Modulation analysis of boundary induced motion of optical solitary waves in a nematic liquid crystal

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1 Universit of Wollongong Research Online Facult of Informatics - Papers Archive) Facult of Engineering and Information Sciences 29 Modulation analsis of boundar induced motion of optical solitar waves in a nematic liquid crstal Alessandro Alberucci Universit of Rome Gaetano Assanto Universit of Rome Daniel Buccoliero Australian National Universit Anton Desatnikov Australian National Universit Timoth R. Marchant Universit of Wollongong, tim@uow.edu.au See net page for additional authors Publication Details Alberucci, A., Assanto, G., Buccoliero, D., Desatnikov, A. S., Marchant, T. R. & Smth, N. F. 29). Modulation analsis of boundar induced motion of optical solitar waves in a nematic liquid crstal. Phsical Review A Atomic, Molecular and Optical Phsics), ), Research Online is the open access institutional repositor for the Universit of Wollongong. For further information contact the UOW Librar: research-pubs@uow.edu.au

2 Modulation analsis of boundar induced motion of optical solitar waves in a nematic liquid crstal Disciplines Phsical Sciences and Mathematics Publication Details Alberucci, A., Assanto, G., Buccoliero, D., Desatnikov, A. S., Marchant, T. R. & Smth, N. F. 29). Modulation analsis of boundar induced motion of optical solitar waves in a nematic liquid crstal. Phsical Review A Atomic, Molecular and Optical Phsics), ), Authors Alessandro Alberucci, Gaetano Assanto, Daniel Buccoliero, Anton Desatnikov, Timoth R. Marchant, and Noel Smth This journal article is available at Research Online:

3 Modulation Analsis of Boundar-Induced Motion of Optical Solitar Waves in a Nematic Liquid Crstal Alessandro Alberucci, 1 Gaetano Assanto, 1 Daniel Buccoliero, 2 Anton S. Desatnikov, 2 Timoth R. Marchant, 3 and Noel F. Smth 4 1 NooEL Nonlinear Optics and OptoElectronics Lab, Department of Electronic Engineering, CNISM Universit of Rome Roma Tre, Via della Vasca Navale 84, 146 Rome, Ital 2 Nonlinear Phsics Center, Research School of Phsics and Engineering, The Australian National Universit, Canberra ACT 2, Australia 3 School of Mathematics and Applied Statistics, Universit of Wollongong, Wollongong, NSW 2522, Australia 4 School of Mathematics and Mawell Institute for Mathematical Sciences, Universit of Edinburgh, Edinburgh, Scotland EH9 3JZ, U.K. We consider the motion of a solitar wave, a nematicon, in a finite cell filled with a nematic liquid crstal. A modulation theor is developed to describe the boundar induced bouncing of a nematicon in a one dimensional cell and it is found to give predictions in ver good agreement with numerical solutions. The boundar induced motion is then considered numericall for a two dimensional cell and a simple etension of the modulation theor from one to two space dimensions is then made, with good agreement being found with numerical solutions for the nematicon trajector. The role of nematicon shape and relative position to the boundaries in its evolution is discussed. PACS numbers: Tg, 42.7.Df I. INTRODUCTION Spatial solitons are ubiquitous as the are found and studied in fluids, plasmas, Bose-Einstein condensates, electronics and optics [1 7]. In optics, in particular, the have received much attention due to the versatile nature of self-induced waveguides, a concept amenable to applications in applied nonlinear optics and communications, for which all-optical switching and routing could pla an important role in future generations of signal processors [6 9]. In this scenario, nematicons i.e. spatial optical solitons in nematic liquid crstals [1, 11] have stirred attention as a convenient plaground for a number of fundamental and applied properties of optical solitons, including the role of nonlocalit, not onl as a stabiliing mechanism which prevents catastrophic collapse in two transverse dimensions, but also as a long-range link between two or more nematicons, nematicons and etra beams, and nematicons and perturbations, including the boundaries of a cell [12 15]. This latter aspect, i.e. the effect of a nonlocal boundar potential on the propagation of spatial solitons, has been recentl addressed with specific reference to thermo-optical and reorientational solitons in glass and in liquid crstals, respectivel [16 2]. In this paper we undertake the ambitious task of providing a theoretical background to describe the boundarnematicon interaction, analsing the boundar induced motion of these self-localied beams b means of modulation theor, modelling the problem in one transverse dimension, but etending some of the results to the full two-dimensional scenario. II. GOVERNING