Different reorientational regimes in a liquid crystalline medium undergoing multiple irradiation

doi:10.1364/OE.15.001663

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Different reorientational regimes in a liquid crystalline medium undergoing multiple irradiation

A. Veltri, L. Pezzi, A. De Luca, and C. Umeton »View Author Affiliations

Optics Express, Vol. 15, Issue 4, pp. 1663-1671 (2007)
http://dx.doi.org/10.1364/OE.15.001663

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Abstract

We present a numerical approach to the nemato-elasticity differential equation in a nematic liquid crystal cell when irradiated with multiple gaussian beams. Solutions have been carried out on a configuration with two coplanar beams illuminating the sample in order to compare it with particular nonlinear phenomena experimentally studied in the past. A new set of experimental measures were realized confirming the validity of the numerical model. Solutions for an instable case showing nonlocal effects are also presented as an example of the broader class of systems this approach can describe.

1. Introduction

Light propagation in nematic liquid crystals (NLC) and the consequent self-induced effects are, by definition, the basics of nonlinear optics in such materials. In this perspective, orientational interaction phenomena have been widely studied during last two decades, providing a great number of fundamental and applied results [1

P. G. de Gennes, The physics of liquid crystals (Clarendon Press, Oxford, 1993).

, 2

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

, 3

F. Simoni, Nonlinear optical properties of liquid crystals (World Scientific, 1997).

]. In recent years, other nonlinear media have been deeply investigated to time domain phenomena [4

E. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak, and L. Vazquez, “Dissipative optical solitons” Phys. Rev. A 49,2806–2811 (1994). [CrossRef] [PubMed]

, 5

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, “Exact localized solution for nonconservative systems with delayed nonlinear response” Phys. Rev. E 57,3664–3667 (1998). [CrossRef]

], spatial effects in photorefractive materials [6

S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response” Phys. Rev. E 57,6066–6070 (1998). [CrossRef]

, 7

E. DelRe, A. Ciattoni, and A. J. Agranat, “Anisotropic charge displacement supporting isolated photorefractive optical needles” Opt. Lett. 26,908–910 (2001). [CrossRef]

] and spatial optical solitons, intended as nondiffracting light beams that are self-confined by the nonlinearity. Because of their possible applications (related to the exploitation of the wave-guiding character [8

A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer, Dordrecht, 2001). [CrossRef]

]), spatial optical solitons have been intensely investigated also in a new scenario, which describes these effects in a smooth transition from the purely local to the entirely nonlocal response [9

C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium” Phys. Rev. Lett. 92,113902–113902 (2004). [CrossRef] [PubMed]

]. In that case, the physical system was again a nematic liquid crystal in a planar cell exhibiting a nonlocal nonlinearity of orientational molecular origin [2

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

], which had already been used for the observation of (2+1)D spatial solitons [10

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals” Appl. Phys. Lett. 81,3335–3338 (2002). [CrossRef]

]. Furthermore, it has been shown that different solitons created in the same sample by different light beams can interact each other giving rise to a series of interesting effects and leading to intriguing applications like the realization of logic gates [10

]. Recently [11

A. De Luca, S. Nersisyan, and C. Umeton, “Observation of cancellation and second light-induced Frdericksz transition in nematic liquid crystals” Opt. Lett. 28,108–110 (2003). [CrossRef] [PubMed]

], it has been found that the possibility of a nonlinear interaction between different light beams is not limited to the case of spatial solitons, but can take place also in different experimental conditions, giving rise to a series of intriguing effects. In particular, it has been found that the nonlinear response induced by one beam can be partially or totally cancelled by a second beam, with interesting perspectives for possible applications. In this perspective, however, further investigations and availability of a general and complete theoretical description for the interpretation of these effects are mandatory if their application oriented utilization is desired. In this letter, we present a general model which describes the behavior of a highly nonlinear system like a liquid crystal sample when undergoing multiple irradiation; furthermore, we show that, under particular experimental conditions, also nonlocality plays an important role.

