## Breathing solitons in nematic liquid crystals

Optics Express, Vol. 17, Issue 14, pp. 11698-11709 (2009)

http://dx.doi.org/10.1364/OE.17.011698

Acrobat PDF (1158 KB)

### Abstract

Dynamical and steady-state behavior of beams propagating in nematic liquid crystals (NLCs) is analyzed. A well-known model for the beam propagation and the director reorientation angle in a NLC cell is treated numerically in space and time. The formation of steady-state soliton breathers in a threshold region of beam intensities is displayed. Below the region the beams diffract, above the region spatiotemporal instabilities develop, as the input intensity and the material parameters are varied. Curiously, the only kind of solitons we could demonstrate in our numerical studies was the breathers. Despite repeated efforts, we could not find the solitons with a steady profile propagating in the NLC model at hand.

© 2009 Optical Society of America

## 1. Introduction

*θ*. The light incident on a NLC modifies the electric permittivity tensor, leading to the reorientational nonlinearity. For these reasons the propagation of self-focused beams [2] in NLCs have been subjected to increased scrutiny in recent years, in both experimental [3

3. 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]

4. X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, “Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal,” J. Opt. Soc. Am. B **22**, 1424–1431 (2005).
[CrossRef]

5. A.I. Strinić, D.V. Timotijević, D. Arsenović, M.S. Petrović, and M.R. Belic, “Spatiotemporal optical instabilities in nematic solitons,” Opt. Express **13**, 493–504 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-2-493.
[CrossRef] [PubMed]

8. J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. **375**, 631–640 (2002).
[CrossRef]

*θ*̂ (0,y,z) along the propagation axis, in the middle of the crystal. We demonstrate the formation of soliton breathers in a threshold region of beam intensities. Despite repeated efforts and careful numerics, we could not observe steady spatial solitons, propagating with an unchanging transverse profile. The only kind of solitons we could identify in the model at hand was the breathers. We observe self-focusing and modulational instabilities (MI). The effects of self-focusing and breathing we describe resemble the physics of self-written waveguides, first noted in photosensitive glass [9

9. T. M. Monro, C. Martijn de Sterke, and L. Poladian, “Investigation of waveguide growth in photosensitive germanosilicate glass,” J. Opt. Soc. Am. B. **13**, 2824–2832 (1996).
[CrossRef]

12. A. S. Kewitsch and A. Yariv, “Nonlinear optical properties of photoresists for projection lithography,” Appl. Phys. Lett. **68**, 455–457 (1996).
[CrossRef]

14. K. D. Dorkenoo, F. Gillot, O. Crégut, Y. Sonnefraud, A. Fort, and H. Leblond, “Control of the refractive index in photopolymerizable materials for (2+1)D solitary wave guide formation,” Phys. Rev. Lett. **93**, 143905 1–4 (2004).
[CrossRef]

*i.e.*the effects observed in 3 spatial dimensions, after temporal steady state is reached in the system. To ascertain the reliability of our numerical procedure and the reality of effects we report upon, in the end a comparison is presented with the steady-state numerical method, based on a successive over-relaxation (SOR) algorithm.

## 2. The model

*θ*of the director in the transverse plane. In the presence of an external low frequency electric field the spatial evolution of a slowly-varying beam envelope

*A*, linearly polarized along the

*x*axis and propagating along the

*z*axis, is well captured by the dimensionless paraxial wave equation [15

15. 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]

_{x,y}is the transverse Laplacian. The temporal evolution of the angle of reorientation

*θ*is described by the dimensionless diffusion equation [4

4. X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, “Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal,” J. Opt. Soc. Am. B **22**, 1424–1431 (2005).
[CrossRef]

16. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express **12**, 1011–1018 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-6-1011.
[CrossRef] [PubMed]

17. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time-dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. **41**, 735–740 (2005).
[CrossRef]

*γ*is the viscous coefficient and

*K*is Frank’s elastic constant,

*τ*is the director relaxation time, and

*k*=

*k*is the wave vector in the medium,

_{0}n_{0}*x*the transverse scaling length, and Δ

_{0}*ε*=

^{OPT}*n*is the optical permittivity anisotropy of the liquid-crystal molecules.

