Abstract

The propagation of electromagnetic radiation in birefringent layered media is considered. A general formulation of the plane-wave propagation in an arbitrarily birefringent layered medium is presented. The concepts of dynamical matrix and propagation matrix are introduced. A 4 × 4 transfer matrix method is used to relate the field amplitudes in different layers. Our general theory is then applied to the special case of periodic birefringent layered media, especially the Šolc birefringent layered media [ I. Šolc, Cesk. Casopis Fẏs. 3, 366 ( 1953); Cesk. Casopis Fẏs. 10, 16 ( 1960)]. The unit cell translation operator is derived. The band structures as well as the Bloch waves are obtained by diagonalizing the translation operator. Coupled mode theory is extended to the case of birefringent periodic perturbation to explain the exchange Bragg scattering. A general mode dispersion relation for guided waves is also obtained in terms of the transfer matrix elements.

© 1979 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Electromagnetic propagation in periodic stratified media. I. General theory*

Pochi Yeh, Amnon Yariv, and Chi-Shain Hong
J. Opt. Soc. Am. 67(4) 423-438 (1977)

Forbidden gaps in periodic anisotropic layered media

E. Cojocaru
Appl. Opt. 39(25) 4641-4648 (2000)

References

  • View by:
  • |
  • |
  • |

  1. P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977);J. Opt. Soc. Am. 67, 438 (1977).
    [Crossref]
  2. A. Wunsche, Ann. Phys. (Leipz.) 25, 201 (1970).
    [Crossref]
  3. J. Schesser and G. Eichmann, J. Opt. Soc. Am. 62, 786 (1972).
    [Crossref]
  4. C. E. Curry, Electromagnetic Theory of Light, (MacMillan, London, 1950), pp. 356–369.
  5. H. Schopper, Z. Phsik 132, 146 (1952).
    [Crossref]
  6. A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifter 1, 27 (1955);Kgl. Norske Videnskab. Selskab, Skrifter 1, 37 (1955).
  7. A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94 (1962)[Opt. Spectrosc. (USSR) 14, 48 (1963)].
  8. J. Schesser and G. Eichmann, J. Opt. Soc. Am. 62, 786 (1972).
    [Crossref]
  9. D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 56, 1763 (1966).
    [Crossref]
  10. J. J. Stamnes and G. C. Sherman, J. Opt. Soc. Am. 67, 683 (1977).
    [Crossref]
  11. M. Born and E. Wolf, Principles of Optics, (Macmillan, New York, 1964).
  12. See for example, H. Goldstein, Classical Mechanics, (Addison-Wesley, Reading, 1965).
  13. See for example, J. M. Stone, Radiation and Optics, (McGraw-Hill, New York, 1963), Sec. 17–5.
  14. E. O. Ammann, Progress in Optics, IX, edited by E. Wolf (North-Holland, Amsterdam, 1971), pp. 125–177.
  15. This is a generalized Šolc layered medium with alternating layer thicknesses as well as azimuths.
  16. F. Bloch, Z. Phys. 52, 555 (1928).
  17. H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
    [Crossref]
  18. A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
    [Crossref]
  19. ∂2A/∂z2 ≪ k ∂A/∂z.
  20. This periodic function is the first term of the Fourier sines expansion of a unit alternating square wave perturbation.
  21. See, for example, A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 523.
  22. I. Šolc, J. Opt. Soc. Am. 55, 621 (1965).
    [Crossref]
  23. L. Brillouin, Wave Propagation and Group Velocity, (Academic, New York, 1960).
  24. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  25. J. W. Evans, J. Opt. Soc. Am. 48, 142 (1958).
    [Crossref]
  26. I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [Crossref]
  27. R. V. Schmidt and I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
    [Crossref]
  28. S. C. Frautschi, Regge Poles and S-Matrix Theory, (Benjamin, New York, 1963).
  29. T. Regge, Nuov. Cim. 14, 951 (1959).
    [Crossref]
  30. M. J. Lighthill, J. Inst. Maths. Applic. 1, 1 (1965).
    [Crossref]

1977 (2)

1976 (1)

H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
[Crossref]

1974 (1)

R. V. Schmidt and I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[Crossref]

1973 (2)

A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
[Crossref]

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

1972 (2)

1970 (1)

A. Wunsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

1966 (1)

1965 (2)

I. Šolc, J. Opt. Soc. Am. 55, 621 (1965).
[Crossref]

M. J. Lighthill, J. Inst. Maths. Applic. 1, 1 (1965).
[Crossref]

1962 (1)

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94 (1962)[Opt. Spectrosc. (USSR) 14, 48 (1963)].

