Abstract

Problem of anomalous refraction of electromagnetic waves is analyzed in a superlattice which consists of alternating layers of ferromagnetic insulator and nonmagnetic semiconductor. Effective permittivity and permeability tensors are derived in the presence of an external magnetic field parallel to the plane of the layers. It is shown that in the case of the Voigt configuration, the structure behaves as a left-handed medium with respect to TE-type polarized wave, in the low-frequency region of propagation. The relative orientation of the Poynting vector and the refractive wave vector is examined in different frequency ranges. It is shown that the frequency region of existence for the backward mode can be changed using external magnetic field as tuning parameter.

©2006 Optical Society of America

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References

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  1. R.A. Shelby, D.R. Smith, and S. Schultz., “Experimental Verification of a Negative Index of Refraction,” Science,  292, 77, 2001.
    [Crossref] [PubMed]
  2. P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002).
    [Crossref]
  3. D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [Crossref]
  4. S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
    [Crossref] [PubMed]
  5. V.G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. 10, 509 (1968)
    [Crossref]
  6. J.B. Pendry, “Negative refractions makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
    [Crossref] [PubMed]
  7. S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005).
  8. A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).
  9. A.S. Raspopin, A.A. Zharov, and H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005).
    [Crossref]
  10. A. Pimenov, A. Loidl, and P. Przyslupski, “Negative refraction in Ferromagnet-Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
    [Crossref] [PubMed]
  11. R.H. Tarkhanyan, “On the Theory of Surface Waves in a Uniaxial Semiconductor Slab,” Phys. Status Sol.(b) 72, 111 (1975).
    [Crossref]
  12. D. Polder, “Theory of electromagnetic Resonance,” Philos. Mag. 40, N 300, 99 (1949).
  13. V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985).
    [Crossref]
  14. R.H. Tarkhanyan and A.G. Nassiopoulou, “Electromagnetic instability of surface waves in semiconductor superlattices,” J. Nanosci. Nanotech.3, 549 (2003), and “Influence of magnetic field on electromagnetic instabilities in semiconductor superlattices,” J. Nanosci. Nanotech4, 1085 (2004).
    [Crossref]
  15. D.J. Bergman, X. Li, and Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one-or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005).
  16. R.H. Tarkhanyan and D.G. Niarchos, “Wave refraction and backward magnon-polaritons in left-handed antiferomagnetic/semiconductor superlattices,” JMMM (to be published).

2005 (4)

S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005).

A.S. Raspopin, A.A. Zharov, and H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005).
[Crossref]

A. Pimenov, A. Loidl, and P. Przyslupski, “Negative refraction in Ferromagnet-Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[Crossref] [PubMed]

D.J. Bergman, X. Li, and Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one-or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005).

2004 (1)

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

2003 (1)

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

2002 (2)

P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002).
[Crossref]

D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

2001 (1)

R.A. Shelby, D.R. Smith, and S. Schultz., “Experimental Verification of a Negative Index of Refraction,” Science,  292, 77, 2001.
[Crossref] [PubMed]

2000 (1)

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

1985 (1)

V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985).
[Crossref]

1975 (1)

R.H. Tarkhanyan, “On the Theory of Surface Waves in a Uniaxial Semiconductor Slab,” Phys. Status Sol.(b) 72, 111 (1975).
[Crossref]

1968 (1)

V.G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. 10, 509 (1968)
[Crossref]

1949 (1)

D. Polder, “Theory of electromagnetic Resonance,” Philos. Mag. 40, N 300, 99 (1949).

Agranovich, V.M.

V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985).
[Crossref]

Bergman, D.J.

D.J. Bergman, X. Li, and Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one-or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005).

Bertolotti, M.

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

Cui, H.L.

A.S. Raspopin, A.A. Zharov, and H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005).
[Crossref]

Foteinopoulou, S.

S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005).

Gregor, R.B.

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Haus, J.W.

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

Koltenbah, B.E.

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Kravtsov, V.E.

V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985).
[Crossref]

Li, K.

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Li, X.

D.J. Bergman, X. Li, and Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one-or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005).

Loidl, A.

A. Pimenov, A. Loidl, and P. Przyslupski, “Negative refraction in Ferromagnet-Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[Crossref] [PubMed]

Mandatory, A.

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

Markos, P.

P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002).
[Crossref]

D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Nassiopoulou, A.G.

R.H. Tarkhanyan and A.G. Nassiopoulou, “Electromagnetic instability of surface waves in semiconductor superlattices,” J. Nanosci. Nanotech.3, 549 (2003), and “Influence of magnetic field on electromagnetic instabilities in semiconductor superlattices,” J. Nanosci. Nanotech4, 1085 (2004).
[Crossref]

Niarchos, D.G.

R.H. Tarkhanyan and D.G. Niarchos, “Wave refraction and backward magnon-polaritons in left-handed antiferomagnetic/semiconductor superlattices,” JMMM (to be published).

Parazzoli, S.G.

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Pendry, J.B.

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

Pimenov, A.

