## Spatiotemporal optical instabilities in nematic solitons

Optics Express, Vol. 13, Issue 2, pp. 493-504 (2005)

http://dx.doi.org/10.1364/OPEX.13.000493

Acrobat PDF (538 KB)

### 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

## 1. Introduction

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]

*et al.*[2

2. M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E **66**, 036603 (2002). [CrossRef]

*et al.*[3

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]

7. M. Peccianti, C. Conti, and 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]

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, and G. Assanto, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlin. Opt. Phys. Mater. **12**, 525–538 (2003). [CrossRef]

## 2. The model

*θ*of the director, measured from the propagation direction. In the presence of an externally applied (low frequency) voltage the evolution of a slowly-varying beam envelope

*A*, linearly polarized along

*x*axis and propagating along

*z*axis, is well described by the paraxial wave equation [5

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]

*k*=

*k*

_{0}

*n*

_{0}is the wave vector in the medium and

*ε*

_{a}=

*θ*

_{rest}in the presence of a low-frequency electric field is modeled by [5

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]

*θ*

_{0}(V) being the orientation distribution due to the applied voltage far from the input interface,

*θ*

_{in}is the director orientation at the boundary z=0, and

*z̄*is the relaxation distance. Temporal evolution of the angle of reorientation is given by the diffusion equation [12, 13]:

*γ*is the viscous coefficient and

*K*is Frank’s elastic constant. Here

*θ*is the overall tilt angle, due to both light and voltage.

*zk*

*x*x

_{0}, y=

*y*x

_{0}and t=

*t*τ, we transform the equations into a dimensionless form:

_{0}is the transverse scaling length. Equations (4) and (5) form the basis of the model.

15. D. Traeger, A. Strinić, J. Schroeder, C. Denz, M. Belić, M. Petrović, S. Matern, and H.G. Purwins, “Interactions in large arrays of solitons in photorefractive crystals,” J. Opt. A **5**, S518–S523 (2003). [CrossRef]

*θ*and

*A*at each moment of time that will satisfy Eqs. (4) and (5). Starting from a given distribution of

*θ*, the incident field

*A*at

*z*=

*0*is integrated along

*z*, to obtain a distribution of

*A*. Then the distribution of

*θ*is integrated for a time step at each

*z*point. Note that this entails the solution of a partial differential equation. The field

*A*now does not correspond to the new distribution of

*θ*, and it has to be propagated again along

*z*. This two-step procedure is iterated until stable self-consistent distributions of

*θ*and

*A*are obtained. Then the distributions are updated and the temporal loop advanced for a time step. The convergence in both temporal and self-consistency loops signifies that a steady-state solution is found. However, this need not always be the case. Time-dependent, dynamical states can also be observed, when the temporal loop refuses to converge. Our procedure is uniquely suited for observing the dynamical states in slowly-varying physical systems, where the fast optical fields are slaved to the slow change in the nonlinearity.

*Ld*=

*k*

*L*=6.3

*Ld*=0.5 mm, the transverse scaling length x

_{0}=2 µm, the laser wavelength λ=514 nm, the relaxation distance

*z̄*=40 µm, the elastic constant

*K*=0.7 10

^{-11}N, the viscous coefficient

*γ*=0.08 kg/ms, the ordinary refractive index

*n*

_{0}=1.53, the director orientation at the boundary

*θ*

_{in}=π/2, the orientation distribution

*θ*

_{0}=π/4, and the initial beam widths FWHM=4 µm. Two values for the birefringence are used, ε

_{a}=0.5 and ε

_{a}=0.8, and the intensity is varied between

*I*=0.5×10

^{+10}V

^{2}/m

^{2}and

*I*=5×10

^{+13}V

^{2}/m

^{2}. All of these data are consistent with the values reported in experimental investigations [5

**77**, 7 (2000). [CrossRef]

## 3. Numerical results

**77**, 7 (2000). [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]

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

*x*,

*y*) plane are presented at the exit face of the crystal (

*z*=

*L*), and all the pictures and movies in the (

*y*,

*z*) plane are for the

*x*=

*0*plane (in the middle of the crystal).

