Gap Solitons Research Articles

Stability of gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity

We investigate the existence and stability of quiescent gap solitons in a system of two linearly coupled Bragg gratings with cubic-quintic nonlinearity. It is found that the model supports two disjoint families of solitons that fill the entire bandgap. There exist symmetric and asymmetric solitons in each family. Exact analytical solutions of symmetric solitons for both families are found. The stability of solitons is investigated by means of systematic numerical stability analysis. The effect of quintic nonlinearity and coupling coefficient on the stability of solitons and the stability regions are analyzed.

Stable radiating gap solitons and their resonant interactions with dispersive waves in systems with a parametric pump.

We study the formation of gap solitons in the presence of a parametric pump. It is shown that a parametric pump can stabilize stationary solitons continuously emitting dispersive waves. The resonant interactions of the radiation and the solitons are studied and it is shown that the solitons can be effectively controlled by the radiation. In particular it is shown that the solitons can collide or get pinned to inhomogeneities due to the interactions mediated by the resonant radiation.

Gap solitons in PT-symmetric optical lattices with higher-order diffraction.

The existence and stability of gap solitons are investigated in the semi-infinite gap of a parity-time (PT)-symmetric periodic potential (optical lattice) with a higher-order diffraction. The Bloch bands and band gaps of this PT-symmetric optical lattice depend crucially on the coupling constant of the fourth-order diffraction, whereas the phase transition point of this PT optical lattice remains unchangeable. The fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons. Specifically, when the fourth-order diffraction coupling constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons are found to be always linearly stable with arbitrary positive value of the coupling constant. We also investigate nonlinear evolution of the PT solitons under perturbation.

Gap solitons and symmetry breaking in parity-time symmetric microring coupled resonator optical waveguides

The propagation properties of optical fields in linear and nonlinear parity-time symmetric microring coupled resonator optical waveguides are studied. The effects described include the existence of symmetry breaking thresholds, the propagation of gap solitons in nonlinear transmission lines and the existence of quasi stable propagation regimes outside the broken symmetry regions.

Optical simulation of neutrino oscillations in binary waveguide arrays.

We theoretically propose and investigate an optical analogue of neutrino oscillations in a pair of vertically displaced binary waveguide arrays with longitudinally modulated effective refractive index. Optical propagation is modeled through coupled-mode equations, which in the continuous limit converge to two coupled Dirac equations for fermionic particles with different mass states, analogously to neutrinos. In addition to simulating neutrino oscillation in the noninteracting regime, our optical setting enables us to explore neutrino interactions in extreme regimes that are expected to play an important role in massive supernova stars. In particular, we predict the quenching of neutrino oscillations and the existence of topological defects, i.e., neutrino solitons, which in our photonic simulator should be observable as excitation of optical gap solitons propagating along the binary arrays at high excitation intensities.

Spatial solitons supported by localized gain [Invited

The creation of stable 1D and 2D localized modes in lossy nonlinear media is a fundamental problem in optics and plasmonics. This article gives a mini review of theoretical methods elaborated on for this purpose, using localized gain applied at one or several hot spots (HS). The introduction surveys a broad class of models for which this approach was developed. Other sections focus in some detail on basic 1D continuous and discrete systems, where the results can be obtained, partly or fully, in an analytical form (and verified by comparison with numerical results), which provides deeper insight into the nonlinear dynamics of optical beams in dissipative nonlinear media. Considered, in particular, are the single and double HS in the usual waveguide with the self-focusing (SF) or self-defocusing (SDF) Kerr nonlinearity, which gives rise to rather sophisticated results in spite of apparent simplicity of the model, solitons attached to a PT-symmetric dipole embedded into the SF or SDF medium, gap solitons pinned to an HS in a Bragg grating, and discrete solitons in a 1D lattice with a hot site.

Surface defect gap solitons in superlattices with self-defocusing nonlinearity

We put forward the existence and stability of defect surface gap solitons at the interface between uniform media and an superlattice with self-defocusing nonlinearity. We reveal that the defect plays the significant role in controlling the region of solitons existing. Various solitons are found to be existed in different gaps for different defects. For positive defects, fundamental solitons can exist stably in the semi-infinite gap, and dipole solitons can exist stably in the first gap but they are unstable in the second gap. For zero or negative defects, fundamental and dipole solitons can exist stably in the first gap and the second gap, respectively.

