Observation of optical spatial solitons in a highly nonlocal medium

Orbital angular momentum and rotational properties of off-axis vortex beams in two-dimensional media with anisotropic nonlocal nonlinearity

By introducing anisotropy into nonlinear propagations, off-axis vortex beams exhibit significantly different characteristics compared to the isotropic case. The orbital angular momentum (OAM) is non-conservative and can periodically change between positive and negative values. Accordingly, the rotation of phase singularity can transit between clockwise and counterclockwise directions. Furthermore, the phase singularity can move to infinity when the OAM approaches zero. By using the Ehrenfest theorem, the motion of the beam center is obtained. Its trajectory can be circular and parabolic or follow other complex shapes, depending closely on the anisotropy of the nonlinearity. The rotational velocity of the beam center can be modulated by the nonlinearity anisotropy and can far exceed the initial value during its propagation. These results may find potential applications in beam shaping and optical manipulation.

Dual Role of Beam Polarization and Power in Nematic Liquid Crystals: A Comprehensive Study of TE- and TM-Beam Interactions

Optical spatial solitons are self-guided wave packets that maintain their transverse profile due to the self-focusing effect of light. In nematic liquid crystals (NLC), such light beams, called nematicons, can be induced by two principal mechanisms: light-induced reorientation of the elongated molecules and thermal changes in the refractive index caused by partial light absorption. This paper presents a detailed investigation of the propagation dynamics of light beams in nematic liquid crystals (NLCs) doped with Sudan Blue dye. Building on the foundational understanding of reorientational and thermal solitons in NLCs and the effective breaking of the action-reaction principle in spatial solitons, this study examines the interaction of infrared (IR) and visible beams in a [-4-(trans-4'-exylcyclohexyl)isothiocyanatobenzene] (6CHBT) NLC. Our experimental results highlight the intricate interplay of beam polarizations, power levels, and the nonlinear properties of NLCs, offering new insights into photonics and nonlinear optics in liquid crystals.

Polarization state evolution of a twisted vector optical field in a strongly nonlocal nonlinear medium

The evolution of the state of polarization (SoP) in a twisted vector optical field (TVOF) with an astigmatic phase in a strongly nonlocal nonlinear medium (SNNM) is investigated. The effect of an astigmatic phase on the propagation dynamics of the twisted scalar optical field (TSOF) and TVOF during propagation in the SNNM leads to reciprocally periodical evolutions of stretch and shrink, accompanied by the reciprocal transformation of the beam shape between an initial circle shape and threadiness distribution. The TSOF and TVOF rotate along the propagation axis if the beams are anisotropic. In particular, the reciprocal conversions between the linear and circular polarizations occur in the TVOF during propagation, which are strongly related to the initial powers, twisting strength coefficients, and initial beam reshapes. The numerical results confirm the analytical predictions by the moment method for the dynamics of the TSOF and TVOF during propagation in a SNNM. The underlying physics for the polarization evolution of a TVOF in a SNNM are discussed in detail.

Controllable focusing behavior of chirped Pearcey-Gaussian pulses under time-dependent potentials

We investigate the propagation dynamics of the Pearcey-Gaussian (PG) pulses in the presence of time-dependent potentials in a linear medium both theoretically and numerically. We demonstrate that the combination of the linear potential and the initial chirp of PG pulses can flexibly control the propagation trajectory and inherent focusing properties of the PG pulses. When the parabolic potential is taken into account, the chirped PG pulses are periodically focused and reversed. By adjusting the parabolic potential and the pulse chirp, the characteristics of the focal points, such as position, intensity, and spacing between focal points, can be manipulated effectively. The interaction of two temporally separated PG pulses still shows a periodic evolution with controllable focusing characteristics. These results can broaden the application range of PG pulses and provide some inspiration for the control of PG pulses under nonlinear conditions.

Twisted Gaussian Schell-model breathers and solitons in strongly nonlocal nonlinear media

Based on the Snyder-Mitchell linear model and the cross-spectral density (CSD) function, the analytical propagation formula of twisted Gaussian Schell-model (TGSM) beams in strongly nonlocal nonlinear medium (SNNM) is derived. Then the propagation characteristics of TGSM beam are studied. It is found that the soliton radius is jointly determined by the initial power, coherence length, and twist factor; the degree of spatial coherence is adjusted by changing the twist factor without affecting the soliton intensity. In the case of non-soliton properties, there is a threshold of coherence length which makes partially coherent beams have the same evolution law as completely coherent beams. Furthermore, increasing the twist factor, decreasing the coherence length and initial power can improve the beam quality of the beam propagating in SNNM.

Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses

We review the recent investigation of a new form of nonlocally nonlinear system with oscillatory responses. The system has various new features, such as the nonlocality-controllable transition of self-focusing and self-defocusing nonlinearities, a unique modulational instability and new forms of solitons. We also discuss the propagation of the optical beam in a nematic liquid crystal with negative dielectric anisotropy and demonstrate theoretically that propagation can be modelled by the system.

Intermixed Time-Dependent Self-Focusing and Defocusing Nonlinearities in Polymer Solutions

Low-power visible light can lead to spectacular nonlinear effects in soft-matter systems. The propagation of visible light through transparent solutions of certain polymers can experience either self-focusing or defocusing nonlinearity, depending on the solvent. We show how the self-focusing and defocusing responses can be captured by a nonlinear propagation model using local spatial and time-integrating responses. We realize a remarkable pattern formation in ternary solutions and model it assuming a linear combination of the self-focusing and defocusing nonlinearities in the constituent solvents. This versatile response of solutions to light irradiation may introduce a new approach for self-written waveguides and patterns.

Incoherent localized structures and hidden coherent solitons from the gravitational instability of the Schrödinger-Poisson equation

The long-term behavior of a modulationally unstable conservative nonintegrable system is known to be characterized by the soliton turbulence self-organization process. We consider this problem in the presence of a long-range interaction in the framework of the Schrödinger-Poisson (or Newton-Schrödinger) equation accounting for the gravitational interaction. By increasing the amount of nonlinearity, the system self-organizes into a large-scale incoherent localized structure that contains "hidden" coherent soliton states: The solitons can hardly be identified in the usual spatial or spectral domains, but their existence can be unveiled in the phase-space representation (spectrogram). We develop a theoretical approach that provides the coupled description of the coherent soliton component [governed by the Schrödinger-Poisson equation (SPE)] and of the incoherent structure [governed by a wave turbulence Vlasov-Poisson equation (WT-VPE)]. We demonstrate theoretically and numerically that the incoherent structure introduces an effective trapping potential that stabilizes the hidden coherent soliton and we show that the incoherent structure belongs to a family of stationary solutions of the WT-VPE. The analysis reveals that the incoherent structure evolves in the strongly nonlinear regime and that it is characterized by a compactly supported spectral shape. By relating the analytical properties of the hidden soliton to those of the stationary incoherent structure, we clarify the quantum-to-classical (i.e., SPE-to-VPE) correspondence in the limit ℏ/m→0: The hidden soliton appears as the latest residual quantum correction preceding the classical limit described by the VPE. This study is of potential interest for self-gravitating Boson models of fuzzy dark matter. Although we focus our paper on the Schrödinger-Poisson equation, we show that the regime of hidden solitons stabilized by an incoherent structure is general for long-range wave systems featured by an algebraic decay of the interacting potential. This work should stimulate nonlinear optics experiments in highly nonlocal nonlinear (thermal) media that mimic the long-range nature of gravitational interactions.

Propagation of Pearcey Gaussian beams in a strongly nonlocal nonlinear medium

We introduce the propagation of Pearcey Gaussian (PG) beams in a strongly nonlocal nonlinear medium (SNNM) analytically. Our results show that PG beams propagating in the SNNM have two different focusing positions. The intensity peak appears at different focusing positions depending on the selection of the nonlinear parameters. In addition, the effects of the nonlinear parameters and the scaling factor on the trajectory, the position of the intensity focusing, the intensity evolution between focus locations, and the radiation force are studied.

Opto-chemo-mechanical transduction in photoresponsive gels elicits switchable self-trapped beams with remote interactions

