Stable vortex solitons in the two-dimensional Ginzburg-Landau equation

Stable Vortex Solitons Sustained by Localized Gain in a Cubic Medium

We propose a simple dissipative system with purely cubic defocusing nonlinearity and nonuniform linear gain that can support stable localized dissipative vortex solitons with high topological charges without the utilization of competing nonlinearities and nonlinear gain or losses. Localization of such solitons is achieved due to an intriguing mechanism when defocusing nonlinearity stimulates energy flow from the ringlike region with linear gain to the periphery of the medium where energy is absorbed due to linear background losses. Vortex solitons bifurcate from linear gain-guided vortical modes with eigenvalues depending on topological charges that become purely real only at specific gain amplitudes. Increasing gain amplitude leads to transverse expansion of vortex solitons, but simultaneously it usually also leads to stability enhancement. Increasing background losses allows creation of stable vortex solitons with high topological charges that are usually prone to instabilities in conservative and dissipative systems. Propagation of the perturbed unstable vortex solitons in this system reveals unusual dynamical regimes, when instead of decay or breakup, the initial state transforms into stable vortex solitons with lower or sometimes even with higher topological charge. Our results suggest an efficient mechanism for the formation of nonlinear excited vortex-carrying states with suppressed destructive azimuthal modulational instabilities in a simple setting relevant to a wide class of systems, including polaritonic systems, structured microcavities, and lasers.

Control of topology of two-dimensional solitons in a laser with saturable absorption by means of a coherent holding radiation

We propose a simple method to control the topology of laser vortex solitons and their complexes in a wide-aperture laser with saturable absorption by means of weak coherent holding radiation. The holding radiation acting on initial "free" vortex solitons induces the appearance of new peripheral vortices and the splitting of multiple central vortices, as well as reconfiguration of energy flow topology. A wide variety of these stable vortex structures makes the scheme promising for topologically protected information processing.

Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities

Thirty years ago, Coullet et al. proposed that a special optical field exists in laser cavities bearing some analogy with the superfluid vortex. Since then, optical vortices have been widely studied, inspired by the hydrodynamics sharing similar mathematics. Akin to a fluid vortex with a central flow singularity, an optical vortex beam has a phase singularity with a certain topological charge, giving rise to a hollow intensity distribution. Such a beam with helical phase fronts and orbital angular momentum reveals a subtle connection between macroscopic physical optics and microscopic quantum optics. These amazing properties provide a new understanding of a wide range of optical and physical phenomena, including twisting photons, spin-orbital interactions, Bose-Einstein condensates, etc., while the associated technologies for manipulating optical vortices have become increasingly tunable and flexible. Hitherto, owing to these salient properties and optical manipulation technologies, tunable vortex beams have engendered tremendous advanced applications such as optical tweezers, high-order quantum entanglement, and nonlinear optics. This article reviews the recent progress in tunable vortex technologies along with their advanced applications.

Vortex Multistability and Bessel Vortices in Polariton Condensates

Vortices are topological objects formed in coherent nonlinear systems. As such they are studied in a wide number of physical systems and promise applications in information storage, processing, and communication. In semiconductor microcavities, vortices in polariton condensates can be conveniently created, studied, and manipulated using solely optical means. For nonresonant excitation with a ring-shaped pump a stable vortex can be formed, leading to bistability with left- and right-handed vorticity. In the present work we report on a much richer vortex multistability, with optically addressable vortices with topological charges m=±1, ±2, and ±3, all stable for the same system and excitation parameters. This unusual multistable behavior is rooted in the inherent nonlinear feedback between reservoir excitations and condensate in the microcavity. For larger radius of the ring-shaped pump we also find a Bessel vortex with its characteristic spiralling phase in the high density region and pronounced self-stabilization ability. Our theoretical results open up exciting possibilities for optical manipulation of vortex multiplets in a compact semiconductor system.