2. Theoretical model

Fig. 1. Here we represent the system under study that is an NLC cell of tickness L, and width d, crossed by E_j (j=1, …,N) gaussian light beams, each of them impinging on the sample with angle α_j . The director orientation is identified by the angle θ formed by the director n̂ and the z-axis

The physical quantities playing the main roles are the re-orientation angle θ of the NLC molecular director n̂ (the unit vector which describes the mean orientation of molecular axes) with respect to the initial (or steady state) direction and the polarization of the impinging light beams. The system we want to study is, in fact the one presented in fig 1: a NLC cell of thickness L, and width d, crossed by N laser beams with electric field E_j (j=1, …,N) and wavelength λ, linearly polarized in the xz-plane, each of them impinging on the sample with an incidence angle α_j . As for system coordinates, we chose the x-axis along the cell width and the z-axis along the cell length. In fact, even if gaussian beams are three-dimensional, the utilization of beams that are linearly polarized in the xz-plane allow to describe all the physics by considering only two dimensions. For sake of generality, we start from the master equation for the angle θ in the case of N reorienting light beams [12

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, “The orientational optical nonlinearity of liquid crystals” Mol. Cryst. Liq. Cryst. 136,1–139 (1986). [CrossRef]

γ \frac{\partial θ}{\partial t} = K \nabla^{2} θ + \frac{ε_{0} Δε}{4} \sum_{i, j = 1}^{N} E_{i} E_{j}^{*} \sin (2 θ - α_{i} - α_{j}),

(1)

where t is the time, γ the viscosity constant and K the elastic constant of the medium in a “one constant” approximation [12

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, “The orientational optical nonlinearity of liquid crystals” Mol. Cryst. Liq. Cryst. 136,1–139 (1986). [CrossRef]

]; Δε=n ² _e-n ² _o indicates the optical anisotropy, n_e and n_o being the extraordinary and ordinary refractive index respectively; ε ₀ is the electric permittivity of vacuum.

Where light propagation is concerned, considering that investigated phenomena occur in a small fraction of a thin cell, we use the fundamental gaussian beam solution of Maxwell equations [13

A. Yariv, Quantum electronics- 3rd ed . (Jhon Wiley & Sons, 1987).

] instead of solving a new light propagation equation, taking into account that the j-th beam crosses the sample with an angle α_j ; this corresponds to apply the following transformation on the solution propagating along z:

x \to x \cos α_{j} - z \sin α_{j}

z \to x \sin α_{j} + z \cos α_{j}

(2)

While crossing the medium, the j-th beam experiences a refractive index n(α_j ,θ) given by [2

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

n (α_{j}, θ) = \frac{n_{0} n_{e}}{\sqrt{n_{e}^{2}} \cos^{2} (α_{j} - θ) + n_{0}^{2} \sin^{2} (α_{j} - θ)} .

(3)

Where θ is obtained from the solution of Eq. (1). We introduce normalized coordinates ξ=x/L, ζ=z/L and normalized time τ=t/τ_R, where τ_R=γL ²/K is the tipical reorientation time of the NLC director [2

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

]. We also introduce parameters Λ=λ/L, which is the normalized wavelength and w ₀=W ₀/L which is a normalized minimum spot size (W ₀ being the minimum spot size), and a normalized expression for the electric field:

e_{j} = E_{j} \sqrt{\frac{L^{2} ε_{0} Δε}{4 K} .}

. Thus, the normalized electric field of the j-th gaussian beam is written as:

e_{j} = \frac{e_{0} w_{0}}{w_{j}} \exp {- i [k (ξ \sin α_{j} + ζ \cos α_{j}) - η_{j}] - {(ξ \cos α_{j} - ζ \sin α_{j})}^{2} [\frac{1}{w_{j}^{2}} + \frac{ik}{{2 r}_{j}}]}

j = 1, …N

Here

e_{0} = E_{0} \sqrt{\frac{L^{2} ε_{0} Δε}{4 K} .}

(where E ₀ is the field amplitude), η_j=η(ζ,α_j), w_j =W(ζ,α_j,w ₀) and r_j =R(ζ,α_j), where