1959 (1)

T. Regge, Nuov. Cim. 14, 951 (1959).
[Crossref]

1958 (1)

1955 (1)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifter 1, 27 (1955);Kgl. Norske Videnskab. Selskab, Skrifter 1, 37 (1955).

1952 (1)

H. Schopper, Z. Phsik 132, 146 (1952).
[Crossref]

1941 (1)

1928 (1)

F. Bloch, Z. Phys. 52, 555 (1928).

Ammann, E. O.

E. O. Ammann, Progress in Optics, IX, edited by E. Wolf (North-Holland, Amsterdam, 1971), pp. 125–177.

Bloch, F.

F. Bloch, Z. Phys. 52, 555 (1928).

Born, M.

M. Born and E. Wolf, Principles of Optics, (Macmillan, New York, 1964).

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity, (Academic, New York, 1960).

Carruthers, J. R.

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

Curry, C. E.

C. E. Curry, Electromagnetic Theory of Light, (MacMillan, London, 1950), pp. 356–369.

Eichmann, G.

Evans, J. W.

Federov, F. J.

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94 (1962)[Opt. Spectrosc. (USSR) 14, 48 (1963)].

Feucht, D. L.

Frautschi, S. C.

S. C. Frautschi, Regge Poles and S-Matrix Theory, (Benjamin, New York, 1963).

Goldstein, H.

See for example, H. Goldstein, Classical Mechanics, (Addison-Wesley, Reading, 1965).

Goncharenko, A. M.

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94 (1962)[Opt. Spectrosc. (USSR) 14, 48 (1963)].

Holmes, D. A.

Hong, C. S.

Jones, R. C.

Kaminow, I. P.

R. V. Schmidt and I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[Crossref]

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

Kogelnik, H.

H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
[Crossref]

Lighthill, M. J.

M. J. Lighthill, J. Inst. Maths. Applic. 1, 1 (1965).
[Crossref]

Regge, T.

T. Regge, Nuov. Cim. 14, 951 (1959).
[Crossref]

Schesser, J.

Schmidt, R. V.

R. V. Schmidt and I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[Crossref]

Schopper, H.

H. Schopper, Z. Phsik 132, 146 (1952).
[Crossref]

Sherman, G. C.

Šolc, I.

Stamnes, J. J.

Stone, J. M.

See for example, J. M. Stone, Radiation and Optics, (McGraw-Hill, New York, 1963), Sec. 17–5.

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifter 1, 27 (1955);Kgl. Norske Videnskab. Selskab, Skrifter 1, 37 (1955).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Macmillan, New York, 1964).

Wunsche, A.

A. Wunsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

Yariv, A.

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977);J. Opt. Soc. Am. 67, 438 (1977).
[Crossref]

A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
[Crossref]

See, for example, A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 523.

Yeh, P.

Ann. Phys. (Leipz.) (1)

A. Wunsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

Appl. Phys. Lett. (2)

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

R. V. Schmidt and I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[Crossref]

Bell. Syst. Tech. J. (1)

H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
[Crossref]

IEEE J. Quant. Electron. (1)

A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
[Crossref]

J. Inst. Maths. Applic. (1)

M. J. Lighthill, J. Inst. Maths. Applic. 1, 1 (1965).
[Crossref]

J. Opt. Soc. Am. (8)

Kgl. Norske Videnskab. Selskab, Skrifter (1)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifter 1, 27 (1955);Kgl. Norske Videnskab. Selskab, Skrifter 1, 37 (1955).