## 3.1 Solitons

7. M. Peccianti, C. Conti, and G. Assanto, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlin. Opt. Phys. Mater. **12**, 525–538 (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]

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]

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

*I*=1.8×10

^{+10}V

^{2}/m

^{2}to

*I*=3×10

^{+10}V

^{2}/m

^{2}, depicted in Fig. 1(c). It should be mentioned that, with Gaussian input, a slight beam width modulation is always present, as the soliton forms. These width modulations become more pronounced as the intensities are increased. In that range of intensities we see a periodical spot size variation along propagation – the breathing of soliton. The pitch of width modulation increases with intensity increase (Fig. 1(d–f)). Eventually, the soliton is broken up (Fig. 1(h), 1(i)) by joint action of the longitudinal modulation instability, just described, and the transversal one, mixing in at the intensities above

*I*=5×10

^{+11}V

^{2}/m

^{2}, (Fig. 1(g)). Transverse modulational instabilities also cause filamentation, examples of which will be given later (Fig. 6(a)).

_{a}=0.8, a similar behavioral pattern is observed, but this time for different intensities, e.g., stable solitons emerge at a smaller intensity

*I*=1.0×10

^{+10}V

^{2}/m

^{2}(Fig. 2). A comparison between the beam and angle reorientation distributions in the same figure reveals that they change in unison, the angle reorientation distribution being wider. This comes because of the nonlocal nature of nonlinearity. In Fig. 3 a comparison between the optical intensity distribution and the reorientation angle distribution shows this clearly (for the case from Fig. 2(c) and 2(i)). The profiles are made in the (

*x*,

*y*) plane, at the same moment. They exhibit similar behavior to the one reported in [8

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]

19.
On transverse patterns, consult
C. Denz, M. Schwab, and C. Weilnau, *Transverse Pattern Formation in Photorefractive Optics* (Springer, Berlin, 2003). [CrossRef]

_{a}is depicted, ε

_{a}=0.5 (Fig. 5(a–d)) and ε

_{a}=0.8 (Fig. 5(e–h)). One can see that there exist similar phases in the propagation for different ε

_{a}, which happen at different intensities (i.e., the soliton instabilities develop similarly). The values of intensity where the stable soliton propagation is observed are different for the two cases. For higher ε

_{a}this value is smaller and similar phases in filamentation develop at lower intensities.

*I*=5×10

^{+11}V

^{2}/m

^{2}) with a movie presented in the same figure. These two figures represent 3D view at different moments (for t=1.4 τ, and t=2.95 τ). Several isosurfaces of the same soliton are merged onto the same picture, represented by different colors, the smaller intensities are lighter, the higher are darker.

## 3.2 Dipoles

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]

*θ*perturbations overlap. In a local medium the index overlap coincides with the field-intensity overlap. In NLC, which represents an example of strongly nonlocal medium, the perturbation diffuses out of the excitation regions (Fig. 2), providing for interaction between solitons over wider distances. In such a case [8

**12**, 123–134 (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]

**12**, 123–134 (2003). [CrossRef]

**12**, 123–134 (2003). [CrossRef]

*I*=0) between the beam components, the crossing is avoided, and no interlacing occurs.

## 3.3 Arrays

15. D. Traeger, A. Strinić, J. Schroeder, C. Denz, M. Belić, M. Petrović, S. Matern, and H.G. Purwins, “Interactions in large arrays of solitons in photorefractive crystals,” J. Opt. A **5**, S518–S523 (2003). [CrossRef]

23. Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt.Lett. **27**, 2019 (2002). [CrossRef]

## 3.4 Vortices

*I*=5×10

^{+11}V

^{2}/m

^{2}and FWHM=8 µm a stable self-focused structure is obtained. For smaller intensity, the vortex could not get through the medium, i.e., it was diffracted. For the same input intensity and smaller vortex width, FWHM=4 µm, a breathing, almost stable structure is seen. For smaller intensity and the same FWHM=4 µm, vortex did not propagate through. For the same input intensity and two times broader vortex (FWHM=8 µm), we finally got a stable structure.

