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The aim of this work is to explore and find new closed-form nematicon solutions for different nonlinearities which occur in nematic liquid crystals (NLC) along with proposing optical system application that utilizes NLC nonlinearities. In particular, Lie point symmetry method is employed to scrupulously inspect and acquire solutions for some interesting cases of nonlinearities which have not been fully examined in literature such as quadratic, generalized dual-power law, and eighth-order nonlinearities. A variety of different nematicon dynamics are observed, including bright solitons, dark solitons and periodic behaviors. The explicit solution form for each dynamical behavior is obtained and the solution dependence on model parameters is investigated. The proposed optical system enables the flexible realization of different types of NLC nonlinearities. To the best of our knowledge, this is the first time to attain explicit exact nontrivial solutions for the particular cases of generalized dual-power law and eighth-order nonlinearities.

We propose a numerically efficient algorithm for simulating the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear elliptic partial differential equation governing the angle between the crystal and the direction of propagation. Numerically, the paraxial parabolic wave equation is solved by a fast Huygens sweeping method, while the nonlinear elliptic PDE is handled by the alternating direction explicit (ADE) method. The overall algorithm is shown to be numerically efficient for computing high frequency beam propagations.

The study of light beams propagating in the nonlinear, dispersive, birefringent and nonlocal medium of nematic liquid crystals has attracted widespread interest in the last twenty years or so. We review hereby the underlying physics, theoretical modelling and numerical approximations for nonlinear beam propagation in planar cells filled with nematic liquid crystals, including bright and dark solitary waves, as well as optical vortices. The pertinent governing equations consist of a nonlinear Schrodinger-type equation for the light beam and an elliptic equation for the medium response. Since the nonlinear and coupled nature of this system presents difficulties in terms of finding exact solutions, we outline the various approaches used to resolve them, pinpointing the good agreement obtained with numerical solutions and experimental results. Measurement and material details complement the theoretical narration to underline the power of the modelling.

Using modulation theory, we develop a simple ((2+1) dimensional) model to describe the synergy between the thermo-optical and reorientational responses of nematic liquid crystals to light beams in order to describe the routing of spatial optical solitary waves— nematicons— in such a uniaxial environment. Introducing several approximations based on the nonlocal physics of the material, we are able to predict the trajectories of nematicons and their angular steering with temperature, accounting for the energy exchange between the input beam and the medium through one photon absorption. The theoretical results are then compared with experimental data from previous studies, showing excellent agreement.

We report on novel features of random lasers assisted by near-infrared spatial solitons in nematic liquid crystals. Specifically, we study the role of light-induced reorientational waveguides (nematicons) on the spatial and spectral distributions of the laser modes. We show that the spatially spiky character of the laser emission propagating backwards with respect to the pump tends to disappear in the forward direction, due to the soliton confinement of the generated light. Moreover, the spectral features associated with various random laser resonances appear to merge upon guided-wave propagation along the nematicon, making the nematicon-aided random laser a bidirectional device with distinct emission properties at the two opposite outputs.