EQUATIONS We consider a coherent, polarised light beam inputted into a planar liquid crstal cell, with the coordinate along the cell and the, ) coordinates orthogonal to this direction. Let us take the light to be linearl polarisedasanetraordinarwaveinthedirection. Inone of the eperimental scenarios, the optic aisor molecular director) of the nematic liquid crstal nlc) is prepared parallel to and a static electric field is applied in the direction to orient it at an angle ˆθ to the direction in the absence of light. In this case the etraordinar polariation of the input results in a perturbation of the director angle from the pre-tilt θ due to the light beam launched along. In non-dimensional form the equations governing the propagation of light through such a liquid crstal cell are then i E E +Esin2θ =, ν 2 θ qsin2θ = 2 E 2 cos2θ, 1) where the Laplacian 2 is in the,) plane [13, 21]. The variable E is the comple-valued, slowl varing envelope of the optical electric field. The nonlocalit ν measures the strength of the response of the nematic in space, with a highl nonlocal response corresponding to ν large. It should be noted that the electric field E in equation 1) has had a phase factor taken out, this factor accounting for the birefringent walk-off due to the Ponting vector of the etraordinar-wave beam deviating from its wavevector [22]. In the nonlocal, ν large limit, it can be seen from the director equation in 1) that θ, the opticall induced deviation of the director angle from ˆθ, is small. The parameter q is related to the energ squared amplitude) of the static electric field which pre-tilts the

4 2 nematic dielectric [13]. Another eperimental scenario, the one which is subject of this work, consists of a cell for which no static pre-tilt field is applied, preparing rubbing ) the boundariesofthe cell sothat the directormakesanangle ˆθ with respect to the direction in the plane [23 25]. Therefore, at equilibrium in the absence of the optical field, the director is at an angle ˆθ throughout the cell. When a light beam is launched with wavevector along, the optical director is then perturbed b a further angle θ. Hence setting q = no static field) in 1) and also taking θ 1, which is valid in the highl nonlocal limit, the nematicon equations 1) become i E E +2Eθ =, ν 2 θ = 2 E 2. 2) The boundar condition for the nematic director at the cell walls is θ =. E, θ Figure 1: Electric field intensit and director response versus. Shown are numerical solutions for E solid line) and θ dashed line) versus at = 2. The initial values are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. III. ONE SPACE DIMENSION Due to the mismatch between the geometries of the rectangular cell and the usual circular Gaussian light beam, it is etremel difficult to perform an analtical analsis for nematicons propagating in two space dimensions. To obtain insight into nematicon evolution in a finite cell without the presence of a static pre-tilting field, let us consider nematicon evolution in one space dimension, so that the optical field evolution is in the plane. An added complication with studing nematicons is that there is no known eact analtical solitar wave nematicon) solution of the nematicon equations. Due to this, it has been found that a powerful approimate technique for studing this problem is that based on trial functions in variational formulations of the governing equations [26 29], this being an etension of modulation theor [3]. This technique is adapted and used in the present work. The nematic equations 2) have the Lagrangian L = ie E EE ) E 2 +4θ E 2 ν θ 2. 3) The success of variational techniques depends on the choice of trial functions. The appropriate choice for nematicons, especiall in one space dimension, is the hperbolic secant profile E = asech ξ w eiσ+iv ξ) +ige iσ+iv ξ), 4) where the parameters a, w, σ, V, ξ and g are functions of. a is the amplitude of the beam, w its width and σ the phase. g is the amplitude of the radiation bed [31]. The first term in this trial function represents a varing nematicon-like beam, while the second term represents the diffractive radiation of low wavenumber which accumulates under the evolving nematicon [27, 31]. This ξ Figure 2: The position ofthenematicon peakversus. Shown are ξ from the numerical solution solid line) and the modulation solution dashed line). The initial conditions are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. radiation cannot remain flat, so it is assumed that g is non-ero in the interval l/2 l/2 [27, 31]. Substituting the optical field form 4) into the director equation, the second of 2), and using the one space dimensional boundar condition θ = at the cell walls = ±L gives the solution νθ = a 2 w 2 1+ ) lns a 2 w 2 1 ) lns + L L + 2a 2 w 2 lnsech ξ w, 5) where s + = sech L+ξ w, s = sech L ξ w, for the director. The peak of the director distribution 5) does not, in general, occur at the location of the electric