One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential

We propose a new mechanism for the stabilization of confined modes in lasers and semiconductor microcavities holding exciton-polariton condensates, with spatially uniform linear gain, cubic loss, and cubic self-focusing or defocusing nonlinearity. We demonstrated that the commonly known background instability driven by the linear gain can be suppressed by a combination of a harmonic-oscillator trapping potential and effective diffusion. Systematic numerical analysis of one- and two-dimensional (1D and 2D) versions of the model reveals a variety of stable modes, including stationary ones, breathers, and quasi-regular patterns filling the trapping area in the 1D case. In 2D, the analysis produces stationary modes, breathers, axisymmetric and rotating crescent-shaped vortices, stably rotating complexes built of up to 8 individual vortices, and, in addition, patterns featuring vortex turbulence. Existence boundaries for both 1D and 2D stationary modes are found in an exact analytical form, and an analytical approximation is developed for the full stationary states.

Stable dissipative optical vortex clusters by inhomogeneous effective diffusion

We numerically show the generation of robust vortex clusters embedded in a two-dimensional beam propagating in a dissipative medium described by the generic cubic-quintic complex Ginzburg-Landau equation with an inhomogeneous effective diffusion term, which is asymmetrical in the two transverse directions and periodically modulated in the longitudinal direction. We show the generation of stable optical vortex clusters for different values of the winding number (topological charge) of the input optical beam. We have found that the number of individual vortex solitons that form the robust vortex cluster is equal to the winding number of the input beam. We have obtained the relationships between the amplitudes and oscillation periods of the inhomogeneous effective diffusion and the cubic gain and diffusion (viscosity) parameters, which depict the regions of existence and stability of vortex clusters. The obtained results offer a method to form robust vortex clusters embedded in two-dimensional optical beams, and we envisage potential applications in the area of structured light.

Spontaneous Formation of Vector Vortex Beams in Vertical-Cavity Surface-Emitting Lasers with Feedback

The spontaneous emergence of vector vortex beams with nonuniform polarization distribution is reported in a vertical-cavity surface-emitting laser (VCSEL) with frequency-selective feedback. Antivortices with a hyperbolic polarization structure and radially polarized vortices are demonstrated. They exist close to and partially coexist with vortices with uniform and nonuniform polarization distributions characterized by four domains of pairwise orthogonal polarization. The spontaneous formation of these nontrivial structures in a simple, nearly isotropic VCSEL system is remarkable and the vector vortices are argued to have solitonlike properties.

Non-unique results of collisions of quasi-one-dimensional dissipative solitons

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.

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.

Solitary vortices supported by localized parametric gain

We demonstrate the existence and stability of bright vortex solitons sustained by a ring-shaped parametric gain, for both focusing and defocusing Kerr nonlinearities in lossy optical media. With the defocusing nonlinearity, the vortices are stable at all values of the detuning parameter, while under the focusing nonlinearity their stability region is limited to some positive values of the detuning. Unstable vortices in the focusing medium transform into stable rotating azimuthons.

Continuous emission of fundamental solitons from vortices in dissipative media by a radial-azimuthal potential

We report novel dynamical regimes of dissipative vortices supported by a radial-azimuthal potential (RAP) in the 2D complex Ginzburg-Landau (CGL) equation with the cubic-quintic nonlinearity. First, the stable solutions of vortices with intrinsic vorticity S = 1 and 2 are obtained in the CGL equation without potential. The RAP is a model of an active optical medium with respective expanding anti-waveguiding structures with m (integer) annularly periodic modulation. If the potential is strong enough, m jets fundamental of solitons are continuously emitted from the vortices. The influence of m, diffusivity term (viscosity) β, and cubic-gain coefficient ε on the dynamic region is studied. For a weak potential, the shape of vortices are stretched into the polygon, such as square for m = 4. But for a stronger potential, the vortices will be broke into m fundamental solitons.

Pattern formation by kicked solitons in the two-dimensional Ginzburg-Landau medium with a transverse grating

We consider the kick- (tilt-) induced mobility of two-dimensional (2D) fundamental dissipative solitons in models of bulk lasing media based on the 2D complex Ginzburg-Landau equation including a spatially periodic potential (transverse grating). The depinning threshold, which depends on the orientation of the kick, is identified by means of systematic simulations and estimated by means of an analytical approximation. Various pattern-formation scenarios are found above the threshold. Most typically, the soliton, hopping between potential cells, leaves arrayed patterns of different sizes in its wake. In the single-pass-amplifier setup, this effect may be used as a mechanism for the selective pattern formation controlled by the tilt of the input beam. Freely moving solitons feature two distinct values of the established velocity. Elastic and inelastic collisions between free solitons and pinned arrayed patterns are studied too.