W (z) = W_{0} \sqrt{\frac{1 + z^{2}}{z_{0}^{2}}}

is the spot size, z ₀=πn(θ)W ² ₀/λ the confocal parameter, η(z)=arctan(z/z ₀) and R(z)=z+z ² ₀/z is the radius of curvature of a generic gaussian beam. In normalized coordinates, Eq. (1) becomes:

\frac{\partial θ}{\partial τ} = \nabla^{2} θ + \sum_{i, j = 1}^{N} e_{i} e_{j}^{*} \sin (2 θ - α_{i} - α_{j}),

(4)

where ∇² indicates now (∂²/∂ζ²+∂²/∂ζ²).

3. Numerical solutions and results

Equation 4 can be used to study a large number of experimental configurations, but it is highly nonlinear and, in general, cannot be solved by analytical methods. We have chosen a numerical approach by developing a c++ code compiled under Linux using the open source GNU compiler gcc 3.3.5. In particular we have used a second order Runge Kutta time scheme and a central difference scheme for spatial derivatives. In order to test our model a number of simulations have been carried out reproducing experimental configuration presented in [11

] that is: utilizing n_o =1.5216 and n_e =1.7462, corresponding to values of the E7 commercial NLC, λ=532 nm, considering two beams of equal intensity, and using θ(ξ,ζ,0)=0 as the initial condition in time and θ(ξ,0,τ)=θ(ξ,1,τ)=0 as boundary conditions in space; this accounts for strong anchoring in a homeotropic cell, were the molecular director is oriented perpendicularly to the cell plates. As in [11

], at first only one beam crosses the sample, so that a director reorientation is induced. The system proceeds this way until ∂θ_c /∂τ≤0.01, where θ_c indicates the reorientation angle in the center of the sample; then the second beam is switched on at the symmetric incident angle (α₁=-α₂), again until ∂θ_c /∂ τ≤0.01. After that, two different behaviors can be observed (in agreement with experimental results presented in [11

]), depending both on light intensity and incidence angle:

Fig. 2. Simulation carried out for an angle of incidence α=1 rad and a normalized field amplitude e ₀=3. The first and second frames show the orientation of the molecular director in the sample at τ=0.25, and τ=0.56; the third is the temporal behavior of the reorientation angle θ_c in the center of the sample. This case represents a complete cancel-lation effect induced by the competition of the two beams.

Fig. 3. Simulation carried out for α=0.2 rad, e ₀=3. The first and second frames show the orientation of the molecular director in the sample at τ=0.34, and τ=0.76; the third is the temporal behavior of the reorientation angle θ_c in the center of the sample. This case represents a critical reorientation effect due to the second light-induced Fréedericksz transition (LIFT II)

1. CAW (Competingly Acting Waves): The second beam competes with the first one and the final effect is a reorganization of the NLC director in such a way that the reorientation effect produced by the first beam is almost completely cancelled by the second one, as presented in Fig. 2;
2. LIFT II (Second Light Induced Fréedericksz Transition): above a given intensity threshold, the second beam adds its reorientational effect to the first one, thus causing a critical director reorientation (Fig. 3).

Fig. 4. The L ²-Norm of a cut in ξ of θ(ξ,ζ) at the center of the sample (ζ=1/2), as a function of the control parameters e ₀ and α, is presented as a measure of the director reorientation. The dark zone corresponds to a critical reorientation, the white one to a cancellation effect. The map refers to the case in which the second beam is switched on the sample well after the reorientation process induced by the first one has been completed.

Fig. 5. Equilibrium states in the case of equal impinging intensities; vectors e _1,2 represent the electric fields of light beams, n is the molecular director. (a) initial homeotropic state, or consequence of a complete cancellation effect. (b) complete planar reorientation.

