Abstract

We present a detailed theoretical calculation, with experimental verification, of the nonlocal molecular reorientation of the nematic-liquid-crystal director axis induced by a cw Gaussian laser beam. The natures of the torque balance equations and the solutions are significantly different for normally and nonnormally incident laser beams. The nonlocal effects resulting from molecular correlation effects are particularly important for laser spot sizes that are different (smaller or larger) from the sample thickness. Experimental measurements for the transverse dependence of the molecules and the dependence of the Freedericksz threshold as a function of the laser beam sizes are in excellent agreement with theoretical results. We also comment on the effect of these nonlocal effects on transverse optical bistability.

© 1987 Optical Society of America

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  1. See, for example, the review paper by I. C. Khoo and Y. R. Shen, Opt. Eng. 24, 579 (1985).
  2. N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
    [CrossRef]
  3. B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981); I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982); I. C. Khoo and S. L. Zhuang, IEEE J. Quantum Electron. QE-18, 246 (1981).
    [CrossRef]
  4. L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
    [CrossRef]
  5. E. Santamato and Y. R. Shen, Opt. Lett. 9, 564 (1984).
    [CrossRef] [PubMed]
  6. A. E. Kaplan, Opt. Lett. 6, 360 (1981); M. LeBerre, E. Ressayre, A. Tallet, K. Tai, and H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1985); J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, and A. E. Kaplan, Opt. Lett. 6, 345 (1981); J. E. Bjorkholm, P. W. Smith, and W. J. Tomlinson, IEEE J. Quantum. Electron. QE-18, 2016 (1982).
    [CrossRef] [PubMed]
  7. I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
    [CrossRef]
  8. I. C. Khoo, T. H. Liu, and R. Normandin, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
    [CrossRef]
  9. See, for example, H. L. Ong, Phys. Rev. A 28, 2393 (1983); B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981).
    [CrossRef]
  10. See, for example, P. G. deGennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1984).

1985 (2)

See, for example, the review paper by I. C. Khoo and Y. R. Shen, Opt. Eng. 24, 579 (1985).

I. C. Khoo, T. H. Liu, and R. Normandin, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

1984 (2)

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

E. Santamato and Y. R. Shen, Opt. Lett. 9, 564 (1984).
[CrossRef] [PubMed]

1983 (1)

See, for example, H. L. Ong, Phys. Rev. A 28, 2393 (1983); B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981).
[CrossRef]

1982 (1)

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

1981 (3)

A. E. Kaplan, Opt. Lett. 6, 360 (1981); M. LeBerre, E. Ressayre, A. Tallet, K. Tai, and H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1985); J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, and A. E. Kaplan, Opt. Lett. 6, 345 (1981); J. E. Bjorkholm, P. W. Smith, and W. J. Tomlinson, IEEE J. Quantum. Electron. QE-18, 2016 (1982).
[CrossRef] [PubMed]

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
[CrossRef]

B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981); I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982); I. C. Khoo and S. L. Zhuang, IEEE J. Quantum Electron. QE-18, 246 (1981).
[CrossRef]

Chilingaryan, Yu. S.

B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981); I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982); I. C. Khoo and S. L. Zhuang, IEEE J. Quantum Electron. QE-18, 246 (1981).
[CrossRef]

Csillag, L.

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

deGennes, P. G.

See, for example, P. G. deGennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1984).

Hou, J. Y.

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Janossy, J.

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

Kaplan, A. E.

Khoo, I. C.

I. C. Khoo, T. H. Liu, and R. Normandin, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

See, for example, the review paper by I. C. Khoo and Y. R. Shen, Opt. Eng. 24, 579 (1985).

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Kitaeva, V. F.

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

Kroo, N.

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

Liu, T. H.

I. C. Khoo, T. H. Liu, and R. Normandin, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Normandin, R.

I. C. Khoo, T. H. Liu, and R. Normandin, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

Ong, H. L.

