Abstract

The behavior of counterpropagating self-trapped optical beam structures in nematic liquid crystals is investigated. A time-dependent model for the beam propagation and the director reorientation in a nematic liquid crystal is numerically treated in three spatial dimensions and time. We find that the stable vector solitons can only exist in a narrow threshold region of control parameters. Bellow this region the beams diffract, above they self-focus into a series of focal spots. Spatiotemporal instabilities are observed as the input intensity, the propagation distance, and the birefringence are increased. We demonstrate undulation, filamentation, and convective dynamical instabilities of counterpropagating beams. Qualitatively similar behavior as of the copropagating beams is observed, except that it happens at lower values of control parameters.

© 2006 Optical Society of America

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References

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  1. P. G. De Gennes and G. Prost, The Physics of Liquid Crystals, (Oxford, Clarendon, 1993).
  2. C. KhooLiquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995)
  3. D. W. McLaughlin, D. J. Muraki, and M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
    [CrossRef]
  4. G. D’Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities," Phys. Rev. A 67, 023816 1-12 (2003).
  5. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
    [CrossRef]
  6. M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystals," Opt. Lett. 26, 1791-1793 (2001).
    [CrossRef]
  7. G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
    [CrossRef]
  8. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
    [CrossRef]
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    [CrossRef]
  11. A.I. Strinić, D.V. Timotijević, D. Arsenović, M.S. Petrović, and M.R. Belić, "Spatiotemporal optical instabilities in nematic solitons," Opt. Express 13, 493-498 (2005).
    [CrossRef] [PubMed]
  12. Y.S. Kivshar and G.P. AgrawalOptical Solitons (Academic Press, San Diego) (2003).
  13. D. Jović, M. Petrović, M. Belić, J. Schroeder, Ph. Jander, and C. Denz, "Dynamics of counterpropagating multipole vector solitons," Opt. Express 13, 10717-10728 (2005).
    [CrossRef] [PubMed]

2005 (3)

2004 (1)

2003 (3)

G. D’Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities," Phys. Rev. A 67, 023816 1-12 (2003).

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

2001 (1)

2000 (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

1996 (1)

D. W. McLaughlin, D. J. Muraki, and M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Arsenovic, D.

Assanto, G.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystals," Opt. Lett. 26, 1791-1793 (2001).
[CrossRef]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Beeckman, J.

Belic, M.

Belic, M.R.

Brzdakiewicz, K.

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

Cambournac, C.

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

D’Alessandro, G.

G. D’Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities," Phys. Rev. A 67, 023816 1-12 (2003).

De Luca, A.

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

De Rossi, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Denz, C.

Haelterman, M.

Hutsebaut, X.

Jander, Ph.

Jovic, D.

Khoo, I.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

McLaughlin, D. W.

D. W. McLaughlin, D. J. Muraki, and M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Muraki, D. J.

D. W. McLaughlin, D. J. Muraki, and M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Neyts, K.

Peccianti, M.

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystals," Opt. Lett. 26, 1791-1793 (2001).
[CrossRef]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Petrovic, M.

Petrovic, M.S.

Schroeder, J.

Shelley, M. J.

D. W. McLaughlin, D. J. Muraki, and M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Strinic, A.I.

Timotijevic, D.V.

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Wheeler, A. A.

G. D’Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities," Phys. Rev. A 67, 023816 1-12 (2003).

Appl. Phys. Lett. (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton and I. Khoo, "Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

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

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

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

G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, "Nonlinear wave propagation and spatial solitons in nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

G. D’Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities," Phys. Rev. A 67, 023816 1-12 (2003).

Physica D (1)

D. W. McLaughlin, D. J. Muraki, and M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Other (3)

P. G. De Gennes and G. Prost, The Physics of Liquid Crystals, (Oxford, Clarendon, 1993).

C. KhooLiquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995)

Y.S. Kivshar and G.P. AgrawalOptical Solitons (Academic Press, San Diego) (2003).

Supplementary Material (27)

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Figures (6)

Fig. 1.
Fig. 1.