5 3 field peak = ξ [14], but at w m = ξ +wtanh 1 2L ln s ). 6) s + Let us set α 1 = θ m,) as the amplitude of the director solution 5) at, which occurs at 6). When the nematicon is at the centre of the cell ξ =, then the director beam has a smmetric form too with m =. In cases when ξ, however, the peak of the director beam is a little closer to the the centre of the cell than the electric field peak, as m < ξ. Appling the modulation method [28, 3, 31], based on the trial function 4) and the director solution 5), then gives the modulation equations governing the evolution of the nematicon. These equations and a short discussion of them are given in the Appendi. IV. RESULTS AND DISCUSSION A. 1D nematicon bouncing in a cell In this section approimate and numerical solutions of the nematicon equations 2) for a 1 + 1) dimensional nematic cell are compared. The 1+1)D numerical solutions were found using the Dufort-Frankel finite differenceschemetosolvetheelectricfieldequation,thefirstof 2). For the director equation, the second of 2), Gauss- Seidel iteration was used with successive over relaation. An advantage of the Dufort-Frankel and Gauss-Seidel schemes is that the are both eplicit methods with low storage costs. The step sies used were =.4 and = Note that / must be small to ensure consistenc of the DuFort-Frankel finite difference scheme. Figure 1 shows the electric field intensit, E, and the director response, θ, versus at = 2. The initial values are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. This figure shows that the electric field has the form of a localised beam, while the director response is not localised and has a near triangular form, as found b Alberucci and Assanto [18]. The initial location of the beam is not at the centre of the cell, =, but is at = ξ = 5. The nematicon then oscillates about the centre line of the cell, as found in previous work [14, 19, 2, 22]. At = 2the electric field beam is peaked at = 3.55, while the director beam is peaked at = 3.24, a little closer to the centre of the cell than the electric field peak. The electric field amplitude is a =.219 and the director amplitude is α 1 =.188. Using the initial values for the parameters and ξ = 3.55 in 6) gives m = Hence the analtical epression for the location of the director beam gives an ecellent prediction which closel matches the numericall obtained location. Qualitativel this solution is quite different to the nematicon that occurs in the presence of a static pre-tilting electric field applied to overcome the Freédericks thresh- E, θ Figure 3: Peak electric field amplitude and director amplitude versus. Shown are a upper dashed line) and α lower dashed line) from the modulation solution and E upper solid line) and θ lower solid line) from the numerical solution. The initial conditions are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. old [13]. In that case both the electric field and the director response take the form of beams [13], while for the rubbed case the director response is not localised. The reason for these differing behaviours can be deduced from the director equation, the second of 1), applicable when there is a pre-tilting field. We consider the region where the optical electric field is ero E = ) and assume that θ is small. In this limit the director equation is θ 2q/ν)θ =, which has the bounded solution θ = e 2q ν, 7) for >, describing the eponential deca of the director tail in the nonlocal regime for a nematicon in the presence of a pre-tilting field. In the rubbed case q = the director equation becomes θ = far from the optical field, which has a linear solution. The linear regions of the director solution far from the optical field can be clearl seen in Fig. 1. Figure 2 shows the location of the nematicon peak versus. Shown are ξ from the modulation solution and the corresponding numerical solution. The initial conditions are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. The nematicon oscillates about the centre line of the cell, this behaviour being previousl shown theoreticall and numericall for liquid crstals [18, 19, 22] and thermo-optic media [17]. The modulation solution is an undamped periodic solution nonlinear oscillator) with a wavelength of = 497. Numericall the position oscillates with a wavelength of = 5 and is decaing to ξ = on a ver long scale. Assuming eponential deca, the position oscillation would take until = 27, to reach 1% of its original value. The comparison between the two solutions is ecellent for shorter values, but for etremel large values of the modulation solution deviates from the numerical one as diffractive radiation loss has

