Stable dissipative optical vortex clusters by inhomogeneous effective diffusion

Huishan Li; Shiquan Lai; Yunli Qui; Xing Zhu; Jianing Xie; Dumitru Mihalache; Yingji He

doi:10.1364/OE.25.027948

1. Introduction

Of much interest from both theoretical and experimental points of view is the study of spatial optical soliton dynamics in conservative and dissipative media [1–9], because the spatial solitons have a great potential for applications to all-optical switching, pattern recognition, and parallel data processing [3].

It is well known that complex Ginzburg-Landau (CGL) equations represent a broad class of models that support spatial patterns due to the simultaneous balance of gain versus loss, and self-focusing nonlinearity versus diffraction or dispersion. In addition, the CGL-type equations play an important role in many areas, such as superconductivity and superfluidity, fluid dynamics, reaction-diffusion phenomena, nonlinear optics, Bose–Einstein condensate, quantum field theories, and biology and medicine [10, 11]. The generic CGL equation also finds a direct realization in nonlinear optics as a realistic dynamical model of laser cavities, for forming stable fundamental solitons, vortex solitons, and various types of solitons clusters [12–37].

Optical vortex solitons (OVS) are another important class of self-trapped beams that present robust propagation in defocusing nonlinear optical media [38, 39]. These beams have a spiral profile that leads to a zero-field amplitude, at the vortex center [40]. In general, the optical nonlinearity offers the stability of a single OVS; however, its robust propagation is always limited by the finiteness of the background beam, which severely limits the applications of single OVSs and arrays of OVSs [41–46]. One of the main proposals for the use of vortex solitons in diverse applications is the transfer of the optical angular moment from light to matter [47]. Also, vortex solitons can be used for guiding light [48]. They are of particular interest for manipulation and tweezing of nanoparticles and nanorods in various colloidal suspensions [49].

Many innovative methods or configurations have been used to meet the applications of OVSs. The optical vortex motion and related phenomena in a slowly diffracting background field was studied both theoretically and experimentally in [50]. The vortex drift and rotation in a diffracting Gaussian beam propagation through a nonlinear saturable medium was experimentally put forward, allowing a simple scheme for steering optical vortex solitons [50]; see also [51–54]. The evolution and robustness of ensembles (clusters) of OVSs in homogeneous and isotropic bulk Kerr self-defocusing nonlinear media have been investigated in [54]. It was found that the coherent interaction of OVSs leads to vortex motion and to the possibility to steer the motion of such complex vortex ensembles [54].

Ring-shaped localized OVSs supported by localized parametric gain [55] have been revealed, and localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity have been recently investigated [56]. Also, a study of guiding and confinement of light induced by vortex solitons in cubic-quintic (CQ) nonlinear optical media has been recently reported [57]. These techniques are quite limited due to the occurrence of a single vortex in the core of the ring-shaped soliton.

However, although stable embedding of a single OVS or of several OVSs in various nonlinear media have been investigated in different physical settings, such studies are still rare. We mention here that twin-vortex solitons in nonlocal nonlinear media have been studied in thermal nonlinear media bounded by rectangular cross sections [58]. It was found that the double-charge states are remarkably robust despite their shape asymmetry and phase-singularity splitting [58].

The dynamics of vortex solitons embedded in top-hat nonlinear Bessel beams in self-defocusing media was recently investigated [59]. These vortex solitons survive unaltered for propagation distances that are one order of magnitude larger than in the case of the usual Gaussian or super-Gaussian backgrounds, but quickly spreading the backgrounds [59].

In this work, we reveal the occurrence of stable optical vortex clusters in the two-dimensional (2D) CQ CGL model by imposing inhomogeneous effective diffusion in the optical medium. Extensive numerical simulations demonstrate that in this generic model, an input 2D Gaussian beam, with an embedded screw dislocation characterized by the integer winding number M, may evolve into a cluster of M separated vortex solitons. We have obtained the domains of stability of these vortex clusters that depend on the specific values of amplitudes and periods of the inhomogeneous effective diffusion and on the region of variation of the parameters that govern the nonlinear dissipative optical medium.