Using an external low-frequency electric field applied to dye-doped nematic liquid crystals, we demonstrate that random lasing obtained by optical pumping can be steered in angular direction by routing an all-optical waveguide able to collect the emitted light. By varying the applied voltage from 0 to 2 V, we reduce the walk-off and sweep the random laser guided beam over 7 degrees. Full Text: PDF References V. S. Letokhov, "Generation of light by a scattering medium with negative resonance absorption," Sov. Phys. JETP 26 (4), 835 (1968). DirectLink H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84 (24), 5584 (2000). CrossRef D. S. Wiersma, "The physics and applications of random lasers," Nature Phys. 4 (5) 359-367 (2008). CrossRef D. Wiersma and S. Cavalieri, "A temperature-tunable random laser," Nature 414, 708-709 (2001). CrossRef G. Strangi, S. Ferjani, V. Barna, A. De Luca, N. Scaramuzza, C. Versace, C. Umeton, and R. Bartolino, "Random lasing and weak localization of light in dye-doped nematic liquid crystals," Opt. Express 14 (17), 7737 (2006). CrossRef G. Strangi, S. Ferjani, V. Barna, A. De Luca, C. Versace, N. Scaramuzza, and R. Bartolino, "Random lasing in dye doped nematic liquid crystals: the role of confinement geometry," SPIE 6587, 65870P (2007) doi: 10.1117/12.722887 CrossRef S. Ferjani, V. Barna, A. De Luca, C. Versace, and G. Strangi, "Random lasing in freely suspended dye-doped nematic liquid crystals," Opt. Lett. 33(6), 557-559 (2008). CrossRef S. Ferjani, L-V. Sorriso, V. Barna, A. De Luca, R. De Marco, and G. Strangi, "Statistical analysis of random lasing emission properties in nematic liquid crystals," Phys. Rev. E 78 (1) 011707 (2008). CrossRef H. Bian, F. Yao, H. Liu, F. Huang, Y. Pei, C. Hou, and X. Sun, "Optically controlled random lasing based on photothermal effect in dye-doped nematic liquid crystals," Liq. Cryst. 41 (10), 1436-1441 (2014) CrossRef C. R. Lee, S. H. Lin, C. H. Guo, S. H. Chang, T. S. Mo, and S. C. Chu, "All-optically controllable random laser based on a dye-doped polymer-dispersed liquid crystal with nano-sized droplets," Opt. Express 18 (3), 2406-2412 (2010) CrossRef S. Perumbilavil, A. Piccardi, O. Buchnev, M. Kauranen, G. Strangi, and G. Assanto, "Soliton-assisted random lasing in optically-pumped liquid crystals," Appl. Phys. Lett. 109(16), 161105 (2016); ibid. 110(1), 1019902 (2017). CrossRef S. Perumbilavil, A. Piccardi, O. Buchnev, M. Kauranen, G. Strangi, and G. Assanto, "All-optical guided-wave random laser in nematic liquid crystals", Opt. Express 25 (5), 4672-4679 (2017). CrossRef S. Perumbilavil, A. Piccardi, R. Barboza, O. Buchnev, M. Kauranen, G. Strangi, and G. Assanto, "Beaming random laser with soliton control," Nature Comm., in press (2018) CrossRef M. Peccianti, C. Conti, G. Assanto, A. De Luca and C. Umeton, "Routing of Anisotropic Spatial Solitons and Modulational Instability in liquid crystals," Nature 432, 733-737 (2004). CrossRef J. Beeckman, K. Neyts and M. Haeltermann, "Patterned electrode steering of nematicons," J. Opt. A - Pure Appl. Opt. 8 (2), 214-220 (2006). CrossRef A. Piccardi, M. Peccianti, G. Assanto, A. Dyadyusha and M. Kaczmarek, "Voltage-driven in-plane steering of nematicons," Appl. Phys. Lett. 94, 091106 (2009). CrossRef R. Barboza, A. Alberucci, and G. Assanto, "Large electro-optic beam steering with Nematicons", Opt. Lett. 36 (14), 2611–2613 (2011). CrossRef A. Piccardi, A. Alberucci, R. Barboza, O. Buchnev, M. Kaczmarek, and G. Assanto, "In-plane steering of nematicon waveguides across an electrically adjusted interface", Appl. Phys. Lett. 100 (25), 251107 (2012). CrossRef Y. V. Izdebskaya, "Routing of spatial solitons by interaction with rod microelectrodes," Opt. Lett. 39(6), 1681-1684 (2014). CrossRef A. Pasquazi, A. Alberucci, M. Peccianti, and G. Assanto, "Signal processing by opto-optical interactions between self-localized and free propagating beams in liquid crystals," Appl. Phys. Lett. 87, 261104 (2005). CrossRef S. V. Serak, N. V. Tabiryan, M. Peccianti and G. Assanto, "Spatial Soliton All-Optical Logic Gates", IEEE Photon. Technol. Lett. 18 (12), 1287-1289 (2006). CrossRef 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(18), 3335-3337 (2002). CrossRef A. Fratalocchi, A. Piccardi, M. Peccianti and G. Assanto, "Nonlinearly controlled angular momentum of soliton clusters," Opt. Lett. 32(11), 1447-1449 (2007). CrossRef Y. Izdebskaya, V. Shvedov, G. Assanto, and W. Krolikowski, Nat. Comm. 8, 14452 (2017). CrossRef M. Peccianti and G. Assanto, "Nematicons," Phys. Rep. 516, 147-208 (2012). CrossRef Y. Izdebskaya, A. Desyatnikov, G. Assanto and Y. Kivshar, "Deflection of nematicons through interaction with dielectric particles," J. Opt. Soc. Am. B 30(6), 1432-1437 (2013). CrossRef U. Laudyn, M. Kwasny, F. Sala, M. Karpierz, N. F. Smyth, and G. Assanto,"Curved solitons subject to transverse acceleration in reorientational soft matter," Sci. Rep. 7, 12385 (2017). CrossRef A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek and G. Assanto, "Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity", Phys. Rev. A 82, 023806 (2010). CrossRef

While curved waveguides are fundamental elements in photonics, those induced all-optically in nonlinear uniform dielectrics tend to be straight. In uniaxial soft matter with a reorientational response, such as nematic liquid crystals, light beams in the extraordinary polarization undergo self-focusing via an increase in refractive index and eventually form spatial solitons, i.e., self-induced waveguides. Hereby we investigate the bending of such waveguides by analyzing the trajectory of solitons in nematic liquid crystals— nematicons— in the presence of a linearly varying transverse orientation of the optic axis. Tothis extent we use and compare two approaches: i) a slowly varying (adiabatic) approximation based on momentum conservation of the nematicon in a Hamiltonian sense; ii) the Frank-Oseen elastic theory coupled with a fully vectorial and nonlinear beam propagation method. The models provide comparable results in such a non-homogeneously oriented uniaxial medium and predict bent soliton paths with either monotonic or non-monotonic curvatures, enabling the design of curved channel waveguides induced by light beams.

