Abstract

We investigate numerically the propagation of self-trapped optical beams in nematic liquid crystals. Our analysis includes both spatial and temporal behavior. We display the formation of stable solitons in a narrow threshold region of beam intensities for fixed birefringence, and depict their spatiotemporal instabilities as the input intensity and the birefringence are increased. We demonstrate the breathing and filamentation of solitons above the threshold with increasing input intensity, and discover a convective instability with increasing birefringence. We consider the propagation of complex beam structures in nematic liquid crystals, such as dipoles, beam arrays, and vortices.

© 2005 Optical Society of America

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References

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  1. E. Braun, L. Faucheux, A. Libchaber, D. McLaughlin, D. Muraki, and M. Shelley, �??Filamentation and undulation of self-focused laser beams in liquid crystals,�?? Europhys. Lett. 23, 239 (1993).
    [CrossRef]
  2. M. A. Karpierz, �??Solitary waves in liquid crystalline waveguides,�?? Phys. Rev. E 66, 036603 (2002).
    [CrossRef]
  3. M. Warenghem, J. F. Henninot, and G. Abbate, �??Non linearly induced self waveguiding structure in dye doped nematic liquid crystals confined in capillaries,�?? Opt. Express 2, 483 (1998).
    [CrossRef] [PubMed]
  4. F. Derrien, J. F. Henninot, M. Warenghem, and G. Abbate, "A thermal (2D+1) spatial optical soliton in a dye doped liquid crystal,�?? J. Opt. A: Pure Appl. Opt. 2, 332 (2000).
    [CrossRef]
  5. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, �??Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,�?? Appl. Phys. Lett. 77, 7 (2000).
    [CrossRef]
  6. M. Peccianti and G. Assanto, �??Incoherent spatial solitary waves in nematic liquid crystals,�?? Opt. Lett. 26, 1791 (2001).
    [CrossRef]
  7. M. Peccianti, C. Conti, G. Assanto, �??Nonlocal optical propagation in nonlinear nematic liquid crystals,�?? J. Nonlin. Opt. Phys. Mater. 12, 525-538 (2003).
    [CrossRef]
  8. G. Assanto, M. Peccianti, K. Brzdakiewicz, A. De Luca, and C. Umeton, �??Nonlinear wave propagation and spatial solitons in nematic liquid crystals,�?? J. Nonlin. Opt. Phys. Mater. 12, 123-134 (2003).
    [CrossRef]
  9. G. Assanto, M. Peccianti, and C. Conti, �??Optical Spatial Solitons in Nematic Liquid Crystals,�?? Opt. Phot. News 14, No. 2, 45 (2003).
  10. Yuri S. Kivshar and Govind P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).
  11. P. G. De Gennes and G. Prost, The Physics of Liquid Crystals, 2nd edn (Oxford, Clarendon, 1993).
  12. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995).
  13. G. D�??Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities,�?? (December 2, 2002).
  14. M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaiser, �??Spatiotemporal effects in double phase conjugation, �?? J. Opt. Soc. Amer. B 12, 1602 (1995).
  15. D. Traeger, A. Strini�?, J. Schroeder, C. Denz, M. Beli�?, M. Petrovi�?, S. Matern, H.G. Purwins, �??Interactions in large arrays of solitons in photorefractive crystals,�?? J. Opt. A 5, S518-S523 (2003).
    [CrossRef]
  16. W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, �??Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,�?? J. Opt. B 6, S288-S294 (2004).
    [CrossRef]
  17. C. Conti, M. Peccianti, and G. Assanto, �??Route to Nonlocality and Observation of Accessible Solitons,�?? PRL 91, 073901 (2003).
    [CrossRef]
  18. M. Peccianti, C. Conti, and G. Assanto, �??Optical modulational instability in a nonlocal medium,�?? PRE 68, 025602(R) (2003).
    [CrossRef]
  19. On transverse patterns, consult C. Denz, M. Schwab, and C. Weilnau, Transverse Pattern Formation in Photorefractive Optics (Springer, Berlin, 2003).
    [CrossRef]
  20. A. W. Snyder and D. J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997).
    [CrossRef]
  21. D. Mitchell and A. Snyder, �??Soliton dynamics in a nonlocal medium,�?? JOSA B 16, 236-239 (1999).
    [CrossRef]
  22. M. Petrovi�? , D. Traeger, A. Strini�?, M. Beli�?, J. Schroeder, C. Denz, �??Solitonic lattices in photorefractive crystals,�?? Phys. Rev. E 68, 055601(R) (2003).
  23. Z. Chen and K. McCarthy, �??Spatial soliton pixels from partially incoherent light,�?? Opt. Lett. 27, 2019 (2002).
    [CrossRef]