_{e}^{2}-n_{0}^{2}*E*=

^{DC}*V*/

*D*is the applied field strength (

*V*is the applied bias voltage,

*D*is the cell thickness) and Δε

^{DC}is the static permittivity anisotropy of the liquid-crystal molecules.

*θ*is the overall tilt angle (the total orientation of the molecules with respect to the

*z*axis), owing to both light and voltage:

*θ*=

*θ*

_{0}+

*θ*̂, where the angle

*θ*accounts for the molecular orientation induced by the static electric field only, while the quantity

_{0}*θ*̂ corresponds to the optically induced molecular reorientation. Hard boundary conditions on the molecular orientation at the transverse borders of the liquid-crystal cell are introduced by relation (3).

5. A.I. Strinić, D.V. Timotijević, D. Arsenović, M.S. Petrović, and M.R. Belic, “Spatiotemporal optical instabilities in nematic solitons,” Opt. Express **13**, 493–504 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-2-493.
[CrossRef] [PubMed]

*Ld*=

*k*x

_{0}

^{2}=76.6 µm, the propagation distance (

*i.e.*the cell length)

*L*=20

*Ld*=1.5 mm, the transverse scaling length x

_{0}=2 µm, the laser wavelength λ=514 nm, the elastic constant

*K*=12 10

^{-12}N, the viscous coefficient γ=0.08 kg/ms, the ordinary refractive index

*n*=1.53, the cell thickness

_{0}*D*=75 µm, and the bias voltage

*V*=1V. All of these data are consistent with the values reported in experimental investigations [4

4. X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, “Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal,” J. Opt. Soc. Am. B **22**, 1424–1431 (2005).
[CrossRef]

16. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express **12**, 1011–1018 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-6-1011.
[CrossRef] [PubMed]

*I*=1.0×10

^{9}V

^{2}/m

^{2}and

*I*=5×10

^{12}V

^{2}/m

^{2}.

5. A.I. Strinić, D.V. Timotijević, D. Arsenović, M.S. Petrović, and M.R. Belic, “Spatiotemporal optical instabilities in nematic solitons,” Opt. Express **13**, 493–504 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-2-493.
[CrossRef] [PubMed]

18. A. Strinić, D. Jović, M. Petrović, D. Timotijević, N. Aleksić, and M. Belić, “Counterpropagating beams in nematic liquid crystals,” Opt. Express **14**, 12310–12315 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12310.
[CrossRef] [PubMed]

*θ*=

*θ*

_{0}+

*θ*̂. Previously

*θ*was given by a simple approximate formula or held constant,

_{0}*θ*=

_{0}*π/4*. Here, the initial distribution

*θ*is determined in the beginning, independent of the solution of the full system of Eqs. (1) and (2), using boundary conditions and a SOR algorithm for solving PDEs. Such a procedure is preferable when one deals with a steady-state boundary-value problem, as is the case of

_{0}*θ*

_{0}. Only after

*θ*

_{0}is determined the main integration of Eqs. (1) and (2) begins. This provides for a more realistic physical modeling of the system.

19. 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]

23. G. Assanto, M. Peccianti, and C. Conti, “One dimensional transverse modulational instability in nonlocal media witha reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. **10**, 862–869 (2004).
[CrossRef]

20. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. **91**, 073901 1–4 (2003).
[CrossRef]

23. G. Assanto, M. Peccianti, and C. Conti, “One dimensional transverse modulational instability in nonlocal media witha reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. **10**, 862–869 (2004).
[CrossRef]

## 3. Results of numerical simulation

### 3.1 Propagation of Gaussian beam in three spatial dimensions and time

#### 3.1.1 Influence of input beam intensity

^{11}V

^{2}/m

^{2}[24].