Starting from the de Gennes theory of director fluctuations in nematics, we report on a model of the spatial fluctuations in nematicon propagation. We demonstrate that, when the long-range correlation that characterizes nematic liquid crystals is taken into account in the thermal noise, it is possible to account for the spatial oscillations and propagation losses experienced by nematicons. Increasing the power of the nematicon, the oscillation amplitudes increase and the propagation losses decrease. The nematicon is then more strongly confined and deviates more, but is less scattered by the thermally induced perturbations of the refractive index. All the results are in good agreement with the experimental observations.

The equations governing optical solitary waves in nonlinear nematic liquid crystals are investigated in both ( 1 + 1 ) and ( 2 + 1 ) dimensions. An isolated exact solitary wave solution is found in ( 1 + 1 ) dimensions and an isolated, exact, radially symmetric solitary wave solution is found in ( 2 + 1 ) dimensions. These exact solutions are used to elucidate what is meant by a nematic liquid crystal to have a nonlocal response and the full role of this nonlocal response in the stability of ( 2 + 1 ) dimensional solitary waves. General, approximate solitary wave solutions in ( 1 + 1 ) and ( 2 + 1 ) dimensions are found using variational methods and they are found to be in excellent agreement with the full numerical solutions. These variational solutions predict that a minimum optical power is required for a solitary wave to exist in ( 2 + 1 ) dimensions, as confirmed by a careful examination of the numerical scheme and its solutions. Finally, nematic liquid crystals subjected to two different external electric fields can support the same solitary wave, exhibiting a new type of bistability.

We discuss, both theoretically and numerically, the propagation of self-confined beams in uniaxial nematic liquid crystals. Our model accounts for optical anisotropy and goes beyond the perturbative regime, that is, it allows addressing the unique properties of nematicons, including their tunable nonlinearity, high nonlocality, and ability to spatial self-steer.

Light localization via reorientation in nematic liquid crystals supports multi-component optical spatial solitons, i.e., vector nematicons. By launching three optical beams of different wavelengths and the same input polarization in a bias-free planar cell, we demonstrate a three-color vector nematicon which is self-trapped thanks to its incoherent nature. Full Text: PDF References G. I. Stegeman and M. Segev, "Optical Spatial Solitons and Their Interactions: Universality and Diversity", Science 286 (5444), 1518 (1999) CrossRef W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions", Phys. Rev. E 63, 016610 (2000) CrossRef D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium", Phys. Rev. A 48, 4583 (1993) CrossRef G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals", IEEE J. Quantum Electron. 39 (1), 13 (2003). CrossRef G. Assanto and M. Karpierz, "Nematicons: self-localised beams in nematic liquid crystals", Liq. Cryst. 36 (10), 1161 (2009) CrossRef M. Peccianti and G. Assanto, "Nematicons", Phys. Rep. 516, 147 (2012). CrossRef M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystals", Opt. Lett. 26 (22), 1791 (2001) CrossRef M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light", Phys. Rev. E Rap. Commun. 65, 035603 (2002) CrossRef G. Assanto, M. Peccianti, C. Umeton, A. De Luca and I. C. Khoo, "Coherent and Incoherent Spatial Solitons in Bulk Nematic Liquid Crystals", Mol. Cryst. Liq. Cryst. 375, 617 (2002) CrossRef A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha and M. Kaczmarek, "Two-Color Vector Solitons In Nonlocal Media", Phys. Rev. Lett. 97, 153903 (2006) CrossRef G. Assanto, N. F. Smyth and A. L. Worthy, "Two-color, nonlocal vector solitary waves with angular momentum in nematic liquid crystals", Phys. Rev. A 78 (1), 013832 (2008) CrossRef G. Assanto, K. Garcia-Reimbert, A. A. Minzoni, N. F. Smyth and A. Worthy, "Lagrange solution for three wavelength solitary wave clusters in nematic liquid crystals", Physica D 240, 1213 (2011) CrossRef G. Assanto, A. A. Minzoni and N. F. Smyth, "Vortex confinement and bending with nonlocal solitons", Opt. Lett. 39 (3), 509 (2014) CrossRef G. Assanto, A. A. Minzoni and N. F. Smyth, "Deflection of nematicon-vortex vector solitons in liquid crystals", Phys. Rev. A 89, 013827 (2014) CrossRef G. Assanto and N. F. Smyth, "Soliton Aided Propagation and Routing of Vortex Beams in Nonlocal Media", J. Las. Opt. Photon. 1, 105 (2014) CrossRef Y. V. Izdebskaya, G. Assanto and W. Krolikowski, "Observation of stable-vector vortex solitons", Opt. Lett. 40 (17), 4182 (2015) CrossRef Y. V. Izdebskaya, W. Krolikowski, N. F. Smyth and G. Assanto, "Vortex stabilization by means of spatial solitons in nonlocal media", J. Opt. 18 (5), 054006 (2016) CrossRef J. F. Henninot, J. Blach and M. Warenghem, "Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal", J. Opt. A 9, 20 (2007) CrossRef Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Z. Krolikowski, M. Belic, G. Assanto and Y. S. Kivshar, "Counterpropagating nematicons in bias-free liquid crystals", Opt. Express 18 (4), 3258 (2010) CrossRef N. Karimi, A. Alberucci, M. Virkki, M. Kauranen and G. Assanto, "Phase-front curvature effects on nematicon generation", J. Opt. Soc. Am. B 5 (33), 903 (2016) CrossRef P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Oxford Science Publications (Clarendon Press, 2nd edition, 1993) I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995) A. Piccardi, M. Trotta, M. Kwasny, A. Alberucci, R. Asquini, M. Karpierz, A. d'Alessandro and G. Assanto, "Trends and trade-offs in nematicon propagation", Appl. Phys. B 104 (4), 805 (2011) CrossRef M. Kwasny, U. A. Laudyn, F. A. Sala, A. Alberucci, M. A. Karpierz and G. Assanto, "Self-guided beams in low-birefringence nematic liquid crystals", Phys. Rev. A 86 (1), 01382 (2012) CrossRef M. Peccianti, A. Fratalocchi and G. Assanto, "Transverse dynamics of nematicons", Opt. Express 12 (26), 6524 (2004) CrossRef C. Conti, M. Peccianti and G. Assanto, "Observation of Optical Spatial Solitons in a Highly Nonlocal Medium", Phys. Rev. Lett. 92 (11), 113902 (2004) CrossRef A. Alberucci, C.-P. Jisha and G. Assanto, "Breather solitons in highly nonlocal media", J. Opt. 18, 125501 (2016) CrossRef

