Spatial solitons in linearly and nonlinearly amplitude-modulated optical lattices

We demonstrated that linearly and nonlinearly amplitude-modulated (chirped) harmonic lattices can support odd and even solitons in both focusing and defocusing saturable media. The modulated lattice modifies the profiles and enlarges the stability domains of solitons, comparing with the unchirped one. Twisted solitons, or “soliton trains” whose profiles exhibit multi-peak structures can also be supported by linearly and nonlinearly chirped lattices. In sharp contrast with periodic lattices, chirped lattices remarkably broaden the existence and stability domains of twisted solitons, especially for solitons with more components. While even solitons in focusing media and twisted solitons in defocusing media are unstable, odd and twisted solitons in focusing media are stable in relatively wide parameter windows. Chirped lattice can be used as a linear guidance to realize the oscillation of solitons which is impossible in unchirped lattice.     

Solitons in a system of three linearly coupled fiber gratings

We introduce a model of three parallel-coupled nonlinear waveguiding cores equipped with Bragg gratings (BGs), which form an equilateral triangle. The most promising way to create multi-core BG configuration is to use inverted gratings, written on internal surfaces of relatively broad holes embedded in a photonic-crystal-fiber matrix. The objective of the work is to investigate solitons and their stability in this system. New results are also obtained for the earlier investigated dual-core system. Families of symmetric and antisymmetric solutions are found analytically, extending beyond the spectral gap in both the dual- and tri-core systems. Moreover, these families persist in the case (strong coupling between the cores) when there is no gap in the systems linear spectrum. Three different types of asymmetric solitons are found (by means of the variational approach and numerical methods) in the tri-core system. They exist only inside the spectral gap, but asymmetric solitons with nonvanishing tails are found outside the gap as well. Stability of the solitons is explored by direct simulations, and, for symmetric solitons, in a more rigorous way too, by computation of eigenvalues for small perturbations. The symmetric solitons are stable up to points at which two types of asymmetric solitons bifurcate from them. Beyond the bifurcation, one type of the asymmetric solitons is stable, and the other is not. Then, they swap their stability. Asymmetric solitons of the third type are always unstable. When the symmetric solitons are unstable, their instability is oscillatory, and, in most cases, it transforms them into stable breathers. In both the dual- and tri-core systems, the stability region of the symmetric solitons extends far beyond the gap, persisting in the case when the system has no gap at all. The whole stability region of antisymmetric solitons (a new type of solutions in the tri-core system) is located outside the gap. Thus, solitons in multi-core BGs can be observed experimentally in a much broader frequency band than in the single-core one, and in a wider parameter range than it could be expected. Asymmetric delocalized solitons, found outside the spectral gap, can be stable too.Received: 13 August 2003PACS: 42.81.Dp Propagation, scattering, and losses; solitons - 42.65.Tg Optical solitons; nonlinear guided waves - 05.45.Yv Solitons     

Surface solitons in trilete lattices

Fundamental solitons pinned to the interface between three semi-infinite one-dimensional nonlinear dynamical chains, coupled at a single site, are investigated. The light propagation in the respective system with the self-attractive on-site cubic nonlinearity, which can be implemented as an array of nonlinear optical waveguides, is modeled by the system of three discrete nonlinear Schrödinger equations. The formation, stability and dynamics of symmetric and asymmetric fundamental solitons centered at the interface are investigated analytically by means of the variational approximation (VA) and in a numerical form. The VA predicts that two asymmetric and two antisymmetric branches exist in the entire parameter space, while four asymmetric modes and the symmetric one can be found below some critical value of the inter-lattice coupling parameter—actually, past the symmetry-breaking bifurcation. At this bifurcation point, the symmetric branch is destabilized and two new asymmetric soliton branches appear, one stable and the other unstable. In this area, the antisymmetric branch changes its character, getting stabilized against oscillatory perturbations. In direct simulations, unstable symmetric modes radiate a part of their power, staying trapped around the interface. Highly unstable asymmetric modes transform into localized breathers traveling from the interface region across the lattice without significant power loss.     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

Gray spatial solitons in biased two-photon photovoltaic photorefractive crystals

This paper predicts that gray spatial solitons can exist in biased two-photon photovoltaic photorefractive crystals. Under appropriate conditions and in the steady state, the gray spatial solitons solution of the optical evolution equation is obtained. The properties associated with these solitons, such as their intensity profile, intensity full width at half-maximum, width, transverse velocity and phase distribution, are discussed as functions of their normalized intensity and degree of “grayness”. Relevant examples are provided.     