Nuov. Cim. (1)

T. Regge, Nuov. Cim. 14, 951 (1959).
[Crossref]

Opt. Spectrosk. (1)

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94 (1962)[Opt. Spectrosc. (USSR) 14, 48 (1963)].

Z. Phsik (1)

H. Schopper, Z. Phsik 132, 146 (1952).
[Crossref]

Z. Phys. (1)

F. Bloch, Z. Phys. 52, 555 (1928).

Other (11)

L. Brillouin, Wave Propagation and Group Velocity, (Academic, New York, 1960).

S. C. Frautschi, Regge Poles and S-Matrix Theory, (Benjamin, New York, 1963).

C. E. Curry, Electromagnetic Theory of Light, (MacMillan, London, 1950), pp. 356–369.

∂2A/∂z2 ≪ k ∂A/∂z.

This periodic function is the first term of the Fourier sines expansion of a unit alternating square wave perturbation.

See, for example, A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 523.

M. Born and E. Wolf, Principles of Optics, (Macmillan, New York, 1964).

See for example, H. Goldstein, Classical Mechanics, (Addison-Wesley, Reading, 1965).

See for example, J. M. Stone, Radiation and Optics, (McGraw-Hill, New York, 1963), Sec. 17–5.

E. O. Ammann, Progress in Optics, IX, edited by E. Wolf (North-Holland, Amsterdam, 1971), pp. 125–177.

This is a generalized Šolc layered medium with alternating layer thicknesses as well as azimuths.

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 (8)

FIG. 1
FIG. 1

Graphic method to determine the propagation constants from the normal surface.

FIG. 2
FIG. 2

Diagram representation of the matrix method.

FIG. 3
FIG. 3

Geometry of a periodic birefringent layered medium.

FIG. 4
FIG. 4

Band struclure of a Šolc layered medium with a = 0.6Λ, b = 0.4Λ. The dotted curves are the imaginary parts of the Bloch wave numbers. The dispersion curves consist of a fast wave branch (solid curves without dots) and a slow wave branch (solid waves with dots).

FIG. 5
FIG. 5

Band structure of a Šolc layered medium with a = b = 0.5Λ. The dotted curves are the imaginary parts of the Bloch wave numbers. The dispersion curves consist of a fast wave branch (solid curves without dots) and an slow wave branch (solid curves with dots). This is a degenerate case when the direct bandgaps at KΛ = π disappear.

FIG. 6
FIG. 6

Coherent reflection spectrum of a typical Šolc layered structure. ω+ = 2π/(ns + nf)Λ.

FIG. 7
FIG. 7

Coherent transmission spectrum of a typical Šolc layered structure. ω = 2π/(ns + nf)Λ.

FIG. 8
FIG. 8

Transmission of a Šolc filter. This figure compares coupled mode theory (dashed curve) and Jones calculus (dotted curve) with the exact 4 × 4 matrix method (solid curve). The horizontal axis is the phase mismatch factor ΔK = (k1k2) − 2π/Λ.

Tables (1)

Tables Icon

TABLE 1 A Sŏlc layered medium

Equations (146)

