Abstract

The explicit dispersion relation for one-dimensional photonic crystals consisting of alternative layers of indefinite metamaterial and ordinary positive-index material is obtained and analyzed in detail. It is shown that such a photonic crystal possesses an omnidirectional zero-averaged refractive-index gap when the indefinite metamaterial is a cutoff or anticutoff medium, yet it possesses an omnidirectional zero-effective phase gap when the photonic crystal contains two different always-cutoff media. The appearance of these two omnidirectional gaps does not require that all tensor elements of permittivity and permeability have negative values, quite different from that for the photonic crystal containing isotropic negative-index or single-negative material. Moreover, the new zero-averaged refractive-index gap and zero-effective phase gap are also insensitive to the incident angle and polarization of light and are invariant upon the change of scale length. These properties provide more degrees of freedom to design new types of optical devices.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  3. 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).
    [CrossRef] [PubMed]
  4. D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
    [CrossRef] [PubMed]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  6. P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
    [CrossRef]
  7. R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design--theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
    [CrossRef]
  8. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
    [CrossRef]
  9. D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef] [PubMed]
  10. D. Schurig and D. R. Smith, "Spatial filtering using media with indefinite permittivity and permeability tensors," Appl. Phys. Lett. 82, 2215-2217 (2003).
    [CrossRef]
  11. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, "Partial focusing of radiation by a slab of indefinite media," Appl. Phys. Lett. 84, 2244-2246 (2004).
    [CrossRef]
  12. L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
    [CrossRef]
  13. T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, "Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials," Appl. Phys. Lett. 86, 251909 (2005).
    [CrossRef]
  14. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
    [CrossRef] [PubMed]
  15. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573-1575 (1998).
    [CrossRef]
  16. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from a one-dimensional dielectric lattice," Appl. Phys. A 68, 25-28 (1999).
    [CrossRef]
  17. A. Mir, A. Akjouj, E. H. El Boudouti, B. Djafari-Ronhani, and L. Dobrzynski, "Large omnidirectional band gaps and selective transmission in one-dimensional multilayer photonic structures," Vacuum 63, 197-203 (2001).
    [CrossRef]
  18. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  19. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  20. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef] [PubMed]
  21. D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
    [CrossRef]
  22. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials," Appl. Phys. Lett. 83, 5386-5388 (2003).
    [CrossRef]
  23. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
    [CrossRef]
  24. L. G. Wang, H. Chen, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
    [CrossRef]
  25. I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
    [CrossRef]
  26. I. Abdulhalim, "Reflective polarization conversion Fabry-Pérot resonator using omnidirectional mirror of periodic anisotropic stack," Opt. Commun. 215, 225-230 (2003).
    [CrossRef]
  27. H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, "Beam shifting of an anisotropic negative refractive medium," Phys. Rev. E 71, 066612 (2005).
    [CrossRef]
  28. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  29. Y. J. Feng, X. H. Teng, Y. Chen, and T. Jiang, "Electromagnetic wave propagation in anisotropic metamaterials created by a set of periodic inductor-capacitor circuit networks," Phys. Rev. B 72, 245107 (2005).
    [CrossRef]
  30. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
    [CrossRef]

2005

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, "Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials," Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, "Beam shifting of an anisotropic negative refractive medium," Phys. Rev. E 71, 066612 (2005).
[CrossRef]

Y. J. Feng, X. H. Teng, Y. Chen, and T. Jiang, "Electromagnetic wave propagation in anisotropic metamaterials created by a set of periodic inductor-capacitor circuit networks," Phys. Rev. B 72, 245107 (2005).
[CrossRef]

2004

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L. G. Wang, H. Chen, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, "Partial focusing of radiation by a slab of indefinite media," Appl. Phys. Lett. 84, 2244-2246 (2004).
[CrossRef]

2003

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design--theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

D. Schurig and D. R. Smith, "Spatial filtering using media with indefinite permittivity and permeability tensors," Appl. Phys. Lett. 82, 2215-2217 (2003).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials," Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

I. Abdulhalim, "Reflective polarization conversion Fabry-Pérot resonator using omnidirectional mirror of periodic anisotropic stack," Opt. Commun. 215, 225-230 (2003).
[CrossRef]

2002

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
[CrossRef]

L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

2001

A. Mir, A. Akjouj, E. H. El Boudouti, B. Djafari-Ronhani, and L. Dobrzynski, "Large omnidirectional band gaps and selective transmission in one-dimensional multilayer photonic structures," Vacuum 63, 197-203 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [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).
[CrossRef] [PubMed]

D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
[CrossRef]

1999

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from a one-dimensional dielectric lattice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

1998

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573-1575 (1998).
[CrossRef]

1987

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1968

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Appl. Phys. A

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from a one-dimensional dielectric lattice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Appl. Phys. Lett.

D. Schurig and D. R. Smith, "Spatial filtering using media with indefinite permittivity and permeability tensors," Appl. Phys. Lett. 82, 2215-2217 (2003).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, "Partial focusing of radiation by a slab of indefinite media," Appl. Phys. Lett. 84, 2244-2246 (2004).
[CrossRef]

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, "Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials," Appl. Phys. Lett. 86, 251909 (2005).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials," Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

IEEE Trans. Antennas Propag.

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design--theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Opt. Commun.

I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
[CrossRef]

I. Abdulhalim, "Reflective polarization conversion Fabry-Pérot resonator using omnidirectional mirror of periodic anisotropic stack," Opt. Commun. 215, 225-230 (2003).
[CrossRef]

Opt. Lett.

