Abstract

The dynamics of the petal-like patterns observed in a liquid crystal light valve with optical feedback are investigated both experimentally and numerically. Based on a phenomenological model, the linear stability analysis predicts the bistability of the liquid crystal tilt angle for several voltage ranges that agrees with the experimental results. The static and dynamical features of the petal patterns obtained in the numerical simulation are very close to the experimental observations. When increasing the voltage applied to the light valve, the numerical simulations reveal a scenario of bifurcation from the static to a fluctuating petal pattern, which is confirmed by the experimental observations.

© 2008 Optical Society of America

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References

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  1. S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, "Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of 'optical turbulence'," JETP Lett. 47, 707-711 (1988).
  2. S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, A. V. Larichev, and N. I. Zheleznykh, "Controlling transverse-wave interactions in nonlinear optics: Generation and interaction of spatiotemporal structures," J. Opt. Soc. Am. B 9, 78-90 (1992).
    [CrossRef]
  3. M. A. Vorontsov and W. B. Miller, "Self-organization in nonlinear optics--kaleidoscope of patterns," in Self-Organization in Optical Systems and Applications in Information Technology, M.A.Vorontsov and W.B.Miller, eds. (Springer, 1995), pp. 1-27.
    [CrossRef]
  4. H. Adachihara and H. Faid, "Two-dimensional nonlinear-interferometer pattern analysis and decay of spirals," J. Opt. Soc. Am. B 10, 1242-1253 (1993).
    [CrossRef]
  5. W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151-153 (1990).
    [CrossRef]
  6. G. D'Alessandro and W. J. Firth, "Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror," Phys. Rev. Lett. 66, 2597-2600 (1991).
    [CrossRef] [PubMed]
  7. M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with two-dimensional feedback," Phys. Rev. A 49, 2891-2906 (1994).
    [CrossRef] [PubMed]
  8. F. T. Arecchi, S. Boccaletti, S. Ducci, E. Pampaloni, P. L. Ramazza, and S. Residori, "The liquid crystal light valve with optical feedback: A case study in pattern formation," J. Nonlinear Opt. Phys. Mater. 9, 183-204 (2000).
  9. R. Neubecker, G. L. Oppo, B. Thuering, and T. Tschudi, "Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects," Phys. Rev. A 52, 791-808 (1995).
    [CrossRef] [PubMed]
  10. S. Residori, "Patterns, fronts, and structures in a liquid-crystal-light-valve with optical feedback," Phys. Rep. 416, 201-272 (2005).
    [CrossRef]
  11. S. Residori, T. Nagaya, and A. Petrossian, "Optical localised structures and their dynamics," Europhys. Lett. 63, 531-537 (2003).
    [CrossRef]
  12. P. L. Ramazza, S. Residori, E. Pampaloni, and A. V. Larichev, "Transition to space-time chaos in a nonlinear optical system with two-dimensional feedback," Phys. Rev. A 53, 400-407 (1996).
    [CrossRef] [PubMed]
  13. Y. Iino and P. Davis, "Switching of self-organized patterns in mutually modulating liquid crystal devices for beam control," J. Appl. Phys. 85, 3399-3405 (1999).
    [CrossRef]
  14. Y. Iino and P. Davis, "Controlling spontaneous generation of optical beam spots in a liquid crystal device," J. Appl. Phys. 87, 8251-8258 (2000).
    [CrossRef]
  15. M. G. Clerc, A. Petrossian, and S. Residori, "Bouncing localized structures in a liquid-crystal light-valve experiment," Phys. Rev. E 71, 015205 (2005).
    [CrossRef]
  16. P. Aubourg, J. P. Huignard, M. Hareng, and R. A. Mullen, "Liquid crystal light valve using bulk monocrystalline Bi12SiO20 as the photoconductive material," Appl. Opt. 21, 3706-3712 (1982).
    [CrossRef] [PubMed]
  17. P. G. de Gennes and J. Prost, "Magnetic field effects," in The Physics of Liquid Crystals, 2nd ed. (Oxford Science Publications, 1993), pp. 117-133.
  18. I. C. Khoo, "Molecular reorientations in the nematic phase," in Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley-Interscience, 1995), pp. 132-139.
  19. I. C. Khoo, "Thermal and density optical nonlinearities of nematic liquid crystals in the visible-infrared spectrum," in Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley-Interscience, 1995), pp. 163-169.