2. The model

We consider the CQ CGL equation of the general form [11] with the inhomogeneous effective diffusion:

i u_{z} + (1 / 2) Δ u + {| u |}^{2} u + ν {| u |}^{4} u = i R [u] .

where

Δ = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

is the transverse Laplacian (x and y are the transverse coordinates, and z is the propagation distance), and the coefficient in front of the cubic self-focusing term is scaled to be 1.

Further, ν is the quintic self-defocusing coefficient, and the combination of the CQ nonlinear terms is R[u] = δu + ε|u|² u + μ|u|⁴ u + Eu, with −δ the linear loss coefficient, −μ the quintic-loss parameter, ε the cubic-gain coefficient, and the term Eu accounts for the inhomogeneous effective diffusion, which is periodically modulated in the propagation direction and has different amplitudes in the transverse x and y directions:

E u = β [A_{x} | \sin (z / T_{x}) | \frac{\partial^{2} u}{\partial x^{2}} + A_{y} | \sin (z / T_{y}) | \frac{\partial^{2} u}{\partial y^{2}}] .

Here β is accounting for the effective diffusion (viscosity), which can produce a friction force to inhibit soliton transverse drift. In Eq. (2),

A_{x}

and

A_{y}

are the amplitudes corresponding to the x- and y-parts of the effective diffusion term, respectively, and

T_{x}

and

T_{y}

are the associated modulation periods in the propagation direction.

The inhomogeneous diffusion in the propagation direction z and the effective diffusion anisotropy with respect to the transverse coordinates x and y might be implemented by using an appropriate doping profile of the nonlinear optical medium. However, in order to stabilize the vortex clusters embedded in the two-dimensional optical beam, the effective diffusion coefficient, the cubic gain coefficient, and the amplitude values corresponding to the modulation of x- and y-diffusion coefficients must belong to certain parameter domains.

3. Results and analysis

The generic results of the propagation outcomes may be adequately represented for the following set of parameters: δ = −0.3, ν = 0.1, and μ = 1, which is considered below. The selected set of parameters is commonly used in many earlier studies of the formation and robustness of different types of dissipative optical solitons; see, for example, Ref [22].

The simulations are performed by means of the split-step Fourier method. The input optical field is a Gaussian beam with an embedded screw dislocation characterized by the integer winding number M:

u = A (z) \exp [- \frac{x^{2} + y^{2}}{2 w^{2} (z)} + i M θ] .

Here, A, w, and M represent the amplitude, width, and topological charge (winding number), respectively. The term exp [iMθ] in Eq. (3) produces an angular momentum for the input optical beam, where θ is the azimuthal angle.

The soliton solutions were obtained numerically and the robustness of the stable solitons was additionally tested in direct numerical simulations by multiplying Eq. (3) with [1 + ρ(x, y)], where ρ(x, y) is a Gaussian random function, whose maximum is 10% of the soliton’s amplitude.

Firstly, we investigate the influence of the cubic-gain coefficient ε and of the amplitude $A_{x}$ on the vortical optical field propagation. We have performed extensive numerical simulations for the following values of the topological charge of the input optical beam: M = 1, 2, 3, and 4, and we have revealed the various regimes of propagation.

For small values of the cubic-gain coefficient ε, in the parameter domain below the line 1 in Fig. 1(a), we have found that for M = 1 the input vortical field becomes elliptically-shaped during propagation and splits into two elliptical fundamental (vorticityless) solitons, see Fig. 1(b).

Fig. 1 (a) Different propagation scenarios in the plane of parameters (ε, $A_{x}$ ) for M = 1. Below line 1: the input vortical waveform evolves into two elliptically-shaped fundamental (vorticityless) solitons; between lines 1 and 2: the generation of a single vortex soliton; between lines 2 and 3: the generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical field splits and then evolves into two elliptically-shaped fundamental solitons for $A_{x}$ = 1.8 and ε = 2.0 [corresponding to point M indicated in panel (a)]. (c) The generation of a single vortex soliton for $A_{x}$ = 1.4 and ε = 2.4 [corresponding to point N indicated in panel (a)]. (d) The generation of a cluster containing several vortex solitons for $A_{x}$ = 1.6 and ε = 2.8 [corresponding to point O indicated in panel (a)]. (e) The input vortical field evolves into a wavefield displaying excess gain for $A_{x}$ = 1.4 and ε = 3.4 [corresponding to point P indicated in panel (a)]. The other parameters are β = 0.6, $A_{y}$ = 1.0, and $T_{x}$ = $T_{y}$ = 4.