Appl. Phys. Lett. (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, �??Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,�?? Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Europhys. Lett. (1)

E. Braun, L. Faucheux, A. Libchaber, D. McLaughlin, D. Muraki, and M. Shelley, �??Filamentation and undulation of self-focused laser beams in liquid crystals,�?? Europhys. Lett. 23, 239 (1993).
[CrossRef]

J. Nonlin. Opt. Phys. Mater. (2)

M. Peccianti, C. Conti, G. Assanto, �??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. Nonlin. Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

J. Opt. A (1)

D. Traeger, A. Strini�?, J. Schroeder, C. Denz, M. Beli�?, M. Petrovi�?, S. Matern, H.G. Purwins, �??Interactions in large arrays of solitons in photorefractive crystals,�?? J. Opt. A 5, S518-S523 (2003).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

F. Derrien, J. F. Henninot, M. Warenghem, and G. Abbate, "A thermal (2D+1) spatial optical soliton in a dye doped liquid crystal,�?? J. Opt. A: Pure Appl. Opt. 2, 332 (2000).
[CrossRef]

J. Opt. B (1)

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, �??Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,�?? J. Opt. B 6, S288-S294 (2004).
[CrossRef]

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

M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaiser, �??Spatiotemporal effects in double phase conjugation, �?? J. Opt. Soc. Amer. B 12, 1602 (1995).

JOSA B (1)

D. Mitchell and A. Snyder, �??Soliton dynamics in a nonlocal medium,�?? JOSA B 16, 236-239 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Phot. News (1)

G. Assanto, M. Peccianti, and C. Conti, �??Optical Spatial Solitons in Nematic Liquid Crystals,�?? Opt. Phot. News 14, No. 2, 45 (2003).

Phys. Rev. E (2)

M. A. Karpierz, �??Solitary waves in liquid crystalline waveguides,�?? Phys. Rev. E 66, 036603 (2002).
[CrossRef]

M. Petrovi�? , D. Traeger, A. Strini�?, M. Beli�?, J. Schroeder, C. Denz, �??Solitonic lattices in photorefractive crystals,�?? Phys. Rev. E 68, 055601(R) (2003).

PRE (1)

M. Peccianti, C. Conti, and G. Assanto, �??Optical modulational instability in a nonlocal medium,�?? PRE 68, 025602(R) (2003).
[CrossRef]

PRL (1)

C. Conti, M. Peccianti, and G. Assanto, �??Route to Nonlocality and Observation of Accessible Solitons,�?? PRL 91, 073901 (2003).
[CrossRef]

Science (1)

A. W. Snyder and D. J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997).
[CrossRef]

Other (5)

On transverse patterns, consult C. Denz, M. Schwab, and C. Weilnau, Transverse Pattern Formation in Photorefractive Optics (Springer, Berlin, 2003).
[CrossRef]

Yuri S. Kivshar and Govind P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).

P. G. De Gennes and G. Prost, The Physics of Liquid Crystals, 2nd edn (Oxford, Clarendon, 1993).

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

G. D�??Alessandro and A. A. Wheeler, "Bistability of liquid crystal micro-cavities,�?? (December 2, 2002).

Supplementary Material (17)

» Media 1: MOV (134 KB)     
» Media 2: MOV (252 KB)     
» Media 3: MOV (508 KB)     
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» Media 5: MOV (144 KB)     
» Media 6: MOV (42 KB)     
» Media 7: MOV (2486 KB)     
» Media 8: MOV (381 KB)     
» Media 9: MOV (89 KB)     
» Media 10: MOV (753 KB)     
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» Media 14: MOV (859 KB)     
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» Media 16: MOV (1532 KB)     
» Media 17: MOV (1521 KB)     

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Figures (11)

Fig. 1.
Fig. 1.

Soliton propagation, shown in the (y,z) plane, for different input intensities: a) I=0.5×10+10 V2/m2, b) movie for I=1.8×10+10 V2/m2 (134 KB) c) I=3×10+10 V2/m2, d) I=4×10+10 V2/m2, e) I=5×10+10 V2/m2, f) I=1×10+11 V2/m2, g) I=5×10+11 V2/m2, h) I=5×10+12 V2/m2, i) I=5×10+13 V2/m2. For all simulations FWHM=4 µm, L=0.5 mm and εa=0.5. Stationary situation is depicted, at t=5 τ.

