Abstract

A theoretical treatment is given of the problem of transmission of light through a slab of cholesteric liquid crystal. The axially symmetric arrangement is considered in which the faces of the crystal are perpendicular to the axis and the light propagation is parallel to the axis. The linear problem, as governed by Maxwell’s equations and the usual boundary conditions, is treated. Explicit solutions for the transmitted fields are obtained, for either circular polarization of the incident wave. Several limiting cases are discussed, in which especially simple formulas apply and in which there are physical pictures of the transmission phenomena.

© 1994 Optical Society of America

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  1. M. C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 71–117 (1911).
  2. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
    [CrossRef]
  3. H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
    [CrossRef]
  4. P. G. de Gennes, The Physics of Liquid Crystals (Oxford U. Press, London, 1974), Chap. 6.
  5. S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sec. 4.1.3.
    [CrossRef]
  6. V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, New York, 1992), Sec. 1.3.
    [CrossRef]
  7. D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970);“Reflection and transmission by single-domain cholesteric liquid crystal films, theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970)
    [CrossRef]
  8. C. Oldano, E. Miraldi, P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
    [CrossRef]
  9. Ref. 4, p. 221.
  10. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Sec. 1.5.
  11. Ref. 4, p. 231.
  12. H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).
  13. S. Chandrasekhar, J. Shashidhara Prasad, “Theory of rotatory dispersion of cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 14, 115–128 (1971).
    [CrossRef]
  14. J. C. Martin, R. Cano, “Light propagation in cholesteric liquid crystals in a domain including inversion range,” Nouv. Rev. Opt. 7, 265–273 (1976).
    [CrossRef]
  15. S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystals Conference, Bangalore, S. Chandrasekhar, ed. (Pramana, 1973), Supp. I, p. 341.
  16. R. Dreher, G. Meier, A. Saupe, “Selective reflection by cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 13, 17–26.

1983

C. Oldano, E. Miraldi, P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

1976

J. C. Martin, R. Cano, “Light propagation in cholesteric liquid crystals in a domain including inversion range,” Nouv. Rev. Opt. 7, 265–273 (1976).
[CrossRef]

1971

S. Chandrasekhar, J. Shashidhara Prasad, “Theory of rotatory dispersion of cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 14, 115–128 (1971).
[CrossRef]

1970

H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970);“Reflection and transmission by single-domain cholesteric liquid crystal films, theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970)
[CrossRef]

1951

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[CrossRef]

1933

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

1911

M. C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 71–117 (1911).

Baessler, H.

H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).

Belyakov, V. A.

V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, New York, 1992), Sec. 1.3.
[CrossRef]

Berreman, D. W.

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970);“Reflection and transmission by single-domain cholesteric liquid crystal films, theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970)
[CrossRef]

Cano, R.

J. C. Martin, R. Cano, “Light propagation in cholesteric liquid crystals in a domain including inversion range,” Nouv. Rev. Opt. 7, 265–273 (1976).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, J. Shashidhara Prasad, “Theory of rotatory dispersion of cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 14, 115–128 (1971).
[CrossRef]

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sec. 4.1.3.
[CrossRef]

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystals Conference, Bangalore, S. Chandrasekhar, ed. (Pramana, 1973), Supp. I, p. 341.

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Oxford U. Press, London, 1974), Chap. 6.

de Vries, H.

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[CrossRef]

Dreher, R.

R. Dreher, G. Meier, A. Saupe, “Selective reflection by cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 13, 17–26.

Labes, M. M.

H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).

Malya, P. A. G.

H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).

Martin, J. C.

J. C. Martin, R. Cano, “Light propagation in cholesteric liquid crystals in a domain including inversion range,” Nouv. Rev. Opt. 7, 265–273 (1976).
[CrossRef]

Mauguin, M. C.

M. C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 71–117 (1911).

Meier, G.

R. Dreher, G. Meier, A. Saupe, “Selective reflection by cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 13, 17–26.

Miraldi, E.

C. Oldano, E. Miraldi, P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Nes, W. R.

H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).

Oldano, C.

