Abstract

Light propagation along the helical axis of cholesteric liquid crystals, whose structure has been distorted by a magnetic or electric field perpendicular to the helix axis, is theoretically investigated. The solutions show several reflection bands whose centers are given by the Bragg condition mλm = 2S n (m is an integer, S is the period of the distorted structure, and n is the average refractive index of the material). The bands with m ≥ 2 consist of three subbands, each characterized by the dependence of the reflection on the polarization of the incident beam. Thus, for example, an incident beam linearly polarized in the direction of the distorting field will be reflected at only two of these subbands. Except for very strong applied fields, the band m = 1 is composed of two subbands only. Outside the reflection bands, the modes of propagation are orthogonal linear polarizations.

© 1974 Optical Society of America

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References

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  1. C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
    [Crossref]
  2. Hl. de Vries, Acta Cryst. 4, 219 (1951).
    [Crossref]
  3. I. G. Chistyakov, Soviet Phys. Uspekhi 9, 551 (1967).
    [Crossref]
  4. G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968).
    [Crossref]
  5. L. Melamed and D. Rubin, Appl. Opt. 10, 231 (1971).
    [Crossref]
  6. D. W. Berreman and T. J. Scheffer, Mol. Cryst. Liq. Cryst. 11, 395 (1970).
    [Crossref]
  7. R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971).
    [Crossref]
  8. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 28.
  9. P. G. De Gennes, Solid State Commun. 6, 163 (1968).
    [Crossref]
  10. R. B. Meyer, Appl. Phys. Lett. 14, 208 (1969).
    [Crossref]
  11. G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
    [Crossref]
  12. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953), p. 139.
  13. IBM System 360, Scientific Subroutine Package, IBM Publication No. H20-0205-3 (1968), p. 275.
  14. F. J. Kahn, Appl. Phys. Lett. 18, 231 (1971).
    [Crossref]
  15. C. Kittel, Introduction to Solid State Physics, 3rd ed. (Wiley, New York, 1967), Ch. 9.
  16. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
    [Crossref]
  17. Reference 15, p. 255.
  18. Reference 15, p. 260.
  19. The matrix Al, Eq. (5), is used to compute the reflection coefficients from a finite slab whose thickness is lS(S is the period), by a method described in Ref. 8, p. 59.
  20. After completion of this work, we noticed that light transmission through a finite slab of distorted cholesteric liquid crystal was also calculated by S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972).
    [Crossref]
  21. L. I. Schiff, Quantum Mechanics (McGraw–Hill, New York, 1968), p. 269.
  22. C. Maugin, Bull. Soc. Franc. Miner. 34, 17 (1911).
  23. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1252 (1972).
    [Crossref]

1972 (2)

After completion of this work, we noticed that light transmission through a finite slab of distorted cholesteric liquid crystal was also calculated by S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1252 (1972).
[Crossref]

1971 (3)

F. J. Kahn, Appl. Phys. Lett. 18, 231 (1971).
[Crossref]

L. Melamed and D. Rubin, Appl. Opt. 10, 231 (1971).
[Crossref]

R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971).
[Crossref]

1970 (1)

D. W. Berreman and T. J. Scheffer, Mol. Cryst. Liq. Cryst. 11, 395 (1970).
[Crossref]

1969 (3)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

R. B. Meyer, Appl. Phys. Lett. 14, 208 (1969).
[Crossref]

G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
[Crossref]

1968 (3)

IBM System 360, Scientific Subroutine Package, IBM Publication No. H20-0205-3 (1968), p. 275.

P. G. De Gennes, Solid State Commun. 6, 163 (1968).
[Crossref]

G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968).
[Crossref]

1967 (1)

I. G. Chistyakov, Soviet Phys. Uspekhi 9, 551 (1967).
[Crossref]

1951 (1)

Hl. de Vries, Acta Cryst. 4, 219 (1951).
[Crossref]

1933 (1)

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[Crossref]

1911 (1)

C. Maugin, Bull. Soc. Franc. Miner. 34, 17 (1911).

Azzam, R. M. A.

Bashara, N. M.

Berreman, D. W.

D. W. Berreman and T. J. Scheffer, Mol. Cryst. Liq. Cryst. 11, 395 (1970).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 28.

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953), p. 139.

Cheung, L.

After completion of this work, we noticed that light transmission through a finite slab of distorted cholesteric liquid crystal was also calculated by S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972).
[Crossref]

Chistyakov, I. G.