Phys. Rev. B

L. G. Wang, H. Chen, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Y. J. Feng, X. H. Teng, Y. Chen, and T. Jiang, "Electromagnetic wave propagation in anisotropic metamaterials created by a set of periodic inductor-capacitor circuit networks," Phys. Rev. B 72, 245107 (2005).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
[CrossRef]

L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

Phys. Rev. E

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, "Beam shifting of an anisotropic negative refractive medium," Phys. Rev. E 71, 066612 (2005).
[CrossRef]

Phys. Rev. Lett.

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [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).
[CrossRef] [PubMed]

D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Science

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vacuum

A. Mir, A. Akjouj, E. H. El Boudouti, B. Djafari-Ronhani, and L. Dobrzynski, "Large omnidirectional band gaps and selective transmission in one-dimensional multilayer photonic structures," Vacuum 63, 197-203 (2001).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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 (10)

Fig. 1
Fig. 1

Unit cell of the one-dimensional photonic crystal consisting of alternating layers of indefinite metamaterial and positive-index material inclusions.

Fig. 2
Fig. 2

Transmittance with different incident angles for case 1, where d A = 12 mm , d B = 6 mm .

Fig. 3
Fig. 3

PBGs as a function of the incident angle in case 1, where d A = 12 mm , d B = 6 mm . The black areas correspond to the forbidden gaps, and the white areas are the allowed bands.

Fig. 4
Fig. 4

Dependence of the PBGs on the ratio of the thicknesses of the two materials under two different lattice constants: d A = ( a ) 10 and (b) 20 mm , for the infinite 1DPC as that of Fig. 3 at normal incidence.

Fig. 5
Fig. 5

PBGs as a function of incident angle for case 2, where d A = 12 mm , d B = 6 mm .

Fig. 6
Fig. 6

Dependence of the PBGs on the ratio of the thicknesses of the two materials under two different lattice constants: d A = ( a ) 10 and (b) 20 mm , for the infinite 1DPC as that of Fig. 5 at normal incidence.

Fig. 7
Fig. 7

PBGs as a function of incident angle for case 3, where d A = 12 mm , d B = 6 mm .

Fig. 8
Fig. 8

Dependence of the PBGs on the ratio of the thicknesses of the two materials under two different lattice constants: d A = ( a ) 10 and (b) 20 mm , for the infinite 1DPC as that of Fig. 7 at normal incidence.

Fig. 9
Fig. 9

PBGs as a function of incident angle for case 4, where d A = 12 mm , d B = 6 mm .

Fig. 10
Fig. 10

Dependence of the PBGs on the ratio of thicknesses of the two materials under two different lattice constants: d A = ( a ) 10 and (b) 20 mm , for the infinite 1DPC as that of Fig. 9 at normal incidence.

Equations (23)

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

ε ̂ A = ( ε A x 0 0 0 ε A y 0 0 0 ε A z ) , μ ̂ A = ( μ A x 0 0 0 μ A y 0 0 0 μ A z ) .
E A y = e i β x ( A e i k A z z + C e i k A z z ) ,
H A x = k A z k 0 μ A x e i β x ( A e i k A z z C e i k A z z ) ,
H A z = β k 0 μ A z e i β x ( A e i k A z z + C e i k A z z ) ,
E B y = e i β x ( B e i k B z z + D e i k B z z ) ,
H B x = k B z k 0 μ B e i β x ( B e i k B z z D e i k B z z ) ,
H B z = β k 0 μ B e i β x ( B e i k B z z + D e i k B z z ) ,
β 2 ε A y μ A z + k A z 2 ε A y μ A x = ω 2 c 2 ,
β 2 ε B μ B + k B z 2 ε B μ B = ω 2 c 2 ,
E A y ( z = 0 ) = E B y ( z = 0 + ) ,
H A x ( z = 0 ) = H B x ( z = 0 + ) .
E B y ( z = d A ) = E A y ( z = d B ) e i q d ,
H B x ( z = d A ) = H A x ( z = d B ) e i q d ,
cos ( q d ) = cos ( k A z d A ) cos ( k B z d B ) 1 2 ( k B z μ A x k A z μ B + k A z μ B k B z μ A x ) sin ( k A z d A ) sin ( k B z d B ) .
cos ( q d ) = cos ( k A z d A ) cos ( k B z d B ) 1 2 ( k B z ε A x k A z ε B + k A z ε B k B z ε A x ) sin ( k A z d A ) sin ( k B z d B ) ,
M A , B ( d A , B , ω ) = ( cos ( k A , B z d A , B ) i p A , B k A , B z sin ( k A , B z d A , B ) i k A , B z p A , B sin ( k A , B z d A , B ) cos ( k A , B z d A , B ) ) ,
t ( ω ) = 2 q 0 ( q s x 11 + q 0 x 22 ) q 0 + ( q 0 q s x 12 + x 21 ) .
cos ( q d ) = cos ( k A z d A + k B z d B ) 1 2 ( k B z μ A x k A z μ B + k A z μ B k B z μ A x 2 ) sin ( k A z d A ) sin ( k B z d B ) .
k A z d A + k B z d B = 0 ,
1 + 1 2 ( k B z μ A x k A z μ B + k A z μ B k B z μ A x 2 ) sin 2 ( k A z d A ) 1 .
θ < θ c = arcsin ( ε A y μ A z ) ,
θ < θ c = arcsin ( μ A y ε A z ) ,
cos ( q d ) = cosh ( k A z d A ) cosh ( k B z d B ) 1 2 ( η B η A + η A η B ) sinh ( k A z d A ) sinh ( k B z d B ) ,

Metrics