C. Oldano, E. Miraldi, P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

Ranganath, G. S.

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystals Conference, Bangalore, S. Chandrasekhar, ed. (Pramana, 1973), Supp. I, p. 341.

Saupe, A.

R. Dreher, G. Meier, A. Saupe, “Selective reflection by cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 13, 17–26.

Scheffer, T. J.

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970);“Reflection and transmission by single-domain cholesteric liquid crystal films, theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970)
[CrossRef]

Shashidhara Prasad, J.

S. Chandrasekhar, J. Shashidhara Prasad, “Theory of rotatory dispersion of cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 14, 115–128 (1971).
[CrossRef]

Suresh, K. A.

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystals Conference, Bangalore, S. Chandrasekhar, ed. (Pramana, 1973), Supp. I, p. 341.

Taverna Valabrega, P.

C. Oldano, E. Miraldi, P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Sec. 1.5.

Acta Crystallogr.

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[CrossRef]

Bull. Soc. Fr. Mineral. Cristallogr.

M. C. Mauguin, “Sur les cristaux liquides de Lehmann,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 71–117 (1911).

Mol. Cryst. Liq. Cryst.

R. Dreher, G. Meier, A. Saupe, “Selective reflection by cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 13, 17–26.

H. Baessler, P. A. G. Malya, W. R. Nes, M. M. Labes, “The absence of helical inversion in single component cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 6, 329–338 (1970).

S. Chandrasekhar, J. Shashidhara Prasad, “Theory of rotatory dispersion of cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 14, 115–128 (1971).
[CrossRef]

Nouv. Rev. Opt.

J. C. Martin, R. Cano, “Light propagation in cholesteric liquid crystals in a domain including inversion range,” Nouv. Rev. Opt. 7, 265–273 (1976).
[CrossRef]

Phys. Rev. A

C. Oldano, E. Miraldi, P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Phys. Rev. Lett.

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970);“Reflection and transmission by single-domain cholesteric liquid crystal films, theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970)
[CrossRef]

Trans. Faraday Soc.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

Other

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystals Conference, Bangalore, S. Chandrasekhar, ed. (Pramana, 1973), Supp. I, p. 341.

Ref. 4, p. 221.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Sec. 1.5.

Ref. 4, p. 231.

P. G. de Gennes, The Physics of Liquid Crystals (Oxford U. Press, London, 1974), Chap. 6.

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sec. 4.1.3.
[CrossRef]

V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, New York, 1992), Sec. 1.3.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Plot of y versus l2, as given by Eq. (13), in the case = 3.06 and = 2.43, so that δ = 0.115. The range 5 > y > 0 is shown. The curve extends slightly into the second quadrant in the range 1.130 > y > 0.897, as shown in the expanded scale in Fig. 2.

Fig. 2
Fig. 2

Plot of y versus l2, as given by Eq. (13), in the case = 3.06 and −2.43, so that δ = 0.115. The forbidden frequency range 1.130 > y > 0.897 is shown.

Fig. 3
Fig. 3

Polarization of the incident wave that is completely reflected, as given by Eq. (61). A thick right-handed crystal with = 3.06 and = 2.43 is considered for various values of ω/cq as indicated in the plot.

Fig. 4
Fig. 4

Polarization of the transmitted wave for frequencies in the forbidden zone in the case = 3.06 and = 2.43 and in which the crystal is right handed and thick. The zone is 0.642 > (ω/cq) > 0.572. The polarization is described by the complex number χtrans, as given approximately by Eq. (64), for various values of ω/cq as indicated in the plot.

Fig. 5
Fig. 5

Polarization χtrans of the transmitted wave in case 1, a right-handed circularly polarized incident wave. The crystal is right handed with = 3.06 and = 2.43. The frequency is at the lower edge of the forbidden zone. Various values of qb, where 2π/q is the pitch of the director and b is the thickness of the slab, are considered, as indicated in the plot. For increasing qb the curve starts at χ = i and goes to χ = 0.57i.