See, for example, H. L. Ong, Phys. Rev. A 28, 2393 (1983); B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981).
[CrossRef]

Pilipetski, N. F.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
[CrossRef]

Santamato, E.

Shen, Y. R.

See, for example, the review paper by I. C. Khoo and Y. R. Shen, Opt. Eng. 24, 579 (1985).

E. Santamato and Y. R. Shen, Opt. Lett. 9, 564 (1984).
[CrossRef] [PubMed]

Shepard, S.

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Sobolev, N. N.

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

Sukhov, A. V.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
[CrossRef]

Tabiryan, N. V.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
[CrossRef]

B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981); I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982); I. C. Khoo and S. L. Zhuang, IEEE J. Quantum Electron. QE-18, 246 (1981).
[CrossRef]

Yan, P. Y.

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Zel’dovich, B. Ya.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
[CrossRef]

B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981); I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982); I. C. Khoo and S. L. Zhuang, IEEE J. Quantum Electron. QE-18, 246 (1981).
[CrossRef]

Mol. Cryst. Liq. Cryst. (2)

L. Csillag, J. Janossy, V. F. Kitaeva, N. Kroo, and N. N. Sobolev, Mol. Cryst. Liq. Cryst. 84, 125 (1982).
[CrossRef]

I. C. Khoo, T. H. Liu, and R. Normandin, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

Opt. Commun. (1)

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya. Zel’dovich, Opt. Commun. 37, 280 (1981); S. R. Galstyan, O. V. Garibyan, N. V. Tabiryan, and Yu. S. Chilingaryan, JETP Lett. 33, 437 (1981); A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, and L. Chillag, JETP Lett. 32, 158 (1980); I. C. Khoo and S. L. Zhuang, Appl. Phys. Lett. 37, 3 (1980); S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Phys. Rev. Lett. 47, 1411 (1981).
[CrossRef]

Opt. Eng. (1)

See, for example, the review paper by I. C. Khoo and Y. R. Shen, Opt. Eng. 24, 579 (1985).

Opt. Lett. (2)

Phys. Rev. A (2)

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, and J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

See, for example, H. L. Ong, Phys. Rev. A 28, 2393 (1983); B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981).
[CrossRef]

Sov. Phys. JETP (1)

B. Ya. Zel’dovich, N. V. Tabiryan, and Yu. S. Chilingaryan, Sov. Phys. JETP 54, 32 (1981); I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982); I. C. Khoo and S. L. Zhuang, IEEE J. Quantum Electron. QE-18, 246 (1981).
[CrossRef]

Other (1)

See, for example, P. G. deGennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1984).

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Figures (7)

Fig. 1
Fig. 1

A linearly polarized laser beam incident upon a homeotropically aligned nematic-liquid-crystal film.

Fig. 2
Fig. 2

(a) Solution of the reorientation transverse profile R(r) for the case w0/d = 0.4 [w0 = 40 μm, d = 100 μm]; (b) R(r) for w0/d = 1; (c) R(r) for w0/d = 2. For all β = 0, d = 0.01 cm.

Fig. 3
Fig. 3

Plot of the width w0 of R(r) as a function of the ratio of incident laser beam waist to the thickness of the film (w0/d). Circles are experimentally observed points. (β = 0.)

Fig. 4
Fig. 4

Plot of the theoretically predicted threshold intensity Ith at which reorientation occurs versus w0/d. The value I(0) corresponds to the case involving an infinite plane wave (β = 0). Circles are experimentally observed points.

Fig. 5
Fig. 5

Plot of the width of the reorientation transverse profile wθ/w0 versus w0/d for the case β ≠ 0 (from previous calculations in Ref. 6). Circles are experimentally observed points.

Fig. 6
Fig. 6

Experimental setup for measuring the width of the director axis reorientation induced by a Gaussian laser beam. BS, beam splitter; L, lens; LC, liquid-crystal film.

