d r k 0 r + γ n ,
k 0 2 s 0 = k 0 2 n 2 1 / 4 r 2
f n ( r ) A n s 0 1 / 4 [ e i ( k 0 r + γ n ) + e i ( k 0 r + γ n + π / 2 ) ] .
i 4 H 0 ( 1 ) ( k 0 | r r 0 | ) = i 4 J n ( k 0 r < ) H n ( 1 ) ( k 0 r > ) e i n ϕ ,
i 4 e in ϕ J n ( k 0 r 0 ) [ H n ( 1 ) ( k 0 r ) + C n H n ( 2 ) ( k 0 r ) ] ,
J n ( k 0 r 0 ) [ H n ( 1 ) ( k 0 r ) + C n H n ( 2 ) ( k 0 r ) ] 2 cos ( k 0 r 0 β n ) k 0 π r r 0 [ e i ( k 0 r β n ) + C n e i ( k 0 r β n ) ] ,
C n e i [ ( n + 1 ) π + 2 γ n ] = ( 1 ) n + 1 e 2 i γ n , A n 2 s 0 1 / 4 k 0 π r 0 cos ( k 0 r 0 β n ) .
C n = ( 1 ) n + 1 e i ϕ ,
κ 0 2 ( ξ ) ξ r 2 κ 2 ( r ) = 1 4 r 2 1 4 ξ 2 ξ r 2 + ξ r 1 / 2 d 2 d r 2 ξ r 1 / 2 ,
κ 2 ( r ) = k 0 2 s ( r ) 1 4 r 2 = 2 r r k 0 2 n 2 r 2 r + 1 r 2 ( 2 r ) 2 ,
d 2 d ξ 2 g n ( ξ ) + [ κ 0 2 ( ξ ) + 1 4 ξ 2 ] g n ( ξ ) = 0 .
f n ( r ) = i D n ξ r 1 / 2 ξ e i ϕ n [ H n ( 1 ) ( ξ ) H n ( 2 ) ( ξ ) e 2 i ϕ n ]
r t r κ ( r ) d r = ξ 2 n 2 n tan 1 ( ξ 2 n 2 ) / n 2 ,
ϕ n = r t r n κ ( r ) d r .
D n = 1 4 J n ( k 0 r 0 ) e i ϕ n ξ r / ξ | r = 1 , C n = e 2 i ϕ n .
ɛ e = ( 1 f ) + ( 1 + f ) ɛ m ( 1 f ) ɛ m + ( 1 + f ) ,
f ( r ) = ( r 1 ) 1 + ɛ m 1 ɛ m .
a p = a 1 ( 1 + π / 30 ) p 1 ,
( a p 1 ) 1 + ɛ m 1 ɛ m π r p 2 a p 2 ( π / 30 ) 2 ,
η = | H ( r 0 ) H ( r 0 ) | 2 ,
d 2 r d ξ 2 = n 2 ( r ) 2 ,
d r d t = c n k k , d k d t = c k n 2 n ( r ) ,
L = r × d r d ξ = n k r × k ,
d L d ξ = r × d 2 r d 2 ξ = 1 2 r × n 2 = d n 2 d r r × r 2 r = 0 ,
d 2 z d ξ 2 = z 2 r d n 2 d r = 1 r 3 z = 1 | z | 3 z ,
z = e i α [ cos ( 2 ξ ) + i sin γ sin ( 2 ξ ) + cos γ ] , d ξ = 2 | z | d ξ ,
E ( x ) = 1 A A E ( x ) d x = f E m ( x ) + ( 1 f ) E d ( x ) ,
P ( x ) = f P m ( x ) + ( 1 f ) P d ( x ) .
P m ( x ) = ɛ 0 ( ɛ m 1 ) E m ( x ) , P d ( x ) = ɛ 0 ( ɛ d 1 ) E d ( x ) ,
P ( x ) = ɛ 0 ( ɛ ¯ e I ¯ ) E ( x ) .
( ɛ ( r ) ϕ ) = 0 .
ɛ e = ɛ d ( 1 f ) ɛ d + ( 1 + f ) ɛ m ( 1 f ) ɛ m + ( 1 + f ) ɛ d ,
f ( r ) = ( ɛ e ɛ d ) ( ɛ m + ɛ d ) ( ɛ e + ɛ d ) ( ɛ m ɛ d ) .

Metrics