The propagation of coherent light with an initial step intensity profile in a nematic liquid crystal is studied using modulation theory. The propagation of light in a nematic liquid crystal is governed by a coupled system consisting of a nonlinear Schrodinger equation for the light beam and an elliptic equation for the medium response. In general, the intensity step breaks up into a dispersive shock wave, or undular bore, and an expansion fan. In the experimental parameter regime for which the nematic response is highly nonlocal, this nematic bore is found to differ substantially from the standard defocusing nonlinear Schrodinger equation structure due to the effect of the nonlocality of the nematic medium. It is found that the undular bore is of Korteweg–de Vries equation-type, consisting of bright waves, rather than of nonlinear Schrodinger equation-type, consisting of dark waves. In addition, ahead of this Korteweg–de Vries bore there can be a uniform wavetrain with a short front which brings the solution down to the initial level ahead. It is found that this uniform wavetrain does not exist if the initial jump is below a critical value. Analytical solutions for the various parts of the nematic bore are found, with emphasis on the role of the nonlocality of the nematic medium in shaping this structure. Excellent agreement between full numerical solutions of the governing nematicon equations and these analytical solutions is found.

We investigate light self-confinement in nematic liquid crystals with homogeneous planar distribution of the molecular director. Using comb-patterned electrodes we are able to vary the initial orientation, exploiting different degrees of nonlinearity by the electro-optic effect. We experimentally demonstrate tuning of the nonlinearity by studying both single nematicon propagation and planar interactions between two nematicons.

We review the mathematical modelling of propagation and specific interactions of solitary beams in nematic liquid crystals — so-called nematicons. The theory is first developed for the evolution of a single nematicon; then it is extended to the interaction of two nematicons of different wavelengths, employing linear momentum conservation equations to predict that two colour nematicons can form a vector bound state. Considering optical vortices, we show that the nonlocal response of liquid crystals stabilises a single vortex, unstable in local media. Moreover, the interaction with a nematicon in another colour can stabilise a vortex for nonlocalities far below those at which an isolated vortex remains unstable. When multiple nematicons of the same wavelength interact, the radiation they shed can join them together, still resulting in a vortex. Finally, we discuss the escape of a nematicon from a nonlinear waveguide, using simple modulation theory based on momentum conservation to model the effect and get excellent agreement with the experimental results.

The addition of orbital angular momentum has been previously shown to stabilise beams of elliptic cross-section. In this article the evolution of such elliptical beams is explored through the use of an approximate methodology based on modulation theory. An approximate method is used as the equations that govern the optical system have no known exact solitary wave solution. This study brings to light two distinct phases in the evolution of a beam carrying orbital angular momentum. The two phases are determined by the shedding of radiation in the form of mass loss and angular momentum loss. The first phase is dominated by the shedding of angular momentum loss through spiral waves. The second phase is dominated by diffractive radiation loss which drives the elliptical solitary wave to a steady state. In addition to modulation theory, the “chirp” variational method is also used to study this evolution. Due to the significant role radiation loss plays in the evolution of an elliptical solitary wave, an attempt is made to couple radiation loss to the chirp variational method. This attempt furthers understanding as to why radiation loss cannot be coupled to the chirp method. The basic reason for this is that there is no consistent manner to match the chirp trial function to the generated radiating waves which is uniformly valid in time. Finally, full numerical solutions of the governing equations are compared with solutions obtained using the various variational approximations, with the best agreement achieved with modulation theory due to its ability to include both mass and angular momentum losses to shed diffractive radiation.