*I*(0,y,z) and the angle reorientation distributions

*θ*̂ (0,y,z) are reached, as functions of the propagation distance, for different input beam intensities. The beam intensity

*I*is proportional to |

*A*|

^{2}. The region of intensities in the figure corresponds to the threshold region in which the breathing solitons (BSs) are found. Figure 1 represents typical behavior during the propagation of input Gaussian beams in NLCs. A comparison between the beam intensity and the reorientation angle distributions shows that they change in unison, as they should; Eq. (2) is solved independently at every

*z*position, using the current propagating value of

*A*from Eq. (1). The reorientation angle distribution follows closely the intensity distribution, but owing to broader nonlocality for the given parameters, it is always wider. The profiles exhibit similar behavior to the experimental profiles reported in [4

**22**, 1424–1431 (2005).
[CrossRef]

17. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time-dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. **41**, 735–740 (2005).
[CrossRef]

25. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. **12**, 525–538 (2003).
[CrossRef]

26. J.F. Henninot, J.F. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” J. Opt. A: Pure Appl. Opt. **10**, 085704 (2008).
[CrossRef]

**22**, 1424–1431 (2005).
[CrossRef]

17. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time-dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. **41**, 735–740 (2005).
[CrossRef]

25. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. **12**, 525–538 (2003).
[CrossRef]

26. J.F. Henninot, J.F. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” J. Opt. A: Pure Appl. Opt. **10**, 085704 (2008).
[CrossRef]

#### 3.1.2 Influence of the input beam width FWHM

^{OPT}=0.4 and Δε

^{DC}=14.5, similar behavior is found. In Fig. 2(a) we show the cases of BS propagation, for FWHM=2 µm, 2.5 µm, 3 µm, 3.5 µm, 4 µm, 4.5 µm and 5 µm. For each input beam width, appropriate input beam intensity can be found to establish the existence of a breather. We also note that the changes in FWHM cause the changes in the period of oscillation. In Fig. 2(b) we depict the intensities of BSs as functions of the input widths (FWHM ranging as in Fig. 2(a)). For smaller FWHM higher input intensities are needed for the emergence of breathers. The curve fitted through the points is

*y*=

*1*/

*x*. Such a functional dependence is predicted by Snyder and Mitchell [27

^{4}27. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science **276**, 1538–1541 (1997).
[CrossRef]

21. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. **92**, 113902 1–4 (2004).
[CrossRef]

25. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. **12**, 525–538 (2003).
[CrossRef]

26. J.F. Henninot, J.F. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” J. Opt. A: Pure Appl. Opt. **10**, 085704 (2008).
[CrossRef]

*I*=8.6×

^{10}10 V

^{2}/m

^{2}, Fig. 2(a)), we present in Fig. 3 the intensity and the optically induced molecular reorientation in the middle of the crystal,

*I*(0,0,

*z*) and

*θ*̂ (

*0*,

*0*,

*z*) and the corresponding FWHM of its transverse profiles, as functions of the propagation distance. In Fig. 3 we also depict the transverse profiles of BS and the induced tilt angle at the exit (

*x*,

*y*) plane. We show how the BS changes its width and intensity, and how the molecular reorientation changes during

*z*propagation. We invariably observe soliton breathing - the beam preserves its new shape but its characteristic width and maximum intensity breathe as it propagates.

#### 3.1.3 Influence of the optical and the static permittivity anisotropy

^{OPT}and Δε

^{DC}for the input Gaussian beam propagation, we observe again stable beams in a narrow threshold region of control parameters. Below the threshold region the beams diffract, above it spatiotemporal instabilities are observed, as the input intensity and the optical and static permittivity anisotropies of the liquid crystal molecules are increased. As before, we only observe spatial soliton breathers.

^{OPT}and repeat the sequence of increasing input intensities for two values of Δε

^{DC}, Δε

^{DC}=10 and Δε

^{DC}=14.5.

^{OPT}and the molecular reorientation induced by the electric field only

*θ*(

_{0}*x*,

*y*), as functions of the transverse variables, for the two mentioned values of Δε

^{DC}. Inset in Fig. 4(A) depicts soliton breathing propagation in

*z*, for Δε

^{OPT}=0.4 and Δε

^{OPT}=0.3 (Δε

^{DC}=14.5). We can see that if we increase birefringence (for fixed ε

^{DC}) the threshold input intensities for soliton propagation are decreasing; for smaller birefringence the appropriate input beam intensity for the emergence of BSs appears at a higher intensity. For smaller Δε

^{DC}=10 we see similarly an increase in the intensity needed for the existence of stable BSs. We conclude that the changes in FWHM lead to the changes in the intensity and in the period of oscillation along

*z*direction.