6 E, ReE), ImE) E, θ Figure 4: Electric field amplitude versus. Shown are a solid line), ReE)large dashes) and ImE)short dashes) from the numerical solution. The initial conditions are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. not been included in the modulation equations. In the present non-dimensional variables tpical cell lengths are around 5, so for most realistic eperimental scenarios these length scales for deca of the position oscillation are far longer than the length of the liquid crstal cell [1]. Figure 3 shows the peak electric field amplitude and director response amplitude versus. Shown are a and α 1 from the modulation solution and the corresponding numerical solution. The parameters are a =.2, w = 5, ξ = 5 with ν = 1 and L = 5. The peak of the numerical electric field oscillates between a =.186 and.224 with a wavelength of = 18. The corresponding director amplitude oscillation is between α 1 =.185 and.189. The modulation solution predicts a periodic solution with the electric field amplitude oscillating between a =.2 and a =.228 with wavelength = 13. The modulation director amplitude oscillates between α 1 =.186 and.189. These comparisons between the solutions are ecellent, ecept that beating of the numerical electric field amplitude occurs. This is due to a second harmonic component being present in the numerical solution for the nematicon, which is not accounted for in the trial function 4). This second harmonic decas on a ver long scale, much larger than a tpical cell length, with the nematicon amplitude oscillation settling down to a harmonic oscillation with a single frequenc. Figure 4 shows the electric field amplitude versus. Shown are a, ReE) and ImE) from the numerical solution. This figure shows that the real and imaginar parts of the electric field oscillate between ±.2 on a much shorter -scale than does the electric field amplitude a for E ), which forms the envelope of the other curves. Numericall, the oscillations in the real and imaginar parts of the electric field have wavelength 17, while the amplitude of E oscillates with wavelength 19. The modulation equations give the propagation constant Figure 5: Electric field amplitude and director amplitude versus. Shown are a upper dashed line) and α lower dashed line) from the modulation solution and E upper solid line) and θ lower solid line) from the numerical solution. The Gaussian initial condition 8) is used with a 1 =.1893, γ =.1274, ξ = 5, ν = 1 and L = 5. E Figure 6: Electric field amplitude E versus. Shown are numerical solutions for the nematicon tail. The Gaussian profile solid line) is at = 435 and the sech profile dashed curve) is at = 428. The parameters are a 1 =.1893, γ =.1274, a =.2, w = 5, ξ =, ν = 1 and L = 5. σ, which generates the oscillations in the real and imaginar parts of the electric field. Assuming that the beam is near the stead-state V = ξ = ), then A14) gives dσ/d =.36 for this eample. This corresponds to a wavelength of 17.5, which is ver close to the numericall obtained wavelength. It can also be seen that the oscillations of the real and imaginar parts of the electric field are not purel harmonic, but undergo beating, which leads to the much smaller amplitude oscillations in the wave envelope E, which represents the modulus of the electric field. Nonlinear effects are generating small contributions from higher harmonics, which lead to the beating of the wave

7 5 envelope, as seen in Figs. 3 and 4. To investigate the beating phenomena further we considered an alternative Gaussian initial condition E = a 1 e γ ξ)2 8) for the numerical solutions. To obtain a sensible comparison with the modulation solution, which is based on the sech initial condition, the difference between the the sech and Gaussian initial profiles was minimised using the method of least squares. For the eample illustrated in Figs. 1 4 the parameters a =.2 and w = 5 were used for the sech profile. The method of least squares gives a 1 =.1893 and γ =.1274 as the parameters of the equivalent Gaussian profile. The Gaussian profile is a little broader at moderate values of, but decas much faster for large values of than does the sech profile. Figure 5 shows the electric field amplitude and director response amplitude versus. Shown are a and α 1 from the modulation solutions and the corresponding numerical solutions. The initial profile is Gaussian with a 1 =.2 and γ = The other parameters are ξ = 5, ν = 1 and L = 5. The numerical electric field amplitude oscillates between a =.19 and.22 