Solitons are self-localized wave packets arising from a robust balance between dispersion and nonlinearity. They are a universal phenomenon, exhibiting properties typically associated with particles. Thus far, interactions between solitons have been observed only between neighbouring solitons at close proximity. Here, we study interactions between solitons in highly non-local nonlinear media, and demonstrate experimentally attraction between solitons propagating in different samples, where their optical fields never overlap and the interaction is mediated solely by a non-optical wire. This increases the soliton interaction range by orders of magnitude, and breaks the close-proximity and nearest-neighbour limitations on soliton interactions. We also experiment with three-dimensional interactions between solitons that are far apart, where the solitons capture each other into a spiralling motion with a circular orbit, and a tangential velocity that does not depend on the separation between solitons. Our study suggests that these phenomena could be used in the construction of novel model systems for studying the behaviour of complex nonlinear networks.

Light filaments or optical spatial solitons are self-confined (non-spreading) beams that originate from the balance between diffraction and self-focusing in nonlinear optical media (those with a response dependent on the level of excitation)(1-3). Owing to their ability to self-trap as well as to guide weaker signals (even if differing in colour or modulation format) within the waveguides or 'light-pipes' they induce, optical spatial solitons could form the basis of future all-optical processing networks(4,5). One of the most interesting challenges in soliton propagation and engineering concerns light. laments incident on linear/nonlinear or nonlinear/nonlinear interfaces. Here we report the robust propagation, refraction and reflection of optical spatial solitons at the interface between two regions of a nematic liquid crystal. The ability to independently tune the optical properties of each region enables us to steer the beams by refraction and total internal reflection by as much as - 18 and +22 degrees, respectively. Moreover, the extended (nonlocal) and anisotropic response of our system supports polarization healing of the solitons across the interface as well as non-specular filament reflection. Finally, exploiting the inherent and all-optically tunable birefringence, we demonstrate unprecedented nonlinear Goos-Hanchen lateral shifts in excess of 0.5 mm.

In certain materials, the spontaneous spreading of a laser beam (owing to diffraction) can be compensated for by the interplay of optical intensity and material nonlinearity. The resulting non-diffracting beams are called ‘spatial solitons’ (refs 1–3), and they have been observed in various bulk media4,5,6. In nematic liquid crystals7,8,9, solitons can be produced at milliwatt power levels10,11,12 and have been investigated for both practical applications13 and as a means of exploring fundamental aspects of light interactions with soft matter14,15. Spatial solitons effectively operate as waveguides, and so can be considered as a means of channelling optical information along the self-sustaining filament. But actual steering of these solitons within the medium has proved more problematic, being limited to tilts of just a fraction of a degree16,17,18,19,20. Here we report the results of an experimental and theoretical investigation of voltage-controlled ‘walk-off’ and steering of self-localized light in nematic liquid crystals. We find not only that the propagation direction of individual spatial solitons can be tuned by several degrees, but also that an array of direction-tunable solitons can be generated by modulation instability21,22,23,24,25. Such control capabilities might find application in reconfigurable optical interconnects, optical tweezers and optical surgical techniques.

Spatial solitons in nematic liquid crystals, ornematicons, can be generated in bulk through a reorientational nonlinearity. We investigate nematicon interactions, showing their behavior related to the inherent nonlocal response of the medium. Solving for both the mechanics and the optics of molecular reorientation in nematic liquid crystals with a nonlinear paraxial version of a Beam Propagator, we demonstrate both in plane and out of plane interactions, disclosing soliton crossing and spiraling.

Nematic liquid crystals exhibit a saturable nonlinearity, nonlocal in time and in space, owing to light-induced reorientation of the molecular director. In such media, (2 + 1)-dimensional spatial solitons can be obtained at milliwatt-power levels and confine weaker co-polarized signals of different wavelengths, employing either coherent or partially incoherent excitations. Our experimental observations of solitons and related phenomena, ranging from angular steering to mutual interactions, are reviewed. Models are briefly discussed and used to predict and interpret the results.

We investigate the role of competing nonlinear responses in the formation and propagation of bright spatial solitons. We use nematic liquid crystals (NLCs) exhibiting both thermo-optic and reorientational nonlinearities with continuous-wave beams. In a suitably prepared dye-doped sample and dual beam collinear geometry, thermal heating in the visible affects reorientational self-focusing in the near infrared, altering light propagation and self-trapping.