Fig. 6
Fig. 6

Polarization χtrans of the transmitted wave in case 2, a left-handed circularly polarized incident wave, where the other conditions are the same as those for Fig. 5. For increasing qb the curve first spirals out from χ = −i but then near qb = 50 it crosses over and spirals in toward χ = 0.57i.

Equations (122)

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α β ( z ) = δ α β + ( ) n α n β ,
n x = cos ( q z ) , n y = sin ( q z ) , n z = 0 .
× H = D / t , × E = B / t , B = 0 , D = 0 .
x α E β x β 2 E β x β x β = ( ω c ) 2 α β E β .
2 E x / z 2 = ( 1 / 2 ) ( ω / c ) 2 { ( + ) E x + ( ) × [ E x cos ( 2 q z ) + E y sin ( 2 q z ) ] } ,
2 E y / z 2 = ( 1 / 2 ) ( ω / c ) 2 { ( + ) E y + ( ) × [ E x sin ( 2 q z ) E y cos ( 2 q z ) ] } ,
0 = E z .
E ± = E x ± i E y .
E = ( E + E 0 ) .
2 E + / z 2 = ( 1 / 2 ) ( ω / c ) 2 [ ( + ) E + + ( ) exp ( 2 iqz ) E ] ,
2 E / z 2 = ( 1 / 2 ) ( ω / c ) 2 [ ( + ) E + ( ) exp ( 2 iqz ) E + ] .
E + = a exp [ i ( l + 1 ) q z ] , E = b exp [ i ( l 1 ) q z ] ,
( l + 1 ) 2 a = y ( a + δ b ) ,
( l 1 ) 2 b = y ( b + δ a ) ,
y = ( ω q c ) 2 + 2 ,
δ = +
( l 2 y ) 2 2 ( l 2 + y ) + 1 δ 2 y 2 = 0 .
y ( 0 ) = 1 / ( 1 δ ) ,
l 2 = y + 1 ± ( 4 y + δ 2 y 2 ) 1 / 2 ,
a b = δ y ( l + 1 ) 2 y ,
a b = ( l 1 ) 2 y δ y .
a b = ( l 1 ) 2 y + δ y ( l + 1 ) 2 y + δ y .
E = 2 A exp ( ilqz ) ( ( 1 ρ ) exp ( iqz ) ( 1 + ρ ) exp ( iqz ) 0 ) , B = 2 iqA ω exp ( ilqz ) ( ( 1 ρ ) ( l + 1 ) exp ( iqz ) ( 1 + ρ ) ( l 1 ) exp ( iqz ) 0 ) ,
ρ = 1 a / b 1 + a / b .
ρ = 2 l l 2 + 1 y + δ y .
B ± = ± ( 1 / ω ) E ± / z ,
E = ( F G 0 ) exp ( i ω z / c ) , B = i c ( F G 0 ) exp ( i ω z / c )
E = ( H I 0 ) exp ( i ω z / c ) , B = i c ( H I 0 ) exp ( i ω z / c ) .
F + H = 2 A ( 1 ρ ) , G + I = 2 A ( 1 + ρ ) , ( i / c ) ( F H ) = 2 A ( i q / ω ) ( 1 ρ ) ( l + 1 ) , ( i / c ) ( G + I ) = 2 A ( i q / ω ) ( 1 + ρ ) ( l 1 ) .
F = ( 1 ρ ) ( 1 + β + β l ) A ,
G = ( 1 + ρ ) ( 1 β + β l ) A ,
H = ( 1 ρ ) ( 1 β β l ) A ,
I = ( 1 + ρ ) ( 1 + β β l ) A ,
β = c q / ω .