#### 3. 1. 4. Influence of the change in boundary condition θ(x=-D/2)=θ(x=D/2)=30^{0}

^{OPT}and

*L*, but change boundary conditions, we find similar behavior as before. For each input beam width the appropriate input beam intensity can be found for the existence of BSs. In Fig. 5 we show the cases of BS propagation, for different FWHM (2.5µm–4µm).

### 3.2. Propagation of input Gaussian beams using a steady-state procedure

*∂θ*/

*∂t*=0 in Eq. (2) from the beginning and by solving Eqs. (1) and (2) as a time-independent system of PDEs. The temporal solution procedure is now eliminated, being substituted by a SOR algorithm. However, the results are very similar to the ones obtained in the limit of long time evolution in the original time-dependent procedure.

#### 3. 2. 1. Influence of input beam intensity

*I*=1.0×10

^{+9}V

^{2}/m

^{2}and

*I*=1.0×10

^{+10}V

^{2}/m

^{2}. While the beam of the smallest intensity (inset) smoothly diminishes in intensity with propagation, the beam next to it is initially greatly reduced in intensity, but then it recovers in an irregular fashion.

*I*=3.5×10

^{+10}V

^{2}/m

^{2}.

#### 3. 2. 2. Comparison between the steady-state propagation procedure and the propagation in time.

_{D}). For longer propagations small quantitative discrepancies appear, however the qualitative behavior in both numerical procedures remains the same. When one compares the spatial distributions obtained by the two procedures, the differences are hardly discernible. It is reassuring that two very different numerical procedures provide very similar results over a broad range of parameters. Still, we believe that the propagation in time procedure provides more reliable and physically more transparent results.

## 4. Summary

^{OPT}, Δε

^{DC}, and boundary conditions. For different combination of these parameters, we always obtained consistently similar results, which present typical behavior during the propagation of input Gaussian beams in NLCs. The typical stable soliton propagation mode is the soliton breathing. Also, we considered the behavior of propagating single input Gaussian beams using a steady state procedure, and found similar results. We could not locate steady (not breathing) spatial solitons in any of our numerical simulations, despite repeated efforts. We conclude that the soliton breathing is the predominant stable way of soliton propagation in NLCs.

## Acknowledgments

## References and links

1. | I. C. Khoo, |

2. | Y. S. Kivshar and G. P. Agrawal, |

3. | 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. |

4. | X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, “Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal,” J. Opt. Soc. Am. B |

5. | A.I. Strinić, D.V. Timotijević, D. Arsenović, M.S. Petrović, and M.R. Belic, “Spatiotemporal optical instabilities in nematic solitons,” Opt. Express |

6. | P. D. Rasmussen, O. Bang, and W. Krolikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E |

7. | G. D’Alessandro and A. A. Wheeler, “Bistability of liquid crystal microcavities,” Phys. Rev. A |

8. | J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. |

9. | T. M. Monro, C. Martijn de Sterke, and L. Poladian, “Investigation of waveguide growth in photosensitive germanosilicate glass,” J. Opt. Soc. Am. B. |

10. | S. LaRochelle, V. Mizrahi, G. I. Stegeman, and J. E. Sipe, “Growth dynamics of photosensitive gratings in optical fibers,” Appl. Phys. Lett. |

11. | T. M. Monro, C. Martijn de Sterke, and L. Poladian, “Catching light in its own trap,” J. Mod. Opt. |

12. | A. S. Kewitsch and A. Yariv, “Nonlinear optical properties of photoresists for projection lithography,” Appl. Phys. Lett. |

13. | A. A. Sukhorukov, S. Shoji, Y. S. Kivshar, and S. Kawata, “Self-written waveguides in photosensitive materials,” J. Nonlinear Opt. Phys. Mater. |