In order to characterize effects of the field amplitude e ₀ and incidence angle α_j, we have evaluated the L ²-Norm of θ(ξ,1/2), defined as L ²[θ(ξ,1/2)]=[∫(θ(ξ,1/2))² dξ]^1/2. This quantity is ~ 0 if cancellation occurs, while it is different from zero in the case of critical reorientation. Results of this characterization are presented in Fig. 4: The critical reorientation occurs only for angles of incidence smaller than the critical value α_th≃0.8 rad, which is independent of the used liquid crystal, and is specific of the system geometry only. For equal intensities of the two beams, two equilibrium states exist, as shown in Fig. 5: At great incidence angles (α>0.78 rad, corresponding to small angles between field vectors e ₁ and e ₂, calculated as 2(π-α)) the can-cellation effect is favored; on the contrary, for small incidence angles (α<0.78 rad) a critical reorientation occurs. Furthermore, for values e ₀<2, the reorienting effect is well balanced by the elastic force; the maximum reorientation angle is small enough and the cancellation effect is in any case favored.

4. Experimental comparison

Theoretical predictions have been confirmed by experimental results obtained in the geometry illustrated in Fig. 6(b), which exploits a particular set of conditions: Average intensities of the two impinging beams are the same, whereas the two angles of incidence are equal and opposite. The scheme of the experimental setup is shown in Fig. 6(a). The continuous source is a diode-pumped solid-state laser (DPSS Ventus 532, by Laser Quantum), which emits light at λ=532 nm. The radiation is split into two beams of equal intensity by a beam splitter. Before reaching the sample, each beam crosses a half-wave plate followed by a polarizer, whose combined action enables varying the total power impinging on the sample; then each beam crosses a spherical lens (f=150 mm) mounted on a translation stage, which allows production of two equal spot sizes (~130 μm in diameter) on the sample. In this way, two equal light intensities impinge on it. The NLC (E7, byMerck) is sandwiched between two glass slabs and is homeotropically aligned. Suitable Mylar spacers ensure a uniform thickness (L=75 μm) of the cell, which is placed on a rotating xyz stage, needed to adjust two equal and opposite angles of incidence. Images after the sample are projected onto a white screen which enables observation of Self Phase Modulation (SPM) rings in the far-field zone, an effect widely investigated in the past [14

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, “Two beam mirrorless optical bistability in nematic liquid crystal film” Mol. Cryst. Liq. Cryst. 143,113–122 (1987). [CrossRef]

, 15

I.C. Khoo, “Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light” Phys. Rev. A 25,1636–1644 (1982). [CrossRef]

, 16

F. Bloisi, L. Vicari, F. Simoni, G. Cipparrone, and C. Umeton, “Self-phase modulation in nematic liquid-crystal films: detailed measurements and theoretical calculations” J. Opt. Soc. Am. B 5,2462–2466 (1988). [CrossRef]

], which can be used to monitor the director reorientation. By varying the incidence angle between 20° and 70°, with steps of 10°, we have characterized the dependence of the ring interference pattern on the impinging power. For angles smaller than 45° (0.78 rad) it is possible to obtain a well-balanced competition of the effects of the two beams. Indeed, when their intensities are equal and the total average intensity remains below a given threshold, SPM rings disappear, indicating an unperturbed orientational state of the director, which we refer to as “spatial cancellation of reorientation”. The situation is shown in Fig. 7(a), where we have used gray dots to refer to the case of no SPMrings observed in the far field zone (45 cm from the sample), while with black dots we have indicated the case of a LIFT II effect. We have chosen to use a field intensity representation for the L ²-Norm values in order to maintain a proportionality with the impinging laser power; normalization relations indicate that the [0 : 21.6] range for e ² ₀ corresponds to the [0 : 160 mW] range for the laser power.

Fig. 6. Sketch of the experimental Set-Up: BS: beam splitter; M, M ₁, M ₂: mirrors,W ₁,W ₂: half-wave plates; P ₁, P ₂: polarizers; L ₁, L ₂: spherical lenses ; S: sample.