Fig. 2.
Fig. 2.

Soliton propagation in the (y,z) plane, for different input intensities and εa=0.8. Upper row I(y,z); lower row θ(y,z) at t=10 τ. a) and g) I=0.5×10+10 V2/m2, b) and h) I=1×10+10 V2/m2, c) and i) I=1.8×10+10 V2/m2, d) and j) I=3×10+10 V2/m2, e) (253 KB) and k) (509 KB), movies for I=4×10+10 V2/m2, f) and l) I=5×10+12 V2/m2, all in V2/m2. For all simulations FWHM=4 µm and L=0.5 mm.

Fig. 3.
Fig. 3.

Soliton at the exit (x,y) plane, for t=5τ. a) I(x,y); b) θ(x,y). The input intensity I=1.8×10+10 V2/m2, FWHM=4 µm, L=0.5 mm, and εa=0.8. Here, x and y axes represent actual points as used in simulation; physical lengths spanned are the same for x and y (NLC film thickness) and amount to 116 µm.

Fig. 4.
Fig. 4.

Snapshots of θ(x,y) for soliton propagation, shown in the (x,y) plane, at different moments. The input intensity I=5×10+11 V2/m2, FWHM=4 µm, L=0.5 mm and εa=0.8. Here, x and y are presented in the same way as in Fig. 3.

Fig. 5.
Fig. 5.

Comparison of beam propagation, shown in the (y,z) plane, for different input intensities and different εa. Upper row εa=0.5, lower row εa=0.8. a) and e) I=0.5×10+10 V2/m2, b) and f) I=1.8×10+10 V2/m2, c) and g) I=4×10+10 V2/m2, d) and h) I=5×10+12 V2/m2. For all simulations FWHM=4 µm, L=0.5 mm, t=5 τ.

Fig. 6.
Fig. 6.

(a) Beam filamentation, shown in 3D, for input intensity I=5×10+11 V2/m2 at t=1.4 τ. FWHM=4 µm, L=0.5 mm and εa=0.5. (A) Movie of the view in the (y,z) plane (236 KB), (B) 3D view using four intensity levels (in relative units: 0.01, 1, 10, 30). (C) 3D view for the highest relative intensity (30).

Fig. 6.
Fig. 6.

(b) Continuation of a), the situation at t=2.95 τ, when the convective instability is about to end. FWHM=4 µm, L=0.5 mm and εa=0.5. (A) (y,z) plane, (B) 3D view, with three levels of intensity (in relative units: 1, 10, 100).

Fig. 7.
Fig. 7.

Dual beam propagation, shown in the (y,z) plane, for different distances between beams, which are in phase. Distances between the components are: (a) 8 µm, (b) 12 µm, (c) 16 µm, (d) 40 µm. Other data: input intensity I=1.8×10+10 V2/m2, FWHM=4 µm, L=0.5 mm and εa=0.5.

Fig. 8.
Fig. 8.

Movies of the dual beam propagation for different phase relations between beams. (a) (144 KB), (b) (42 KB) and (c) (2.486 MB) Components in phase, (d) (382 KB), (e) (89 KB) and (f) (753 KB) phase difference of π between the components. (a) and (d) Intensity in the (y,z) plane. (b) and (e) Intensity in the exit (x,y) plane. (c) and (f) Phase distributions in the (x,y) plane. Other data as in Fig. 7(a).

Fig. 9.
Fig. 9.

Movies of the propagation of 5×5 arrays, L=1mm, initial distance between the array components 20 µm. (a) (1.506 MB) and (b) (1.237 MB) Array in phase; (c) (1.493 MB) and (d) (859 KB) Array out-of-phase, with the chess-board arrangement of input phase. (a) and (c) Intensity in the (y,z) plane; (b) and (d) Intensity in the (x,y) plane. Other data are the same as in Fig 7.

Fig. 10.
Fig. 10.

Movies of the stable vortex propagation, (a) (1.205 MB) Intensity in the (y,z) plane. (b) (1.532 MB) Intensity, and (c) (1.522 MB) phase in the exit (x,y) plane. I=5×10+11 V2/m2, FWHM=8 µm, L=0.5 mm and εa=0.5.

Equations (5)

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

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