Generation of polygonal soliton clusters and fundamental solitons in dissipative systems by necklace-ring beams with radial-azimuthal phase modulation

We demonstrate that, in a two-dimensional dissipative medium described by the cubic-quintic (CQ) complex Ginzburg-Landau (CGL) equation with the viscous (spectral-filtering) term, necklace rings carrying a mixed radial-azimuthal phase modulation can evolve into polygonal or quasipolygonal stable soliton clusters, and into stable fundamental solitons. The outcome of the evolution is controlled by the depth and azimuthal anharmonicity of the phase-modulation profile, or by the radius and number of "beads" in the initial necklace ring. Threshold characteristics of the evolution of the patterns are identified and explained. Parameter regions for the formation of the stable polygonal and quasipolygonal soliton clusters, and of stable fundamental solitons, are identified. The model with the CQ terms replaced by the full saturable nonlinearity produces essentially the same set of basic dynamical scenarios; hence this set is a universal one for the CGL models.

Continuous generation of "light bullets" in dissipative media by an annularly periodic potential

We report novel dynamical regimes of "light bullets" supported by an annularly periodic potential in the three-dimensional (3D) complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. This is a model of an active optical medium with respective expanding anti-waveguiding structures with m≥2 (integer) annularly periodic modulation. If the potentials are strong enough, they give rise to continuous generation of m jets light bullet by an initial light bullet initially placed at the center. The influence of m and diffusivity term (viscosity) β on the corresponding strength of potential is studied. In the case of m = 0 (conical geometry), these are concentric waves expanding in the radial direction.

Vortex twins and anti-twins supported by multiring gain landscapes

We address the properties of multivortex soliton complexes supported by multiring gain landscapes in focusing Kerr nonlinear media with strong two-photon absorption. Stable complexes incorporating two, three, or four vortices featuring opposite or identical topological charges are shown to exist. In the simplest geometries with two amplifying rings vortex twins with equal topological charges exhibit asymmetric intensity distributions, while vortex anti-twins may be symmetric or asymmetric, depending on the gain level and separation between rings.

From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback

We use the cubic complex Ginzburg-Landau equation linearly coupled to a dissipative linear equation as a model for lasers with an external frequency-selective feedback. This system may also serve as a general pattern-formation model in media driven by an intrinsic gain and selective feedback. While, strictly speaking, the approximation of the laser nonlinearity by a cubic term is only valid for small field intensities, it qualitatively reproduces results for dissipative solitons obtained in models with a more complex nonlinearity in the whole parameter region where the solitons exist. The analysis is focused on two-dimensional stripe-shaped and vortex solitons. An analytical expression for the stripe solitons is obtained from the known one-dimensional soliton solution, and its relation with vortex solitons is highlighted. The radius of the vortices increases linearly with their topological charge m, therefore the stripe-shaped soliton may be interpreted as the vortex with m=∞, and, conversely, vortex solitons can be realized as unstable stripes bent into stable rings. The results for the vortices are applicable for a broad class of physical systems.

Rotating vortex solitons supported by localized gain

We show that ringlike localized gain landscapes imprinted in focusing cubic (Kerr) nonlinear media with strong two-photon absorption support new types of stable higher-order vortex solitons containing multiple phase singularities nested inside a single core. The phase singularities are found to rotate around the center of the gain landscape, with the rotation period being determined by the strength of the gain and the nonlinear absorption.

Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain

We discover that a spatially localized gain supports stable vortex solitons in media with cubic nonlinearity and two-photon absorption. The interplay between nonlinear losses and gain in amplifying rings results in the suppression of otherwise ubiquitous azimuthal modulation instabilities of radially symmetric vortex solitons. We find that the topology of the gain profile imposes restrictions on the maximal possible charge of vortex solitons. Symmetry breaking occurs at high gain levels, resulting in the formation of necklace vortex solitons composed of asymmetric bright spots.

Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses

Using a combination of the variation approximation and direct simulations, we consider the model of the light transmission in nonlinearly amplified bulk media, taking into account the localization of the gain, i.e., the linear loss shaped as a parabolic function of the transverse radius, with a minimum at the center. The balance of the transverse diffraction, self-focusing, gain, and the inhomogeneous loss provides for the hitherto elusive stabilization of vortex solitons, in a large zone of the parameter space. Adjacent to it, stability domains are found for several novel kinds of localized vortices, including spinning elliptically shaped ones, eccentric elliptic vortices which feature double rotation, spinning crescents, and breathing vortices.

Vortex lattice solitons supported by localized gain

We show that localized gain supports the existence of dissipative vortex solitons in periodic Kerr media with strong two-photon absorption. Vortex solitons exist in both focusing and defocusing media, with their propagation constants emerging from semi-infinite or finite gaps in the lattice spectrum. Coincidence of the discrete rotational symmetries of the gain landscape and refractive index distribution is a necessary condition for exciting vortex solitons, which otherwise transform into stable dissipative multipoles.

Continuous generation of soliton patterns in two-dimensional dissipative media by razor, dagger, and needle potentials

We report dynamic regimes supported by a sharp quasi-one-dimensional (1D) ("razor"), pyramid-shaped ("dagger"), and conical ("needle") potentials in the 2D complex Ginzburg-Landau (CGL) equation with cubic-quintic nonlinearity. This is a model of an active optical medium with respective expanding antiwaveguiding structures. If the potentials are strong enough, they give rise to continuous generation of expanding soliton patterns by a 2D soliton initially placed at the center. In the case of the pyramidal potential with M edges, the generated patterns are sets of M jets for M < or = 5, or expanding polygonal chains of solitons for M > or = 6. In the conical geometry, these are concentric waves expanding in the radial direction.

Bistable and addressable localized vortices in semiconductor lasers

We demonstrate experimentally that localized emission states in coupled broad-area semiconductor lasers can carry a finite orbital angular momentum. The resulting structures therefore possess the chirality of optical vortices together with the properties of localized structures in dissipative systems, namely, the coexistence with a low intensity homogeneous emission and the mutual independence. These results open the way to the realization of arrays of optically addressable and bistable chiral laser pixels.

Vortex solitons in lasers with feedback

We report on the existence, stability and dynamical properties of two-dimensional self-localized vortices with azimuthal numbers up to 4 in a simple model for lasers with frequency-selective feedback.We build the full bifurcation diagram for vortex solutions and characterize the different dynamical regimes. The mathematical model used, which consists of a laser rate equation coupled to a linear equation for the feedback field, can describe the spatiotemporal dynamics of broad area vertical cavity surface emitting lasers with external frequency selective feedback in the limit of zero delay.

Dissipative ring solitons with vorticity

We study dissipative ring solitons with vorticity in the frame of the (2+1)-dimensional cubic-quintic complex Ginzburg-Landau equation. In dissipative media, radially symmetric ring structures with any vorticity m can be stable in a finite range of parameters. Beyond the region of stability, the solitons lose the radial symmetry but may remain stable, keeping the same value of the topological charge. We have found bifurcations into solitons with n-fold bending symmetry, with n independent on m. Solitons without circular symmetry can also display (m + 1)-fold modulation behaviour. A sequence of bifurcations can transform the ring soliton into a pulsating or chaotic state which keeps the same value of the topological charge as the original ring.

Collisions between counter-rotating solitary vortices in the three-dimensional Ginzburg-Landau equation

We report results of collisions between coaxial vortex solitons with topological charges +/-S in the complex cubic-quintic Ginzburg-Landau equation. With the increase of the collision momentum, merger of the vortices into one or two dipole or quadrupole clusters of fundamental solitons (for S=1 and 2, respectively) is followed by the appearance of pairs of counter-rotating "unfinished vortices," in combination with a soliton cluster or without it. Finally, the collisions become elastic. The clusters generated by the collisions are very robust, while the "unfinished vortices," eventually split into soliton pairs.

Stable solitons of even and odd parities supported by competing nonlocal nonlinearities

We introduce a one-dimensional phenomenological model of a nonlocal medium featuring focusing cubic and defocusing quintic nonlocal optical nonlinearities. By means of numerical methods, we find families of solitons of two types, even-parity (fundamental) and dipole-mode (odd-parity) ones. Stability of the solitons is explored by means of computation of eigenvalues associated with modes of small perturbations, and tested in direct simulations. We find that the stability of the fundamental solitons strictly follows the Vakhitov-Kolokolov criterion, whereas the dipole solitons can be destabilized through a Hamiltonian-Hopf bifurcation. The solitons of both types may be stable in the nonlocal model with only quintic self-attractive nonlinearity, in contrast with the instability of all solitons in the local version of the quintic model.

Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity S=0, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for perturbation eigenmodes. The results provide the first example of stable vortex tori in a 3D dissipative medium, as well as the first example of higher-order tori (with S=2) in any nonlinear medium. It is found that all stable vortical solitons coexist in a large domain of the parameter space; in smaller regions, there coexist stable solitons with either S=0 and S=1, or S=1 and S=2.

Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation

We consider a class of self-trapped spatiotemporal solitons: spatiotemporal necklace-ring solitons, whose intensities are azimuthally periodically modulated. We reveal numerically that the spatiotemporal necklace-ring solitons carrying zero, integer, and even fractional angular momentum can be self-trapped over a huge propagation distance in the three-dimensional cubic-quintic complex Ginzburg-Landau equation, even in the presence of random perturbations.

Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations

The generation and nonlinear dynamics of multidimensional optical dissipative solitonic pulses are examined. The variational method is extended to complex dissipative systems, in order to obtain steady state solutions of the (D + 1)-dimensional complex cubic-quintic Ginzburg-Landau equation (D = 1, 2, 3). A stability criterion is established fixing a domain of dissipative parameters for stable steady state solutions. Following numerical simulations, evolution of any input pulse from this domain leads to stable dissipative solitons.

Stable vortex solitons supported by competing quadratic and cubic nonlinearities

We address the stability problem for vortex solitons in two-dimensional media combining quadratic and self-defocusing cubic [chi(2):chi(3)- ] nonlinearities. We consider the propagation of spatial beams with intrinsic vorticity S in such bulk optical media. It was earlier found that the S=1 and S=2 solitons can be stable, provided that their power (i.e., transverse size) is large enough, and it was conjectured that all the higher-order vortices with S> or =3 are always unstable. On the other hand, it was recently shown that vortex solitons with S>2 and very large transverse size may be stable in media combining cubic self-focusing and quintic self-defocusing nonlinearities. Here, we demonstrate that the same is true in the chi(2):chi(3)- model, the vortices with S=3 and S=4 being stable in regions occupying, respectively, approximately 3% and 1.5% of their existence domain. The vortex solitons with S>4 are also stable in tiny regions. The results are obtained through computation of stability eigenvalues, and are then checked in direct simulations, with a conclusion that the stable vortices are truly robust ones, easily self-trapping from initial beams with embedded vorticity. The dependence of the stability region on the chi(2) phase-mismatch parameter is specially investigated. We thus conclude that the stability of higher-order two-dimensional vortex solitons in narrow regions is a generic feature of optical media featuring the competition between self-focusing and self-defocusing nonlinearities. A qualitative analytical explanation to this feature is proposed.

Spatial resonator solitons

Spatial solitons can exist in various kinds of nonlinear optical resonators with and without amplification. In the past years different types of these localized structures such as vortices, bright, dark solitons, and phase solitons have been experimentally shown to exist. Many links appear to exist to fields different from optics, such as fluids, phase transitions, or particle physics. These spatial resonator solitons are bistable and due to their mobility suggest schemes of information processing not possible with the fixed bistable elements forming the basic ingredient of traditional electronic processing. The recent demonstration of existence and manipulation of spatial solitons in semiconductor microresonators represents a step in the direction of such optical parallel processing applications. We review some proof of principle soliton experiments on slow systems, and describe in more detail the experiments on semiconductor resonator solitons which are aimed at applications.

Discrete Ginzburg-Landau solitons

We demonstrate that discrete solitons are possible in Ginzburg-Landau lattices. As a result of discreteness, we find that this system exhibits a host of features that have no counterpart whatsoever in either the continuous limit or in other conservative discrete models.

Domain-size control by global feedback in bistable systems

We study domain structures in bistable systems such as the Ginzburg-Landau equation. The size of domains can be controlled by a global negative feedback. The domain-size control is applied for a localized spiral pattern.