Fig. 7. (a) Experimental results obtained for different values of impinging power and angles of incidence. Black dots indicate that, at the end of the process, reorientation due to a LIFT II effect has taken place. Gray dots indicate that a a complete cancellation has occurred. (b) Theoretical map of L ²[θ(ξ,1/2)] in the corresponding zone of the α-e ² ₀ plane

Fig. 8. (a) Experimental behavior of the optical divergence as a function of the laser power for different α values that correspond to geometries in which a threshold in the field amplitude exists for the reorientation effect. (b) L ²[θ(ξ,1/2)] as a function of e ² ₀ (in the corresponding range of values of the field intensity) for the same α values.

In Fig. 7(b) theoretical predictions are reported for the same cases. Also in this figure the gray zone refers to the case in which a cancellation of reorientational effects is obtained, while the black zone refers to a critical reorientational effect. Comparison with 7(a) demonstrates the validity of our model.

Starting from evidences of Fig. 7, for a deeper investigation of the obtained results, an analysis of the optical divergence of one beam as a function of the impinging laser power has been realized for α=0.35, α=0.55 and α=0.65 corresponding to the appearing of the LIFT II effect. In fact, L ²[θ(ξ,1/2)] is a measure of the reorientational effect; on the other hand, the optical divergence of one beam is a direct evidence of the director reorientation [12

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, “The orientational optical nonlinearity of liquid crystals” Mol. Cryst. Liq. Cryst. 136,1–139 (1986). [CrossRef]

]. In Fig. 8(a) the optical divergence as a function of the laser power is reported, while Fig. 8(b) represents the theoretical behavior of the L ²-Norm obtained for the same α values in the corresponding range of field intensity. Also in this case, the excellent agreement between the two figures confirms the validity of our model.

5. Non local solutions

Effects discussed up to now are concerned with a local responce of the medium and they can be explained by considering the local equilibrium states presented in fig 5; there is however a particular case in which nonlocality plays a role. In particular, if both laser beams are switched on the sample simultaneously, an apparently stable situation is realized. On the contrary, theoretical simulations show that a different kind of critical reorientation is observed, due to a nonlocal response of the medium. Indeed, characterization of the L ²-Norm presented in Fig. 9 shows that also in this case there is a threshold in e ₀ and α above which a critical reorientation occurs. In our opinion, this effect can be explained by assuming that, from the boundary of the irradiated zone where the two gaussian beams are not perfectly superimposed, nonlocality extends the reorientation towards the center of the spot. If elastic forces are relatively small (high e ₀), molecules can receive enough energy to evolve from the homeotropic equilibrium state (a in Fig. 5) to the planar one (b in Fig. 5). We have called this effect “De-localized Freéderiksz Transition of Second Order” (DELFTII) , which gives evidence of a nonlocal response of the medium. These results show that the interplay between nonlinear and nonlocal effects is an intrinsic feature of the investigated system and open new perspectives for its study not only from a fundamental point of view, but also for applications. In fact, cancellation effects could be utilized in the fields of “optical switching” and “all optical addressing”, while the angular dependence of this effect could be exploited in high resolution “optical goniometer” devices.

Fig. 9. The L ²-Norm of a cut in ξ of θ at the center of the sample (L ²[θ(ξ,1/2)]), as a function of control parameters e ₀ and α. The dark zone corresponds to critical reorientation, the white one to a cancellation effect. The map refers to the case in which the two beams are switched on the sample simultaneously

6. Conclusion

In conclusion we have carried out numerical solutions of a general model for the interaction of N gaussian light beams with a physical system which is a NLC cell with strong anchoring conditions. The model has been experimentally checked for the case of two beams in an opposite incidence angle geometry; in good agreement with theoretical predictions, in the framework of a local response of the medium, it has been found that a series of effects can occur. Furthermore, under particular experimental conditions, nonlocality plays an important role. Results give a new light on these kind of nonlinear media, stimulating investigation of the phenomena that can take place.