A. Pimenov, A. Loidl, and P. Przyslupski, “Negative refraction in Ferromagnet-Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[Crossref] [PubMed]

Polder, D.

D. Polder, “Theory of electromagnetic Resonance,” Philos. Mag. 40, N 300, 99 (1949).

Przyslupski, P.

A. Pimenov, A. Loidl, and P. Przyslupski, “Negative refraction in Ferromagnet-Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[Crossref] [PubMed]

Raspopin, A.S.

A.S. Raspopin, A.A. Zharov, and H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005).
[Crossref]

Scalora, M.

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

Schultz, S.

D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

R.A. Shelby, D.R. Smith, and S. Schultz., “Experimental Verification of a Negative Index of Refraction,” Science,  292, 77, 2001.
[Crossref] [PubMed]

Shelby, R.A.

R.A. Shelby, D.R. Smith, and S. Schultz., “Experimental Verification of a Negative Index of Refraction,” Science,  292, 77, 2001.
[Crossref] [PubMed]

Sibilia, C.

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

Smith, D.R.

D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

R.A. Shelby, D.R. Smith, and S. Schultz., “Experimental Verification of a Negative Index of Refraction,” Science,  292, 77, 2001.
[Crossref] [PubMed]

Soukoulis, C.M.

S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005).

P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002).
[Crossref]

D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Strelniker, Y.M.

D.J. Bergman, X. Li, and Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one-or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005).

Tanielian, M.

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Tarkhanyan, R.H.

R.H. Tarkhanyan, “On the Theory of Surface Waves in a Uniaxial Semiconductor Slab,” Phys. Status Sol.(b) 72, 111 (1975).
[Crossref]

R.H. Tarkhanyan and A.G. Nassiopoulou, “Electromagnetic instability of surface waves in semiconductor superlattices,” J. Nanosci. Nanotech.3, 549 (2003), and “Influence of magnetic field on electromagnetic instabilities in semiconductor superlattices,” J. Nanosci. Nanotech4, 1085 (2004).
[Crossref]

R.H. Tarkhanyan and D.G. Niarchos, “Wave refraction and backward magnon-polaritons in left-handed antiferomagnetic/semiconductor superlattices,” JMMM (to be published).

Veselago, V.G.

V.G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. 10, 509 (1968)
[Crossref]

Zharov, A.A.

A.S. Raspopin, A.A. Zharov, and H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005).
[Crossref]

Zhukovsky, S.

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

J. Appl. Phys. (1)

A.S. Raspopin, A.A. Zharov, and H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005).
[Crossref]

Philos. Mag. (1)

D. Polder, “Theory of electromagnetic Resonance,” Philos. Mag. 40, N 300, 99 (1949).

Phys. Rev. (3)

D.J. Bergman, X. Li, and Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one-or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005).

S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005).

A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, and M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004).

Phys. Rev. B (2)

P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002).
[Crossref]

D.R. Smith, S. Schultz, P. Markos, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Phys. Rev. Lett. (3)

S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, and M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

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

A. Pimenov, A. Loidl, and P. Przyslupski, “Negative refraction in Ferromagnet-Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005).
[Crossref] [PubMed]

Phys. Status Sol.(b) (1)

R.H. Tarkhanyan, “On the Theory of Surface Waves in a Uniaxial Semiconductor Slab,” Phys. Status Sol.(b) 72, 111 (1975).
[Crossref]

Science (1)

R.A. Shelby, D.R. Smith, and S. Schultz., “Experimental Verification of a Negative Index of Refraction,” Science,  292, 77, 2001.
[Crossref] [PubMed]

Sol. St. Commun. (1)

V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985).
[Crossref]

Sov. Phys. Usp. (1)

V.G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. 10, 509 (1968)
[Crossref]

Other (2)

R.H. Tarkhanyan and A.G. Nassiopoulou, “Electromagnetic instability of surface waves in semiconductor superlattices,” J. Nanosci. Nanotech.3, 549 (2003), and “Influence of magnetic field on electromagnetic instabilities in semiconductor superlattices,” J. Nanosci. Nanotech4, 1085 (2004).
[Crossref]

R.H. Tarkhanyan and D.G. Niarchos, “Wave refraction and backward magnon-polaritons in left-handed antiferomagnetic/semiconductor superlattices,” JMMM (to be published).

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Figures (4)

Fig. 1.
Fig. 1. Geometry of the problem. A semiinfinite SL-medium is in the space region y>0; the space y<0 is taken to be a vacuum.
Fig. 2.
Fig. 2. Dispersion curves of TE-wave for the cases: a. ωp 1<ωr , b. ωr <ωp 1<ωs 3, c. <ωp 1>ωs 3.
Fig. 3.
Fig. 3. Refraction of TE-wave in the case of positive index (a) and negative index (b) LHM. k 0, k R and k are wave vectors for incident, reflected and refractive waves, respectively.
Fig. 4.
Fig. 4. Angle dependence of the reflection coefficient from the surface of LHM with positive (a) and negative (b) group refractive index.