Next-generation photonics envisions circuitry-free, rapidly reconfigurable systems powered by solitonic beams of self-trapped light and their particlelike interactions. Progress, however, has been limited by the need for reversibly responsive materials that host such nonlinear optical waves. We find that repeatedly switchable self-trapped visible laser beams, which exhibit strong pairwise interactions, can be generated in a photoresponsive hydrogel. Through comprehensive experiments and simulations, we show that the unique nonlinear conditions arise when photoisomerization of spiropyran substituents in pH-responsive poly(acrylamide-co-acrylic acid) hydrogel transduces optical energy into mechanical deformation of the 3D cross-linked hydrogel matrix. A Gaussian beam self-traps when localized isomerization-induced contraction of the hydrogel and expulsion of water generates a transient waveguide, which entraps the optical field and suppresses divergence. The waveguide is erased and reformed within seconds when the optical field is sequentially removed and reintroduced, allowing the self-trapped beam to be rapidly and repeatedly switched on and off at remarkably low powers in the milliwatt regime. Furthermore, this opto-chemo-mechanical transduction of energy mediated by the 3D cross-linked hydrogel network facilitates pairwise interactions between self-trapped beams both in the short range where there is significant overlap of their optical fields, and even in the long range--over separation distances of up to 10 times the beam width--where such overlap is negligible.

Three-dimensional chirped Airy Complex-variable-function Gaussian vortex wave packets in a strongly nonlocal nonlinear medium

Three-dimensional chirped Airy Complex-variable-function Gaussian vortex (CACGV) wave packets in a strongly nonlocal nonlinear medium (SNNM) are studied. By varying the distribution parameter, CACGV wave packets can rotate stably in a SNNM in different forms, including dipoles, elliptic vortices, and doughnuts. Numerical simulation results for the CACGV wave packets agree well with theoretical analysis results under zero perturbation. The Poynting vector related to the physics of the rotation phenomenon and the angular momentum as a torque corresponding to the force are also presented. Finally, the radiation forces of CACGV wave packets acting on a nanoparticle in a SNNM are discussed.

Solitary wave formation under the interplay between spatial inhomogeneity and nonlocality

The presence of spatial inhomogeneity in a nonlinear medium restricts the formation of solitary waves (SW) on a discrete set of positions, whereas a nonlocal nonlinearity tends to smooth the medium response by averaging over neighboring points. The interplay of these antagonistic effects is studied in terms of SW formation and propagation. Formation dynamics is analyzed under a phase-space approach and analytical conditions for the existence of either discrete families of bright SW or continuous families of kink SW are obtained in terms of Melinikov's method. Propagation dynamics are studied numerically and cases of stable and oscillatory propagation as well as dynamical transformation between different types of SW are shown. The existence of different types and families of SW in the same configuration, under appropriate relations between their spatial width and power with the inhomogeneity and the nonlocality parameters, suggests an advanced functionality of such structures that is quite promising for applications.

Trajectories and rotations controlled off-axis winding beams in nonlocal nonlinear media

Off-axis winding beams are discussed in nonlocal nonlinear media, whose trajectories and rotations can be controlled by the optical power and the orbital angular momentum (OAM). By changing optical power or the OAM, optical beams can propagate along different trajectories including ellipses, circles, rectangles, rhombus and so on. Due to the OAM stemming from a combination of the cross phase and the vortex phase, different parts of the optical beam exhibit distinct rotating characteristics. The elliptic envelope revolves around the beam center resulting from the cross phase, while optical peaks spin about their own axes because of the vortex phase. The two rotating velocities can be adjusted separately by the optical power and the OAM, respectively. Optical power plays a global role on the revolving velocity and the spinning velocity and can enhance the two rotations synchronously. In contrast, the OAM has a local effect and can change the relative magnitudes of these two velocities. Our results may provide a tailoring tool of evolution properties for optical beams. The property of controllable trajectories may find applications in all-optical controlling. While, the controllable rotating property can be used as in optical spanners to operate micro- or nanoparticles.

Self-induced periodic interfering behavior of dual Airy beam in strongly nonlocal medium

Based on a reduction of nonlocal nonlinear Schrödinger equation in strongly nonlocal regime, the linear Schrödinger equation with parabolic potential, analytical results describing the evolution of dual Airy beam are presented. The results show that the dual Airy beam in strongly nonlocal medium exhibits a periodic focusing and defocusing behavior, and forms the interference fringes between the focusing and defocusing positions. The analytical results are verified by numerically solving nonlocal nonlinear Schrödinger equation and shown to be reasonable when the characteristic response width is broader than the width of the dual Airy beam. Furthermore, the characteristics of the interference fringes induced by the dual Airy beam are also investigated in detail, and can be used for the measurement of the system parameters. In addition, we propose a scheme to generate dual Airy beam in strongly nonlocal medium.