Accurate one-dimensional effective description of realistic matter-wave gap solitons

We consider stationary matter-wave gap solitons realized in Bose–Einstein condensates loaded in one-dimensional (1D) optical lattices and investigate whether the effective 1D equation proposed in Muñoz Mateo and Delgado (2008 Phys. Rev. A 77 013617) can be a reliable alternative to the three-dimensional treatment of this kind of system in terms of the Gross–Pitaevskii equation (GPE). Our results demonstrate that, unlike the standard 1D GPE (which is not applicable in most realistic situations), the above effective model is able to correctly predict the distinctive trajectories characterizing the different gap soliton families as well as the corresponding axial wavefunctions along the entire band gaps. It can also predict the stability properties of the different gap soliton families as follows from both a linear stability analysis and a representative set of numerical computations. In particular, by numerically solving the corresponding Bogoliubov–de Gennes equations we show that the effective 1D model gives the correct spectrum of the complex eigenfrequencies responsible for the dynamical stability of the system, thus providing us with a useful tool for the physical description of stationary matter-wave gap solitons in 1D optical lattices.

Antisymmetry breaking of discrete dipole gap solitons induced by a phase-slip defect

In this paper, we study the antisymmetry breaking of discrete dipole soliton induced by a phase-slip one-dimensional discrete lattice, which contains on-site self-repulsive nonlinearity. This system can be realized in waveguide arrays system in optics or Bose–Einstein condensate in optical lattice. Different from the symmetry breaking occurring in the ground-state, antisymmetry breaking occurs in the first excited state, which contains antisymmetry. For this system, stable antisymmetric dipole soliton and anti-asymmetric are found, the symmetry transition between them is supercritical type. It is found that, increasing the total norm of the soliton or decreasing the coupled strength of the defect waveguide can lead to the antisymmetry breaking. Such kind of symmetry breaking can lead to the change of the tendency of some characters of the soliton.

Lattice surface solitons in diffusive nonlinear media driven by the quadratic electro-optic effect.

We study theoretically surface lattice solitons driven by quadratic electro-optic effect at the interface between an optical lattice and diffusive nonlinear media with self-focusing and self-defocusing saturable nonlinearity. Surface solitons originating from self-focusing nonlinearity can be formed in the semi-infinite gap, and are stable in whole domain of their existence. In the case of self-defocusing nonlinearity, both surface gap and twisted solitons are predicted in first gap. We discover that surface gap solitons can propagate stably in whole existence domain except for an extremely narrow region close to the Bloch band, and twisted solitons are linearly unstable in the entire existence domain.

Gap solitons in parity—time complex superlattice with dual periods

A theory is presented to investigate the existence and propagation stability of gap solitons in a parity—time (PT) complex superlattice with dual periods. In this superlattice, the real and imaginary parts are both in the form of superlattices with dual periods. In the self-focusing nonlinearity, PT solitons can exist in the semi-infinite gap. However, only those gap solitons with low powers can propagate stably, whereas the high-power solitons present periodic oscillation and simultaneously suffer energy decay. In the self-defocusing nonlinearity, PT solitons only exist in the first gap and all these solitons are stable.

Gap solitons with null-scattering

In this Letter, we study the excitation of gap solitons under the conditions of coherent perfect absorption. Our system consists of a symmetric periodic structure with alternating Kerr nonlinear and linear layers illuminated from both ends. We show near-total transfer of incident light energy into the gap solitons resulting in null scattering. We also report on the nonlinear super-scattering states. Both the exact and the approximate results, which show good agreement based on nonlinear characteristic matrix methods, are presented.

Defect gap solitons in self-focusing Kerr media with parity-time symmetric linear superlattices and modulated nonlinear lattices

We report the existence and stability of defect solitons (DSs) in a dissipative system with a parity-time (PT) symmetric linear superlattice and modulated periodical nonlinearity based on self-focusing Kerr media. It is found that for positive defects, the DSs can exist stably in the semi-infinite gap. For negative defects, the DSs can exist stably in both the semi-infinite and the first gap. The DSs in high power regions are unstable. If the defect strength increases, the stable regions shrink rapidly and eventually disappear. If the PT-symmetry is broken, all DSs are unstable.

Nonlocal gap solitons inPT-symmetric periodic potentials with defocusing nonlinearity

Existence and stability of PT-symmetric gap solitons in a periodic structure with defocusing nonlocal nonlinearity are studied both theoretically and numerically. We find that, for any degree of nonlocality, gap solitons are always unstable in the presence of an imaginary potential. The instability manifests itself as a lateral drift of solitons due to an unbalanced particle flux. We also demonstrate that the perturbation growth rate is proportional to the amount of gain (loss), thus predicting the observability of stable gap solitons for small imaginary potentials.