I. G. Chistyakov, Soviet Phys. Uspekhi 9, 551 (1967).
[Crossref]

Chou, S. C.

After completion of this work, we noticed that light transmission through a finite slab of distorted cholesteric liquid crystal was also calculated by S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972).
[Crossref]

Conners, G. H.

De Gennes, P. G.

P. G. De Gennes, Solid State Commun. 6, 163 (1968).
[Crossref]

de Vries, Hl.

Hl. de Vries, Acta Cryst. 4, 219 (1951).
[Crossref]

Dreher, R.

R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971).
[Crossref]

Durand, G.

G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
[Crossref]

Kahn, F. J.

F. J. Kahn, Appl. Phys. Lett. 18, 231 (1971).
[Crossref]

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 3rd ed. (Wiley, New York, 1967), Ch. 9.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Leger, L.

G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
[Crossref]

Maugin, C.

C. Maugin, Bull. Soc. Franc. Miner. 34, 17 (1911).

Meier, G.

R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971).
[Crossref]

Melamed, L.

L. Melamed and D. Rubin, Appl. Opt. 10, 231 (1971).
[Crossref]

Meyer, R. B.

After completion of this work, we noticed that light transmission through a finite slab of distorted cholesteric liquid crystal was also calculated by S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972).
[Crossref]

R. B. Meyer, Appl. Phys. Lett. 14, 208 (1969).
[Crossref]

Oseen, C. W.

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[Crossref]

Rondelez, F.

G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
[Crossref]

Rubin, D.

L. Melamed and D. Rubin, Appl. Opt. 10, 231 (1971).
[Crossref]

Saupe, A.

R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971).
[Crossref]

Scheffer, T. J.

D. W. Berreman and T. J. Scheffer, Mol. Cryst. Liq. Cryst. 11, 395 (1970).
[Crossref]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw–Hill, New York, 1968), p. 269.

Veyssie, M.

G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 28.

Acta Cryst. (1)

Hl. de Vries, Acta Cryst. 4, 219 (1951).
[Crossref]

Appl. Opt. (1)

L. Melamed and D. Rubin, Appl. Opt. 10, 231 (1971).
[Crossref]

Appl. Phys. Lett. (2)

R. B. Meyer, Appl. Phys. Lett. 14, 208 (1969).
[Crossref]

F. J. Kahn, Appl. Phys. Lett. 18, 231 (1971).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Bull. Soc. Franc. Miner. (1)

C. Maugin, Bull. Soc. Franc. Miner. 34, 17 (1911).

IBM System 360, Scientific Subroutine Package (1)

IBM System 360, Scientific Subroutine Package, IBM Publication No. H20-0205-3 (1968), p. 275.

J. Opt. Soc. Am. (2)

Mol. Cryst. Liq. Cryst. (2)

D. W. Berreman and T. J. Scheffer, Mol. Cryst. Liq. Cryst. 11, 395 (1970).
[Crossref]

R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971).
[Crossref]

Phys. Rev. Lett. (1)

G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969).
[Crossref]

Solid State Commun. (2)

P. G. De Gennes, Solid State Commun. 6, 163 (1968).
[Crossref]

After completion of this work, we noticed that light transmission through a finite slab of distorted cholesteric liquid crystal was also calculated by S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972).
[Crossref]

Soviet Phys. Uspekhi (1)

I. G. Chistyakov, Soviet Phys. Uspekhi 9, 551 (1967).
[Crossref]

Trans. Faraday Soc. (1)

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[Crossref]

Other (7)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 28.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953), p. 139.

Reference 15, p. 255.

Reference 15, p. 260.

The matrix Al, Eq. (5), is used to compute the reflection coefficients from a finite slab whose thickness is lS(S is the period), by a method described in Ref. 8, p. 59.

L. I. Schiff, Quantum Mechanics (McGraw–Hill, New York, 1968), p. 269.

C. Kittel, Introduction to Solid State Physics, 3rd ed. (Wiley, New York, 1967), Ch. 9.

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Figures (14)

Fig. 1
Fig. 1

Definition of the angle ϕ between the xyz space-fixed coordinate system and the principal axes of the dielectric tensor of molecules at a distance z from the origin. (After Ref. 23.)

Fig. 2
Fig. 2

(a) Angle θ(ζ) = ϕ() that specifies the distorted structure for various values of applied field H in the y direction. (b) Dependence of the period S on the field. S0 is the period when H = 0.