Anomalous interaction of spatial solitons in nematic liquid crystals

We study experimentally the interaction of mutually incoherent bright spatial solitons in dye-doped nematic liquid crystals (LCs). The dye-induced light absorption results in a complex nonlinear optical response of the LC having spatially nonlocal focusing and defocusing contributions. The competition between both nonlinearities leads to the separation-dependent soliton interaction with repulsion of distant and attraction of closely placed solitons.

Interplay of Thermo-Optic and Reorientational Responses in Nematicon Generation

Employing several nematic liquid crystal mixtures, we investigate how the thermo-optic response of nonlinear birefringent soft-matter affects the propagation of light beams and the features of self-induced waveguides. We address the formation of optical spatial solitons and the control of their trajectories versus temperature, comparing the measurements with the expectations based on a simplified model, showing an excellent agreement. Moreover, in a guest⁻host mixture with an absorbing dye dopant, we study the competition between reorientational and thermal nonlinearities, demonstrating that the two processes can be adjusted independently in order to tune the soliton properties, i.e., trajectory and confinement strength. Our results are an important contribution to better comprehend the role played by material properties on linear and nonlinear beam propagation, as well as their exploitation for signal processing and addressing.

Deformation of dark solitons in a PT -invariant variable coefficients nonlocal nonlinear Schrödinger equation

We derive dark and antidark soliton solutions of a parity-time reversal (PT) -invariant variable coefficients nonlocal nonlinear Schrödinger (NNLS) equation. We map the considered equation into a defocusing PT -invariant NNLS equation with a constraint between dispersion, nonlinearity, and gain/loss parameters. We show that the considered system is PT -invariant only when the dispersion and nonlinearity coefficients are even functions and gain/loss coefficient is an odd function. The characteristics of the constructed dark soliton solutions are investigated with four different forms of dispersion parameters, namely, (1) constant, (2) periodically distributed, (3) exponentially distributed, and (4) periodically and exponentially distributed dispersion parameter. We analyze in detail how the nonlocal dark soliton profiles get deformed in the plane wave background with these dispersion parameters.

Thermo-optic soliton routing in nematic liquid crystals

We demonstrate thermo-optic control on the propagation of optical spatial solitons in nematic liquid crystals. By varying the sample temperature, both linear and nonlinear optical properties of the reorientational material are modulated by acting on the refractive indices, the birefringence, and the elastic response. As a result, both the trajectory and transverse confinement of spatial solitons can be adjusted, demonstrating an effective means to tune and readdress self-induced optical waveguides.

Spatiotemporal Airy Ince-Gaussian wave packets in strongly nonlocal nonlinear media

The self-accelerating Airy Ince–Gaussian (AiIG) and Airy helical Ince–Gaussian (AihIG) wave packets in strongly nonlocal nonlinear media (SNNM) are obtained by solving the strongly nonlocal nonlinear Schrödinger equation. For the first time, the propagation properties of three dimensional localized AiIG and AihIG breathers and solitons in the SNNM are demonstrated, these spatiotemporal wave packets maintain the self-accelerating and approximately non-dispersion properties in temporal dimension, periodically oscillating (breather state) or steady (soliton state) in spatial dimension. In particular, their numerical experiments of spatial intensity distribution, numerical simulations of spatiotemporal distribution, as well as the transverse energy flow and the angular momentum in SNNM are presented. Typical examples of the obtained solutions are based on the ratio between the input power and the critical power, the ellipticity and the strong nonlocality parameter. The comparisons of analytical solutions with numerical simulations and numerical experiments of the AiIG and AihIG optical solitons show that the numerical results agree well with the analytical solutions in the case of strong nonlocality.

Nonlinear continuous-wave optical propagation in nematic liquid crystals: Interplay between reorientational and thermal effects

We investigate nonlinear optical propagation of continuous-wave (CW) beams in bulk nematic liquid crystals. We thoroughly analyze the competing roles of reorientational and thermal nonlinearity with reference to self-focusing/defocusing and, eventually, the formation of nonlinear diffraction-free wavepackets, the so-called spatial optical solitons. To this extent we refer to dye-doped nematic liquid crystals in planar cells excited by a single CW beam in the highly nonlocal limit. To adjust the relative weight between the two nonlinear responses, we employ two distinct wavelengths, inside and outside the absorption band of the dye, respectively. Different concentrations of the dye are considered in order to enhance the thermal effect. The theoretical analysis is complemented by numerical simulations in the highly nonlocal approximation based on a semi-analytic approach. Theoretical results are finally compared to experimental results in the Nematic Liquid Crystals (NLC) 4-trans-4'-n-hexylcyclohexylisothiocyanatobenzene (6CHBT) doped with Sudan Blue dye.

