Abstract

In this paper, we present the theoretical analysis of excitation of well-bounded oblique surface electromagnetic waves propagating along an anisotropically conducting interface of two different dielectrics by means of the attenuated-total-reflection method. The flat surface of the interface contains a one-dimensional array of thin metal wires. It has been assumed that both the lattice constant of the array and the diameter of the wires are far less than the wavelengths of the surface waves. The surface electromagnetic waves may propagate along the interface at frequencies that are far lower than the plasma frequency of a metal, and the electric field of the waves is always perpendicular to the wires in any propagation directions. It has been shown that the oblique surface electromagnetic waves can be excited by the TM-polarized electromagnetic wave which becomes partially polarized during the excitation.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  2. R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width,” Opt. Lett. 25, 844–846 (2000).
    [CrossRef]
  3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [CrossRef]
  4. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
    [CrossRef] [PubMed]
  5. H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459–3465 (1993).
    [CrossRef] [PubMed]
  6. H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795–4801 (1994).
    [CrossRef]
  7. H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B 79, 245427 (2009).
    [CrossRef]
  8. C. Hu and D. Liu, “Polarization characteristics of subwavelength aluminum wire grating in near infrared,” Front. Optoelectron. China 2, 187–191 (2009).
    [CrossRef]
  9. A. V. Kats, M. L. Nesterov, and A. Y. Nikitin, “Polarization properties of a periodically-modulated metal film in regions of anomalous optical transparency,” Phys. Rev. B 72, 193405(2005).
    [CrossRef]
  10. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
    [CrossRef] [PubMed]
  11. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A Pure Appl. Opt. 7, S97–S101 (2005).
    [CrossRef]
  12. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
    [CrossRef] [PubMed]
  13. A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
    [CrossRef]
  14. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18, 754–764 (2010).
    [CrossRef] [PubMed]
  15. Y. O. Averkov and V. M. Yakovenko, “Surface electromagnetic waves at an anisotropically conducting artificial interface,” Phys. Rev. B 81, 045427 (2010).
    [CrossRef]
  16. Surface Polaritons, V.M.Agranovich and D.L.Mills eds. (North-Holland, 1982).
  17. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

2010 (2)

2009 (3)

A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
[CrossRef]

H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B 79, 245427 (2009).
[CrossRef]

C. Hu and D. Liu, “Polarization characteristics of subwavelength aluminum wire grating in near infrared,” Front. Optoelectron. China 2, 187–191 (2009).
[CrossRef]

2008 (1)

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

2005 (2)

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A Pure Appl. Opt. 7, S97–S101 (2005).
[CrossRef]

A. V. Kats, M. L. Nesterov, and A. Y. Nikitin, “Polarization properties of a periodically-modulated metal film in regions of anomalous optical transparency,” Phys. Rev. B 72, 193405(2005).
[CrossRef]

2004 (2)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[CrossRef] [PubMed]

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

2000 (1)

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1994 (1)

H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795–4801 (1994).
[CrossRef]

1993 (1)

Averkov, Y. O.

Y. O. Averkov and V. M. Yakovenko, “Surface electromagnetic waves at an anisotropically conducting artificial interface,” Phys. Rev. B 81, 045427 (2010).
[CrossRef]

Barnes, W. L.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

Berini, P.

Berolo, E.

Bozhevolnyi, S. I.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Charbonneau, R.

Depine, R.

Devaux, E.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

Dintinger, J.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Ebessen, T. W.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

Fernandez-Dominguez, A. I.

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18, 754–764 (2010).
[CrossRef] [PubMed]

A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
[CrossRef]

Garcia-Vidal, F. J.

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18, 754–764 (2010).
[CrossRef] [PubMed]

A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
[CrossRef]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A Pure Appl. Opt. 7, S97–S101 (2005).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Hu, C.

C. Hu and D. Liu, “Polarization characteristics of subwavelength aluminum wire grating in near infrared,” Front. Optoelectron. China 2, 187–191 (2009).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Kats, A. V.