Acknowledgments

We acknowledge useful discussions in Mathematics with Dr. Luca Guzzardi

References and links

1.	P. G. de Gennes, The physics of liquid crystals (Clarendon Press, Oxford, 1993).
2.	I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Jhon Wiley & Sons, Inc., New York, 1995).
3.	F. Simoni, Nonlinear optical properties of liquid crystals (World Scientific, 1997).
4.	E. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak, and L. Vazquez, “Dissipative optical solitons” Phys. Rev. A 49,2806–2811 (1994). [CrossRef] [PubMed]
5.	N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, “Exact localized solution for nonconservative systems with delayed nonlinear response” Phys. Rev. E 57,3664–3667 (1998). [CrossRef]
6.	S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response” Phys. Rev. E 57,6066–6070 (1998). [CrossRef]
7.	E. DelRe, A. Ciattoni, and A. J. Agranat, “Anisotropic charge displacement supporting isolated photorefractive optical needles” Opt. Lett. 26,908–910 (2001). [CrossRef]
8.	A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer, Dordrecht, 2001). [CrossRef]
9.	C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium” Phys. Rev. Lett. 92,113902–113902 (2004). [CrossRef] [PubMed]
10.	M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals” Appl. Phys. Lett. 81,3335–3338 (2002). [CrossRef]
11.	A. De Luca, S. Nersisyan, and C. Umeton, “Observation of cancellation and second light-induced Frdericksz transition in nematic liquid crystals” Opt. Lett. 28,108–110 (2003). [CrossRef] [PubMed]
12.	N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, “The orientational optical nonlinearity of liquid crystals” Mol. Cryst. Liq. Cryst. 136,1–139 (1986). [CrossRef]
13.	A. Yariv, Quantum electronics- 3rd ed . (Jhon Wiley & Sons, 1987).
14.	E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, “Two beam mirrorless optical bistability in nematic liquid crystal film” Mol. Cryst. Liq. Cryst. 143,113–122 (1987). [CrossRef]
15.	I.C. Khoo, “Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light” Phys. Rev. A 25,1636–1644 (1982). [CrossRef]
16.	F. Bloisi, L. Vicari, F. Simoni, G. Cipparrone, and C. Umeton, “Self-phase modulation in nematic liquid-crystal films: detailed measurements and theoretical calculations” J. Opt. Soc. Am. B 5,2462–2466 (1988). [CrossRef]

OCIS Codes
(190.4400) Nonlinear optics : Nonlinear optics, materials
(190.5940) Nonlinear optics : Self-action effects

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 23, 2006
Revised Manuscript: October 30, 2006
Manuscript Accepted: November 7, 2006
Published: February 19, 2007

Citation
A. Veltri, L. Pezzi, A. De Luca, and C. Umeton, "Different reorientational regimes in a liquid crystalline medium undergoing multiple irradiation," Opt. Express 15, 1663-1671 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1663

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References

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).
I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (John Wiley & Sons Inc., New York, 1995).
F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, 1997).
E. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak, and L. Vazquez, "Dissipative optical solitons," Phys. Rev. A 49, 2806-2811 (1994). [CrossRef] [PubMed]
N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998). [CrossRef]
S. Abe and A. Ogura, "Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response," Phys. Rev. E 57, 6066-6070 (1998). [CrossRef]
E. DelRe, A. Ciattoni, and A. J. Agranat, "Anisotropic charge displacement supporting isolated photorefractive optical needles," Opt. Lett. 26, 908-910 (2001). [CrossRef]
A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer, Dordrecht, 2001). [CrossRef]
C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004). [CrossRef] [PubMed]
M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002). [CrossRef]
A. De Luca, S. Nersisyan, and C. Umeton, "Observation of cancellation and second light-induced Frdericksz transition in nematic liquid crystals," Opt. Lett. 28, 108-110 (2003). [CrossRef] [PubMed]
N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986). [CrossRef]
A. Yariv, Quantum electronics-3rd ed. (John Wiley & Sons, 1987).
E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987). [CrossRef]
I. C. Khoo, "Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light," Phys. Rev. A 25, 1636-1644 (1982). [CrossRef]
F. Bloisi, L. Vicari, F. Simoni, G. Cipparrone, and C. Umeton, "Self-phase modulation in nematic liquid-crystal films: detailed measurements and theoretical calculations," J. Opt. Soc. Am. B 5, 2462-2466 (1988). [CrossRef]

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