Stable multipole solitons in defocusing saturable media with an annular trapping potential

TL;DR

This paper addresses the stabilization of high-order multipole (necklace-shaped) solitons in a 2D defocusing saturable medium using an annular trapping potential. It models beam propagation with a defocusing saturable nonlinearity and analyzes stationary solutions and linear stability to map existence and stability across parameters, revealing broad stability domains for multipole states from dipole up to very high-N necklaces. Key findings include stable solitons up to $N=48$ (and effectively unlimited high N with larger ring radius), enhanced power relative to prior models, and robust rotation of multipole solitons under phase torque. The results suggest practical routes for manipulating complex light beams and enabling high-power, structured light applications in photonics, including rotation-based beam routing.

Abstract

We systematically investigate the existence, stability, and propagation dynamics of multipole-mode (necklace-shaped) solitons in the two-dimensional model of an optical medium with the defocusing saturable nonlinearity and an annular potential trough. Various families of stable multipole solitons trapped in the trough, from dipole, quadrupole, and octupole ones to multi-lobe complexes, are found. The existence domain remains invariant with the increase of the potential's depth. Solitons with a large number N of lobes are stable in a wide parameter region, up to N=48 and even farther. Actually, stable multipole solitons of an arbitrarily high order N can be found, provided that the trough's radius is big enough. The power of stable multipoles is essentially larger in comparison to previously studied models. It is demonstrated analytically and numerically that the application of a phase torque initiates stable rotation of the multipole complexes. Thus, we put forward an effective scheme for the stabilization of multipole solitons with an arbitrary high number of lobes, including rotating ones, which offers new possibilities for manipulating complex light beams.

Figures (8)

• Figure 1: (a) The annular potential, as defined by Eq. (\ref{['Eq2']}) (it is plotted here with factor $p$). (b) A typical spectrum of discrete eigenvalues $b$ of the linearized version of equation (\ref{['Eq3']}) with this potential. Values of $n$ denote the number of the eigenmode. (c, d) Absolute values of the wave function, $\left\vert \psi (x,y)\right\vert$, in the quadrupole and $16$-pole linear eigenstates with eigenvalues marked by two red dots in (b). The insets display the corresponding phase structures. The parameters of the annular potential are $r_{0}=8,d=2$, and $p=5$.
• Figure 2: (a) Integral power (\ref{['U']}) versus propagation constant $b$ for dipole and quadrupole solitons in the annular potential (\ref{['Eq2']}) with the saturation parameters $s=0.5$ and $1.5$. (b, c) Examples of stable low-power ($b=4.2$) and unstable high-power ($b=2.3$) dipole modes marked by blue dots in (a). (d, e) Examples of stable moderate-power ($b=3.8$) and unstable high-power ($b=3.563$) quadrupoles marked by red dots in (a). (f) The Hamiltonian of multipole modes versus $b$. Solid and dashed lines denote the stable and unstable segments. The parameters are $r_{0}=8,d=2$ and $p=5$. $s=0.5$ in (b, c) and $1.5$ in (d, e).
• Figure 3: (a) Power $U$ vs. propagation constant $b$ for dipole and quadrupole solitons, for $s=0.5$ and varying values of $p$. (b) The existence domain of the dipole solitons for $s=0.5$ and varying $p$, which agrees with relation (\ref{['s']}). (c) The existence domain of the dipole solitons for $p=5$ and varying values of $s$. (d) The peak value $|\psi |_{\text{max}}$ of the dipole and quadrupole solitons vs. $b$ at $s=0.5,p=5$. (e) The instability growth rate $\text{Re}(\lambda )$ vs. $b$ for the dipole solitons at $s=0.5$. (f) The same for the quadrupole solitons at $s=0.5,p=5$. Other parameters are $r_{0}=8,d=2$.
• Figure 4: (a,b) The shapes of 8-pole soliton at $b=2.5$ and 16-pole soliton at $b=2.7$. The corresponding points are marked by dots in panel (c), which displays integral power $U$ vs. propagation constant $b$ for the 8- and 16-pole solitons. (d) The instability growth rate $\text{Re}(\lambda )$ vs. $b$ for the 8-pole and 16-pole solitons. The parameters are $r_{0}=8,d=2,s=0.5,$ and $p=5$.
• Figure 5: (a, b) The field and phase shapes of the $26$-pole soliton at $b=2$. (c) The instability growth rate $\text{Re}(\lambda )$ vs. $b$ for the family of $26$-pole solitons. (d) The existence area (between the $b_{\text{low}}$ and $b_{\text{upp}}$ curves) and the stability region (shaded) vs. the pole number $N$. The parameters are $r_{0}=8,d=2,s=0.5$, and $p=5.$