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When M = 2 and M = 3, and for small values of the cubic-gain coefficient ε, in the parameter domain below the lines 1 in Figs. 1(a) and 2(a), we see that the input vortical waveforms are distorted during propagation and become elliptically-shaped due to the asymmetric (in the transverse directions) inhomogeneous effective diffusion effect. The input vortical fields evolve into rather irregular beams, see Figs. 2(b) and 3(b). However, for M = 4, and in the parameter domain below the line 1 in Fig. 4(a), in input field splits into two fragments in the early stage of evolution and then the two splinters coalesce into an elliptically-shaped spot.

Fig. 2 (a) Different propagation scenarios in the plane of parameters (ε, $A_{x}$ ) for M = 2. Below line 1: the input vortex waveform evolves into an irregular beam; between lines 1 and 2: the generation of a cluster containing two vortex solitons; between lines 2 and 3: the generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical field evolves into an irregular beam for $A_{x}$ = 1.2 and ε = 2.1 [corresponding to point A indicated in panel (a)]. (c) The generation of a vortex cluster composed of two individual vortex solitons for $A_{x}$ = 1.4 and ε = 2.5 [corresponding to point B indicated in panel (a)]. (d) The generation of a vortex cluster containing more than two individual vortex solitons for $A_{x}$ = 1.4 and ε = 3.0 [corresponding to point C indicated in panel (a)]. (e) The input vortical field evolves into a wavefield displaying excess gain for $A_{x}$ = 1.4 and ε = 3.4 [corresponding to point D indicated in panel (a)]. The other parameters are the same as in Fig. 1.

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Fig. 3 (a) Different propagation scenarios in the plane of parameters (ε, $A_{x}$ ) for M = 3. Below line 1: the input optical field evolves into an irregular beam; between lines 1 and 2: the generation of a cluster containing three vortex solitons; between lines 2 and 3: generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical field evolves into an irregular hollow beam for $A_{x}$ = 1.2, ε = 2.3 [corresponding to point E indicated in panel (a)]; (c) The generation of a vortex cluster composed of three individual vortex solitons for $A_{x}$ = 1.2 and ε = 2.5 [corresponding to point F indicated in panel (a)]; (d) The generation of a vortex cluster containing more than three individual vortex solitons for $A_{x}$ = 1.4 and ε = 2.9 [corresponding to point G indicated in panel (a)]. (e) The input vortical field evolves into an optical waveform displaying excess gain for $A_{x}$ = 1.4 and ε = 3.4 [corresponding to point H indicated in panel (a)]. The other parameters are the same as in Fig. 1.

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Fig. 4 (a) Different propagation scenarios in the plane of parameters (ε, $A_{x}$ ) for M = 4. Below line 1: the input vortical field evolves into an elliptically-shaped beam; between lines 1 and 2: the generation of a cluster containing four vortex solitons; between lines 2 and 3: the generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical waveform splits into two fundamental solitons that further coalesce and evolve into an elliptically-shaped beam for $A_{x}$ = 1.4, ε = 2.3 [corresponding to point I indicated in panel (a)]; (c) The generation of a vortex cluster composed of four individual vortex solitons for $A_{x}$ = 1.2 and ε = 2.48 [corresponding to point J indicated in panel (a)]; (d) The generation of a vortex cluster containing more than four individual vortex solitons for $A_{x}$ = 1.4 and ε = 2.8 [corresponding to point K indicated in panel (a)]. (e) The input vortical field evolves into an optical waveform displaying excess gain for $A_{x}$ = 1.2 and ε = 3.4 [corresponding to point L indicated in panel (a)]. The other parameters are the same as in Fig. 1.