E = ( K L 0 ) exp [ i ω ( z + b ) / c ] , B = i c ( K L 0 ) exp [ i ω ( z + b ) / c ]
E = ( M N 0 ) exp [ i ω ( z + b ) / c ] , B = i c ( M N 0 ) exp [ i ω ( z + b ) / c ] .
K = ( 1 ρ ) ( 1 + β + β l ) exp [ i ( l + 1 ) q b ] A ,
L = ( 1 + ρ ) ( 1 β + β l ) exp [ i ( l 1 ) q b ] A ,
M = ( 1 ρ ) ( 1 β β l ) exp [ i ( l + 1 ) q b ] A ,
N = ( 1 + ρ ) ( 1 + β β l ) exp [ i ( l 1 ) q b ] A .
E inc = ( K tot L tot 0 ) exp [ i ω ( z + b ) / c ] ,
K i = K tot , L i = L tot , H i = 0 , I i = 0 .
[ ( 1 ρ 1 ) ( 1 β β l 1 ) ( 1 ρ 2 ) ( 1 β β l 2 ) ( 1 + ρ 1 ) ( 1 β + β l 1 ) ( 1 + ρ 2 ) ( 1 β + β l 2 ) 0 ( 1 + ρ 1 ) ( 1 + β β l 1 ) ( 1 + ρ 2 ) ( 1 + β β l 2 ) ( 1 ρ 1 ) ( 1 + β + β l 1 ) ( 1 ρ 2 ) ( 1 + β + β l 2 ) 0 ( 1 ρ 1 ) ( 1 + β + β l 1 ) exp ( i l 1 q b ) ( 1 ρ 2 ) ( 1 + β + β l 2 ) exp ( i l 2 q b ) ( 1 + ρ 1 ) ( 1 + β β l 1 ) exp ( i l 1 q b ) ( 1 + ρ 2 ) ( 1 + β β l 2 ) exp ( i l 2 q b ) exp ( iqb ) K tot ( 1 + ρ 1 ) ( 1 β + β l 1 ) exp ( i l 1 q b ) ( 1 + ρ 2 ) ( 1 β + β l 2 ) exp ( i l 2 q b ) ( 1 ρ 1 ) ( 1 β β l 1 ) exp ( i l 1 q b ) ( 1 ρ 2 ) ( 1 β β l 2 ) exp ( i l 2 q b ) exp ( iqb ) L tot ] .
( 1 ρ 1 ) ( 1 β β l 1 ) A 1 + ( 1 ρ 2 ) ( 1 β β l 2 ) A 2 + ( 1 + ρ 1 ) ( 1 β + β l 1 ) A 3 + ( 1 + ρ 2 ) ( 1 β + β l 2 ) A 4 = 0.
d 12 = | ( 1 ρ 1 ) ( 1 β β l 1 ) ( 1 ρ 2 ) ( 1 β β l 2 ) ( 1 + ρ 1 ) ( 1 + β β l 1 ) ( 1 + ρ 2 ) ( 1 + β β l 2 ) | .
d 13 = ( 1 ρ 1 ) 2 [ 1 β 2 ( l 1 + 1 ) 2 ] ( 1 + ρ 1 ) 2 × [ 1 β 2 ( l 1 1 ) 2 ] .
D = d 12 2 exp [ i ( l 1 + l 2 ) q b ] d 34 2 exp [ i ( l 1 + l 2 ) q b ] + d 14 2 exp [ i ( l 1 l 2 ) q b ] + d 23 2 exp [ i ( l 1 l 2 ) q b ] 2 d 13 d 24 .
E trans + = i = 1 4 F i exp ( i ω z / c ) = i = 1 4 ( 1 ρ i ) ( 1 + β + β l i ) A i exp ( i ω z / c ) = [ exp ( iqb ) / D ] i = 1 4 ( 1 ρ i ) ( 1 + β + β l i ) × ( minor of 4 i element ) exp ( i ω z / c ) = | ( 1 ρ 1 ) ( 1 β β l 1 ) ( 1 ρ 2 ) ( 1 β β l 2 ) ( 1 + ρ 1 ) ( 1 β + β l 1 ) ( 1 + ρ 2 ) ( 1 β + β l 2 ) ( 1 + ρ 1 ) ( 1 + β β l 1 ) ( 1 + ρ 2 ) ( 1 + β β l 2 ) ( 1 ρ 2 ) ( 1 + β + β l 1 ) ( 1 ρ 2 ) ( 1 + β + β l 2 ) ( 1 ρ 1 ) ( 1 + β + β l 1 ) exp ( i l 1 q b ) ( 1 ρ 2 ) ( 1 + β + β l 2 ) exp ( i l 2 q b ) ( 1 + ρ 1 ) ( 1 + β β l 1 ) exp ( i l 1 q b ) ( 1 + ρ 2 ) ( 1 + β β l 2 ) exp ( i l 2 q b ) ( 1 ρ 1 ) ( 1 + β + β l 1 ) ( 1 ρ 2 ) ( 1 + β + β l 2 ) ( 1 + ρ 1 ) ( 1 + β β l 1 ) ( 1 + ρ 2 ) ( 1 + β β l 2 ) | × exp ( iqb ) exp ( i ω z / c ) D .
E trans ± = 2 D 1 exp ( iqb ) exp ( i ω z / c ) × { ( 1 ρ 1 ) ( 1 + β + β l 1 ) exp ( i l 1 q b ) × [ ( 1 ρ 2 ) d 34 ( 1 ± ρ 1 ) d 24 + ( 1 ± ρ 2 ) d 23 ] ( 1 ρ 2 ) ( 1 + β + β l 2 ) exp ( i l 2 q b ) × [ ( 1 ρ 1 ) d 34 ( 1 ± ρ 1 ) d 14 + ( 1 ± ρ 2 ) d 13 ] + ( 1 + ρ 1 ) ( 1 + β β l 1 ) exp ( i l 1 q b ) × [ ( 1 ρ 1 ) d 24 ( 1 ρ 2 ) d 14 + ( 1 ± ρ 2 ) d 12 ] ( 1 + ρ 2 ) ( 1 + β β l 2 ) exp ( i l 2 q b ) × [ ( 1 ρ 1 ) d 23 ( 1 ρ 2 ) d 13 + ( 1 ± ρ 1 ) d 12 ] } .
E trans ± = 2 D 1 exp ( iqb ) exp ( i ω z / c ) × { ( 1 + ρ 1 ) ( 1 β + β l 1 ) exp ( i l 1 q b ) × [ ( 1 ρ 2 ) d 34 ( 1 ± ρ 1 ) d 24 + ( 1 ± ρ 2 ) d 23 ] + ( 1 + ρ 2 ) ( 1 β + β l 2 ) exp ( i l 2 q b ) × [ ( 1 ρ 1 ) d 34 ( 1 ± ρ 1 ) d 14 + ( 1 ± ρ 2 ) d 13 ] ( 1 ρ 1 ) ( 1 β β l 1 ) exp ( i l 1 q b ) × [ ( 1 ρ 1 ) d 24 ( 1 ρ 2 ) d 14 + ( 1 ± ρ 2 ) d 12 ] + ( 1 ρ 2 ) ( 1 β β l 2 ) exp ( i l 2 q b ) × [ ( 1 ρ 1 ) d 23 ( 1 ρ 2 ) d 13 + ( 1 ± ρ 1 ) d 12 ] }
E trans ± ( 1 , q ) = E trans ( 2 , q ) .
r = δ y / 2 l ,
( l 2 y + 1 ) 2 = 4 l 2 ( 1 + r 2 ) .
l 2 y + 1 = 2 l ( 1 + r 2 / 2 ) .
l 1 / 2 = 1 ± y ± ( l r 2 / 2 y ) .
l 1 / 2 = 1 ± y ± δ 2 y 3 / 2 8 ( 1 y ) δ 2 y 2 8 ( 1 y )
ρ 1 / 2 = 1 [ δ y / 2 ( 1 ± y ) ] ,
γ = β y = ± [ ( + ) / 2 ] 1 / 2 ,
d 12 = d 34 = 0 , d 13 = d 24 = 4 ( 1 γ 2 ) , d 14 = 4 ( 1 γ ) 2 , d 23 = 4 ( 1 + γ ) 2 .
D = 16 [ ( 1 γ ) 2 exp ( i l 1 q b ) ( 1 + γ ) 2 exp ( i l 2 q b ) ] × [ ( 1 + γ ) 2 exp ( i l 1 q b ) ( 1 γ ) 2 exp ( i l 2 q b ) ] .
E trans + = 0 ,
E trans = 64 γ D 1 exp ( i ω z / c ) exp ( iqb ) × [ ( 1 γ ) 2 exp ( i l 1 q b ) ( 1 + γ ) 2 exp ( i l 2 q b ) ] = 4 γ exp ( i ω z / c ) { ( 1 + γ ) 2 exp [ i ( l 1 1 ) q b ] ( 1 γ ) 2 exp [ i ( l 2 1 ) q b ] } 1 .