A study on relativistic lagrangian field theories with non-topological soliton solutions

We perform a general analysis of the dynamic structure of two classes of relativistic lagrangian field theories exhibiting static spherically symmetric non-topological soliton solutions. The analysis is concerned with (multi-) scalar fields and generalized gauge fields of compact semi-simple Lie groups. The lagrangian densities governing the dynamics of the (multi-) scalar fields are assumed to be general functions of the kinetic terms, whereas the gauge-invariant lagrangians are general functions of the field invariants. These functions are constrained by requirements of regularity, positivity of the energy and vanishing of the vacuum energy, defining what we call “admissible” models. In the scalar case we establish the general conditions which determine exhaustively the families of admissible lagrangian models supporting this kind of finite-energy solutions. We analyze some explicit examples of these different families, which are defined by the asymptotic and central behaviour of the fields of the corresponding particle-like solutions. From the variational analysis of the energy functional, we show that the admissibility constraints and the finiteness of the energy of the scalar solitons are necessary and sufficient conditions for their linear static stability against small charge-preserving perturbations. Furthermore, we perform a general spectral analysis of the dynamic evolution of the small perturbations around the statically stable solitons, establishing their dynamic stability. Next, we consider the case of many-components scalar fields, showing that the resolution of the particle-like field problem in this case reduces to that of the one-component case. The study of these scalar models is a necessary step in the analysis of the gauge fields. In this latter case, we add the requirement of parity invariance to the admissibility constraints. We determine the general conditions defining the families of admissible gauge-invariant models exhibiting finite-energy electrostatic spherically symmetric solutions which, unlike the (multi-) scalar case, are not always stable. The variational analysis of the energy functional leads now to supplementary restrictions to be imposed on the lagrangian densities in order to ensure the linear stability of the solitons. We establish a correspondence between any admissible soliton-supporting (multi-) scalar model and a family of admissible generalized gauge models supporting finite-energy electrostatic point-like solutions. Conversely, for each admissible soliton-supporting gauge-invariant model there is an associated unique admissible (multi-) scalar model with soliton solutions. This shows the exhaustive character of the admissibility and stability conditions in determining the class of soliton-supporting generalized gauge models. The usual Born-Infeld electrodynamic theory and its non-abelian extensions are shown to be (very particular) examples of one of these families.     

Stabilization of spatiotemporal solitons in second-harmonic-generating media

We report the results of a systematic analysis of the existence and stability of spatiotemporal (two-dimensional) solitons (STSs) in the model of a planar waveguide with the intrinsic χ⁽²⁾ nonlinearity. Fundamental obstacles to the creation of STSs under physically realistic conditions are the normal sign of the group-velocity dispersion (GVD) at the second harmonic (SH), and the significant group-velocity mismatch (GVM) between the SH and fundamental-frequency (FF) components. To construct STS solutions in a numerical form, we adjust the iterative method, which was recently used for finding temporal (one-dimensional) χ⁽²⁾ solitons in a similar setting. We identify effective existence borders for the STSs, within which the energy loss to the generation of extended “tails” in the SH component (due to the normal sign of the GVD) is negligible. It is demonstrated that the existence region can be made much broader by means of the GVD-management and GVM-management techniques. We also explore interactions between the STSs, and find robust two-soliton bound states, with a moderate separation in the longitudinal (temporal) direction. Head-on collisions between the STSs are always destructive.     

Defect vector gap solitons in photonic lattices

The existence and general properties of different kinds of defect vector gap solitons in one dimensional optically induced photonic defect lattice with focusing saturable nonlinearity in photorefractive crystal are analyzed. The defect is well localized in a single site with two existence forms, namely repulsive and attractive defect. Propagation constants of two beams that compose defect vector gap solitons could be from same gap or from different gaps. We show that some kinds of unstable scalar defect gap solitons could be stabilized by their corresponding vector cases.     

Two-dimensional higher-band vortex lattice solitons

We study self-localized second-band vortex states in two-dimensional photonic lattices and find stable ring solitons whose phase forms an array of counterrotating vortices. We also identify composite solitons in which a second-band vortex is jointly trapped with a mode arising from the first band and study their stability. When such a composite entity is unstable, it disintegrates while exchanging angular momentum between its constituents, eventually stabilizing into another form of composite soliton.     

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrödinger equation

The fractional quadric-cubic coupled nonlinear Schrödinger equation is concerned, and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method. The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated. Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index, respectively. The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied. Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.     

Gap solitons supported by optical lattices in biased centrosymmetric photorefractive crystals

We analyze the existence and stability of gap solitons supported by optical lattices with self-focusing nonlinearity in biased centrosymmetric photorefractive crystals. It is shown that, in first finite bandgap, gap solitons are symmetric in transverse dimension, single humped, entirely positive and linearly stable, while these solitons are antisymmetric with similar profiles, the stable and unstable intervals of the gap solitons are intertwined in the second finite bandgap.     

Gap solitons in defocusing lithium niobate binary waveguide arrays fabricated by proton implantation and selective light illumination

Photovoltaic photorefractive binary waveguide arrays are fabricated by proton implantation and selective light illumination on top of an iron-doped near stoichiometric lithium niobate crystal. Linear discrete diffraction and nonlinear formation of gap solitons were investigated by single-channel excitation using Gaussian light beams coupled into either wide or narrow waveguide channels. The results show that, at low power, linear light propagation leads to discrete diffraction, whilst for higher input power the focusing mechanism dominates, finally leading to the formation of gap solitons in the binary waveguide arrays. Our simulation of light propagation based on a nonlinear beam propagation method confirms the experimental findings.     