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When the parameter ε becomes larger and reach to the region between lines 1 and 2 in Figs. 1(a), 2(a), 3(a), and 4(a), the inhomogeneous effective diffusion effect changes the shape of input vortex background from a circle to an ellipse, and then, for enough gain, the elliptical vortex hole evolves into a vortex cluster nested in the optical beam. The vortex cluster is composed of M = 1, 2, 3, and 4 individual vortices, as shown in Figs. 1(c), 2(c), 3(c), and 4(c). Thus the number of the produced vortex solitons in the vortex cluster is equal to the topological charge M of the input beam, as shown in the final plots corresponding to z = 36 in Figs. 1(c), 2(c), 3(c), and 4(c), respectively. These vortex clusters are robust for relatively large propagation distances up to z = 36. We see from Figs. 1(a), 2(a), 3(a), and 4(a) that the parameter domains where stable vortex clusters can be generated become smaller with the growth of the amplitude $A_{x}$ , because by increasing the magnitude of $A_{x}$ , the vortex cluster stability will suffer from the effect of the inhomogeneous effective diffusion.

Further, if the parameter ε (the cubic-gain coefficient) increases to the domains between lines 2 and 3, although the input waveform can produce more vortex solitons nested in the beam, the vortex clusters are unstable upon propagation. Also, the number of the produced individual vortex solitons that form the vortex cluster is not equal to the topological charge M of the input beam, as shown in Figs. 1(d), 2(d), 3(d), and 4(d).

Finally, when the parameter ε is larger than some threshold (the domains above the lines 3 in Figs. 1(a), 2(a), 3(a), and 4(a)), the optical field spreads and displays excess gain as shown in Figs. 1(e), 2(e), 3(e), and 4(e).

Secondly, we study the influence of the effective diffusion coefficient β and of the modulation amplitude $A_{x}$ on the propagation of the vortical field. We find that for every fixed value of the topological charge M of the input 2D Gaussian beam, with the embedded screw dislocation, there exist four distinct propagation regimes. For small values of β (i.e. for a quite low viscous effect), in the region below the line 1 in Figs. 5(a), 6(a), 7(a), and 8(a), several vortex solitons are created, which are nested in the beam. However, the generated vortex cluster is unstable upon propagation; the number of the produced vortex solitons that belong to the cluster is different from the value of the vorticity number M of the input Gaussian beam, as shown in Figs. 5(b), 6(b), 7(b), and 8(b).

Fig. 5 (a) Different soliton scenarios in the plane of parameters (β, $A_{x}$ ) for M = 1. Below line 1: the input wavefield evolves into several vortex solitons forming a cluster nested in the beam (see in Fig. 5(b) the final plot corresponding to the propagation distance z = 36); between lines 1 and 2: the generation of a single vortex soliton; between lines 2 and 3: the input optical field splits into two fragments in the early stage of evolution and then the two splinters coalesce into a rather irregular beam containing three distinct vortices (see in Fig. 5(d) the final plot corresponding to the propagation distance z = 36); above line 3: the input optical beam is vanishing during propagation. (b) The input beam evolves into a cluster composed of several vortex solitons for $A_{x}$ = 1.4 and β = 0.1 [corresponding to point M indicated in panel (a)]; (c) The generation of a single vortex soliton for $A_{x}$ = 1.4 and β = 1.0 [corresponding to point N indicated in panel (a) ]; (d) The input beam splits and then the two splinters coalesce into a rather irregular beam that contains three distinct vortex solitons for $A_{x}$ = 1.8 and β = 2.5 [corresponding to point O indicated in panel (a)]. (e) The input beam is vanishing during propagation for $A_{x}$ = 1.6 and β = 3.0 [corresponding to point P indicated in panel (a)]. The other parameters are ε = 2.6, $A_{y}$ = 1.0, and $T_{x}$ = $T_{y}$ = 4.

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Fig. 6 (a) Different soliton scenarios in the plane of parameters (β, $A_{x}$ ) for M = 2. Below line 1: the input wavefield evolves into several vortex solitons forming a cluster nested in the beam, see in Fig. 6(b) the final plot corresponding to the propagation distance z = 36; between lines 1 and 2: the generation of a vortex cluster composed of two vortex solitons; between lines 2 and 3: the input optical field evolves into an elliptical beam; above line 3: the input optical beam is vanishing during propagation. (b) The input beam evolves into a cluster composed of several vortex solitons for $A_{x}$ = 1.2 and β = 0.25 [corresponding to point A indicated in panel (a)]; (c) The generation of a cluster formed by two vortex solitons for $A_{x}$ = 1.4 and β = 0.6 [corresponding to point B indicated in panel (a) ]; (d) The input beam evolves into an elliptically-shaped beam for $A_{x}$ = 1.6 and β = 2.25 [corresponding to point C indicated in panel (a)]. (e) The input beam is vanishing during propagation for $A_{x}$ = 1.6 and β = 2.6 [corresponding to point D indicated in panel (a)]. The other parameters are the same as in Fig. 5.

