Two-dimensional Spatial Solitons Research Articles

Soliton dynamics in partially PT -symmetric two-dimensional Bessel lattices

We study the dynamics of two-dimensional spatial solitons supported by partially-parity-time ( PPT )-symmetric optical Bessel lattices. We show that these solitons with an imposed tilt can follow an oscillating or even a whole rotary trajectory in the transverse plane guided by a Bessel optical lattice. Due to the complex optical potential, these rotary solitons can also suffer from amplification or attenuation. Interestingly, we found that it is possible to find quasi-stability regions of parameters where there is a balance between the imposed tilt and the PPT -symmetric potential, allowing the launched spatial dissipative solitons to complete several rotations in their propagation.

Two-dimensional spatial optical solitons in Rydberg cold atomic system under the action of optical lattice

Realizing stable high-dimensional light solitons is a long-standing goal in the study of nonlinear optical physics. However, in high-dimensional space, the light field will inevitably be distorted due to diffraction. In order to solve the diffraction effect in nonlinear Kerr media and achieve the spatial localization of light fields, we propose a scheme to generate stable two-dimensional (2D) solitons in a cold Rydberg atomic system with a Bessel optical lattice, where a three-level atomic structure, a weak probe laser field, and a strong control field constitute the Rydberg-dressed atomic system. When the local nonlinearity, Bessel potential, and nonlocal nonlinearity which is caused by the long-range Rydberg-Rydberg interaction (RRI) between Rydberg atoms are balanced, the probe field can be localized. Under the approximation of electric dipole and rotating wave, the stable solution of probe field is obtained by solving Maxwell-Bloch equations numerically. A cluster of 2D spatial solitons, including fundamental, two-pole, quadrupole and vortex solitons, is found in this system. Among them, the fundamental, dipole and quadrupole have, one, two, and four intensity centers, respectively. Vortex solitons, on the other hand, exhibit vertical characters in profiles and phase structures. The formation and transmission of these solitons can be controlled by system parameters, such as the propagation coefficient, the degree of nonlocal nonlinearity, and Bessel lattice strength. The stable regions of these solitons are determined by anti Vakhitov Kolokolov (anti-VK) criterion and linear stability analysis method. It is found that four kinds of solitons can be generated and stably propagate in space with proper parameters. Owing to the different structures of the poles, the fundamental state and vortex state remain stable, while the quadrupole ones are unstable. In the modulation of solitons, there is a cutoff value of propagation constant <inline-formula><tex-math id="M1">\begin{document}${b_{{\text{co}}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.png"/></alternatives></inline-formula>, only below which value, the solitons can propagate stably. The light intensity of soliton shows a periodic behavior by tuning Bessel lattice strength. The period of the intensity decreases with the order of the solitons as a result of the interaction between the poles. It is also found that the solitons are more stable with weak nonlocal nonlinearity coefficient. This study provides a new idea for the generation and regulation of optical solitons in high dimensional space.

Stability of one and two-dimensional spatial solitons in a cubic–quintic–septimal nonlinear Schrödinger equation with fourth-order diffraction and [formula omitted]-symmetric potentials

In this paper, the existence and stability of solitons in parity-time (PT)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with fourth-order diffraction (FOD) coefficient and higher-order nonlinearities have been investigated. For the linear case, we have demonstrated numerically that, the FOD parameter can alter the PT-breaking points. Exact analytical expressions of the localized modes are obtained respectively, in one and two dimensional nonlinear Schrödinger (NLS) equation with both self-focusing and self-defocusing Kerr, and higher-order nonlinearities for nonlinear case. The effects of both FOD and higher-order nonlinearities on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Some stable and unstable solutions have been given and, it has been seen how higher-order self-focusing and self-defocusing nonlinearities can influence the stability/instability of the system.