E trans + = 4 γ exp ( i ω z / c ) { ( 1 + γ ) 2 exp [ i ( l 2 1 ) q b ] ( 1 γ ) 2 exp [ i ( l 1 1 ) q b ] } 1 ,
E trans = 0 .
E trans = exp ( i ω z / c ) cos θ [ ( 1 + γ 2 ) / 2 γ ] i sin θ ( exp ( i ψ ) K tot exp ( i ψ ) L tot ) ,
θ = q b y { 1 + [ δ 2 y / ( 1 y ) ] } ,
ψ = q b δ 2 y 2 / [ 8 ( 1 y ) ] .
ψ / b = q δ 2 y 2 / 8 ( 1 y ) ,
E x inc = ( cos α ) exp [ i ω ( z + b ) / c ] , E y inc = ( sin α ) exp [ i ω ( z + b ) / c ] .
E inc = exp [ i ω ( z + b ) / c ] ( exp ( i α ) exp ( i α ) 0 ) .
( exp [ i ( α + ψ ) ] exp [ i ( α + ψ ) ] 0 ) ,
T = | 1 cos θ [ ( 1 + γ 2 ) / 2 γ ] i sin θ | 2 = { 1 2 [ 1 + ( 1 + γ 2 2 γ ) 2 ] + 1 2 [ 1 ( 1 + γ 2 2 γ ) 2 ] × cos ( 2 θ ) } 1 ,
2 ( | γ | ω / c ) Δ b = 2 π .
l 1 / 2 = ( 1 ± δ ) 1 / 2 y .
ω c | q | > 10 .
ρ 1 = 0 , of order ( ω / c q ) 1 ,
ρ 2 = δ y / ( 1 δ ) 1 / 2 .
β l 1 = ( 1 + δ ) 1 / 2 γ = ± ,
β l 2 = ( 1 δ ) 1 / 2 γ = ± ,
d 12 = 2 ρ 2 ( 1 ) ( 1 ) , d 13 = 0 , of order ( ω / c q ) 1 , d 14 = 2 ρ 2 ( 1 ) ( 1 + ) , d 23 = 2 ρ 2 ( 1 + ) ( 1 ) , d 24 = 4 ρ 2 ( ρ 2 β + 1 ) , d 34 = 2 ρ 2 ( 1 + ) ( 1 + ) .
D = 4 ρ 2 2 [ ( 1 ) 2 exp ( i l 1 q b ) ( 1 + ) 2 × exp ( i l 1 q b ) ] [ ( 1 ) 2 exp ( i l 2 q b ) ( 1 + ) 2 exp ( i l 2 q b ) ] .
E trans ± = 2 exp ( i ω z / c ) × { ± [ K tot exp ( iqb ) L tot exp ( iqb ) ] ( 1 ) 2 exp ( i l 2 q b ) ( 1 + ) 2 exp ( i l 2 q b ) + [ K tot exp ( iqb ) + L tot exp ( iqb ) ] ( 1 ) 2 exp ( i l 1 q b ) ( 1 + ) 2 exp ( i l 1 q b ) } .
K tot = exp ( iqb ) , L tot = exp ( iqb ) ,
E x inc = cos ( q b ) , E y inc = sin ( q b ) ,
E x trans = 4 exp ( i ω z / c ) ( 1 ) 2 exp ( i l 1 q b ) ( 1 + ) 2 exp ( i l 1 q b ) , E y trans = 0 .
T = | 4 ( 1 ) 2 exp ( i l 1 q b ) ( 1 + ) 2 exp ( i l 1 q b ) | 2 = 8 1 + 6 + 2 ( 1 ) 2 cos ( 2 l 1 q b ) .
Δ b = ( 1 / 2 l 1 ) ( 2 π / q ) ( 1 / 2 ) ( 2 π c / ω ) ,
K tot = i exp ( iqb ) , L tot = i exp ( iqb ) ,
E x inc = sin ( q b ) , E y inc = cos ( q b ) ,