Temperature dependence of the nonlinear optical response of smectic liquid crystals containing gold nanorods

The present study is devoted to the investigation of the nonlinear optical properties of a smectic liquid crystal doped with gold nanorods. Using the Z-scan technique, we investigate the changes in the optical birefringence of a homeotropic sample upon laser exposure, considering the configurations of normal and oblique incidence. Our results reveal that the birefringence variations may be governed by distinct physical mechanisms, depending on the relative angle between the far-field director and the wave vector of the excitation laser beam. In particular, we observe that the position dependence of the far-field transmittance exhibits different behaviors as the incidence angle is changed, indicating that distortions in the beam wavefront may be associated with the thermal lens phenomenon or an optically induced reorientation of the nematic director. The temperature dependence of the nonlinear refractive and absorptive coefficients is investigated close to the smectic-A-nematic phase transition. A detailed analysis of the interplay between smectic order and plasmon resonance is performed, thus unveiling the capability of plasmonic liquid crystal to be used in optical devices.

Coupling nonlinear optical waves to photoreactive and phase-separating soft matter: Current status and perspectives

Nonlinear optics and polymer systems are distinct fields that have been studied for decades. These two fields intersect with the observation of nonlinear wave propagation in photoreactive polymer systems. This has led to studies on the nonlinear dynamics of transmitted light in polymer media, particularly for optical self-trapping and optical modulation instability. The irreversibility of polymerization leads to permanent capture of nonlinear optical patterns in the polymer structure, which is a new synthetic route to complex structured soft materials. Over time more intricate polymer systems are employed, whereby nonlinear optical dynamics can couple to nonlinear chemical dynamics, opening opportunities for self-organization. This paper discusses the work to date on nonlinear optical pattern formation processes in polymers. A brief overview of nonlinear optical phenomenon is provided to set the stage for understanding their effects. We review the accomplishments of the field on studying nonlinear waveform propagation in photopolymerizable systems, then discuss our most recent progress in coupling nonlinear optical pattern formation to polymer blends and phase separation. To this end, perspectives on future directions and areas of sustained inquiry are provided. This review highlights the significant opportunity in exploiting nonlinear optical pattern formation in soft matter for the discovery of new light-directed and light-stimulated materials phenomenon, and in turn, soft matter provides a platform by which new nonlinear optical phenomenon may be discovered.

Spiraling elliptic Hermite-Gaussian solitons in nonlocal nonlinear media without anisotropy

We introduce a kind of the spiraling elliptic Hermite-Gaussian solitons in nonlocal nonlinear media without anisotropy, which carries the orbital angular momentum and can rotate in the transverse. The n-th mode of the spiraling elliptic Hermite-Gaussian solitons has n holes nested in the elliptic profile. The analytical spiraling elliptic Hermite-Gaussian solitons solutions are obtained based on the variational approach, which agree well with the numerical simulations. It is found that the critical power and the critical angular velocity for the spiraling elliptic Hermite-Gaussian solitons are the same as the counterpart of the ground mode.

Spiraling elliptic solitons in lossy nonlocal nonlinear media

We address the propagation dynamics of the spiraling elliptic beams in nonlocal nonlinear media with losses based on the variational approach. It is found that the spiraling elliptic beams exhibit complicated behaviors, which result from the combined effects of the losses and orbital angular momentum (OAM). The OAM brings in an effective anisotropic diffraction and rotation for the spiraling elliptic beams. However, due to the losses, the rotation of the spiraling beams slows down. Besides, the ellipticity of the spiraling elliptic beams is greatly affected by the lossesand the OAM. When the OAM is not equal to its critical value, a periodic oscillation of the ellipticity is found in the presence of losses. However, when the OAM is equal to the critical one, the ellipticity of the spiraling elliptic beam remains unchanged during propagation regardless of the loss factor. The comparisons between our approximate analytic solutions and numerical simulations confirm our results.