2005 (2)

S. Residori, "Patterns, fronts, and structures in a liquid-crystal-light-valve with optical feedback," Phys. Rep. 416, 201-272 (2005).
[CrossRef]

M. G. Clerc, A. Petrossian, and S. Residori, "Bouncing localized structures in a liquid-crystal light-valve experiment," Phys. Rev. E 71, 015205 (2005).
[CrossRef]

2003 (1)

S. Residori, T. Nagaya, and A. Petrossian, "Optical localised structures and their dynamics," Europhys. Lett. 63, 531-537 (2003).
[CrossRef]

2000 (2)

Y. Iino and P. Davis, "Controlling spontaneous generation of optical beam spots in a liquid crystal device," J. Appl. Phys. 87, 8251-8258 (2000).
[CrossRef]

F. T. Arecchi, S. Boccaletti, S. Ducci, E. Pampaloni, P. L. Ramazza, and S. Residori, "The liquid crystal light valve with optical feedback: A case study in pattern formation," J. Nonlinear Opt. Phys. Mater. 9, 183-204 (2000).

1999 (1)

Y. Iino and P. Davis, "Switching of self-organized patterns in mutually modulating liquid crystal devices for beam control," J. Appl. Phys. 85, 3399-3405 (1999).
[CrossRef]

1996 (1)

P. L. Ramazza, S. Residori, E. Pampaloni, and A. V. Larichev, "Transition to space-time chaos in a nonlinear optical system with two-dimensional feedback," Phys. Rev. A 53, 400-407 (1996).
[CrossRef] [PubMed]

1995 (1)

R. Neubecker, G. L. Oppo, B. Thuering, and T. Tschudi, "Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects," Phys. Rev. A 52, 791-808 (1995).
[CrossRef] [PubMed]

1994 (1)

M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with two-dimensional feedback," Phys. Rev. A 49, 2891-2906 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

1991 (1)

G. D'Alessandro and W. J. Firth, "Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror," Phys. Rev. Lett. 66, 2597-2600 (1991).
[CrossRef] [PubMed]

1990 (1)

W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151-153 (1990).
[CrossRef]

1988 (1)

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, "Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of 'optical turbulence'," JETP Lett. 47, 707-711 (1988).

1982 (1)

Appl. Opt. (1)

Europhys. Lett. (1)

S. Residori, T. Nagaya, and A. Petrossian, "Optical localised structures and their dynamics," Europhys. Lett. 63, 531-537 (2003).
[CrossRef]

J. Appl. Phys. (2)

Y. Iino and P. Davis, "Switching of self-organized patterns in mutually modulating liquid crystal devices for beam control," J. Appl. Phys. 85, 3399-3405 (1999).
[CrossRef]

Y. Iino and P. Davis, "Controlling spontaneous generation of optical beam spots in a liquid crystal device," J. Appl. Phys. 87, 8251-8258 (2000).
[CrossRef]

J. Mod. Opt. (1)

W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151-153 (1990).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

F. T. Arecchi, S. Boccaletti, S. Ducci, E. Pampaloni, P. L. Ramazza, and S. Residori, "The liquid crystal light valve with optical feedback: A case study in pattern formation," J. Nonlinear Opt. Phys. Mater. 9, 183-204 (2000).

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, "Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of 'optical turbulence'," JETP Lett. 47, 707-711 (1988).