A. V. Kats, M. L. Nesterov, and A. Y. Nikitin, “Polarization properties of a periodically-modulated metal film in regions of anomalous optical transparency,” Phys. Rev. B 72, 193405(2005).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Lisicka-Shrzek, E.

Liu, D.

C. Hu and D. Liu, “Polarization characteristics of subwavelength aluminum wire grating in near infrared,” Front. Optoelectron. China 2, 187–191 (2009).
[CrossRef]

Lochbihler, H.

H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B 79, 245427 (2009).
[CrossRef]

H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795–4801 (1994).
[CrossRef]

H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459–3465 (1993).
[CrossRef] [PubMed]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Martin-Cano, D.

Martin-Moreno, L.

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18, 754–764 (2010).
[CrossRef] [PubMed]

A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
[CrossRef]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A Pure Appl. Opt. 7, S97–S101 (2005).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[CrossRef] [PubMed]

Moreno, E.

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18, 754–764 (2010).
[CrossRef] [PubMed]

A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
[CrossRef]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Murray, W. A.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

Nesterov, M. L.

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18, 754–764 (2010).
[CrossRef] [PubMed]

A. V. Kats, M. L. Nesterov, and A. Y. Nikitin, “Polarization properties of a periodically-modulated metal film in regions of anomalous optical transparency,” Phys. Rev. B 72, 193405(2005).
[CrossRef]

Nikitin, A. Y.

A. V. Kats, M. L. Nesterov, and A. Y. Nikitin, “Polarization properties of a periodically-modulated metal film in regions of anomalous optical transparency,” Phys. Rev. B 72, 193405(2005).
[CrossRef]

Pendry, J. B.

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A Pure Appl. Opt. 7, S97–S101 (2005).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[CrossRef] [PubMed]

Rodrigo, S. G.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Yakovenko, V. M.

Y. O. Averkov and V. M. Yakovenko, “Surface electromagnetic waves at an anisotropically conducting artificial interface,” Phys. Rev. B 81, 045427 (2010).
[CrossRef]

Appl. Opt. (1)

Front. Optoelectron. China (1)

C. Hu and D. Liu, “Polarization characteristics of subwavelength aluminum wire grating in near infrared,” Front. Optoelectron. China 2, 187–191 (2009).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A Pure Appl. Opt. 7, S97–S101 (2005).
[CrossRef]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (5)

A. V. Kats, M. L. Nesterov, and A. Y. Nikitin, “Polarization properties of a periodically-modulated metal film in regions of anomalous optical transparency,” Phys. Rev. B 72, 193405(2005).
[CrossRef]

H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795–4801 (1994).
[CrossRef]

H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B 79, 245427 (2009).
[CrossRef]

Y. O. Averkov and V. M. Yakovenko, “Surface electromagnetic waves at an anisotropically conducting artificial interface,” Phys. Rev. B 81, 045427 (2010).
[CrossRef]

A. I. Fernandez-Dominguez, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79, 233104 (2009).
[CrossRef]

Phys. Rev. Lett. (2)

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebessen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401(2004).
[CrossRef] [PubMed]

Science (1)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[CrossRef] [PubMed]

Other (3)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Surface Polaritons, V.M.Agranovich and D.L.Mills eds. (North-Holland, 1982).

J. D. Jackson, Classical Electrodynamics (Wiley, 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 (2)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Dependencies of the reflection coefficients R ( TM ) and R ( TE ) on the angle of incidence for a number of ϑ values. Reflection coefficient curve 1 corresponds to ϑ = 20 ° , curve 2 is for ϑ = 40 ° , curve 3 is for ϑ = 50 ° , and curve 4 is for ϑ = 60 ° .