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

= A ( 1 0 0 0 2 0 0 0 3 ) A 1
A = ( cos ψ cos ϕ cos θ sin ϕ sin ψ sin ψ cos ϕ cos θ sin ϕ cos ψ sin θ sin ϕ cos ψ sin ϕ + cos θ cos ϕ sin ψ sin ψ sin ϕ + cos θ cos ϕ cos ψ sin θ cos ϕ sin θ sin ψ sin θ cos ψ cos θ ) .
k × ( k × E ) + ω 2 μ E = 0 ,
( ω 2 μ x x β 2 γ 2 ω 2 μ x y + α β ω 2 μ x z + α γ ω 2 μ y x + α β ω 2 μ y y α 2 γ 2 ω 2 μ y z + β γ ω 2 μ z x + α γ ω 2 μ z y + β γ ω 2 μ z z α 2 β 2 ) × ( E x E y E z ) = 0 .
p ̂ = N σ ( ( ω 2 μ y y α 2 γ σ 2 ) ( ω 2 μ z z α 2 β 2 ) ( ω 2 μ y z + β γ σ ) 2 ( ω 2 μ y z + β γ σ ) ( ω 2 μ x z + α γ σ ) ( ω 2 μ x y + α β ) ( ω 2 μ z z α 2 β 2 ) ( ω 2 μ x y + α β ) ( ω 2 μ y z + β γ σ ) ( ω 2 μ x z + α γ σ ) ( ω 2 μ y y α 2 γ σ 2 ) )
E = σ = 1 4 A σ p ̂ σ e i ( α x + β y + γ σ z ω t ) .
= { ( 0 ) z < z 0 ( 1 ) z 0 < z < z 1 ( 2 ) z 1 < z < z 2 ( N ) z N 1 < z < z N ( s ) z N < z .
E = σ = 1 4 A σ ( n ) p ̂ σ ( n ) e i [ α x + β y + γ σ ( n ) ( z z n ) ω t ] .
H = σ = 1 4 A σ ( n ) q ̂ σ ( n ) e i [ α x + β y + γ σ ( n ) ( z z n ) ω t ] ,
q ̂ σ ( n ) = c k σ ( n ) ω μ × p ̂ σ ( n )
k σ ( n ) = α x ̂ + β ŷ + γ σ ( n ) .
σ = 1 4 A σ ( n 1 ) p ̂ σ ( n 1 ) x ̂ = σ = 1 4 A σ ( n ) p ̂ σ ( n ) x ̂ e i γ σ ( n ) t n ,
σ = 1 4 A σ ( n 1 ) p ̂ σ ( n 1 ) ŷ = σ = 1 4 A σ ( n ) p ̂ σ ( n ) ŷ e i γ σ ( n ) t n ,
σ = 1 4 A σ ( n 1 ) q ̂ σ ( n 1 ) x ̂ = σ = 1 4 A σ ( n ) q ̂ σ ( n ) x ̂ e i γ σ ( n ) t n ,
σ = 1 4 A σ ( n 1 ) q ̂ σ ( n 1 ) ŷ = σ = 1 4 A σ ( n ) q ̂ σ ( n ) ŷ e i γ σ ( n ) t n ,
( A 1 ( n 1 ) A 2 ( n 1 ) A 3 ( n 1 ) A 4 ( n 1 ) ) = D 1 ( n 1 ) D ( n ) P ( n ) ( A 1 ( n ) A 2 ( n ) A 3 ( n ) A 4 ( n ) ) ,
D ( n ) = ( x ̂ p ̂ 1 ( n ) x ̂ p ̂ 2 ( n ) x ̂ p ̂ 3 ( n ) x ̂ p ̂ 4 ( n ) ŷ q ̂ 1 ( n ) ŷ q ̂ 2 ( n ) ŷ q ̂ 3 ( n ) ŷ q ̂ 4 ( n ) ŷ p ̂ 1 ( n ) ŷ p ̂ 2 ( n ) ŷ p ̂ 3 ( n ) ŷ p ̂ 4 ( n ) x ̂ q ̂ 1 ( n ) x ̂ q ̂ 2 ( n ) x ̂ q ̂ 3 ( n ) x ̂ q ̂ 4 ( n ) )
P ( n ) = ( e i γ 1 ( n ) t n 0 0 0 0 e i γ 2 ( n ) t n 0 0 0 0 e i γ 3 ( n ) t n 0 0 0 0 e i γ 4 ( n ) t n ) .
T n 1 , n = D 1 ( n 1 ) D ( n ) P ( n ) .
( A 1 ( n 1 ) A 2 ( n 1 ) A 3 ( n 1 ) A 4 ( n 1 ) ) = T n 1 , n ( A 1 ( n ) A 2 ( n ) A 3 ( n ) A 4 ( n ) ) .