E x trans = 0 , E y trans = 4 exp ( i ω z / c ) ( 1 ) 2 exp ( i l 2 q b ) ( 1 + ) 2 exp ( i l 2 q b ) .
T = 8 1 + 6 + 2 ( 1 ) 2 cos ( 2 l 2 q b ) .
E trans ± = 2 D 1 exp ( i ω z / c ) × [ ( 1 ρ 1 ) d 34 ( 1 ± ρ 1 ) d 14 + ( 1 ± ρ 2 ) d 13 ] × [ ( 1 + ρ 2 ) ( 1 β + β l 2 ) exp ( iqb ) K tot ( 1 ρ 2 ) ( 1 + β + β l 2 ) exp ( iqb ) L tot ] ,
D = d 14 2 exp ( i l 1 q b ) d 34 2 exp ( i l 1 q b ) .
χ = E y / E x ,
χ trans = i ( ρ 1 d 34 + ρ 1 d 14 ρ 2 d 13 ) d 34 d 14 + d 13 .
K ¯ tot = K tot exp ( iqb ) , L ¯ tot = L tot exp ( iqb ) .
K ¯ tot L ¯ tot = ( 1 ρ 2 ) ( 1 + β + β l 2 ) ( 1 + ρ 2 ) ( 1 β + β l 2 ) ,
χ ¯ inc = β ρ 2 ρ 2 β l 2 1 + β l 2 ρ 2 ( i ) .
d 13 = 4 ( 1 β 2 ) , d 14 = 2 ( 1 + ρ 2 ) ( 1 β ) 2 , d 34 = 2 ( 1 ρ 2 ) ( 1 + β ) 2 , d 12 = 2 ( 1 ρ 2 ) ( 1 β ) 2 , d 23 = 2 ( 1 + ρ 2 ) ( 1 + β ) 2 , d 24 = 4 ρ 2 ( 1 β 2 ) .
E trans ± = 2 β [ ( β ρ 2 ) ± ( β ρ 2 1 ) ] [ ( 1 + ρ 2 ) ( 1 β ) exp ( iqb ) K tot ( 1 ρ 2 ) ( 1 + β ) exp ( iqb ) L tot ] ( 1 + ρ 2 ) 2 ( 1 β ) 4 exp ( i l 1 q b ) ( 1 ρ 2 ) 2 ( 1 + β ) 4 exp ( i l 1 q b ) .
χ trans = i ( 1 β ρ 2 ) / ( β ρ 2 ) .
D = 16 ρ 2 [ ( β 1 ) 2 exp ( i l 1 q b / 2 ) + ( β + 1 ) 2 × exp ( i l 1 q b / 2 ) ] 2 .
E trans = 4 β ( β + 1 ) 2 exp ( i l 1 q b / 2 ) ( β 1 ) 2 exp ( i l 1 q b / 2 ) × ( exp ( i ψ ) K tot exp ( i ψ ) L tot 0 ) ,
ψ = ( 1 l 1 / 2 ) q b .
T = 16 β 2 ( β + 1 ) 4 + ( β 1 ) 4 2 ( β 2 1 ) 2 cos ( l 1 q b ) .
Δ b = 2 π / l 1 q .
E + = a exp ( iqz ) , E = b exp ( iqz )
E ± = a ( q z ± i η ) exp ( ± iqz ) ,
η = 2 / ( ) ,
E ± = a ( ± q z + i η ¯ ) exp ( ± iqz ) ,
η ¯ = 2 / ( ) ,
E = 2 A 4 ( ( q z + i η ) exp ( iqz ) ( q z i η ) exp ( iqz ) 0 ) ,
= 2 i q ω A 4 ( [ q z + i ( η 1 ) ] exp ( iqz ) [ q z i ( η 1 ) ] exp ( iqz ) 0 ) .
F 4 = I 4 = i [ ( β + 1 ) η β ] A 4 ,
G 4 = H 4 = i [ ( β 1 ) η β ] A 4 .
K 4 = [ ( q b + i η ) ( β + 1 ) i β ] exp ( iqb ) A 4 , L 4 = [ ( q b + i η ) ( β 1 ) i β ] exp ( iqb ) A 4 , M 4 = [ ( q b i η ) ( β 1 ) + i β ] exp ( iqb ) A 4 , N 4 = [ ( q b i η ) ( β + 1 ) + i β ] exp ( iqb ) A 4 .