Equations (30)

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

E = E 0 exp [ i ( κ ρ + k y y ω t ) ] ,
k y 1 = k 1 cos φ , k y = ( ω 2 ε / c 2 κ 2 ) 1 / 2 , = 2 , 3 .
E x 1 = E x 1 ( inc ) + E x 1 ( TM ) + E x 1 ( TE ) ,
E y 1 = κ 2 k y 1 k x ( E x 1 ( inc ) E x 1 ( TM ) ) ,
E z 1 = k z k x ( E x 1 ( inc ) + E x 1 ( TM ) ) k x k z E x 1 ( TE ) ,
H x 1 = ω ε 1 k z c k x k y 1 ( E x 1 ( inc ) E x 1 ( TM ) ) + c k x k y 1 ω k z E x 1 ( TE ) ,
H y 1 = H y 1 ( TE ) = c κ 2 ω k z E x 1 ( TE ) ,
H z 1 = ω ε 1 c k y 1 ( E x 1 ( inc ) E x 1 ( TM ) ) + c k y 1 ω E x 1 ( TE ) ,
E x 1 ( inc ) = E x 0 exp ( i k y 1 y ) ,
E x 1 ( TM , TE ) = A ( TM , TE ) exp ( i k y 1 y ) .
E x 2 ( TM ) = B 1 exp ( i k y 2 y ) + B 2 exp ( i k y 2 y ) ,
E y 2 ( TM ) = κ 2 k y 2 k x [ B 1 exp ( i k y 2 y ) B 2 exp ( i k y 2 y ) ] ,
E z 2 ( TM ) = k z k x E x 2 ( TM ) ,
H x 2 ( TM ) = ω ε 2 k z c κ 2 E y 2 ( TM ) , H z 2 ( TM ) = ω ε 2 k x c κ 2 E y 2 ( TM ) ,
E x 2 ( TE ) = C 1 exp ( i k y 2 y ) + C 2 exp ( i k y 2 y ) ,
E z 2 ( TE ) = k x k z E x 2 ( TE ) ,
H x 2 ( TE ) = c k x k y 2 ω k z [ C 1 exp ( i k y 2 y ) C 2 exp ( i k y 2 y ) ] ,
H y 2 ( TE ) = c κ 2 ω k z E x 2 ( TE ) , H z 2 ( TE ) = k z k x H x 2 ( TE ) .
E x 3 = F exp ( i k y 3 y ) , E y 3 = k x k y 3 E x 3 ,
H x 3 = c k x k z ω k y 3 E x 3 , H y 3 = c k z ω E x 3 ,
H z 3 = c ( k x 2 + k y 3 2 ) ω k y 3 E x 3 .
E z 2 ( TM ) ( h ) + E z 2 ( TE ) ( h ) = 0 .
R ( TM ) = | 1 i P 1 + i P | 2 , P = g 0 P 1 P 2 ,
P 1 = k y 1 k y 2 k z 2 + κ 2 cosh 2 ψ 2 × [ k y 1 tanh ψ 2 + k y 2 ] [ a tanh ψ 2 + b ] ,
P 2 = k y 2 2 k z 2 κ 2 cosh 2 ψ 2 × [ k y 1 tanh ψ 2 + k y 2 ] [ a + b tanh ψ 2 ] ,
a = 1 k y 2 ( ω 2 c 2 ε 2 k z 2 ) , b = 1 k y 3 ( ω 2 c 2 ε 3 k z 2 ) ,
R ( TE ) = 4 k x 2 k z 2 k y 1 2 ω 2 c 2 ε 2 2 ε 1 1 | P 2 + i g 0 P 1 | 2 ,
Im ( a ) + Im ( b ) tanh ψ 2 = 0 .
φ ( ϑ ) = arcsin [ 2 ε 2 ε 3 / [ ε 1 Λ ( ϑ ) ] ] 1 / 2 ,
ϑ max = arcsin [ 1 [ ε 1 ( ε 2 + ε 3 ) ε 2 ε 3 ] / ε 1 2 ] 1 / 4 .

Metrics