with a wavelength of = 11. The corresponding director amplitude oscillation is between α 1 =.185 and.189. The modulation solution predicts a periodic solution with the electric field amplitude oscillating between a =.2 and a =.228 with wavelength = 13. The director amplitude given b the modulation solution oscillates between α 1 =.186 and.189. As for the sech profile used for Fig. 3, the comparisons between these solutions are ecellent. For the Gaussian profile, however, the beating is much reduced, with the oscillator pattern of the numerical solution much closer to the constant amplitude oscillations of the modulation solution. The position of the nematicon peak for the Gaussian case is the same, to graphical accurac, as the position in the sech case illustrated in Fig. 2. Figure 6 shows the numerical nematicon tail versus for both the Gaussian at = 435) and sech at = 428) profiles. The parameters are a 1 =.1893, γ =.1274, a =.2, w = 5, ξ =, ν = 1 and L = 5. The values correspond to minima of the nematicon amplitude, so that the values differ slightl. Also ξ = initiall was chosen so that both nematicons are located at the centre of the cell. The figure shows that the sech profile generates much more diffractive radiation, as indicated b the more pronounced oscillations in its tail, while the Gaussian profile shows little shed diffractive radiation. It should be noted, however, that in absolute terms both profiles shed little radiation, in agreement with eperiments [25]. The enhanced shedding of diffractive radiation b the sech profile is the reason for the enhanced beating of the amplitude of the sech profile compared with that of the Gaussian profile. This enhanced radiation shedding is associated with a greater evolution of the beam profile as it propagates a) b) c) d) Trajector b) magnified d) a) b) c) e) Coordinates for a) f) Amplitude g) Width e) Figure 7: Color online. a c) Intensit distributions left) and nematicon trajectories right) for three different initial positions of a Gaussian input beam, the colours for intensit are mapped from ero blue) to maimal value red). d) Magnified trajector b). e) Coordinates, f) amplitude, and g) widths of the nematicon on the trajector c); red lines: -direction, black lines: -direction. B. 2D soliton bouncing in a rectangular cell Let us now stud the propagationofatwo dimensional beam in a rectangular cell. To solve the nematicon equations 2) and simulate beam propagation, we emploed the split-step fast Fourier transform algorithm, with the director equation, a Poisson equation, being solved at each propagation step. The condition θ = was applied to all four boundaries. We investigated the trajectories of a nematicon in a rectangular shaped sample with aspect ratio two for three different cases, i.e. we launched the nematicon from three different positions, ). In all cases we launched the beam adjacent to the elongated boundar, keeping = 18, see Fig. 7a) c), and varied the position in the horiontaldirection, with =, 2and 4. When the beam was placed in the center of the cell with =, as shown in Fig. 7a), the induced refractive inde profile was smmetric in the -direction, see Fig. 8a), and thus the effective forces from the opposing boundaries compensated each other. However, due to the induced asm-

8 6 2 a) b) c) Figure 9: Comparison between trajectories of a nematicon in a two dimensional cell for the initial conditions of Fig. 7c). Numerical solution: solid line, modulation solution: dashed line Figure 8: Color online. Refractive inde profiles induced b the soliton at various input positions as in Fig. 7. metr [18, 19] in the gradient of the transverse profile of the director response in the direction, see Fig. 8a), the beam is repelled b the boundar and the nematicon moves in the vertical, direction, periodicall bouncing from one side of the sample to the other as it propagates. This behaviour changes markedl, however, when we launch the input beam shifted slightl off the centre with = 2, see Fig. 7b). As a consequence of this displacement, the induced potential well, the director profile in Fig. 8b), is now both horiontall and verticall asmmetric and eerts a net force in both transverse directions. The force eerted b the farther awa, vertical boundar is correspondingl smaller and the velocit of the nematicon is therefore much higher in the vertical direction. This, in turn, causes the nematicon to follow a steep trajector, the blue line in Fig. 7b) and the inset, which clearl shows that despite being launched ver near the center, the nematicon can reach and bounce off the upper, horiontal boundar before the middle of the cell is reached. Lastl the nematicon was launched near the corner of the