Fig. 7
Fig. 7

(a) Transverse optical bistability for the case β = 22°, showing how the switching changes as the width of the reorientation profile is varied. Curve I: width of θ(r) = laser width, i.e., local response; curve II: width of θ(r) = 1.5-laser width; curve III: width of θ(r) = 2-laser width; curve IV: width of θ(r) = 4-laser width. (b) Transverse optical bistability for the case β = 0°. Curve I: local response, width of θ(r) = laser width; curve II: width of θ(r) = 1/3 laser width showing markedly different switching characteristics.

Equations (23)

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F 1 = k 1 2 ( · n ) 2 = k 1 2 ( cos 2 θ cos 2 ϕ θ r 2 2 sin θ cos θ cos ϕ θ r θ z + sin 2 θ θ z 2 ) ,
F 2 = = k 2 2 ( · × n ) 2 = k 2 2 sin 2 ϕ θ r 2 ,
F 3 = k 3 2 ( × × n ) 2 = ( sin θ sin ϕ cos θ cos ϕ θ r cos 2 θ cos ϕ θ z sin θ cos θ θ r ) 2 + ( cos ϕ sin θ cos θ cos ϕ θ z + cos ϕ sin 2 θ θ r + sin θ sin ϕ cos θ sin ϕ θ Z ) 2 + ( cos 2 θ sin ϕ θ Z + sin θ cos ϕ cos θ sin ϕ θ r ) 2 ,
F c = r d r d Z 0 2 π ( F 1 + F 2 + F 3 ) d ϕ = r d r d Z { [ F 1 ( r ) + F 2 ( r ) + F 3 ( r ) ] } ,
F 1 ( r ) = k 1 2 ( cos 2 θ θ r 2 + 2 sin 2 θ θ Z 2 ) ,
F 2 ( r ) = k 2 2 θ r 2 ,
F 3 ( r ) = k 3 2 ( 2 cos 2 θ θ Z 2 + sin 2 θ θ r 2 ) .
F 4 ( r ) = Δ 8 π / E op 2 ( r ) sin 2 θ .
Δ = .
f = r d r d z { [ F 1 ( r ) + F 2 ( r ) + F 3 ( r ) + F 4 ( r ) ] } .
r ( f θ r ) + Z ( f θ Z ) f θ = 0 ,
f = r [ F 1 ( r ) + F 2 ( r ) + F 3 ( r ) + F 4 ( r ) ] .
[ ( K 1 K 3 ) sin 2 θ + K 3 ] θ z z + 1 2 ( K 1 K 3 ) sin 2 θ ( θ z ) 2 + Δ 8 π E op 2 ( r ) sin 2 θ + 1 2 [ ( K 1 K 3 ) cos 2 θ + K 3 + K 2 ] θ r r + 1 4 ( K 3 K 1 ) sin 2 θ ( θ r ) 2 + ( K 1 K 3 ) cos 2 θ + K 2 + K 3 ) θ r / 2 r = 0.
θ ( r , z ) = R ( r ) sin ( π z d ) ,
F e 1 = 2 π k 0 d d z 0 r d r [ K 2 ( θ r 2 + θ Z 2 ) ] ,
F 1 = 2 π 0 d d z 0 r d r [ K 2 ( θ r 2 + θ Z 2 ) Δ 8 π E op 2 ( r ) sin 2 θ ] .
E op 2 ( r ) = E op 2 e a r 2 ,
F 1 = π k d 2 0 r d r [ ( d R / d r ) 2 + ( π d ) 2 + R 2 Δ 4 π k E op 2 e a r 2 R 2 + Δ E op 2 16 π K e a r 2 R 3 ] .
d d r [ I ( R , R ) R I ( R , R ) R ] = 0 ,
R + R r + [ b e a r 2 ( π d ) 2 ] R b 2 e a r 2 R 3 = 0 ,
R ( 0 ) = 0 , R ( ) = 0.
b th = π 2 / d 2 ( E th 2 = 4 π 3 K Δ 1 d 2 ) .
R 1 ( r ) + R 1 r + [ b cos 2 β e a r 2 ( π 2 ) 2 ] R 1 + b 2 e a r 2 sin 2 β = 0.

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