The interaction of an accelerating Airy beam and a solitary wave is investigated for integrable and non-integrable equations governing nonlinear optical propagation in various media. For the integrable nonlinear Schrodinger equation, by way of a Backlund transformation, we show that no momentum exchange takes place, as the only effect of the interaction is to modulate the amplitude of the solitary wave. The latter result also holds for propagation in anisotropic media with birefringent walkoff and nonlocality, as specifically addressed with reference to uniaxial nematic liquid crystals in the absence of beam curvature. When the wavefront curvature characteristic of accelerating Airy beams is accounted for, both asymptotic and numerical solutions show that a small amount of momentum is initially exchanged, with the solitary wave rapidly settling to a state of constant momentum. (C) 2014 Elsevier B.V. All rights reserved.

Optical spatial solitons are self-trapped optical beams that exist by virtue of the balance between diffraction and nonlinearity. They propagate and interact with one another while displaying properties that are normally associated with real particles. Solitons, in general, manifest themselves in a large variety of wave/particle systems in nature: practically in any system that possesses both dispersion (in time or space) and nonlinearity. Solitons have been identified in optics, plasmas, fluids, condensed matter, particle physics, and astrophysics. Yet over the past decade, the forefront of soliton research has shifted to optics. In the paper, we describe the historical evolution of spatial solitons from speculative creatures to one of the most fascinating features optics has to offer.

Spatial solitons, beams that do not spread owing to diffraction when they propagate, have been demonstrated to exist by virtue of a variety of nonlinear self-trapping mechanisms. Despite the diversity of these mechanisms, many of the features of soliton interactions and collisions are universal. Spatial solitons exhibit a richness of phenomena not found with temporal solitons in fibers, including effects such as fusion, fission, annihilation, and stable orbiting in three dimensions. Here the current state of knowledge on spatial soliton interactions is reviewed.

We demonstrate some basic all-optical (electrically unbiased) logic gates in azobenzene liquid crystalline cells, exploiting their large nonlinearity for light localization and the trans-cis photoisomerization for all-optical external control. Spatial solitons were excited at microwatt power levels at 632.8 nm, whereas gating and switching were achieved with milliwatt beams at 409 nm

The deflection of a vector soliton formed by a solitary wave and an optical vortex in nematic liquid crystals is investigated upon interaction with a localized refractive index defect. The azimuthal instability of the vortex can be triggered by the index perturbation and enhanced by the distortion of the copropagating solitary wave when in the vicinity of the defect. A modulation theory is developed to study the refraction of the vector soliton and is found to be in good agreement with numerical solutions. This model reveals the crucial role of the diffractive radiation shed by both beam components as they evolve, showing that radiation reduces the destabilizing effect of the solitary wave interaction with the vortex, thus enlightening the effect of this continuous spectrum on the evolution of the nonlinear wave packets.

Two-dimensional optical vortex beams can propagate stably and undergo bending in nonlocal media with the aid of a spatial soliton, the latter preventing their destabilization and breakup. By colaunching a suitable pair of collinear soliton and vortex beams in nematic liquid crystals, we conduct a series of numerical experiments to demonstrate that the vortex beam can be made to propagate in the nematicon waveguide and follow its trajectory, even in the cases of refraction and total internal reflection at a dielectric interface. Modulation theory supports these findings and provides an excellent theoretical framework.

We investigate nonlinear self-focusing and nematicon generation in low birefringencenematic liquid crystals (NLCs), where n=0.04. Nematicons are obtained for larger optical powers but they are more than in NLCs with standard birefringence, of the order of n=0.2 Moreover, in low-birefringent NLCs, polarization components exchange energy in a nonlinear regime. Such behaviour is not observed in typical NLCs, where ordinary and extraordinary waves tend to propagate independently due to large walk off.The results of our experimental rest are also compared with numerical simulations, using a fully vectorial BPM. Calculations and experimental results are in good qualitative and quantitative agreement. Full Text: PDF References G. Assanto and M. A. Karpierz, "Nematicons: self-localised beams in nematic liquid crystals", Liq. Cryst. 36, 1161 (2009). CrossRef M. Peccianti and G. Assanto, "Nematicons", Phys. Rep. 516, 147(2012). CrossRef M. Peccianti, 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. A. Karpierz, M. Sierakowski, M. Swillo and T. Wolinski, "Self Focusing in Liquid Crystalline Waveguides", Mol. Cryst. Liq. Cryst. 320, 157 (1998). CrossRef U. A. Laudyn, M. Kwasny, M. A. Karpierz, "Nematicons in chiral nematic liquid crystals", Appl. Phys. Lett. 94, 091110 (2009). CrossRef U. A. Laudyn, M. Kwasny, K. Jaworowicz, K. A. Rutkowska, M. A. Karpierz and G. Assanto, "Properties of spatial solitons in chiral nematic liquid crystal cells", Photon. Lett. Poland 1, 7 (2009). CrossRef I. C. Khoo, "Liquid Crystals, Second Edition", Wiley & Sons, New Jersey (2007). CrossRef A. Piccardi, A. Alberucci, G. Assanto, "Soliton self-deflection via power-dependent walk-off", Appl. Phys. Lett.96, 061105 (2010). CrossRef F. A. Sala and M. A. Karpierz, "Modeling of molecular reorientation and beam propagation in chiral and non-chiral nematic liquid crystals", Opt. Express 20, 13923 (2012). CrossRef A. Piccardi, M. Trotta, M. Kwasny, A. Alberucci, R. Asquini, M. Karpierz, A. D'Alessandro and G. Assanto, "Trends and trade-offs in nematicon propagation", Appl. Phys. B 104, 805 (2011). CrossRef R. Dąbrowski, J. Dziaduszek and T. Szczucinski, "Mesomorphic Characteristics of Some New Homologous Series with the Isothiocyanato Terminal Group", Mol. Cyst. Liq. Cryst. 124, 241 (1985). CrossRef