Traveling Solitary Waves in the Periodic Nonlinear Schrödinger Equation with Finite Band Potentials

This paper studies asymptotics of moving gap solitons in nonlinear periodic structures of finite contrast (``deep grating'') within the one dimensional periodic nonlinear Schrodinger equation (PNLS). Periodic structures described by a finite band potential feature transversal crossings of band functions in the linear band structure and a periodic perturbation of the potential yields new small gaps. Novel gap solitons with $O(1)$ velocity despite the deep grating are presented in these gaps. An approximation of gap solitons is given by slowly varying envelopes which satisfy a system of generalized coupled mode equations (gCME) and by Bloch waves at the crossing point. The eigenspace at the crossing point is two dimensional and it is necessary to select Bloch waves belonging to the two band functions. This is achieved by an optimization algorithm. Traveling solitary wave solutions of the gCME then result in nearly solitary wave solutions of the PNLS moving at an $O(1)$ velocity across the periodic structure...

Gap solitons supported by parity-time-symmetric optical lattices with defocusing saturable nonlinearity

We show that the parity-time (PT)-symmetric optical lattices can support stable fundamental, in-phase multipeak gap solitons in defocusing saturable media. These stable solitons exist in the first finite gap and the saturation parameter can affect the existence of the solitons. The out-of-phase multipeak solitons can also exist in the first gap, but they are unstable. We also investigate the transverse energy flow of these solitons in saturable media. The imaginary part of the PT-symmetric periodic potentials will affect the stability of these solitons.

Effective equations for matter-wave gap solitons in higher-order transversal states

We demonstrate that an important class of nonlinear stationary solutions of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) exhibiting nontrivial transversal configurations can be found and characterized in terms of an effective one-dimensional (1D) model. Using a variational approach we derive effective equations of lower dimensionality for BECs in (m,n(r)) transversal states (states featuring a central vortex of charge m as well as n(r) concentric zero-density rings at every z plane) which provides us with a good approximate solution of the original 3D problem. Since the specifics of the transversal dynamics can be absorbed in the renormalization of a couple of parameters, the functional form of the equations obtained is universal. The model proposed finds its principal application in the study of the existence and classification of 3D gap solitons supported by 1D optical lattices, where in addition to providing a good estimate for the 3D wave functions it is able to make very good predictions for the μ(N) curves characterizing the different fundamental families. We have corroborated the validity of our model by comparing its predictions with those from the exact numerical solution of the full 3D GPE.

Dynamics of interacting solitons in dual core Bragg gratings with dispersive reflectivity

Interactions of in-phase and out-of-phase quiescent gap solitons in a system of two linearly-coupled Bragg gratings with dispersive reflectivity are studied. By means of systematic numerical simulations, we show that the interaction of the in-phase solitons may lead to merger, repulsion, destruction, or separation of solitons which may be symmetric or asymmetric. A key feature of the interactions is that even in the absence of dispersive reflectivity the interaction of solitons may result in the formation of two moving solitons and one quiescent one. To the best of our knowledge, such outcomes have not been observed in the standard models of gap solitons (i.e., single core Bragg gratings without dispersive reflectivity). Another interesting finding is that in the region where solitons do not have sidelobes, the outcomes of the interactions are weakly dependent on the initial separation of the solitons. On the other hand, the presence of sidelobes, which occur for larger values of dispersive reflectivity, results in more complex interactions. The π-out-of-phase solitons without sidelobes always repel each other. On the other hand, the interaction of solitons with sidelobes is affected by the initial separation. In this case, the interactions may either result in the repulsion of solitons or the formation of a temporary bound state that subsequently splits into two separating solitons.

Modulation theory inPT-symmetric magnetic metamaterial arrays in the continuum limit

We present results on the dynamics of split-ring dimers having both gain and loss in one dimensional nonlinear parity-time- (PT-)symmetric magnetic metamaterials. For the longwave (continuum) limit approximation and in the weakly nonlinear limit, we show analytic results on the existence of gap soliton solutions and on symmetry breaking phenomenon at a critical value of the gain/loss term.

Scattering of gap solitons byPT-symmetric defects

The resonant scattering of gap solitons (GS) of the periodic nonlinear Schr\"odinger equation with a localized defect which is symmetric under the parity and the time-reversal (PT) symmetry, is investigated. It is shown that for suitable amplitudes ratios of the real and imaginary parts of the defect potential the resonant transmission of the GS through the defect becomes possible. The resonances occur for potential parameters which allow the existence of localized defect modes with the same energy and norm of the incoming GS. Scattering properties of GSs of different band-gaps with effective masses of opposite sign are investigated. The possibility of unidirectional transmission and blockage of GSs by PT defect, as well as, amplification and destruction induced by multiple reflections from two PT defects, are also discussed.