## 4. Conclusions

## Acknowledgments

## References and links

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. |

2. | M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E |

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 |

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. |

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. |

6. | M. Peccianti and G. Assanto, “Incoherent spatial solitary waves in nematic liquid crystals,” Opt. Lett. |

7. | M. Peccianti, C. Conti, and G. Assanto, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlin. Opt. Phys. Mater. |

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. |

9. | G. Assanto, M. Peccianti, and C. Conti, “Optical Spatial Solitons in Nematic Liquid Crystals,” Opt. Phot. News |

10. | Yuri S. Kivshar and Govind P. Agrawal, |

11. | P. G. De Gennes and G. Prost, |

12. | I. C. Khoo, |

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. |

15. | D. Traeger, A. Strinić, J. Schroeder, C. Denz, M. Belić, M. Petrović, S. Matern, and H.G. Purwins, “Interactions in large arrays of solitons in photorefractive crystals,” J. Opt. A |

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 |

17. | C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” PRL |

18. | M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” PRE |

19. |
On transverse patterns, consult
C. Denz, M. Schwab, and C. Weilnau, |

20. | A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science |

21. | D. Mitchell and A. Snyder, “Soliton dynamics in a nonlocal medium,” JOSA B |

22. | M. Petrović, D. Traeger, A. Strinić, M. Belić, J. Schroeder, and C. Denz, “Solitonic lattices in photorefractive crystals,” Phys. Rev. E |

23. | Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt.Lett. |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 19, 2004

Revised Manuscript: January 12, 2005

Published: January 24, 2005

**Citation**

A. Strini�?, D. Timotijevi�?, D. Arsenovi�?, M. Petrovi�?, and M. Beli�?, "Spatiotemporal optical instabilities in nematic solitons," Opt. Express **13**, 493-504 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-493

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### References

- 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]
- M. A. Karpierz, �??Solitary waves in liquid crystalline waveguides,�?? Phys. Rev. E 66, 036603 (2002). [CrossRef]
- 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]
- 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]
- 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]
- M. Peccianti and G. Assanto, �??Incoherent spatial solitary waves in nematic liquid crystals,�?? Opt. Lett. 26, 1791 (2001). [CrossRef]
- 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]
- G. Assanto, M. Peccianti, and C. Conti, �??Optical Spatial Solitons in Nematic Liquid Crystals,�?? Opt. Phot. News 14, No. 2, 45 (2003).
- 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).
- M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaiser, �??Spatiotemporal effects in double phase conjugation, �?? J. Opt. Soc. Amer. B 12, 1602 (1995).
- 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]
- 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]
- C. Conti, M. Peccianti, and G. Assanto, �??Route to Nonlocality and Observation of Accessible Solitons,�?? PRL 91, 073901 (2003). [CrossRef]
- M. Peccianti, C. Conti, and G. Assanto, �??Optical modulational instability in a nonlocal medium,�?? PRE 68, 025602(R) (2003). [CrossRef]
- On transverse patterns, consult C. Denz, M. Schwab, and C. Weilnau, Transverse Pattern Formation in Photorefractive Optics (Springer, Berlin, 2003). [CrossRef]
- A. W. Snyder and D. J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997). [CrossRef]
- D. Mitchell and A. Snyder, �??Soliton dynamics in a nonlocal medium,�?? JOSA B 16, 236-239 (1999). [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).
- Z. Chen and K. McCarthy, �??Spatial soliton pixels from partially incoherent light,�?? Opt. Lett. 27, 2019 (2002). [CrossRef]

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