Instability suppression of vector vortex solitons in nonlocal nonlinear media

We studied the instability suppression of vector vortex solitons (VVS), comprised of two incoherently coupled vortices with different topological charges $m$ in thermal nonlocal nonlinear media with cylindrical symmetry. Using linear stability analysis, we found that the azimuthal instability of the higher-order ($\left|m\right|\ge 3$) vortex can be suppressed and even eliminated because of the presence of the other lower-order ($\left|m\right|\le 2$) vortex, including the fundamental beam. The VVS will be stable when the power ratio of the lower-order vortex exceeds a threshold. We also found that stable VVS exists in some combinational states with opposite-sign charges ($m_1=-1$, $m_2\ge 3$) when the two vortex components have equal Gaussian beam widths.

Figure 1

The Gaussian beam width ratio $\sigma$ versus the power ratio $\eta$ for VVS. (a) ($-1$, 5) state, (b) ($-1$, 6) state. Here $R=40$ and $b_2=10$ in all cases. The blue solid point denotes the solution plotted in Fig. 2 with $b_1=11.82$, $b_2=10$, $\eta =0.30$, and $\sigma =1.06$.

Figure 2

Solutions of VVS with fixed power ratio $\eta =0.30$ and beam width ratio $\sigma =1.06$ in ($-1$, 5) state, labeled as a solid point in Fig. 1. (a) Profiles of VVS before ($b_1=11.82$ and $b_2=10$, denoted as blue points in other three figures) and after ($b_1^{\prime }=4.89$ and $b_2^{\prime }=4$, denoted as green points) the scale transformations, where the factor $\nu =0.7$. (b)–(c) Beam power and width versus propagation constant $b_2$. (d) Beam width versus $b_1-b_2$.

Figure 3

The ${\delta }_r$ of different $k$ versus $\eta$ for different combinational states. The vertical dashdot lines denote the equal beam width points. The solid points labeled as I, II, and III in (2, 5) state and IV, V, and VI in (1, 6) state correspond to those in Figs. 5 and 6, respectively.

Figure 4

The ${\delta }_r$ of different $k$ versus $\eta$ for (0, $m_2$) states. The stable power ratio regions exist in the range $\eta \in$ [0.08, 0.35], [0.15, 0.28], [0.17, 0.22], and [0.51, 0.69] for (0, 3), (0, 4), (0, 5), and (0, 6) states, respectively. The horizontal dashed line indicates the MIGR of the scalar higher-order vortex.

Figure 5

Evolutions of the perturbed vector vortex beams with $m_1=2$, $m_2=5$. (a, b) $\eta =0.008$, $\sigma =1.11$, $M_1=44\%$, $M_2=50\%$; (c, d) $\eta =0.55$, $\sigma =1.04$, $M_1=16\%$, $M_2=30\%$; (e, f) $\eta =19.0$, $\sigma =0.95$, $M_1=1\%$, $M_2=1\%$.

Figure 6

Evolutions of the perturbed vector vortex beams with $m_1=1$, $m_2=6$. (a, b) $\eta =0.007$, $\sigma =1.25$, $M_1=18\%$, $M_2=50\%$; (c, d) $\eta =0.48$, $\sigma =1.00$, $M_1=1\%$, $M_2=1\%$; (e, f) $\eta =100$, $\sigma =0.75$, $M_1=8\%$, $M_2=50\%$.