The forward beam propagation, shown in the (y, z) plane (the first and third rows) and in the (x, y) plane (the second row), for different input intensities. a), f) and k) Diffracting beam, I=6×10+9 V2/m2. b), g) and l) Soliton, I=7×10+9 V2/m2. c), h) and m) One beam shift, I=8×10+9 V2/m2. d) (632 KB), i) (334 KB) and n) (1.381 MB) Two beam shifts, I=1×10+10 V2/m2. e) (131 KB), j) (229 KB) and o) (760 KB) Convective instability, I=9×10+10 V2/m2. The first and second rows depict intensity distributions, the third row θ(y,z). For all the simulations FWHM=4 µm, L=0.5 mm and εa=0.5.

Fig. 2.
Fig. 2.

Beam propagation, shown in 3D, for different input intensities: a) I=7×10+9 V2/m2 at t=15 τ, using two intensity levels (0.1 and 0.2 of I); b) I=1×10+10 V2/m2 at t=3.5 τ and t=4.8 τ, the same intensity levels.; c) I=9×10+10 V2/m2 at t=1 τ, using eight intensity levels (in relative units: 0.0001, 0.001, 0.01,0.1, 0.5, 1.0, 3.0 and 5.0). Parameters: FWHM=4 µm, L=0.5 mm and εa=0.5.

Fig. 3.
Fig. 3.

Comparison between the copropagating (the first row (708 KB, 591 KB, 296 KB)) and the counterpropagating (the second row (521 KB, 406 KB, 184 KB)) beams in NLC, for different input intensities, indicated in the figures. For all the simulations input FWHM=20 µm, L=0.5 mm and εa=0.5.

Fig. 4.
Fig. 4.

Intensity distributions at the output face of the crystal for two values of intensities: I=1×10+9 V2/m2 (the first row (71 KB, 1.428 MB, 1.303 MB, 1.749 MB)) and I=7×10+9 V2/m2 (the second row (116 KB, 108 KB, 873 KB, 1.123 MB, 1.290 MB)), for different input FWHM of beams (indicated in each figure). In all the simulations L=0.5 mm and εa=0.5.

Fig. 5.
Fig. 5.

The intensity distributions for εa=0.8 and for different input intensities: a) I=2×10+9 V2/m2, b) I=2.7×10+9 V2/m2, c) I=5×10+9 V2/m2 (757 KB), d) I=7×10+9 V2/m2 (518 KB). In all simulations FWHM=4 µm and L=0.5 mm.

Fig. 6.
Fig. 6.

CP vortices in the (y, z) plane (the first row) and in the (x, y) plane (the second row), for different input FWHM: 8 µm (first, second and third columns (125 KB, 82 KB)) and 23 µm (fourth, fifth and sixth columns (94 KB, 169 KB)), and for different intensities, noted in each figure. The first and fourth columns depict stable vortex propagation. The second and fifth columns display the filamentation of vortices. The third and sixth columns display dynamical instabilities. In all the simulations L=0.5 mm and εa=0.5.

Equations (7)

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2 i k A z + Δ x , y A + k 0 2 ε a ( sin 2 θ sin 2 ( θ rest ) ) A = 0 ,
2 i k B z + Δ x , y B + k 0 2 ε a ( sin 2 θ sin 2 ( θ rest ) ) B = 0 ,
θ rest ( z , V ) = θ 0 ( V ) + [ θ in θ 0 ( V ) ] [ exp ( z z ¯ ) + exp ( L z z ¯ ) ] ,
γ θ t = K Δ x , y θ + 1 4 ε 0 ε a sin ( 2 θ ) { A 2 + B 2 } ,
2 i A z + Δ x , y A + k 0 2 x 0 2 ε a ( sin 2 ( θ ) sin 2 ( θ rest ) ) A = 0 ,
2 i B z + Δ x , y B + k 0 2 x 0 2 ε a ( sin 2 ( θ ) sin 2 ( θ rest ) ) B = 0 ,
θ t = K τ γ x 0 2 Δ x , y θ + ε 0 ε a τ 4 γ sin ( 2 θ ) { A 2 + B 2 } ,

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