Investigation of the interaction of a pair of bright optical spatial pyroelectric solitons during their in-phase propagation in an undoped lithium niobate crystal at a wavelength of 532 nm

The formation of two-dimensional bright spatial solitons by in-phase laser beams with a wavelength of 532 nm propagating in the bulk of photorefractive lithium niobate crystal under the reversal of the sign of the nonlinear optical response of the material with the contribution of the pyroelectric effect has been experimentally demonstrated. It is shown that varying the distance between the beams at the input face of the crystal leads to the interaction of pyroelectric solitons, which allows the formation of non-rectilinear propagation paths of light beams.

Pyroelectric generation of 2D spatial soliton sets in a bulk of lithium niobate crystal

The generation of two-dimensional bright spatial soliton sets in lithium niobate sample has been experimentally demonstrated at light wavelength of 532 nm, contribution of pyroelectric effect into nonlinear optical response of the crystal, and spatial modulation of one-dimensional beam along direction normal to the crystal optical axis. Diameters of soliton beams and channel waveguides formed within the crystal bulk by these solitons are near to 20 μm at light polarization corresponding to extraordinary wave of the crystal.

The stability of two-dimensional spatial solitons in cubic–quintic–septimal nonlinear media with different diffractions and $${\varvec{\mathcal {PT}}}$$ PT -symmetric potentials

A (2+1)-dimensional nonlinear Schrodinger equation in cubic–quintic–septimal nonlinear media with different diffractions and \({\mathcal {PT}}\)-symmetric potentials is studied, and (2+1)-dimensional spatial solitons are derived. The stable region of analytical spatial solitons is discussed by means of the eigenvalue method. The direct numerical simulation indicates that analytical spatial soliton solutions stably evolve within stable region in the media of focusing septimal and focusing or defocusing cubic nonlinearities with disappearing quintic nonlinearity under the 2D extended Scarf II potential. However, under the extended \({\mathcal {PT}}\)-symmetric potential with \(p=2\) and \(p=3\), analytical spatial soliton solutions stably evolve within stable region in the media of focusing quintic and septimal nonlinearities with defocusing cubic nonlinearity. In other cases, analytical spatial soliton solutions cannot sustain their original shapes, and they are distorted and broken up and finally decay into noise.

Stable spatial and spatiotemporal optical soliton in the core of an optical vortex

We demonstrate a robust, stable, mobile, two-dimensional (2D) spatial and three-dimensional (3D) spatiotemporal optical soliton in the core of an optical vortex, while all nonlinearities are of the cubic (Kerr) type. The 3D soliton can propagate with a constant velocity along the vortex core without any deformation. Stability of the soliton under a small perturbation is established numerically. Two such solitons moving along the vortex core can undergo a quasielastic collision at medium velocities. Possibilities of forming such a 2D spatial soliton in the core of a vortical beam are discussed.

Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: I. One-dimensional dark solitons

Quantum fluctuations of one-dimensional dark dissipative solitons sustained by an external radiation in an interferometer with a Kerr nonlinearity are analyzed theoretically. The stability region of classical solitons in this interferometer is studied. The boundaries of this region are determined, and types of excited solitons are classified. Quantum fluctuations of solitons are analyzed in an approximation linear in fluctuations. This problem was solved by linearizing the quantum Langevin equation in a neighborhood of a classical solution for the main type of a soliton from the obtained stability region. The main attention has been paid to studying quantum fluctuations of collective variables of dissipative solitons, namely, the coordinate of the center and momentum of the soliton. Based on the expansion of solutions of the linearized equation in eigenfunctions of the discrete spectrum of this equation, a solution describing quantum fluctuations of these variables is constructed. Using this expansion scheme made it possible to give a rigorous definition of the dissipative soliton position fluctuation operator. The study performed based on this scheme has made it also possible to construct a solution for a one-dimensional dark relaxing dissipative soliton. This soliton generalizes the stationary soliton with allowance for the shift of its center and deformation of its profile followed by the recovery of its initial shape. Average squares of quantum fluctuations of collective variables are calculated. A domain of parameters in which there exist quantum states of solitons with an initially high degree of squeezing with respect to the momentum is found. It is shown that such states are in correspondence with significantly higher velocities of soliton center drift. An experiment that could detect the relative squeezing with respect to the momentum due to the soliton center drift is discussed.

Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: II. Two-dimensional light solitons

Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: II. Two-dimensional light solitons

Spatial solitons in a medium with lumped amplification and dissipation

This work analyzes the dynamics of two-dimensional spatial solitons in dissipative graded index Kerr media, analytically and numerically. The dynamics of two-dimensional spatial solitons has been studied in a medium with lumped amplification. The presence of lumped amplification in the medium results in the formation of soliton like waves called similaritons. The spatial soliton dynamics has been studied in a dissipative medium. In a dissipative graded index Kerr medium, the balancing between the coefficients of constant dissipation and lumped amplification results in the stabilization of the beam. The beam dynamics has also been studied by varying amplification coefficients with propagation distance. Hyperbolic, linear, and exponential amplification profiles are considered. When amplification coefficient varies with propagation distance, the beam gets compressed. The beam compression is higher in the case of exponential amplification profile than that of hyperbolic and linear amplification profiles.

Two-dimensional solitons in a quintic-septimal medium

We report an observation of spatial solitons in a medium managed to present fifth-seventh (focusing-defocusing) refractive nonlinearities with suppressed third-order nonlinearity. Propagation of two-dimensional bright spatial solitons for \ensuremath{\sim}10 Rayleigh lengths was observed and characterized in a suspension of silver nanoparticles in acetone using the scattered light imaging method. Numerical calculations based on a nonlinear Schr\odinger-type equation, including contributions up to the seventh-order susceptibility, were performed showing good agreement with the experimental results.

Spatial weak-light ring soliton in self-assembled quantum dots

By using semiclassical theory combined with multiple-scale method, we analytically study the linear absorption and the nonlinear dynamical properties in a lifetime broadened Λ-type three-level self-assembled quantum dots. It is found that this system can exhibit the transparency, and the width of the transparency window becomes wider with the increase of control light field. Interestingly, a weak probe light beam can form spatial weak-light dark solitons. When it propagates along the axial direction, the soliton will transform into a steady spatial weak-light ring dark soltion. In addition, the stability of two-dimensional spatial optical solitons is testified numerically.

Stabilization of two-dimensional spatial solitons in dissipative media

This work analyzes the dynamics of two-dimensional spatial solitons in dissipative media. Stable solitons are formed in a two-dimensional medium with constant dissipation due to the balancing between dissipation and the instabilities due to nonlinearity. The dynamics have also been studied for linear, quadratic and exponential loss profiles. When a beam propagates in media with dissipation, where dissipation is a function of distance, it becomes compressed. A linear loss profile results in higher compression, while compression is minimal for an exponential profile.

Two-dimensional solitons and clusters in dissipative lattices

We study the dynamics of two-dimensional spatial solitons in the structured optical medium modeled by the complex Ginzburg–Landau equation with cubic–quintic nonlinearity and a spatially periodic modulation of the local gain–loss coefficient [a dissipative lattice (DL)]. The analysis, following the variation of the DL’s amplitude and period, reveals several dynamical scenarios: stable or unstable propagation of a single dissipative soliton (the unstable propagation entails generation of an irregular multisoliton cluster), transformation of the input soliton into stable or unstable regular clusters patterned as the underlying DL, and decay of the input. Most results are obtained by means of systematic simulations, but the boundary of the single-soliton stability domain is explained analytically.

Self-phase modulation spectral broadening in two-dimensional spatial solitons: toward three-dimensional spatiotemporal pulse-train solitons

We demonstrate self-phase modulation (SPM) spectral broadening in two-dimensional solitons in homogeneous media using two different schemes. In the active mode, a train of pulses are collectively trapped and form a spatial soliton through a photorefractive, slowly responding, and electronically controlled self-focusing nonlinearity, and each pulse experiences spectral broadening by the fast SPM nonlinearity. In the passive mode, the pulse-train beam is guided in a waveguide that is optically induced by a continuous-wave thermal spatial soliton. The soliton formation increased the normalized spectral broadening factor from 0.5% up to 197%. This experiment presents significant progress toward the experimental demonstration of three-dimensional spatiotemporal pulse-train solitons.