Tripole-mode and quadrupole-mode solitons in (1 + 1)-dimensional nonlinear media with a spatial exponential-decay nonlocality

The approximate analytical expressions of tripole-mode and quadrupole-mode solitons in (1 + 1)-dimensional nematic liquid crystals are obtained by applying the variational approach. It is found that the soliton powers for the two types of solitons are not equal with the same parameters, which is much different from their counterparts in the Snyder-Mitchell model (an ideal and typical strongly nolocal nonlinear model). The numerical simulations show that for the strongly nonlocal case, by expanding the response function to the second order, the approximate soliton solutions are in good agreement with the numerical results. Furthermore, by expanding the respond function to the higher orders, the accuracy and the validity range of the approximate soliton solutions increase. If the response function is expanded to the tenth order, the approximate solutions are still valid for the general nonlocal case.

Chaoticons described by nonlocal nonlinear Schrödinger equation

It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).

Spatial solitons with complicated structure in nonlocal nonlinear media

In nonlocal nonlinear media with a sine-oscillation response function, two kinds of spatial solitons with complicated structure, in-phase and out-of-phase bound-state solitons, are obtained numerically in the case of the weak nonlocality. The in-phase bound-state soliton exhibits the symmetrical profile and the nonzero central value, and the out-of-phase bound-state soliton has the antisymmetrical profile and the zero central value. The two kinds of bound-state solitons form the degenerate modes subject to the same dependance of soliton power on the propagation constant. For those solitons there exist two abnormal properties: both the soliton propagation constants and the slope of the power versus propagation constants are negative. Both the in-phase and the out-of-phase bound-state solitons are stable by the linear stability analysis, and the stability of the two kinds of solitons obey an inverted Vakhitov-Kolokolov stability criterion. We also discuss the way how to obtain a set of the soliton solutions from one numerical soliton solution via the transform invariance of the nonlocal nonlinear Schrödinger equation and the nonlinear Schrödinger equation.

Two-dimensional solitons in parity-time-symmetric optical lattices with nonlocal defocusing nonlinearity

We investigate the evolution of two-dimensional solitons in PT-symmetric optical lattices with defocusing nonlocal nonlinearity. We discuss the existence of 2D fundamental solitons, and we numerically stimulate the propagation of these solitons in the optical lattice. We also discuss how different degrees of nonlocality can affect the evolution of solitons, and discover that nonlocality can have significant impacts on the formation and propagation of the solitons. In addition, we find that PT-symmetry can lead to fission and shifting of solitons, and this phenomenon is also affected by the degree of nonlocality.

Self-organization of frozen light in near-zero-index media with cubic nonlinearity

Optical beams are generally unbound in bulk media, and propagate with a velocity approximately amounting to the speed of light in free-space. Guidance and full spatial confinement of light are usually achieved by means of waveguides, mirrors, resonators, and photonic crystals. Here we theoretically demonstrate that nonlinear self-organization can be exploited to freeze optical beams in bulk near-zero-index media, thus enabling three-dimensional self-trapping of still light without the need of optical resonators. Light is stopped to a standstill owing to the divergent wavelength and the vanishing group velocity, effectively rendering, through nonlinearity, a positive-epsilon trapping cavity carved in an otherwise slightly-negative-epsilon medium. By numerically solving Maxwell’s equations, we find a soliton-like family of still azimuthal doughnuts, which we further study through an adiabatic perturbative theory that describes soliton evaporation in lossy media or condensation in actively pumped materials. Our results suggest applications in optical data processing and storage, quantum optical memories, and soliton-based lasers without cavities. Additionally, near-zero-index conditions can also be found in the interplanetary medium and in the atmosphere, where we provide a complementary explanation to the rare phenomenon of ball-lightning.

Nonlinear competition in nematicon propagation

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.

Symmetries and exact solutions of a class of nonlocal nonlinear Schrödinger equations with self-induced parity-time-symmetric potential

A class of nonlocal nonlinear Schrödinger equations (NLSEs) is considered in an external potential with a space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable nonlocal NLSE with self-induced potential that is parity-time-symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or nonautonomous nonlocal NLSE by using similarity transformation, and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the nonlocal NLSE without the external potential and a (d+1)-dimensional generalization of it admits all the symmetries of the (d+1)-dimensional Schrödinger group. The conserved Noether charges associated with the time translation, dilatation, and special conformal transformation are shown to be real-valued in spite of being non-Hermitian. Finally, the dynamics of different moments are studied with an exact description of the time evolution of the "pseudowidth" of the wave packet for the special case in which the system admits a O(2,1) conformal symmetry.