We investigate light self-trapping and nematicon generation in reorientional nematic liquid crystals, addressing the role of phase-front curvature and size of the TEM00 input beam excitation. The experimental results in planar cells are in excellent agreement with a semi-analytical model and numerical simulations. These findings can substantially aid characterization and generation of spatial solitons in liquid crystals and be readily generalized to other nonlocal nonlinear media and soft matter systems.

We use modulation theory to analyze the interaction of optical solitons and vortices with a dielectric interface between two regions of nematic liquid crystals. In the analysis we consider the role of nonlocality, anisotropy and nonlinear reorientation and compare modulation theory results with numerical results. Upon interacting with the interface, nematicons undergo transverse distortion but remain stable and eventually return to a steady state, whereas vortices experience an enhanced instability and can break up into bright beams or solitary waves.

We discuss the interaction of spatial solitary waves in nematic liquid crystals, termed nematicons, with localized inhomogeneities in the distribution of the optic axis. For beam waists small compared with the defect width, self-localization is preserved; in the opposite limit the spatial solitary wavesundergo significant attenuation due to diffractive losses, to the extent that nematicons are essentially destroyed. Owing to their power-waist dependence spatial solitary waves interacting with defects in nematic liquid crystals are subject to an excitation dependent transition from particle-like to wave-like behaviour. In the latter regime their trajectory varies with input power, leading to novel approaches to the design and realization of all-optical circuits.

Using mW light beams to generate spatial solitons in nematic liquid crystals, all-optical switching/logic can be performed on a signal launched in the soliton-induced waveguides. Through the collisional behavior of solitons in a nonlocal medium, the signal can be steered in angle and output position. A power-dependent X junction, AND, and NOR gates are demonstrated.

The propagation of a bulk optical solitary wave in a rectangular cell filled with a nematic liquid crystal-a nematicon-is mathematically modelled. In order to overcome the Freedricksz threshold the cell walls are rubbed to pretilt the nematic. A modulation theory, based on a Lagrangian formulation, is developed for the (2 + 1)-dimensional propagation of the solitary wave beam down the cell. This modulation theory is based on two different formulations of the director distribution. The relative advantages and disadvantages of these two methods are discussed. A previously unexplored method based on images is found to possess significant advantages. Excellent agreement with full numerical solutions of the nematicon equations is found for both methods. Finally, the implications of the results obtained for some widely used approximations to the nematicon equations are discussed, particularly their use in comparisons with experimental results.

Nematicons, i.e., optical spatial solitons in nematic liquid crystals (NLC), have been attracting a great deal of attention due to their unique properties such as, for example, excitability at powers of a few hundred µW and the possibility to be electrically and/or optically (by other light beams) bent.[1] In this work we investigate, both experimentally and theoretically, the nematicon behavior for different degrees of nonlinearity, discussing how the latter affects the beam width (self-focusing) and trajectory (self-steering). Having defined the angle between the molecular director n (i.e., the local optic axis) and the beam wavevector k = n 0 k 0 z (k 0 is the vacuum wavenumber, n 0 the linear refractive index), the propagation of the extraordinary wave in the plane yz in a homogeneous NLC cell of thickness L across x is ruled by the equivalent 2D model [2,3] equation equation where Φ is the beam magnetic field and ψ is the all-optical perturbation on θ, with θ 0 being the unperturbed θ, i.e., θ 0 = Φ (F = 0). In Eqs. (1–2), D y is the diffraction coefficient along y, δ(b) the walk-off of the soliton, γ = [e 0 /(4K)](n ∥ 2 − n ⊥ 2)[Z 0 /(n 0 cosδ)]2, and Δn e 2 is the nonlinear change in the extraordinary refractive index n e . Eq. (2) is a reorientational equation which allows to compute the dielectric properties of the medium (δ and n e ) once it is known the torque exerted by light on the NLC molecules, whereas (1) determines the beam profile once the n-distribution is known. It is clear from Eq. (2) that the nonlinear response, determined by the optical torque, depends on the initial angle θ 0 : hence, by changing θ 0 it is possible to easily modify the nonlinear response of the sample, the latter feasible via an applied bias in a planar cell with interdigitated comb-like electrodes. [3] In the limit ψ θ ≪ 0 , using Eqs. (1–2) it is possible to define two scalar parameters to investigate self-focusing (ruled by Δn e 2) and self-steering (determined by δ) versus θ 0 : a nonlocal Kerr coefficient n 2 , given by n 2 (θ 0 ) = 2γsin[2(θ 0 −δ)]n e 2(θ 0 ) tanδ, and equation, proportional to the sensitivity of δ to the light intensity (Fig. 1), respectively. Numerical simulations of Eqs. (1–2) via BPM confirm the soliton behavior versus θ 0 (Fig. 1). Figure 1 also shows the corresponding experimental results, with an excellent agreement with the theoretical predictions.