Fig. 3
Fig. 3

Dispersion relations near ωs = π (m = 1) as obtained from the numerical solution of Sec. II for H = 0.9Hc, δ = 0.05. The solid curves describe ωs vs Re[K] (the real part of K). The dashed curves describe the attenuation (reflection) coefficients μ = Im[K] (the imaginary part of K) in the bands. Outside the bands μ = 0. λ′ = π/ωs.

Fig. 4
Fig. 4

Dispersion relations (numerical solution, Sec. II) near ωs = 2π (m = 2), H = 0.9Hc, δ = 0.05. Solid curves—Re[K]; dashed curves—μ = Im[K].

Fig. 5
Fig. 5

Dispersion relations (numerical solution, Sec. II) near ωs = 3π (m = 3), H = 0.9Hc, δ = 0.05. Solid curves—Re[K]; dashed curves—μ = Im[K].

Fig. 6
Fig. 6

Dispersion relations (numerical solution Sec. II) of undistorted liquid crystal (H = 0) near ωs = π; δ = 0.05. Solid curves—Re[K]; dashed curves—μ = Im[K].

Fig. 7
Fig. 7

Solid curves are boundaries of the three subbands around ωs = 2π as functions of the applied field H (numerical solution, Sec. II). The crosses, circles, and squares are, respectively, the results of the approximations Eqs. (13), (14), and (15).

Fig. 8
Fig. 8

Dispersion relations (numerical solution, Sec. II) for a wide range of ωs when H = 0.9Hc; δ = 0.05 (solid curves). The dotted lines are the results of Eqs. (10) (C0 = −0.31, see Fig. 9).

Fig. 9
Fig. 9

Fourier coefficients of e2(ζ).

Fig. 10
Fig. 10

Dispersion relations: Eqs. (10) (H = 0.9Hc, C0 = −0.31, δ = 0.05), around ωs = 2π in the reduced-zone scheme. K± belongs to solutions that propagate in the ±ζ direction.

Fig. 11
Fig. 11

Reflection coefficients (numerical solution, Sec. II) from a finite slab whose thickness is 64S, around ω s = 2 π ( λ = 1 2 ), H = 0.9Hc, δ = 0.05. The dielectric constant of the medium surrounding the liquid-crystal slab is assumed to be equal to the average dielectric constant of the liquid crystal. The polarizations of the incident waves are indicated in the figure.

Fig. 12
Fig. 12

Same as Fig. 11 for wavelength region around λ = 1 3 ( ω s = 3 π ).

Fig. 13
Fig. 13

Attenuation coefficients near ωs = π (numerical solution, Sec. II). (a) H = 0.962Hc, (b) H = 0.99Hc. In both cases, δ = 0.05.

Fig. 14
Fig. 14

Same as Fig. 11 for the wavelength region around λ′ = 1 (ωs = π). (a) Incident wave is right-handed circularly polarized. (b) Incident wave is left-handed circularly polarized.

Tables (2)

Tables Icon

Table I Comparison between results of numerical method of Sec. II and the approximate solutions of Sec. III for the bands m = 2, 3. H = 0.9Hc. δ = 0.05.

Tables Icon

Table II Comparison between results of the numerical method of Sec. II and the approximate solutions of Sec. IV for the m = 1 band. H = 0.8Hc, 0.9Hc. δ = 0.05.

Equations (63)