Higher-order-effects management of soliton interactions in the Hirota equation

The study of soliton interactions is of significance for improving pulse qualities in nonlinear optics. In this paper, interaction between two solitons, which is governed by the Hirota equation, is considered. Via use of the Hirota method, an analytic soliton solution is obtained. Then a two-period vibration phenomenon is observed. Moreover, turning points of the coefficients of higher-order terms, which are related with sudden delaying or leading, are found and analyzed. With different coefficient constraints, soliton interactions are discussed by different frequency separation with the split-step Fourier method, and characteristics of soliton interactions are exhibited. Through turning points, we get a pair of solitons which tend to be bound solitons but not exactly. Furthermore, we control a pair of solitons to emit at different emission angles. The stability of the two-period vibration is analyzed. Results in this paper may be helpful for the applications of optical self-routing, waveguiding, and faster switching.

Variational and accessible soliton approximations to multidimensional solitons in highly nonlocal nonlinear media

We apply the variational approach to solitons in highly nonlocal nonlinear media in D = 1, 2, 3 dimensions. We compare results obtained by the variational approach with those obtained by the accessible soliton approximation, by considering the same system of equations in the same spatial region and under the same boundary conditions. To assess the accuracy of these approximations, we also compare them with the numerical solution of the equations. We discover that the accessible soliton approximation suffers from systematic errors, when compared to the variational approach and the numerical solution. The errors increase with the dimension of the system. The variational highly nonlocal approximation provides more accurate results in any dimension and as such is more appropriate solution than the accessible soliton approximation.

Power-induced evolution and increased dimensionality of nonlinear modes in reorientational soft matter

We demonstrate the evolution of higher order one-dimensional guided modes into two-dimensional solitary waves in a reorientational medium. The observations, carried out at two different wavelengths in chiral nematic liquid crystals, are in good agreement with a simple nonlocal nonlinear model.

Beam hysteresis via reorientational self-focusing

We theoretically investigate light self-trapping in nonlinear dielectrics with a reorientational response subject to threshold, specifically nematic liquid crystals. Beyond a finite excitation, two solitary waves exist for any given power, with an hysteretic dynamics due to feedback between beam size, self-focusing and the nonlinear threshold. Soliton stability is discussed on the basis of the system free energy.

Collapse arrest in instantaneous Kerr media via parametric interactions

We demonstrate, theoretically and experimentally, that a four-wave mixing parametric interaction is able to arrest the collapse of a two-dimensional multicolor beam in an instantaneous Kerr medium. We consider two weak idlers interacting via a third order nonlinearity with two pump beams and we show that a class of collapse-free quasisolitary solutions can be experimentally observed in a normal dispersion Kerr glass. This observation is sustained by rigorous theoretical analysis demonstrating the stability of the observed self-trapped beams.

Accessible solitons in diffusive media

We investigate spatial solitons in nonlocal media with a nonlinear index well modeled by a diffusive equation. We address the role of nonlocality and its interplay with boundary conditions, shedding light on the behavior of accessible solitons in real samples and discussing the accuracy of the highly nonlocal approximation. We find that symmetric solitons exist only above a power threshold, with an existence region that grows larger and larger with nonlocality in the plane width power.

Bistability with optical beams propagating in a reorientational medium

We investigated bistability with light beams in reorientational nematic liquid crystals. For a range of input powers, beams can propagate as either diffracting or self-trapped, the latter corresponding to spatial solitons. The first-order transition in samples exhibiting abrupt self-focusing with a threshold is in agreement with a simple model.

Observation of migrating transverse Anderson localizations of light in nonlocal media

We report the experimental observation of the interaction and attraction of many localized modes in a two-dimensional system realized by a disordered optical fiber supporting transverse Anderson localization. We show that a nonlocal optically nonlinear response of thermal origin alters the localization length by an amount determined by the optical power and also induces an action at a distance between the localized modes and their spatial migration. Evidence of a collective and strongly interacting regime is given.

Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities

We investigate the dynamical behavior of continuous and discrete Schrödinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear Schrödinger counterparts. In particular, the PT-symmetric nonlinear Schrödinger equation can simultaneously support both bright and dark soliton solutions. In addition, we study a discretized version of this PT-nonlinear Schrödinger equation on a lattice. When only two elements are involved, by obtaining the underlying invariants, we show that this system is fully integrable and we identify the PT-symmetry-breaking conditions. This arrangement is unique in the sense that the exceptional points are fully dictated by the nonlinearity itself.