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Fig. 7 (a) Different propagation scenarios in the plane of parameters (β, $A_{x}$ ) for M = 3. Below line 1: the input wavefield evolves into several vortex solitons forming a cluster nested in the beam, see in Fig. 7(b) the final plot corresponding to the propagation distance z = 36; between lines 1 and 2: the generation of a vortex cluster composed of three vortex solitons; between lines 2 and 3: the input optical field evolves into an elliptically-shaped beam containing a nested vortex; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for $A_{x}$ = 1.4 and β = 0.25 [corresponding to point E indicated in panel (a)]. (c) The generation of a cluster formed of three vortex solitons for $A_{x}$ = 1.2 and β = 0.7 [corresponding to point F indicated in panel (a)]. (d) The input optical field evolves into an elliptic beam that contains a nested vortex soliton for $A_{x}$ = 1.4 and β = 1.0 [corresponding to point G indicated in panel (a)]. (e) The input beam is vanishing during propagation for $A_{x}$ = 1.2 and β = 1.3 [corresponding to point H indicated in panel (a)]. The other parameters are the same as in Fig. 5.

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Fig. 8 (a) Different propagation scenarios in the plane of parameters (β, $A_{x}$ ) for M = 4. Below line 1: the input optical field evolves into several vortex solitons forming a cluster nested in the beam; between lines 1 and 2: the generation of a vortex cluster composed of four vortex solitons, see in Fig. 8(c) the final plot corresponding to the propagation distance z = 36; between lines 2 and 3: the input optical field evolves into an irregular beam; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for $A_{x}$ = 1.2, β = 0.2 [corresponding to point I indicated in panel (a)]. (c) The generation of a cluster formed of four vortex solitons for $A_{x}$ = 1.4 and β = 0.7 [corresponding to point J indicated in panel (a)]. (d) The input optical field splits into two fragments and then the splinters coalesce into a rather irregular beam for $A_{x}$ = 1.6 and β = 0.8 [corresponding to point K indicated in panel (a)]. (e) The input beam is vanishing during propagation for $A_{x}$ = 1.6 and β = 1.1 [corresponding to point L indicated in panel (a)]. The other parameters are the same as in Fig. 5.

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When the parameter β increases and reach to the region between curves 1 and 2 in Figs. 5(a), 6(a), 7(a), and 8(a), the inhomogeneous effective diffusion changes the shape of the optical field from a circle to an ellipse, as shown in Figs. 5(c), 6(c), 7(c), and 8(c). The number of the produced vortex solitons in the nested clusters is equal to the input topological charge M, as shown in the final plots corresponding to the propagation distance z = 36, in Figs. 5(c), 6(c), 7(c), and 8(c), for M = 1, 2, 3, and 4, respectively. We see from Figs. 5(a), 6(a), 7(a), and 8(a) that the regions where the stable vortex clusters can be generated become smaller with the growth of the amplitude $A_{x}$ , because by increasing the parameter $A_{x}$ , the inhomogeneous effective diffusion more easily destroys the stability and robustness of vortex clusters.

When the parameter β becomes larger and reach to the region between curves 2 and 3 in Fig. 5(a) for the case M = 1, we see from Fig. 5(d) that the input vortical field splits into two fundamental (vorticityless) solitons in the first stage of the evolution and then the two fragments coalesce into a rather irregular wavefield containing three distinct vortex solitons, see the final plot corresponding to the propagation distance z = 36 in Fig. 5(d). When M = 2, in the domain between the curves 2 and 3 in Fig. 6(a), the input field evolves into an elliptically-shaped beam, see the final plot corresponding to z = 36 in Fig. 6(d). For M = 3, in the region between the curves 2 and 3 in Fig. 7(a), the input waveform evolves into an elliptically-shaped beam containing a single nested vortex, see the final plot corresponding to z = 36 in Fig. 7(d). When M = 4, in the domain between the curves 2 and 3 in Fig. 8(a), the input field splits in the early stage of propagation and then the two fragments coalesce and evolve into a rather irregular beam, see Fig. 8(d).