sample, see Fig. 7c). We immediatel notice that the repelling effect of the vertical boundar is considerabl stronger, which is evident when comparing Figs. 8b) and c). In contrast to the previous case, the nematicon actuall crosses the center of the cell before the opposite hori- -5 ontal boundar is reached. The dnamics of the beam in this case is further demonstrated in Figs. 7e) g), where we plot the two spatial positions in the red line) and black line) directions e), the beam amplitude f), and the two full nematicon widths in both directions, red, and, black) g). Note that the degree of ellipticit of the beam is comparable with the ratio of the lengths of the cell boundaries. We note that, for both non-smmetric launch conditions, once the nematicon crosses the middle of the cell it slows down and eventuall turns around due to the repelling force of the opposite boundar [18]. This repulsion is a direct consequence of the nonlocal nature of the nematic response, combined with the intricate dependence of the response on the specific properties of the boundaries. In general, boundaries affect not onl the position, but also the internal dnamics [2] of more comple soliton of different smmetries [32]. The modulation theor derived in Section 3 and the Appendi was for the case of a one dimensional nematicon. Equivalent modulation equations cannot be derived in 2+1 dimensions due to the mismatch in smmetr between the rectangular cell and the elliptical beam. However, a simple etension of the one dimensional modulation equations can be made which gives unepectedl good agreement with numerical solutions and gives insightintotheoscillationofthebeaminthecell. Thissimple etension of the one dimensional equations is made b adding a momentum equation equivalent to the momentum equation A4). The resulting trajector is compared with the numerical one in Fig. 9 for the case shown in Fig. 7c). In comparing the numerical and theoretical trajectories it should be remembered that the numerical trajector is for a Gaussian beam and the theoretical one for a sech beam. It can be seen that while there is not detailed agreement, the overall trends and

9 7 shapes of the trajectories are consistent. In particular the positions of maimum deviation from the middle of the cell have a good match. The differences in detail between the trajectories are due, in part, to the sech beam having a much slower deca than a Gaussian beam, so that it has a stronger interaction with the boundaries. CONCLUSIONS In conclusion, the dnamics of a nematicon influenced b the effective forces resulting from the boundaries of a liquid crstal cell was studied both analticall and numericall. Particular emphasis was placed on this dnamics when the position of the nematicon within the cell was non-smmetric. The phsical reason for this boundar effect lies in the interaction of the nematicon tails with the boundaries and the consequent creation of an effective repulsive potential. For a one-dimensional nematicon we developed a modulation theor and verified its predictions b direct numerical simulations. In particular, we discussed the influence of the nematicon profile, namel the deca rate of its tails, on the motion of a nematicon, as well as on its oscillation dnamics. We etended our investigation to the two-dimensional case with equal boundar conditions on all the rectangular boundaries of the cell, for which the results of numerical simulations were compared with the 2 dimensional generaliation of the modulational analsis. We found good qualitative agreement in the description of the 2 dimensional nematicon motion and discussed the reasons for the quantitative discrepancies. This research was supported b the Engineering and Phsical Sciences Research Council EPSRC) under grant EP/D75947/1 and b the Australian Research Council. Appendi: Shelf Radius Appling the modulation method [28, 3, 31], based on the trial function 4) and the director solution 5), gives the modulation equations d [ 2a 2 w+lg 2] =, A1) d π d aw) = lg d dg d = 2 3π dσ d 1 ) 2 V 2, A2) ) 2 4a3 wl ξ) a w 2 2a3 w 3 πνl ln s ) 2 s2 s πν 4a3 wl+ξ) 1 1 πν 2 s2 + )+ 2a3 w 2 ) πνl ln s+ [L ξ)t L+ξ)t + ] s 4a3 w 2 [ ln1 t ) lns + 1 πν 2 t +ln1 t + ) lns ] 2 t +, d [ 2a 2 w+lg 2) V ] = 2a4 w 3 ) d νl ln s+ [t +t + ]+ 2a4 w 2 ν dξ d = V, dh d = d d 8a4 w 3 ν { 2 3 dσ d 1 2 V 2 = 1 + 2a2 w 2 ν + 2a2 wl ξ) ν s a 2 w + 8a4 w 2 L + 2a4 w 4 ln s ) a 2 w+lg 2) V 2 ν νl s ]} [ lns ln1 t ) 1 2 t ln1 t + )+lns t + s+ ) [L ξ)t L+ξ)t + ] A3) [ s 2 s 2 ] +, A4) A5) =, A6) 2w 2 4a2 wl a2 w 2 ν νl ln s [ ln1 t )+lns 1 2 t ln1 t + )+lns + 1 ] 2 t ) 2 s2 + 2a2 wl+ξ) 1 1 ) ν 2 s2 +, where A7) s + = sech L+ξ w, s = sech L ξ w, t + = tanh L+ξ w, t = tanh L ξ w.