Quasi-stable propagation of vortices and soliton clusters in saturable Kerr media with square-root nonlinearity

Analytical and numerical investigation of the propagation of optical beams in Kerr-like saturable photorefractive media is carried out, utilizing a novel model for the local isotropic part of the space-charge field generated in the medium. Using a variational technique, optimal propagation parameters for the most stable propagation of otherwise unstable single Gaussian, single vortex, and optical soliton cluster beams are determined. Analogy between a ring of identical weakly overlapping solitons and a vortex of the same topological charge is explored.     

Photorefractive spatial solitons as waveguiding elements for optical telecommunication

Spatial solitons permit optical waveguiding. This holds true for the soliton write beam (i.e. the driving laser beam), as well as for additional probe beams, which may carry optically encoded information. This feature of spatial solitons is of significant interest for applications in optical telecommunication. We present systematic experimental investigations on single and multiple spatial solitons in the infrared spectral regime (i.e. around optical telecommunication wavelengths), applied as controllable all-optical devices. In particular, we present the implementations of a Y-coupler as an optical signal divider, a switchable Y-coupler as an optical add multiplexer, and a novel design for a 1 × 3 optical beam switch, i.e. applied as a router for infrared signal beams. We report large waveguiding efficiencies up to 40% and transmission rates of 90 Tbit/s in our setups. The presented experimental data are confirmed by numerical simulations.     

Modulation instability: The beginning

We discuss the early history of an important field of “sturm and drang” in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.     

Stabilization of three-dimensional light bullets by a transverse lattice in a Kerr medium with dispersion management

We demonstrate a possibility to stabilize three-dimensional spatiotemporal solitons (“light bullets”) in self-focusing Kerr media by means of a combination of dispersion management in the longitudinal direction (with the group-velocity dispersion alternating between positive and negative values) and periodic modulation of the refractive index in one transverse direction (out of the two). Assuming the usual model based on the paraxial nonlinear Schrödinger equation for the local amplitude of the electromagnetic field, the analysis relies upon the variational approximation (results of direct three-dimensional simulations will be reported in a follow-up). A predicted stability area is identified in the model’s parameter space. It features a minimum of the necessary strength of the transverse modulation of the refractive index, and finite minimum and maximum values of the soliton’s energy. The former feature is also explained analytically.     

Trapped states and transitions between them in double-well pseudopotentials

We analyze a model of a double-well pseudopotential (DWPP), based in the 1D Gross-Pitaevskii equation with a spatially modulated self-attractive nonlinearity. In the limit case when the DWPP structure reduces to the local nonlinearity coefficient represented by a set of two delta-functions, analytical solutions are obtained for symmetric, antisymmetric and asymmetric states. In this case, the transition from symmetric to asymmetric states, i.e., a spontaneous-symmetry-breaking (SSB) bifurcation, is subcritical. Numerical analysis demonstrates that the symmetric states are stable up to the SSB point, while emerging asymmetric states (together with all antisymmetric solutions) are unstable in the delta-function model. In a general model, which features a finite width of the nonlinear-potential wells, the asymmetric states quickly become stable, simultaneously with the switch of the bifurcation into the supercritical type. Antisymmetric solutions may also enjoy stabilization in the finite-width DWPP structure, demonstrating a bistability involving the asymmetric states. The symmetric states require a finite norm for their existence. A full diagram for the existence and stability of the trapped states is produced for the general model.     

Gap solitons and their linear stability in one-dimensional periodic media

An analytical theory utilizing exponential asymptotics is presented for one-dimensional gap solitons that bifurcate from edges of Bloch bands in the presence of a general periodic potential. It is shown that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self-defocusing nonlinearity, and an asymptotic expression for the eigenvalues associated with the linear stability of these solitons is derived. The locations of these solitons relative to the underlying potential are determined from a certain recurrence relation, that contains information beyond all orders of the usual perturbation expansion in powers of the soliton amplitude. Moreover, this same recurrence relation decides which of the two soliton families is unstable. The analytical predictions for the stability eigenvalues are in excellent agreement with numerical results.     

Gap and dark solitons in discrete photorefractive media with?intensity-resonant nonlinearity

Light propagation in one-dimensional nonlinear waveguide arrays with self-defocusing intensity-resonant nonlinearity is investigated theoretically. We study thoroughly conditions for existence and stability of both gap and discrete dark solitons. According to the linear stability analysis both fundamental types (on-site and intersite) of gap solitons may be stable. Discrete dark solitons are unstable except in the low-power regime and, depending on system parameters, evolve into either gray solitons, breathers, or background radiation. Mobility of these solitons is analyzed by the free energy concept: gap solitons are immobile but dark solitons can be easily set in motion.