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ϕ = 2 π P z .
z [ E x i H y E y - i H x ] = ω c [ 0 1 0 0 ( - x x + x z 2 z z ) 0 ( x z y z z z - x y ) 0 0 0 0 1 ( x z y z z z - x y ) 0 ( y z 2 z z - y y ) 0 ] · [ E x i H y E y - i H x ] .
ɛ = [ ( 1 + δ cos 2 ϕ ) δ sin 2 ϕ 0 δ sin 2 ϕ ( 1 - δ cos 2 ϕ ) 0 0 0 z z 0 ] ,
= x x 0 + y y 0 2 ;             δ = x x 0 - y y 0 2 ;
ζ = z S .
ζ ψ ( ζ ) = ω s D ( ζ ) ψ ( ζ ) ,
ω s = ω S c ;             θ ( ζ ) = ϕ ( S ζ )
ψ ( ζ ) = [ E x i H y E y - i H x ] ;     D ( ζ ) = [ 0 1 0 0 - ( 1 + δ cos 2 θ ( ζ ) ) 0 - δ sin 2 θ ( ζ ) 0 0 0 0 1 - δ sin 2 θ ( ζ ) 0 - ( 1 - δ cos 2 θ ( ζ ) ) 0 ] .
D ( ζ + 1 ) = D ( ζ ) .
ψ j B ( ζ + d ) = e i K j d ψ j B ( ζ ) ,             j = 1 , 4.
ψ ˆ i j ( ζ = 0 ) = δ i j ,             i = 1 , 4 ,             j = 1 , 4 ;
ψ B ( ζ ) = B i ψ ˆ i ( ζ ) , ψ B ( ζ + 1 ) = B i ψ ˆ i ( ζ + 1 ) = C B i ψ ˆ i ( ζ ) ;
ψ ˆ j ( ζ + 1 ) = i = 1 4 A i j ψ ˆ i ( ζ ) .
j = 1 4 A i j B j = C B i ,             i = 1 , 4
A · B = C B ,             ( ( A ) i j = A i j ; ( B ) i = B i ) .
ψ ˆ j k ( 1 ) = i A i j ψ ˆ i k ( 0 ) = i A i j δ i k = A k j , A = [ ψ ˆ 1 ( 1 ) ψ ˆ 2 ( 1 ) ψ ˆ 3 ( 1 ) ψ ˆ 4 ( 1 ) ] .
ω m π c S .
E x ( ζ ) + ω s 2 [ E x ( ζ ) + δ 2 ( e 2 i θ ( ζ ) + e - 2 i θ ( ζ ) ) E x ( ζ ) - i δ 2 ( e 2 i θ ( ζ ) - e - 2 i θ ( ζ ) ) E y ( ζ ) ] = 0 , E y ( ζ ) + ω s 2 [ E y ( ζ ) - δ 2 ( e 2 i θ ( ζ ) + e - 2 i θ ( ζ ) ) E y ( ζ ) - i δ 2 ( e 2 i θ ( ζ ) - e - 2 i θ ( ζ ) ) E x ( ζ ) ] = 0.
e 2 i θ ( ζ ) = - C m e 2 π i m ζ ;
E x ( ζ ) + ω s 2 [ E x ( ζ ) + δ ( a m e 2 π i m ζ ) E x ( ζ ) - i δ ( b m e 2 π i m ζ ) E y ( ζ ) ] = 0 , E y ( ζ ) + ω s 2 [ E y ( ζ ) - δ ( a m e 2 π i m ζ ) E y ( ζ ) - i δ ( b m e 2 π i m ζ ) E x ( ζ ) ] = 0 ,
a m = C m + C - m 2 ;             b m = C m - C - m 2 .
E x ( ζ ) + ω s 2 ( 1 + δ C 0 ) E x ( ζ ) = 0 , E y ( ζ ) + ω s 2 ( 1 - δ C 0 ) E y ( ζ ) = 0.
E x ( ζ ) e ± i K 1 ζ , E y ( ζ ) = 0 , E x ( ζ ) = 0 , E y ( ζ ) e ± i K 2 ζ .
ω s = ± K 1 ( 1 + δ C 0 ) 1 2 , ω s = ± K 2 ( 1 - δ C 0 ) 1 2 .
K 2 + - K 2 - = 4 π .
E x = 0 , E y e i K 2 ζ ,
E x = 0 , E y e - i K 2 ζ .
m π ( 1 - δ C 0 ) 1 2 ,             2 m π ( 1 + δ C 0 ) 1 2 + ( 1 - δ C 0 ) 1 2 ,             m π ( 1 + δ C 0 ) 1 2 ,
E x ( ζ ) = I 1 e i γ ζ e i K 1 ζ + R 1 e i γ ζ e i ( K 1 - 2 π m ) ζ ,             E y ( ζ ) = 0 ,
K 1 = ω s ( 1 + δ C 0 ) 1 2 .
[ - 2 K 1 γ δ ω s 2 a m δ ω s 2 a m 4 π m ( K 1 - π m ) - 2 ( K 1 - 2 π m ) γ ] · [ I 1 R 1 ] = 0.