Finally, when β is too large, corresponding to parameter domains situated above the curve 3 in Figs. 5(a), 6(a), 7 (a), and 8(a), the vortex soliton cluster disappears due to a too weak optical gain since the parameter β produces a large viscous effect (or friction force) in the system, which leads to the complete loss of the optical field energy as shown in Figs. 5(e), 6(e), 7(e), and 8(e).

Thirdly, we study the influence of the effective diffusion coefficient β and of the parameter $T_{x}$ on the optical field propagation. We found four generic propagation regimes as well. For small values of β, in the domains below curves 1 in Figs. 9(a), 10(a), 11(a), and 12(a), during propagation, many vortex solitons are created, which are nested in the beam, but the generated cluster is unstable upon propagation. The number of the produced vortex solitons differs from the topological charge M of the input beam, as shown in Figs. 9(b), 10(b), 11(b), and 12(b), since for small values of β, the viscous effect becomes weaker, thus the vortex solitons can be more easily generated.

Fig. 9 (a) Different propagation scenarios in the plane of parameters (β, $T_{x}$ ) for M = 1. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a single vortex soliton; between lines 2 and 3: the input optical field splits into two fragments in the early stage of evolution and then the two splinters coalesce into a rather irregular beam containing three distinct vortices (see in Fig. 9(d) the final plot corresponding to the propagation distance z = 36); above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for $T_{x}$ = 7 and β = 0.15 [corresponding to point M indicated in panel (a)]. (c) The generation of a single vortex soliton for $T_{x}$ = 5 and β = 1.5 [corresponding to point N indicated in panel (a)]. (d) The input optical field splits and then the two splinters evolve into a rather irregular beam containing three distinct vortex solitons for $T_{x}$ = 1 and β = 2.1 [corresponding to point O indicated in panel (a)]. (e) The input beam is vanishing during propagation for $T_{x}$ = 3 and β = 4.0 [corresponding to point P indicated in panel (a)]. The other parameters are ε = 2.6, $A_{x}$ = $A_{y}$ = 1.0, and $T_{y}$ = 4.

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Fig. 10 (a) Different propagation scenarios in the plane of parameters (β, $T_{x}$ ) for M = 2. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a vortex cluster composed of two vortex solitons; between lines 2 and 3: the input optical field splits into two fragments, which then coalesce during further evolution; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for $T_{x}$ = 7 and β = 0.25 [corresponding to point A indicated in panel (a)]. (c) The generation of a cluster formed of two vortex solitons for $T_{x}$ = 5 and β = 0.75 [corresponding to point B indicated in panel (a)]. (d) The evolution of the input optical field for $T_{x}$ = 2 and β = 2.5 [corresponding to point C indicated in panel (a)]. (e) The input beam is vanishing during propagation for $T_{x}$ = 5 and β = 4.0 [corresponding to point D indicated in panel (a)]. The other parameters are the same as in Fig. 9.

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Fig. 11 (a) Different propagation scenarios in the plane of parameters (β, $T_{x}$ ) for M = 3. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a cluster formed of three vortex solitons; between lines 2 and 3: the input optical field splits into two fundamental vortex solitons that further coalesce into a rather irregular beam having a nested vortex soliton; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for $T_{x}$ = 5 and β = 0.23 [corresponding to point E as indicated in panel (a)]. (c) The generation of a cluster formed of three vortex solitons for $T_{x}$ = 5 and β = 0.75 [corresponding to point F as indicated in panel (a)]. (d) The evolution of the input optical field for $T_{x}$ = 5 and β = 1.4 [corresponding to point G as indicated in panel (a)]. (e) The input beam is vanishing during propagation for $T_{x}$ = 5 and β = 1.6 [corresponding to point H as indicated in panel a)]. The other parameters are the same as in Fig. 9.