Phys. Rep. (1)

S. Residori, "Patterns, fronts, and structures in a liquid-crystal-light-valve with optical feedback," Phys. Rep. 416, 201-272 (2005).
[CrossRef]

Phys. Rev. A (3)

M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with two-dimensional feedback," Phys. Rev. A 49, 2891-2906 (1994).
[CrossRef] [PubMed]

P. L. Ramazza, S. Residori, E. Pampaloni, and A. V. Larichev, "Transition to space-time chaos in a nonlinear optical system with two-dimensional feedback," Phys. Rev. A 53, 400-407 (1996).
[CrossRef] [PubMed]

R. Neubecker, G. L. Oppo, B. Thuering, and T. Tschudi, "Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects," Phys. Rev. A 52, 791-808 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. G. Clerc, A. Petrossian, and S. Residori, "Bouncing localized structures in a liquid-crystal light-valve experiment," Phys. Rev. E 71, 015205 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

G. D'Alessandro and W. J. Firth, "Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror," Phys. Rev. Lett. 66, 2597-2600 (1991).
[CrossRef] [PubMed]

Other (4)

M. A. Vorontsov and W. B. Miller, "Self-organization in nonlinear optics--kaleidoscope of patterns," in Self-Organization in Optical Systems and Applications in Information Technology, M.A.Vorontsov and W.B.Miller, eds. (Springer, 1995), pp. 1-27.
[CrossRef]

P. G. de Gennes and J. Prost, "Magnetic field effects," in The Physics of Liquid Crystals, 2nd ed. (Oxford Science Publications, 1993), pp. 117-133.

I. C. Khoo, "Molecular reorientations in the nematic phase," in Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley-Interscience, 1995), pp. 132-139.

I. C. Khoo, "Thermal and density optical nonlinearities of nematic liquid crystals in the visible-infrared spectrum," in Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley-Interscience, 1995), pp. 163-169.

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Figures (17)

Fig. 1
Fig. 1

Basic structure of the LCLV. LC, nematic liquid crystal layer; M, dielectric mirror; PC, photoconductor layer; ITO, indium thin oxide transparent electrode.

Fig. 2
Fig. 2

Experimental setup to observe self-organized patterns. P, polarizer; O, microscope objective; PH, pinhole; BS 1 , 2 , beam splitters; L 1 , 2 , lenses; M, mirror; A, polarizer; S, screen; f, focal length of L 1 , 2 ; z, focal plane for L 2 ; L, free propagation length; Δ, feedback rotation angle.

Fig. 3
Fig. 3

Bifurcation diagrams of ϕ ( V ) for (a) numerical simulations and (b) experiment. The black and gray curves in (a) are the stable, and unstable branches, respectively. The black and gray curves in (b) correspond to increasing, and decreasing the voltage, respectively. Most of the black curve, except for the three bistable regions, is overlaid by the gray curve.

Fig. 4
Fig. 4

(a) Numerical simulations and (b) experimental snapshots of the static petal patterns observed under commensurate feedback rotation angles, Δ = 60 ° , 45 ° , 36 ° , and 30 ° . The diameter of the system is l 0 = 10 mm .

Fig. 5
Fig. 5

(a) Numerical simulations and (b) experimental snapshots showing the dependence of the sixfold petal pattern on the applied voltage.

Fig. 6
Fig. 6

Time evolution of the fluctuating sixfold pattern: (a) numerical simulations for V = 1.970 V , (b) experiment for V = 2.07 V .

Fig. 7
Fig. 7

Angular velocity measured at r = 2.5 mm as a function of the feedback rotation angle. The lines are a guide for the eyes. (a) Numerical simulations for V = 1.90 V , (b) experiment for V = 2.05 V .

Fig. 8
Fig. 8

Deformation process from the fourfold to the fivefold petals states under Δ = 37 ° : (a) numerical simulations for V = 1.970 V , (b) experiment for V = 2.07 V .

Fig. 9
Fig. 9

Numerical results for the sixfold petal pattern. Upper images: spatiotemporal plot. Lower images: pattern snapshots. V = ( a ) 1.925, (b) 1.950, (c) 1.960, and (d) 1.966 V .