10 8 These modulation equations do not include loss to diffractive radiation [28, 31] as it has been observed eperimentall [25] and from theoretical solutions [29] that on the length scales of a tpical cell this loss is ver small and can be ignored. This point is taken up further in the Results and Discussions section. The modulation equation A1) is the equation for conservation of mass, A4) is the momentum equation and A6) is the equation for conservation of energ in the sense of invariances of the Lagrangian for the nematicon equations. The modulation equations A1) A7) have a fied point for a stead nematicon. Denoting fied point values with aˆsuperscript, ˆξ =, ˆV = and ĝ =. The modulation equation A3) gives the stead amplitude as ν 6ŵ 4 â 2 = 2ln 1 tanh L ) 2lnsech L + tanh L+2 Ltanh 2 L+ Lsech 2 L, A8) here L = L/ŵ. It should be noted that when the nematicon is far from the cell walls L ŵ, and it can be easil found that in this limit the fied point relation has a phsical solution. Using this relation between â and ŵ, the fied point is determined from the boundar condition at = on using the conserved energ A6). The final parameter to determine is the shelf length l. The usual method of determining this parameter is to linearise the modulation equations about their fied point, which results in a simple harmonic oscillator equation [28, 31]. The frequenc of this oscillator is then matched to the solitar wave frequenc, determining l [28, 31]. However this method was found not to work for the present modulation equations. This is because in the previous cases the perturbation in the nematic formed a beam, which is not the case here, as can be seen from Fig. 1. The shelf length l was then found from numerical solutions b matching the oscillation frequenc of the solution of the modulation equations A1) A7) to the numerical oscillation frequenc for a particular initial condition, giving l =.5ŵ. This value was found to be robust for other initial conditions. If the nematicon is far from the cell walls, L w. In this limit the modulation equations A1) A7) simplif greatl to become d [ 2a 2 w+lg 2] =, A9) d d d aw) = lg dσ π d 1 ) 2 V 2, A1) dg d = 2 a 3π w 2 4a3 w 2 6a3 wξ 2 πν πνl, d [ 2a 2 w+lg 2) V ] = 8a4 w 2 ξ d νl, dξ d = V, dσ d V 2 2 = 1 2w 2 2a2 w 2 2Lw 8ξ2 [1 + ν L = dh d = d d + 8a4 w 2 ξ 2 νl A11) A12) A13) ], A14) [ 2a 2 3w + 8a4 w 3 8a4 w 2 L ν ν + 2a 2 w +lg 2) V 2 ]. A15) to eponentiall small terms. In this limit the fied point can also be easil found to be ŵ 6 = 2νL 9 H, with â2 = ν 6ŵ 4. A16) [1] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York 1982). [2] P. G. Drain and R. S. Johnson, Solitons: an Introduction, Cambridge Universit Press, New York 1989). [3] L. Lamb and J. Prost, Solitons in Liquid Crstals, Springer, New York 1991). [4] M. J. Ablowit and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge Universit Press, New York 1991). [5] A. C. Newell, Solitons in Mathematics and Phsics, Soc. Ind. Mathem., New York 1991). [6] F. K. Abdullaev, S. Darmanan, and P. Khabibulaev, Optical Solitons, Springer, Berlin 1993). [7] Yu.S.KivsharandG.P.Agrawal, Optical Solitons: From Fibers to Photonic Crstals Academic Press, San Diego 23). [8] A. W. Snder and D. J. Mitchell, Accessible Solitons, Science 276, ). [9] G. I. Stegeman, D. N. Christodoulides, and M. Segev, Optical Spatial Solitons: a historical perspective, IEEE J. Sel. Top. Quantum Electron. 6, ). [1] M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Electricall Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crstal cells, Appl. Phs. Lett. 77, 7 9 2). [11] G. Assanto, M. Peccianti, and C. Conti, Nematicons: optical spatial solitons in nematic liquid crstals, Opt. Photon. News 14, ). [12] M. Peccianti, K. A. Brdakiewic, and G. Assanto, Nonlocal spatial soliton interactions in bulk nematic liquid crstals, Opt. Lett. 27, ). [13] C. Conti, M. Peccianti and G. Assanto, Route to nonlocalit and observation of accessible solitons, Phs. Rev. Lett. 91, ). [14] C. Conti, M. Peccianti and G. Assanto, Observation of optical spatial solitons in a highl nonlocal medium, Phs. Rev. Lett. 92, ). [15] C. Conti, M. Peccianti, and G. Assanto, Comple d-

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