X-shape two-dimensional photovoltaic dark spatial solitons with fully incoherent white light

We report on the first successful experimental observation of X-shape two-dimensional photovoltaic dark spatial solitons in an anisotropic LiNbO3: Fe crystal with fully incoherent white light beams under odd or even initial conditions. This kind of solitons results from the photovoltaic self-defocusing nonlinearity, which depends on the orientation of the principal axes of the crystal.

Dynamics of two-dimensional dissipative spatial solitons interacting with an umbrella-shaped potential

We numerically study the dynamics of two-dimensional spatial solitons on the top of an external umbrella-shaped potential in the cubic-quintic complex Ginzburg-Landau model. Unique scenarios of the dynamics of dissipative spatial solitons interacting with this potential are put forward, such as generation of straight-lined arrays (or “jets”), emission of either one necklace-shaped soliton array or several such soliton arrays, soliton evolution into an oscillatory mode, and soliton spreading. In addition, by changing the number of lateral planes of the external potential, keeping fixed the other parameters of the potential, the various scenarios of soliton dynamics can transform into each other. These results suggest possible applications to signal routing in all-optical information processing devices.

Splitting broad beams into arrays of dissipative spatial solitons by material and virtual gratings

We elaborate two generic methods for producing two-dimensional (2D) spatial soliton arrays (SSAs) in the framework of the cubic–quintic (CQ) complex Ginzburg–Landau (CGL) model. The first approach deals with a broad beam launched into the dissipative nonlinear medium which is equipped with an imprinted grating of a sufficiently sharp form. The beam splits into a cluster of jets, each subsequently self-trapping into a stable soliton, if the power is sufficient. We consider two kinds of sharp gratings—‘raised-cosine’ (RC) and Kronig–Penney (KP) lattices—and two types of input beams: fundamental and vortical. By selecting appropriate parameters, this method makes it possible to create various types of soliton arrays, such as solid, annular (with single and double rings) and cross-shaped ones. The second method uses a ‘virtual lattice’, in the form of a periodic transverse phase modulation imprinted into the broad beam which is passed through an appropriate phase mask and then shone into a uniform nonlinear medium. Two different types of masks are considered; in the form of a ‘checkerboard’ or ‘tilings’. In these cases, broad fundamental and vortical beams may also evolve into stable SSAs if the beam power and spacing of the virtual phase lattice are large enough. By means of the latter technique, square-shaped, hexagonal and quasi-crystalline SSAs can be created.

Bistable Helmholtz bright solitons in saturable materials

We present, to the best of our knowledge, the first exact analytical solitons of a nonlinear Helmholtz equation with a saturable refractive-index model. These new two-dimensional spatial solitons have a bistable characteristic in some parameter regimes, and they capture oblique (arbitrary-angle) beam propagation in both the forward and backward directions. New conservation laws are reported, and the classic paraxial solution is recovered in an appropriate multiple limit. Analysis and simulations examine the stability of both solution branches, and stationary Helmholtz solitons are found to emerge from a range of perturbed input beams.

Gap vortex solitons in periodic media with quadratic nonlinearity

We explore the existence and stability of two-dimensional spatial gap solitons with embedded vorticity in a bulk medium with the quadratic $({\ensuremath{\chi}}^{(2)})$ nonlinearity and a transverse grating represented by periodic modulation of the refractive index. Gap-vortex solitons (GVSs) can be found in total gaps of the underlying spectra of the fundamental-frequency and second-harmonic waves. We demonstrate the existence of a family of stable GVSs, which are built as four-peak complexes, in the lowest total gap. We also consider dynamical effects, such as self-trapping of GVSs from input beams, and delocalization transitions.