Fig. 10
Fig. 10

Time evolution of the squared intensity of the (a) k θ = 6 and (b) k θ = 3 modes at r = 3 mm .

Fig. 11
Fig. 11

Experimental results for the sixfold petal pattern for V = ( a ) 1.950, (b) 2.020, (c) 2.0343 , and (d) 2.100 V .

Fig. 12
Fig. 12

Numerical results for the fivefold petal pattern. Upper images: spatiotemporal plot. Lower images: pattern snapshots. V = ( a ) 1.925, (b) 1.950, and (c) 1.966 V .

Fig. 13
Fig. 13

Experimental results for the fivefold petal pattern under V = ( a ) 2.000, (b) 2.030, and (c) 2.040 V .

Fig. 14
Fig. 14

Experimental setup to evaluate the parameters a ( V ) , b ( V ) , and s ( V ) . O, microscope objective; BS, nonpolarizing beam splitter; PBS, polarizing beam splitter; M, mirror; ND, neutral density filter.

Fig. 15
Fig. 15

Dependence of I on the applied voltage V and writing light intensity I w . The abscissa and ordinate correspond to the applied voltage 0 V < V < 10 V and the intensity of the writing light 0.0027 mW cm 2 < I w < 0.3072 mW cm 2 , respectively. The gray level is proportional to I .

Fig. 16
Fig. 16

Nonlinear response function F ( I w ) for V = 2.00 , 2.51, and 3.02 V . The marks and the solid curves show the experimental data and the fitting curves based on Eqs. (2, A1), respectively.

Fig. 17
Fig. 17

Dependence of the parameters (a) a ( V ) and (b) b ( V ) , s ( V ) on the applied voltage. The marks and the solid curves show the experimental data and the fitting curves, respectively. (a) a ( V ) = 120.07 + 263.88 V ̃ 179.361 V ̃ 2 + 56.146 V ̃ 3 10.396 V ̃ 4 + 1.0989 V ̃ 5 6.2024 × 10 2 V ̃ 6 + 1.4559 × 10 3 V ̃ 7 , (b) b ( V ) = 1.0 0.104 exp ( 0.0146 V ̃ ) 1.31 exp ( 0.371 V ̃ ) , s ( V ) = 1.0 0.156 exp ( 0.0556 V ̃ ) 1.64 exp ( 0.639 V ̃ ) , where V ̃ is the dimensionless normalized voltage defined by V ̃ = V V F , V F = 0.9759 V .

Equations (16)

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τ u ( r , t ) t = u ( r , t ) + l d 2 2 u ( r , t ) + R { F [ I w ( r , t ) ] } ,
F ( I w ) = s a I w ˜ + b a I w ˜ + 1 ,
I w ˜ = I w I ref ,
ϕ = ϕ 0 ( 1 u ) ,
ϕ 0 = 2 d k δ n ,
δ n ( α ) = n e n o n o 2 cos 2 α + n e 2 sin 2 α n o ,
δ n ( α ) δ n cos 2 α .
I w = e γ I 0 2 ( 1 cos ϕ ) ,
τ ϕ ( r , t ) t = ϕ ( r , t ) + ϕ 0 + l d 2 2 ϕ ( r , t ) ϕ 0 R { F [ I w ( r , t ) ] } .
r ̃ = r l 0 ,
t ̃ = t τ 0 ,
V ̃ = V V F ,
τ ̃ = τ τ 0 ,
l d ̃ = l d l 0 ,
τ ̃ ϕ ( r ̃ , t ̃ ) t ̃ = ϕ ( r ̃ , t ̃ ) + ϕ 0 + l d ̃ 2 ̃ 2 ϕ ( r ̃ , t ̃ ) ϕ 0 R { F [ I w ( r ̃ , t ̃ ) ] } ,
I 1 cos ϕ .

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