4 K 1 ( K 1 - 2 π m ) γ 2 - 8 π m K 1 ( K 1 - π m ) γ - δ 2 ω s 4 a m 2 = 0.
m π ( 1 + δ C 0 ) 1 2 - δ π m a m 2 < ω s < m π ( 1 + δ C 0 ) 1 2 + δ π m a m 2 .
Δ ω + = δ π m a m .
μ + max = δ π m a m 2 .
Δ ω 0 = δ π m b m , μ 0 max = δ π m b m 2 ,
Δ ω - = δ π m a m , μ - max = δ π m a m 2 ,
Δ ω s = π δ 2 m 4 C 1 4 ( m - 1 ) ( m + 1 ) .
U 1 = E x + i E y ,             U 2 = E x - i E y ,
U 1 ( ζ ) + ω s 2 U 1 + δ ω s 2 e 2 i θ ( ζ ) U 2 ( ζ ) = 0 , U 2 ( ζ ) + ω s 2 U 2 + δ ω s 2 e - 2 i θ ( ζ ) U 1 ( ζ ) = 0.
U j = e 2 π i a ζ V j ( ζ ) ,             j = 1 , 2
V j ( ζ ) = m = - V j ( m ) e - 2 π i m ζ ,             j = 1 , 2.
[ ω s 2 - 4 π 2 ( a - m ) 2 ] V 1 ( m ) + δ ω s 2 q C q V 2 ( m + q ) = 0 , [ ω s 2 - 4 π 2 ( a - m ) 2 ] V 2 ( m ) + δ ω s 2 q C q V 1 ( m - q ) = 0 ,
ω s 2 - 4 π 2 a 2 = 0.
V 2 ( m ) = δ ω s 2 C m V 1 ( 0 ) [ ω s 2 - 4 π 2 ( a - m ) 2 ] ,             m = - , V 1 ( m ) = δ ω s 2 C - m V 2 ( 0 ) [ ω s 2 - 4 π 2 ( a - m ) 2 ] ,             m = - , .
m ( ω s π - m ) 0.
U 1 ( ζ ) = V 1 ( 0 ) e i ω s ζ , U 2 ( ζ ) = V 2 ( m ) e i ( ω s - 2 π m ) ζ ,
U 1 ( ζ ) = V 1 ( 0 ) e i π m ζ , U 2 ( ζ ) = V 2 ( m ) e - i π m ζ .
[ ω s 2 - 4 π 2 a 2 δ ω s 2 C 0 0 δ ω s 2 C m δ ω s 2 C 0 ω s 2 - 4 π 2 a 2 δ ω s 2 C m 0 0 δ ω s 2 C - m ω s 2 - 4 π 2 ( a - m ) 2 δ ω s 2 C 0 δ ω s 2 C m 0 δ ω s 2 C 0 ω s 2 - 4 π 2 ( a - m ) 2 ] · [ V 1 ( 0 ) V 2 ( 0 ) V 1 ( m ) V 2 ( m ) ] = 0
M ( a , ω s , m ) · O = 0.
det ( M ( a , ω s , m ) ) = 0.
B = ( a - 1 2 ) 2 , det ( M ( a , ω s , 1 ) ) = i = 0 4 g i ( ω s ) B 4 - i .
Broad band             π 2 1 + δ D 1 < ω s 2 < π 2 1 - δ D 1 , Narrow band             π 2 ( 1 - δ F 1 D 1 ) < ω s 2 < π 2 ( 1 + δ F 1 D 1 ) , D 1 2 = 2 C 0 2 + C 1 2 + C - 1 2 ;             F 1 2 = ( C 0 2 - C 1 C - 1 ) 2 .
μ Broad = π δ 2 · [ G 1 2 2 + 1 2 ( G 1 4 - 4 F 1 2 ) 1 2 ] 1 2 , μ Narrow = π δ 2 · [ G 1 2 2 - 1 2 ( G 1 4 - 4 F 1 2 ) 1 2 ] 1 2 , G 1 2 = C 1 2 + C - 1 2 - 2 C 0 2 .
U ˆ 1 ( ζ ) = A U ˆ 2 ( ζ ) = i B } e i K ζ { e i θ ( ζ ) e - i θ ( ζ ) .
[ ω s 2 - K 2 + i θ - 2 K θ - ( θ ) 2 δ ω s 2 δ ω s 2 ω s 2 - K 2 - i θ + 2 K θ - ( θ ) 2 ] · [ A B ] = 0.
| θ ζ · λ | 1
ω s 2 - K 2 + i θ ± 2 K θ - ( θ ) 2 ω s 2 - K 2
[ ω s 2 - K 2 δ ω s 2 δ ω s 2 ω s 2 - K 2 ] · [ A B ] = 0.
Mode 1             K 1 = ω s ( 1 + δ ) 1 2 ,             B A = 1
Mode 2             K 2 = ω s ( 1 - δ ) 1 2 ,             B A = - 1.
ζ 2 ( ϕ z ) 2 + sin 2 ϕ = 1 k 2 = const , ξ = [ K 22 χ a ] 1 2 H - 1 ;
H c = π 2 2 S 0 [ K 22 χ a ] 1 2 .