Abstract

Leading term approximative analytical formulas are derived for the powers of the fringes formed by a periodically distorted homogeneous nematic layer, illuminated by a monochromatic light beam at normal incidence. The agreement between the predictions of the derived formulas and those of the direct numerical computation for the odd-order fringes is restricted to very small distortions and, thus, to very low fringe power values. On the other hand, the respective comparison for the even-order fringes gives quite satisfactory results provided the sample is sufficiently thick and the order of the fringe is low.

© 1989 Optical Society of America

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References

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  1. W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
    [Crossref]
  2. S. Lu, D. Jones, “Light Diffraction Phenomena in an a.c.-Excited Nematic Liquid Crystal,” J. Appl. Phys. 42, 2138 (1971).
    [Crossref]
  3. P. A. Penz, “Voltage Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
    [Crossref]
  4. J. A. Kosmopoulos, H. M. Zenginoglou, “Geometrical Optics Approach to the Nematic Liquid Crystal Grating: Numerical Results,” Appl. Opt. 26, 1714 (1987).
    [Crossref] [PubMed]
  5. K. Rokushima, J. Yamakita, “Analysis of Anisotropic Dielectric Gratings,” J. Opt. Soc. Am. 73, 901 (1983).
    [Crossref]
  6. R. A. Kashnow, J. E. Bigelow, “Diffraction from a Liquid Crystal Phase Grating,” Appl. Opt. 12, 2302 (1973).
    [Crossref] [PubMed]
  7. T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
    [Crossref]

1987 (1)

1983 (1)

1973 (1)

1972 (1)

T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
[Crossref]

1971 (2)

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

S. Lu, D. Jones, “Light Diffraction Phenomena in an a.c.-Excited Nematic Liquid Crystal,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

1970 (1)

P. A. Penz, “Voltage Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
[Crossref]

Bigelow, J. E.

Carroll, T. O.

T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
[Crossref]

Greubel, W.

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

Jones, D.

S. Lu, D. Jones, “Light Diffraction Phenomena in an a.c.-Excited Nematic Liquid Crystal,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

Kashnow, R. A.

Kosmopoulos, J. A.

Lu, S.

S. Lu, D. Jones, “Light Diffraction Phenomena in an a.c.-Excited Nematic Liquid Crystal,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

Penz, P. A.

P. A. Penz, “Voltage Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
[Crossref]

Rokushima, K.

Wolf, H.

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

Yamakita, J.

Zenginoglou, H. M.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

W. Greubel, H. Wolf, “Electrically Controllable Domains in Nematic Liquid Crystals,” Appl. Phys. Lett. 19, 213 (1971).
[Crossref]

J. Appl. Phys. (2)

S. Lu, D. Jones, “Light Diffraction Phenomena in an a.c.-Excited Nematic Liquid Crystal,” J. Appl. Phys. 42, 2138 (1971).
[Crossref]

T. O. Carroll, “Liquid Crystal Diffraction Grating,” J. Appl. Phys. 43, 767 (1972).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. Lett. (1)

P. A. Penz, “Voltage Induced Vorticity and Optical Focusing in Liquid Crystals,” Phys. Rev. Lett. 24, 1405 (1970).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Distortion amplitude dependence of the power of the ninth-order fringe. The small circles represent points computed numerically. Curve (a) represents the predictions of Eq. (19a) whereas curve (b) represents the predictions of the solution described by Eqs. (15)(18) where the expansion is extended up to tenth-order Bessel functions in l and m.

Fig. 2
Fig. 2

Distortion amplitude dependence of the power of the first five even-order fringes. The small circles represent points computed numerically and the solid curve is the plot of Eq. (19b).

Fig. 3
Fig. 3

Distortion amplitude dependence of the power of the eighth-order fringe. The small circles represent points computed numerically. Curves (a) and (b) represent the predictions of eq. (19b) and Eqs. (15)(18), respectively, where, in the second case, the expansion is extended up to fourth-order Bessel functions in l and m.

Equations (22)

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= m sin ( q x ) sin ( π y / D ) ,
L = n e 0 D d y { [ 1 α sin 2 ( θ + ) ] 1 / 2 / cos θ }
d 2 Δ ξ d η 2 β S η β S 2 ξ β S ( 2 2 + 3 θ ) .
θ = tan 1 ( 1 S d Δ ξ d η ) 1 S d Δ ξ d η 1 3 ( 1 S d Δ ξ d η ) 3 .
Δ ξ D = 2 σ sin ξ 0 τ sin 2 ξ 0 + β S m 3 × ( { ( 2 / 3 + β ) + β S 2 [ ( β + 1 ) π 2 / 16 ( 5 β ) / 9 ] } sin ξ 0 + { ( 2 / 3 + β ) / 3 + β S 2 [ 1 + 5 β / 3 3 ( β + 1 ) π 2 / 16 ] } sin 3 ξ 0 ) ,
σ S β m ,
τ β [ ( β + 1 ) π 2 / 8 β ] S 2 m 2 .
Δ ϕ = Δ ϕ m ( cos 2 ξ 0 1 ) + σ Δ ϕ m ( cos 3 ξ 0 cos ξ 0 ) + 1 4 Δ ϕ m m 2 ( 1 + 9 β / 4 + 5 β 2 S 2 ) ( 1 cos 2 ξ 0 ) + 1 8 Δ ϕ m m 2 { ( 1 + 9 β / 4 ) / 2 + β S 2 × [ 5 ( 13 β + 1 ) / 8 2 π 2 ( β + 1 ) / 3 ] } ( cos 4 ξ 0 1 ) ,
Δ ϕ m = 1 8 β n e k 0 D m 2 .
θ D = const m 2 sin 2 ξ 0 + ( terms of higher order ) .
P n = 1 cos γ n | 1 2 π 0 π d x 0 | 1 + d Δ ξ D / d ξ 0 | 1 / 2 H cos γ n + cos θ t 2 × exp [ j ( n ξ 0 + n Δ ξ D Δ ϕ ) ] | 2 ,
F n = 1 2 π 0 2 π 1 + d Δ ξ D / d ξ 0 exp [ j ( n ξ 0 + n Δ ξ D Δ ϕ ) ] d ξ 0 .
exp ( j x sin θ ) = k = k = + J k ( x ) exp ( i k θ ) ,
1 2 π 0 2 π exp [ j ( k θ x cos 2 θ ) ] d θ = { 0 , k : odd , ( j ) k / 2 J k / 2 ( x ) , k : even ,
F n = G n + σ 2 ( G n 1 + G n + 1 ) ,
G n l m j l exp ( j m ψ ) J m ( c 1 ) J l ( c 3 ) R ( n + m 3 l , ω , c 2 ) ,
c 1 σ 4 n 2 + Δ ϕ m 2 and ψ tan 1 ( Δ ϕ m / 2 n ) ,
c 2 σ Δ ϕ m 2 + n 2 τ 2 and ω tan 1 ( n τ / Δ ϕ m ) ,
c 3 σ Δ ϕ m ,
R ( n , x , y ) { 0 , n : odd , j n / 2 exp ( j n x / 2 ) J n / 2 ( x ) , n : even .
P n = n 2 σ 2 4 { [ J ( n 1 ) / 2 ( Δ ϕ m ) ] 2 + [ J ( n + 1 ) / 2 ( Δ ϕ m ) ] 2 } , n : odd ,
P n = [ J n / 2 ( Δ ϕ m ) ] 2 , n : even .

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