Stable high-charge vortex dissipative solitons in azimuthally modulated waveguide arrays with localized gain

TL;DR

This work addresses the existence and stability of dissipative vortex solitons in Kerr media with azimuthally modulated ring-shaped waveguide arrays that include localized linear gain and nonlinear loss. Using a 2D nonlinear Schrödinger framework with a complex potential, stationary vortices are computed via Newton's method and analyzed with linear stability theory, revealing that the maximum achievable topological charge $m$ is constrained by the ring symmetry $n$. The results show that higher-$m$ vortices can be more stable and narrower, with stability windows suppressed for low-$m$ states, and that larger $n$ enables realization of high-charge vortices; unstable states tend to relax into stable, lower-power high-$m$ configurations. Overall, the study expands the family of dissipative vortex solitons in structured gain landscapes and suggests feasible pathways to realize stable high-symmetry vortex states in optical lattices.

Abstract

We study the existence and dynamical properties of vortex solitons in Kerr media supported by azimuthally modulated waveguide lattices with localized gain and nonlinear loss. In this dissipative system, we find that the accessible topological charge of vortex solitons is strongly determined by the number of waveguide channels, with higher-order charges requiring progressively larger arrays. Power curves of vortex solitons with different charges exhibit clear separation in large arrays but become less distinguishable in smaller ones. Furthermore, these robust vortex solitons can be excited with nearly vanishing power thresholds, and higher-charge vortices display enhanced propagation stability compared with lower-charge states. These findings expand the family of dissipative vortex solitons supported by waveguide lattices and provide a route to the realization of stable high-symmetry vortex states.

Figures (6)

• Figure 1: Representative distributions of ring-shaped waveguide arrays. Waveguide indices for (a) $n = 6$ , (b) $n = 10$, and (c) $n = 20$. The red dashed line marks the ring at $r_c = 5.25$. The optical lattice and linear localized gain exhibit similar distributions. $x,y\in [-12,+12]$ in all panels.
• Figure 2: Typical field modulus $|\phi|$ and phase distributions (a-e), and real and imaginary parts $\phi_\text{r,i}$ (f-h) of vortex solitons. (a,f) $m=1$; (b) $m=3$; (c) $m=5$; (d) $m=7$; (e,h) $m=9$; (g) $m=4$. $n=20$, $p_i=2$ and $x,y\in [-12,+12]$ in all panels.
• Figure 3: Power $U$, propagation constant $\beta$ and effective width $w_\text{eff}$ of vortex solitons with different topological charges $m$ under varying linear localized gain levels $p_\text{im}$. Panels (a), (c), and (e) correspond to $n=10$, while (b), (d), and (f) display results for $n=20$. Arrows in panels (a-d) indicate the direction of increasing $m$. The insets in (a) and (c) show magnified views of the gray-shaded regions.
• Figure 4: Dependence of maximum instability growth rate $\lambda_{rm}$ on gain parameter $p_i$ for distinct topological charges. (a) $m=1\sim 5$, and (b) $m=6\sim 9$. Eigenvalue spectra of (c) stable ($p_i=2.0$) and (d) unstable ($p_i=4.5$) vortex solitons with topological charge $m=9$. In panel (d), the maximum $\lambda_r$ is marked by black solid markers and annotated with bidirectional arrows. The dashed black lines in (c) and (d) indicate $\lambda_{r}=0$. $n=20$ in all panels.
• Figure 5: Evolutions of amplitude $A$ (left panels) and topological charge $m$ (right panels) of dissipative vortices versus propagation distance $z$. Rows $1\sim2$ display unstable propagation dynamics, while rows $3\sim4$ exhibit stable evolutions. Parameter configurations: $n=20,m=1,p_i=2$ in (a1) and (a2); $n=20,m=5,p_i=2$ in (b1) and (b2); $n=20,m=9,p_i=2$ in (c1) and (c2); and $n=10,m=3,p_i=2$ in (d1) and (d2). Insets in right panels show phase distributions at $z=0$ and $z=1000$.