14. | K. D. Dorkenoo, F. Gillot, O. Crégut, Y. Sonnefraud, A. Fort, and H. Leblond, “Control of the refractive index in photopolymerizable materials for (2+1)D solitary wave guide formation,” Phys. Rev. Lett. |

15. | 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. |

16. | J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express |

17. | J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time-dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. |

18. | A. Strinić, D. Jović, M. Petrović, D. Timotijević, N. Aleksić, and M. Belić, “Counterpropagating beams in nematic liquid crystals,” Opt. Express |

19. | 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 |

20. | C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. |

21. | C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. |

22. | M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E |

23. | G. Assanto, M. Peccianti, and C. Conti, “One dimensional transverse modulational instability in nonlocal media witha reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. |

24. | A. I. Strinic and M. R. Belic, “Beam propagation in nematic liquid crystals,” Acta Phys. Pol. A , |

25. | M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Nonlocal optical propagation in nonlinear nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. |

26. | J.F. Henninot, J.F. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” J. Opt. A: Pure Appl. Opt. |

27. | A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 27, 2009

Revised Manuscript: June 6, 2009

Manuscript Accepted: June 9, 2009

Published: June 26, 2009

**Citation**

A. I. Strinic, M. Petrovic, D. V. Timotijevic, N. B. Aleksic, and M. R. Belic, "Breathing solitons in nematic liquid crystals," Opt. Express **17**, 11698-11709 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11698

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### References

- I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995).
- Y. S. Kivshar and G. P. Agrawal, Optical solitons (Academic Press, San Diego, 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]
- X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, "Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal," J. Opt. Soc. Am. B 22, 1424-1431 (2005). [CrossRef]
- A. I. Strinić, D. V. Timotijević, D. Arsenović, M. S. Petrović, and M. R. Belic, "Spatiotemporal optical instabilities in nematic solitons," Opt. Express 13, 493-504 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-2-493. [CrossRef] [PubMed]
- P. D. Rasmussen, O. Bang, and W. Krolikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 1-7 (2005). [CrossRef]
- G. D’Alessandro and A. A. Wheeler, "Bistability of liquid crystal microcavities," Phys. Rev. A 67, 023816 1-12 (2003).
- J. F. Henninot, M. Debailleul, and M. Warenghem, "Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal," Mol. Cryst. Liq. Cryst. 375, 631-640 (2002). [CrossRef]
- T. M. Monro, C. Martijn de Sterke, and L. Poladian, "Investigation of waveguide growth in photosensitive germanosilicate glass," J. Opt. Soc. Am. B. 13, 2824 - 2832 (1996). [CrossRef]
- S. LaRochelle, V. Mizrahi, G. I. Stegeman, and J. E. Sipe, "Growth dynamics of photosensitive gratings in optical fibers," Appl. Phys. Lett. 57, 747-749 (1990). [CrossRef]
- T. M. Monro, C. Martijn de Sterke, and L. Poladian, "Catching light in its own trap," J. Mod. Opt. 48, 191-238 (2001).
- A. S. Kewitsch and A. Yariv, "Nonlinear optical properties of photoresists for projection lithography," Appl. Phys. Lett. 68, 455-457 (1996). [CrossRef]
- A. A. Sukhorukov, S. Shoji, Y. S. Kivshar, and S. Kawata, "Self-written waveguides in photosensitive materials," J. Nonlinear Opt. Phys. Mater. 11, 391-407 (2002). [CrossRef]
- K. D. Dorkenoo, F. Gillot, O. Crégut, Y. Sonnefraud, A. Fort, and H. Leblond, "Control of the refractive index in photopolymerizable materials for (2 + 1)D solitary wave guide formation," Phys. Rev. Lett. 93, 143905 1-4 (2004). [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]
- J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, "Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells," Opt. Express 12, 1011-1018 (2004),http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-6-1011. [CrossRef] [PubMed]
- J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, "Time-dependence of soliton formation in planar cells of nematic liquid crystals," IEEE J. Quantum Electron. 41, 735-740 (2005). [CrossRef]
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