( A 1 ( 0 ) A 2 ( 0 ) A 3 ( 0 ) A 4 ( 0 ) ) = T 0 , 1 T 1 , 2 T 2 , 3 T N 1 , N T N , s ( A 1 ( s ) A 2 ( s ) A 3 ( s ) A 3 ( s ) ) .
( z ) = { a a < z < 0 b 0 < z < b
( z + Λ ) = ( z ) ,
E = { σ = 1 4 B σ ( n ) p ̂ σ ( b ) e i [ α x + β y + γ σ ( b ) ( z + a n Λ ) ω t ] ( n 1 ) Λ < z < n Λ a , σ = 1 4 A σ ( n ) p ̂ σ ( a ) e i [ α x + β y + γ σ ( a ) ( z n Λ ) ω t ] n Λ a < z < n Λ ,
T = D a 1 D b P b D b 1 D a P a ,
γ 1 ( a ) = γ 1 ( b ) = ( 2 π / λ ) n s , γ 2 ( a ) = γ 2 ( b ) = ( 2 π / λ ) n s , γ 3 ( a ) = γ 3 ( b ) = ( 2 π / λ ) n f , γ 4 ( a ) = γ 4 ( b ) = ( 2 π / λ ) n f ,
p ̂ 1 ( a ) = p ̂ 2 ( a ) = ( cos ρ , sin ρ , 0 ) , p ̂ 3 ( a ) = p ̂ 4 ( a ) = ( sin ρ , cos ρ , 0 ) , q ̂ 1 ( a ) = q ̂ 2 ( a ) = ( n s sin ρ , n s cos ρ , 0 ) , q ̂ 3 ( a ) = q ̂ 4 ( a ) = ( n f cos ρ , n f sin ρ , 0 ) ,
p ̂ 1 ( b ) = p ̂ 2 ( b ) = ( cos ρ , sin ρ , 0 ) , p ̂ 3 ( b ) = p ̂ 4 ( b ) = ( sin ρ , cos ρ , 0 ) , q ̂ 1 ( b ) = q ̂ 2 ( b ) = ( n 2 sin ρ , n s cos ρ , 0 ) , q ̂ 3 ( b ) = q ̂ 4 ( b ) = ( n f cos ρ , n f sin ρ , 0 ) ,
D ( n ) = ( cos ρ cos ρ sin ρ sin ρ n s cos ρ n s cos ρ n f sin ρ n f sin ρ sin ρ sin ρ cos ρ cos ρ n s sin ρ n s sin ρ n f cos ρ n f cos ρ )
P ( t ) = ( e i ( 2 π / λ ) n s t 0 0 0 0 e i ( 2 π / λ ) n s t 0 0 0 0 e i ( 2 π / λ ) n f t 0 0 0 0 e i ( 2 π / λ ) n f t ) ,
T = D 1 ( ρ ) D ( ρ ) P ( b ) D 1 ( ρ ) D ( ρ ) P ( a ) .
T 11 = e i k s a { e i k s b cos 2 Δ + [ cos k f b ( i / 2 ) ( e + 1 / e ) sin k f b ] sin 2 Δ } , T 12 = e i k s a [ ( i / 2 ) ( e 1 / e ) sin k f b ] sin 2 Δ , T 13 = e i k f a [ ( 1 / 2 ) ( 1 + e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 14 = e i k f a [ ( 1 / 2 ) ( 1 e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 21 = e i k s a [ ( i / 2 ) ( e 1 / e ) sin k f b ] sin 2 Δ , T 22 = e i k s a { e i k s b cos 2 Δ + [ cos k f b + ( i / 2 ) ( e + 1 / e ) sin k f b ] sin 2 Δ } , T 23 = e i k f a [ ( 1 / 2 ) ( 1 e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 24 = e i k f a [ ( 1 / 2 ) ( 1 + e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 31 = e i k s a [ ( 1 / 2 ) ( 1 + 1 / e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 32 = e i k s a [ ( 1 / 2 ) ( 1 1 / e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 33 = e i k f a { e i k f b