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Fig. 12 (a) Different propagation scenarios in the plane of parameters (β, $T_{x}$ ) for M = 4. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a cluster formed of four vortex solitons; between lines 2 and 3: the input optical field splits into two fundamental solitons that further coalesce into a rather irregular beam; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for $T_{x}$ = 3 and β = 0.31 [corresponding to point I indicated in panel (a)]. (c) The generation of a cluster formed of four vortex solitons for $T_{x}$ = 5 and β = 0.7 [corresponding to point J indicated in panel (a)]. (d) The evolution of the input optical field for $T_{x}$ = 5 and β = 1.2 [corresponding to point K indicated in panel (a)]. (e) The input beam is vanishing during propagation for $T_{x}$ = 3 and β = 1.3 [corresponding to point L indicated in panel (a)]. The other parameters are the same as in Fig. 9.

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When β increases larger and reach to the region between curves 1 and 2 in Figs. 9(a), 10(a), 11(a), and 12(a), the inhomogeneous effective diffusion makes the input vortical field to evolve into stable vortex clusters with the gain and loss effects properly balanced, as shown in Figs. 9(c), 10(c), 11(c), and 12(c). The number of the produced vortex solitons in the corresponding soliton clusters is equal to the topological charge M of the input beam, as shown in Figs. 9(c), 10(c), 11(c), and 12(c), for M = 1, 2, 3, and 4, respectively. The vortex clusters can be robust upon propagation up to z = 36. We see from Figs. 9(a), 10(a), 11(a), and 12(a) that the regions where stable vortex clusters can be generated become slightly larger with the growth of the parameter $T_{x}$ .

When the parameter β increases and reach to the region between curves 2 and 3 in Figs. 9(a), 10(a), 11(a), and 12(a) we have revealed the following propagation scenarios. For M = 1 the input optical field splits and then the two splinters evolve into a rather irregular beam containing three distinct vortex solitons, see the final plot in Fig. 9(d). For M = 2, the input field is splitting into two fragments, which then coalesce during further evolution into a rather regular spot, see the final plot in Fig. 10(d). For M = 3, the input optical field splits into two fundamental vortex solitons that further coalesce into a rather irregular beam having a nested vortex soliton, see the last plot corresponding to z = 36 in Fig. 11(d), whereas for M = 4, the two corresponding splinters coalesce into a rather irregular beam, see the final plot, corresponding to z = 36 in Fig. 12(d).

Finally, when the coefficient β is much larger, belonging to the regions above curves 3 in Figs. 9(a), 10(a), 11(a), and 12(a), the input optical field is vanishing as shown in Figs. 9(e), 10(e), 11(e), and 12(e), due to a too weak energy gain in the system since the parameter β produces, in this case, a large viscous effect, i.e., a kind of a friction force.

4. Summary

In conclusion, we have revealed the occurrence of stable multivortex clusters in nonlinear optical media described by the two-dimensional cubic-quintic complex Ginzburg-Landau equation, by imposing an inhomogeneous effective diffusion term in the nonlinear propagation model. The numerical simulations show that, in this physical setting, an input two-dimensional Gaussian beam, with an embedded screw dislocation characterized by the integer winding number M, may evolve into robust vortex clusters nested in the optical beam. We have obtained the domains of stability of these vortex clusters that depend on the magnitudes of the amplitudes and the modulation periods of the inhomogeneous effective diffusion process in the nonlinear optical medium and on the governing parameters of the dissipative system. These results offer a method for generating stable multivortex beams in nonlinear optical media, which can be used in structured light techniques and applications.

Funding

National Natural Science Foundation of China (NSFC) (11174061, 61675001, 61575041, 11774068); Guangdong Province Nature Foundation of China (2017A030311025); Guangdong Province Education Department Foundation of China (2014KZDXM059).

Acknowledgments

The first two authors of this paper, Huishan Li and Shiquan Lai, equally contributed to this work.

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Stable dissipative optical vortex clusters by inhomogeneous effective diffusion

Abstract

1. Introduction

2. The model

3. Results and analysis

4. Summary

Funding

Acknowledgments

References and links

Cited By

Figures (12)

Equations (3)

Optics Express