Equations (42)

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ε ̂ = ( ε x x ε x y 0 ε x y ε x x 0 0 0 ε z z ) ,
ε x x = ε ( 1 ω p 2 ω 2 ω c 2 ) ,
ε x y = i ε ω c ω p 2 ω ( ω 2 ω c 2 ) , ε z z = ε ( 1 ω p 2 ω 2 ) ,
μ ̂ = ( μ x x μ x y 0 μ x y μ x x 0 0 0 μ z z ) ,
μ x x = 1 ω 0 ω s ω 2 ω 0 2 , μ x y = i ω ω s ω 2 ω 0 2 , μ z z = const ,
D = ε 0 ε ̂ e f E , B = μ 0 μ ̂ e f H ,
ε ̂ e f = ( ε 1 i ε a 0 i ε a ε 2 0 0 0 ε 3 ) ,
ε 1 = ε [ 1 ω p 1 2 ( ω 2 ω p 2 2 ) ω 2 ( ω 2 ω c 2 ω p 2 2 ) ] , ε a = ε ω c ω p 1 2 ω ( ω 2 ω c 2 ω p 2 2 ) ,
ε 2 = ε ( 1 ω p 1 2 ω 2 ω c 2 ω p 2 2 ) , ε 3 = ε ( 1 ω p 1 2 ω 2 ) ,
ω p 1 2 = l 1 d ω p 2 , ω p 2 2 = l 2 d ω p 2 ,
μ ̂ e f = ( μ 1 i μ α 0 i μ α μ 2 0 0 0 μ 3 ) ,
μ 1 = ω 2 ω σ 2 ω 2 ω 0 ( ω 0 + ω s 1 ) , μ a = ω ω s 2 ω 2 ω 0 ( ω 0 + ω s 1 ) ,
μ 2 = ω 2 ω 0 ( ω 0 + ω s ) ω 2 ω 0 ( ω 0 + ω s 1 ) , μ 3 = 1 d ( l 1 + l 2 μ z z ) ,
ω s 1 = l 1 d ω s , ω s 2 = l 2 d ω s . ω σ 2 = ω s 1 ω s 2 + ω 0 ( ω 0 + ω s ) .
rot H = D t , rot E = B t ,
ε 1 k x 2 + ε 2 k y 2 = μ 3 ( ε 1 ε 2 ε a 2 ) k 0 2 , k 0 ω c ,
H = { i μ a k x + μ 2 k y , μ 1 k x + i μ a k y , 0 } ( ω μ 0 μ 2 μ v ) 1 E ,
μ v μ 1 μ a 2 μ 2 = ω 2 ω s 3 2 ω 2 ω 0 ( ω 0 + ω s )
ω s 3 = γ ( H 0 + M 0 ) ( H 0 + M 0 l 2 d 1 )
μ 1 k x 2 + μ μ k y 2 = ε 3 μ 2 μ ν k 0 2 ,
min { ω r , ω p 1 } < ω < max { ω r , ω p 1 } , ω > ω s 3 ,
ω r < ω < ω s 3 , ω > ω p 1 ,
ω r = ( ω 0 2 + ω 0 ω s + ω s 1 ω s 2 sin 2 β ) 1 2
S = { μ 1 μ 2 k x , k y , 0 } E 2 2 ω μ 0 μ v .
k S = ( μ 1 k x 2 + μ 2 k y 2 ) E 2 2 ω μ 0 μ 2 μ ν = 1 2 ω ε 0 ε 3 E 2 ,
ε 3 < 0 , μ v < 0 , 1 + μ 1 μ 2 1 tan 2 β > 0 .
ω r < ω < min { ω s 3 , ω p 1 } ,
cos φ = ε 3 μ 2 μ v n p n g μ 1 ,
n p = c k ω = sin α sin β
n g = sin α sin γ
n p = [ ε 3 μ v + ( 1 μ 1 μ 2 ) sin 2 α ] 1 2 ,
n g = [ ε 3 μ v ( μ 2 μ 1 ) 2 + ( 1 μ 2 μ 1 ) sin 2 α ] 1 2 Sgn ( μ 1 μ v ) .
I . ω r < ω < ω σ
II . ω σ < ω < min { ω p 1 , ω s 3 } ,
sin α c = ε 3 μ 2 μ v μ 1 .
R = ( μ v cos α + n p 2 sin 2 α ) 2 + ( μ a μ 2 ) 2 sin 2 α ( μ v cos α n p 2 sin 2 α ) 2 + ( μ a μ 2 ) 2 sin 2 α ,
R m = BC A BC + A ,
tan 2 α m = μ 2 ( C ε 3 μ v B ) AB ,
A = ε 3 μ 2 μ v μ 1 , B = 1 ε 3 μ 2 , C = A μ v + ε 3 ( μ 1 μ v ) .
μ v ε 3 > 1 + 2 μ a 2 A μ 2
k y = k 0 C B 1 cos α m .
R 0 = ( ε 3 μ v ε 3 + μ v ) 2 ,

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