cos 2 Δ + [ cos k s b ( i / 2 ) ( e + 1 / e ) sin k s b ] sin 2 Δ } , T 34 = e i k f a [ ( i / 2 ) ( e 1 / e ) sin k s b ] sin 2 Δ , T 41 = e i k s a [ ( 1 / 2 ) ( 1 1 / e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 42 = e i k s a [ ( 1 / 2 ) ( 1 + 1 / e ) ( e i k f b e i k s b ) ] sin Δ cos Δ , T 43 = e i k f a [ ( i / 2 ) ( e 1 / e ) sin k s b ] sin 2 Δ , T 44 = e i k f a { e i k f b cos 2 Δ + [ cos k s b + ( i / 2 ) ( e + 1 / e ) sin k s b ] sin 2 Δ } ,
k s = ( 2 π / λ ) n s , k f = ( 2 π / λ ) n f ,
Δ = ρ ρ ,
e = n f / n s .
T 21 = T 12 * T 22 = T 11 * T 23 = T 14 * T 24 = T 13 * T 41 = T 32 * T 42 = T 31 * T 43 = T 34 * T 44 = T 33 * .
T 31 = T 13 ( f s ) T 32 = T 14 ( f s ) T 33 = T 11 ( f s ) T 34 = T 12 ( f s ) T 41 = T 23 ( f s ) T 42 = T 24 ( f s ) T 43 = T 21 ( f s ) T 44 = T 22 ( f s ) ,
T E ( z ) = E ( T 1 z ) = E ( z l Λ ) .
E K ( x , y , z , t ) = E K ( z ) e i K z e i ( α x + β y ω t ) ,
E K ( z + Λ ) = E K ( z ) .
A σ ( n ) = e i K Λ A σ ( n 1 ) , σ = 1 , 2 , 3 , 4 .
T ρ σ A σ ( n ) = e i K Λ A ρ ( n ) ,
det | T ρ σ ξ δ ρ σ | = 0 .
ω = ω ( α , β , K ) .
ω ( α , β , K ) = ω ( α , β , K ) .
ω ( α , β , K ) = ω ( α , β , K ) .
K 1 Λ = m π , K 2 Λ = l π , K 1 Λ = K 2 Λ = 2 n π ,
= 0 + Δ ,
= 0 ( n 1 2 0 0 0 n 2 2 0 0 0 n 3 2 ) ,
{ 0 ( σ κ 0 κ σ 0 0 0 0 ) , a < z < 0 Δ = 0 ( σ κ 0 κ σ 0 0 0 0 ) , 0 < z < b ,
σ = ( n 2 2 n 1 2 ) sin 2 Δ / 2 ,
κ = ½ ( n 2 2 n 1 2 ) sin Δ .
Δ = ( σ κ 0 κ σ 0 0 0 0 ) P ( z ) ,
P ( z ) = ( 4 0 / π ) cos [ ( 2 π / Λ ) z ] .
E ( z , t ) = [ A ( z ) e i k 1 z + B ( z ) e i k 1 z ] x ̂ + [ C ( z ) e i k 2 z + D ( z ) e i k 2 z ] ŷ ,
2 z 2 E + ω 2 μ 0 E + ω 2 μ Δ E = 0 .
2 i k 1 ( d A d z e i k 1 z d B d z e i k 1 z ) + ω 2 μ σ ( A e i k 1 z + B e i k 1 z ) P + ω 2 μ κ ( C e i k 2 z + B e i k 2 z ) P = 0 2 i k 2 ( d C d z e i k 2 z d D d z e i k 2 z ) ω 2 μ σ ( C e i k 2 z + D e i k 2 z ) P + ω 2 μ κ ( A e i k 1 z + B e i k 1 z ) P = 0 .
P ( z ) = 2 0 π ( e i ( 2 π / Λ ) z + e i ( 2 π / Λ ) z ) .
d A d z = i ω 2 μ 0 π k 1 σ B , d B d z = i ω 2 μ 0 π k 1 σ A ,
d C d z = i ω 2 μ 0 π k 2 σ D , d D d z = i ω 2 μ 0 π k 2 σ C ,
d A d z = i ω 2 μ 0 π k 1 κ D , d D d z = i ω 2 μ 0 π k 2 κ A .
d B d z = i ω 2 μ 0 π k 1 κ C , d C d z = i ω 2 μ 0 π k 2 B ,