[ ( 1 1 2 l 1 ) ( 1 + β l 1 β 1 ) 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β 1 ) i η i β β 1 0 ( 1 + 1 2 l 1 ) ( 1 β l 1 β + 1 ) 1 ( 1 1 2 l 1 ) ( 1 + β l 1 β + 1 ) i η + i β β + 1 0 ( 1 1 2 l 1 ) ( 1 + β l 1 β + 1 ) exp ( i l 1 q b ) 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β + 1 ) exp ( i l 1 q b ) q b + i η i β β + 1 exp ( iqb ) K tot 1 + β ( 1 + 1 2 l 1 ) ( 1 β l 1 β 1 ) exp ( i l 1 q b ) 1 ( 1 1 2 l 1 ) ( 1 + β l 1 β 1 ) exp ( i l 1 q b ) q b i η + i β β 1 exp ( iqb ) L tot β 1 ] .
E ± = exp ( iqb ) exp ( i ω z / c ) ( 1 β ) D lwr × [ ( 1 1 2 l 1 ) ( 1 + β l 1 β 1 ) 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β 1 ) i η i β β 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β + 1 ) 1 ( 1 1 2 l 1 ) ( 1 + β l 1 β + 1 ) i η + i β β + 1 ( 1 1 2 l 1 ) ( 1 + β l 1 β + 1 ) exp ( i l 1 q b ) 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β + 1 ) exp ( i l 1 q b ) q b + i η i β β + 1 ( 1 1 2 l 1 ) ( 1 ± β + β l 1 ) 1 ± β ( 1 ± 1 2 l 1 ) ( 1 ± β β l 1 ) i ( β η β ± η ) ] ,
E ± = exp ( iqb ) exp ( i ω z / c ) ( 1 β ) D lwr × [ ( 1 1 2 l 1 ) ( 1 + β l 1 β 1 ) 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β 1 ) + i η i β β 1 ( 1 + 1 2 l 1 ) ( 1 β l 1 β + 1 ) 1 ( 1 1 2 l 1 ) ( 1 + β l 1 β + 1 ) i η + i β β + 1 ( 1 1 2 l 1 ) ( 1 ± β + β l 1 ) 1 ± β ( 1 ± 1 2 l 1 ) ( 1 ± β β l 1 ) i ( β η β ± η ) ( 1 + 1 2 l 1 ) ( 1 β l 1 β 1 ) exp ( i l 1 q b ) 1 ( 1 1 2 l 1 ) ( 1 + β l 1 β 1 ) exp ( i l 1 q b ) q b i η + i β β 1 ] ,
E trans ± = exp ( iqb ) exp ( i ω z / c ) D lwr 1 [ ( 29.30 ± 16.75 ) q b + ( 23.50 ± 27.77 ) i exp ( i l 1 q b ) ( 12.41 ± 13.19 ) i exp ( i l 1 q b ) ( 428.73 ± 407.42 ) i ] ,
D lwr = [ 0.38 exp ( i l 1 q b ) 49.86 exp ( i l 1 q b ) ] q b + 2.68 i exp ( i l 1 q b ) + 891.60 i × exp ( i l 1 q b ) 108.62 i .
E trans ± = exp ( iqb ) exp ( i ω z / c ) D lwr 1 [ ( 7.99 ± 4.57 ) q b ( 1.96 ± 2.32 ) i exp ( i l 1 q b ) + ( 405.46 ± 431.05 ) i exp ( i l 1 q b ) ( 14.59 ± 35.90 ) i ] .

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