Stabilization of nonlocal solitons by boundary conditions

We discovered that boundary conditions can stabilize nonlocal solitons with an oscillatory periodic response function. These solitons are the equivalent of quadratic solitons consisting of fundamental waves and oscillatory second harmonics, which are unstable unless subject to boundary confinement.

Soliton controlling and steering in asymmetric nonlocal media with optical lattices

Existence and stability of the fundamental and higher-order solitons, which exist in nonlinear media with asymmetric response and periodic linear refractive index modulation, are presented. It is found that the existence of solitons results in the balance between linear refractive index modulation (optical lattices) and nonlinear refractive index induced by incident optical field. In addition, Dynamical properties of fundamental mode solitons are also investigated in detail, and may be applied in the fields of soliton controlling and steering.

Nematicons and their electro-optic control: light localization and signal readdressing via reorientation in liquid crystals

Liquid crystals in the nematic phase exhibit substantial reorientation when the molecules are driven by electric fields of any frequencies. Exploiting such a response at optical frequencies, self-focusing supports transverse localization of light and the propagation of self-confined beams and waveguides, namely “nematicons”. Nematicons can guide other light signals and interact with inhomogeneities and other beams. Moreover, they can be effectively deviated by using the electro-optic response of the medium, leading to several strategies for voltage-controlled reconfiguration of light-induced guided-wave circuits and signal readdressing. Hereby, we outline the main features of nematicons and review the outstanding progress achieved in the last twelve years on beam self-trapping and electro-optic readdressing.

Self-induced mode transformation in nonlocal nonlinear media

We report on the first experimental observation of self-induced optical mode transformations in nonlocal nonlinear media. We show that the quadrupole Hermite-Gaussian mode experiences complex nonlinear dynamics in a nematic liquid crystal, including power-dependent conversion into a radially symmetric Laguerre-Gaussian mode. The physical mechanism responsible for self-induced transformation is the excitation of internal modes of a metastable quadrupole nonlocal soliton and its subsequent transmutation into a robust soliton with a bright peak surrounded by a bright ring. We also observe the onset of transformations of higher-order modes, proving the generic character of this nonlinear phenomenon.

Shock waves in disordered media

We experimentally investigate the interplay between spatial shock waves and the degree of disorder during nonlinear optical propagation in a thermal defocusing medium. We characterize the way the shock point is affected by the amount of disorder and scales with wave amplitude. Evidence for the existence of a phase diagram in terms of nonlinearity and amount of randomness is reported. The results are in quantitative agreement with a theoretical approach based on the hydrodynamic approximation.

Measurement of scaling laws for shock waves in thermal nonlocal media

We are able to detect the details of spatial optical collisionless wave breaking through the high aperture imaging of a beam suffering shock in a fluorescent nonlinear nonlocal thermal medium. This allows us to directly measure how nonlocality and nonlinearity affect the point of shock formation and compare results with numerical simulations.

Optomechanical self-channeling of light in a suspended planar dual-nanoweb waveguide

It is shown that optomechanical forces can cause nonlinear self-channeling of light in a planar dual-slab waveguide. A system of two parallel silica nanowebs, spaced ~100 nm and supported inside a fiber capillary, is studied theoretically and an iterative scheme developed to analyze its nonlinear optomechanical properties. Steady-state field distributions and mechanical deformation profiles are obtained, demonstrating that self-channeling is possible in realistic structures at launched powers as low as a few mW. The differential optical nonlinearity of the self-channeled mode can be as much as 10×10(6) times higher than the corresponding electronic Kerr nonlinearity. It is also intrinsically broadband, does not utilize resonant effects, can be viewed as a consequence of the extreme nonlocality of the mechanical response, and in fact is a notable example of a so-called accessible soliton.

Incoherent soliton turbulence in nonlocal nonlinear media

The long-term behavior of a modulationally unstable nonintegrable system is known to be characterized by the soliton turbulence self-organization process: It is thermodynamically advantageous for the system to generate a large-scale coherent soliton in order to reach the ("most disordered") equilibrium state. We show that this universal process of self-organization breaks down in the presence of a highly nonlocal nonlinear response. A wave turbulence approach based on a Vlasov-like kinetic equation reveals the existence of an incoherent soliton turbulence process: It is advantageous for the system to self-organize into a large-scale, spatially localized, incoherent soliton structure.