d A d z = i ω 2 μ 0 π k 1 κ C , d C d z = i ω 2 μ 0 π k 2 κ A .
d B d z = i ω 2 μ 0 π k 1 κ D , d D d z = i ω 2 μ 0 π k 2 κ B ,
A ( o ) = 1 ,
C ( o ) = 0 .
A ( z ) = cos { [ ω 2 μ 0 / π ( k 1 k 2 ) 1 / 2 ] κ z } ,
C ( z ) = i ( k 1 / k 2 ) 1 / 2 sin { [ ω 2 μ 0 / π ( k 1 k 2 ) 1 / 2 ] κ z } .
[ ω 2 μ 0 / π ( k 1 k 2 ) 1 / 2 ] κ N Λ = π / 2 ,
[ ( n 2 2 n 1 2 ) / π ( n 1 n 2 ) 1 / 2 ] N Λ sin Δ = λ / 2 .
( 2 π / λ ) ( n 2 n 1 ) = 2 π / Λ
N Δ = π / 4 .
E K ( z ) = n e K ( n ) e i n ( 2 π / Λ ) z .
E = n e K ( n ) e i [ K + n ( 2 π / Λ ) ] z e i ( α x + β y ω t ) ,
| e K ( o ) | | e K ( n ) |
1 Λ o Λ E K ( z ) d z E K
V p = ω / ( K 2 + α 2 + β 2 ) 1 / 2 ,
E = E K e i ( α x + β y + K z ω t ) .
V g = ( ω α ) β , K x ̂ + ( ω β ) α , K ŷ + ( ω K ) α , β .
V e = 1 Λ o Λ ( Poynting vector ) d z 1 Λ o Λ ( energy density ) d z
r = ( r x x r x y r y x r y y ) ,
t = ( t x x t x y t y x t y y ) .
E = [ A ( o ) e i k 1 z + B ( o ) e i k 1 z ] x ̂ + [ C ( o ) e i k 2 z + D ( o ) e i k 2 z ] ŷ , z < 0 ,
E = [ A ( s ) e i k 1 ( z L ) + B ( s ) e i k 1 ( z L ) ] x ̂ + [ C ( s ) e i k 2 ( z L ) + D ( s ) e i k 2 ( z L ) ] y , L < z ,
( A ( o ) B ( o ) C ( o ) D ( o ) ) = ( M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ) ( A ( s ) B ( s ) C ( s ) D ( s ) ) .
B ( s ) = D ( s ) = 0 .
r x x = ( B ( o ) A ( o ) ) C ( o ) = 0 = M 21 M 33 M 23 M 31 M 11 M 33 M 13 M 31
r x y = ( D ( o ) A ( o ) ) C ( o ) = 0 = M 41 M 33 M 43 M 31 M 11 M 33 M 13 M 31
r y x = ( B ( o ) C ( o ) ) A ( o ) = 0 = M 11 M 23 M 21 M 13 M 11 M 33 M 13 M 31
r y y = ( D ( o ) C ( o ) ) A ( o ) = 0 = M 11 M 43 M 41 M 13 M 11 M 33 M 13 M 31
t x x = ( A ( s ) A ( o ) ) C ( o ) = 0 = M 33 M 11 M 33 M 13 M 31
t x y = ( C ( s ) A ( o ) ) C ( o ) = 0 = M 31 M 11 M 33 M 13 M 31
t y x = ( A ( s ) C ( o ) ) A ( o ) = 0 = M 13 M 11 M 33 M 13 M 31
t y y = ( C ( s ) C ( o ) ) A ( o ) = 0 = M 11 M 11 M 33 M 13 M 31 ,
M 11 M 33 M 13 M 31 = 0 .
ω = ω ( α , β )
ω = ω θ p ( β p )
β p = ( α 2 , β 2 ) 1 / 2 ,
θ p = tan 1 ( β / α ) .
( n 0 2 0 0 0 n 0 2 0 0 0 n e 2 ) .
n = ( 0 , cos ϕ , sin ϕ )
( n 0 2 0 0 0 n 0 2 cos 2 ϕ + n e 2 sin 2 ϕ ( n e 2 n 0 2 ) sin ϕ cos ϕ 0 ( n e 2 n 0 2 ) sin ϕ cos ϕ n 0 2 sin 2 ϕ + n e 2 cos 2 ϕ ) .