We report on the first (to our knowledge) experimental observation of higher-order modes guided by soliton-induced waveguides in nematic liquid crystals. We find that the nematicon waveguides operate in a bounded power region specific to each guided mode. Below this region, the guided beams diffract; above this region, the mode mixing and coupling give rise to an unstable output.

We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and grey optical spatial solitary waves for both the defocusing nonlinear Schrodinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used as a test bed for the modulation theory applied to the nematicon equations, which have no exact solitary wave solution. We find that the evolution of dark and grey NLS solitons, as well as nematicons, is entirely driven by the emission of diffractive radiation, in contrast to the evolution of bright NLS solitons and bright nematicons. Moreover, the steady nematicon profile is non-monotonic due to the long range nonlocality associated with the perturbation of the optic axis. Excellent agreement isobtained with numerical solutions of both the defocusing NLS and nematicon equations. The comparisons for the nematicon solutions raise a number of subtle issues relating to the definition and measurement of the width of adark or grey nematicon.

The work of Kavitha et al. [1] uses Hirota's method to derive N solitary wave solutions of the equations governing the propagation of coherent light through a nematic liquid crystal. However, Hirota's method implicitly assumes that the governing equations possess an inverse scattering solution, which is not the case for the equations governing light propagation in nematic liquid crystals. The solutions derived in [1] are then invalid. Furthermore, these solutions do not satisfy the assumed boundary conditions and have a complex angle of rotation of the nematic molecules, which is unphysical.

We demonstrate power-dependent steering of self-confined light beams propagating in the nonparaxial regime in a guest-host nonlinear medium. Exploiting the Janossy enhancement in dye-doped liquid crystals with reorientational response, we observe sub-mW spatial soliton self-steering as large as 39 degrees, accompanied by trapping of the light spontaneously emitted by the excited dye. The self-turning is modeled in terms of an effective transverse force stemming from the longitudinal electric field in the tightly confined beams, with good agreement between data and simulations.

We experimentally investigate the interaction of two counter-propagating spatial optical solitons (nematicons) in bias-free nematic liquid crystals. We demonstrate the existence of vector solitons composed of two nematicons propagating in opposite directions and analyze their stability versus relative distance and input power.We observe the dynamic instability of two counterpropagating nematicons in the form of time-dependent splitting and spatial entanglement.

We study experimentally the propagation dynamics and interaction of a pair of mutually incoherent nematicons: spatial optical solitons in nematic liquid crystals. In contrast to earlier studies, we consider a bias-free liquid-crystal cell and compare the soliton interaction in copropagating and counterpropagating geometries. We analyze the dependence of nematicon interaction on input power and observe a direct manifestation of a long-range nonlocal nonlinearity. Attraction of counterpropagating solitons requires higher powers and longer relaxation times than that of copropagating nematicons due to losses-induced power asymmetry of counterpropagating nematicons., Comment: 5 pages, z figures

We investigate spatial optical solitons propagating in a medium with a saturable but adjustable nonlinearity and a fixed degree of nonlocality. We employ nematic liquid crystals in a planar cell with optical properties tuned by an external voltage and solitons excited in the near infrared. We also demonstrate soliton self-bending versus excitation due to nonlinear variations in walk-off. A theoretical model accounting for the longitudinal derivatives is employed to compute the refractive index distribution and is found in excellent agreement with the experimental data.

The propagation of solitary waves in nematic liquid crystals in the presence of localized nonuniformities is studied. These nonuniformities can be caused by external electric fields, other light beams, or any other mechanism which results in a modified director orientation in a localized region of the liquid-crystal cell. The net effect is that the solitary wave undergoes refraction and trajectory bending. A general modulation theory for this refraction is developed, and particular cases of circular, elliptical, and rectangular perturbations are considered. The results are found to be in excellent agreement with numerical solutions.

The steering of a self-guided beam in a dye-doped nematic liquid crystal caused by an external illumination (control beam) that induces changes in the refractive index of the medium is theoretically analyzed. The interaction between the control beam and the dye molecules modifies the anchoring of the nematic molecules, so changing the director orientation in the bulk of the medium. Beam evolution is investigated by use of a modulation theory approach. It is found that the beam trajectory is independent of the beam profile, as long as this profile is self-similar. Solutions obtained from the modulation theory approach are in excellent agreement with numerical solutions.

We demonstrate, both experimentally and theoretically, excitation-dependent self-bending of spatial solitons in nematic liquid crystals. The observed deflection is explained by nonlinear changes in walk-off, as induced by the rotation of the optic axis via power-driven reorientation.