k = ( 0 , β , γ ) ,
| n o 2 β 2 γ 2 0 0 , 0 n y 2 γ 2 n y z 2 + β γ 0 n y z 2 + β γ 2 n z 2 β 2 | = 0 ,
ω / c = 1
n y 2 = n o 2 cos 2 ϕ + n e 2 sin 2 ϕ
n z 2 = n o 2 sin 2 ϕ + n e 2 cos 2 ϕ
n y z 2 = ( n e 2 n o 2 ) sin ϕ cos ϕ ,
γ 0 = ± ( n o 2 β 2 ) 1 / 2 ,
γ e = β ( n y z 2 / n z 2 ) ± ( n o n e / n z 2 ) ( n z 2 β 2 ) 1 / 2 ,
p ̂ o = ( 1 0 0 ) ,
p ̂ e = ( 0 β 2 n z 2 β γ e + n y z 2 ) ,
E = A p ̂ e e i ( β y + γ z ω t ) .
H = A c k ω μ × p ̂ e e i ( β y + γ z ω t ) .
S = | A | 2 2 Re [ p ̂ e × ( c k ω μ × p ̂ e ) * e i ( γ γ * ) z ] .
S z = | A | 2 2 Re [ ( n z 2 β 2 ) ( β n y z 2 + γ * n z 2 ) e i ( γ γ * ) z ] .
S z = | A | 2 2 Re [ ± ( n z 2 β 2 ) 3 / 2 ( n o n e / n z 2 ) e i ( γ γ * ) z ] .
i j ( x ) = i j ( x + a ) , μ i j ( x ) = μ i j ( x + a ) ,
× H = i ω E ,
× E = i ω μ H ,
E = E k ( x ) e iK x ,
H = H k ( x ) e iK x ,
E K ( x ) = E K ( x + a ) , H K ( x ) = H K ( x + a ) .
ω = ω ( K ) .
S = ( 1 / 2 ) Re [ E × H * ] .
U = ( 1 / 4 ) [ E E * + H μ H * ] .
V e = ( 1 / V ) S d 3 x ( 1 / V ) U d 3 x S U ,
V e = ( 1 / 2 ) Re [ E k × E k * ] ( 1 / 4 ) ( E k E k * + H k μ H k * ) ,
V g = K ω = ω K ,
× H K + i K × H K = i ω E K
× E K + i K × E K = i ω μ H K .
× δ H K + i δ K × H K + i K × δ H K = i E K δ ω i ω δ E K ,
× δ E K + i δ K × E K + i K × δ E K = i μ H K δ ω + i ω μ δ H K .
H K * × δ E K + E K × δ H K * + i δ K ( E K × H K * ) + i δ K ( E K × H K * ) + i K ( δ E K × H K * ) + i K ( E K × δ H K * ) = i ω [ H K * μ δ H K + E K δ E K * ] + i δ ω [ H K * μ H K + E K δ E K * ] .
H K * μ δ H K = δ H K * μ H K * ,
E K δ E K * = δ E K * E K .
i ω δ E K * E K = δ E K * × H K i K ( H K × δ E K * ) ,
i ω δ H K μ H K * = δ H K × E k * + i k ( E k * × δ H K ) .
H K * × δ E K + E K × δ H K * + δ E K * × H K + δ H K × E K * × 2 i δ K ( E K × H K * ) + i K ( δ E K × H K * + H K × δ E K * + E K × δ H K * + δ H K × E K * ) = i δ ω [ H K * μ H K + E K * E K ] .
F + 2 i δ K 2 Re [ E K × H K * ] = 2 i δ ω [ H K μ H K * + E K E K * ] ,
F = δ E K × H K * + δ H K * × E K + H K × δ E K * + E K * × δ H K .
F + 4 i δ K Re ( E K × H K * ) = 2 i δ ω H K μ H K * + E K E K * .
F = F d 3 x = 0 .
δ ω = V e δ K .
δ ω = ( K ω ) δ K = V g δ K .
V e = V g .