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vortices with *m* = 3. Since the peaks of lobes grow with the propagation constant, the net force becomes stronger and the lobes cannot be trapped on the waveguide sites, which leads to the shrinkage of vortices with *m* = 1 and expansion of vortices with *m* = 3. The lobes of vortices with *m* = 2 do not change their positions with the variation of *b*, just similar to the vortices with *m* = 1 in waveguide array with *n* = 4 [Fig. 10]. Thus, we draw a conclusion that the expansion or shrinkage of vortices is solely determined by the radio between the

To comprehensively understand the stability characteristics of vortex solitons, we conduct linear stability analysis on the stationary solutions with different *m* in circular waveguide array with different *n* for varying nonlinearity modulation depth *δ*. Representative stability and instability domains on the (*δ*, *b*) plane are illustrated in Fig. 13. At larger *b* where *dU*/*db* < 0 (*b* > *bin*), vortices are expected to be linearly unstable according to the Vakhitov-Kolokolov (V-K) criterion [44]. This is in good agreement to the numerical analysis results. However, other types of instability (e.g. oscillatory instability with complex growth rates) may arise

For vortex solitons with *m* = 1, *n* = 4 and *m* = 3, *n* = 8, the stability domains shrink with the growth of nonlinearity modulation depth. Note again vortex solitons with *m* = 1, *n* = 8 are completely unstable. Comparing Fig. 13(a) with Fig. 13(b), one can immediately find that the stability area of vortices with *m* = 3, *n* = 8 is broader than that of *m* = 1, *n* = 4. Thus, increasing waveguide number can significantly suppress the instability of vortex solitons with higher charges. There is an instability band near *blow* for vortex solitons with *m* = 2, *n* = 8 [Fig. 12(d)]. We stress that vortex solitons with higher topological charges are stable in a wide parameter window which is hardly realized in other settings except in Bessel-like lattice

**Figure 15.** Propagation simulations of stable (b-d, f) and unstable (a, e) vortex solitons. (a) Difference between a vortex soliton at *z* = 120 and *z* = 0. *m* = 1, *b* = 4.5, *n* = 4. Propagation results of vortex solitons at *m* = 2, *b* = 3.6, *n* = 8 (b), *m* = 3, *b* = 5.16, *n* = 8 (c), *m* = 3, *b* = 2.1, *n* = 12, *δ* = 0.5, *z* = 69 (e), and *m* = 5, *b* = 5, *n* = 12 (f). (d) Phase structure of (c). *δ* = 0.7 except for (e) and *z* = 512 except for (a, e).

topological charge and the number of waveguides.

when *b* < *bin*.

modulated media [25, 26].

In all cases *p* = 7.

Liangwei Dong *Institute of Information Optics, Zhejiang Normal University, Jinhua, 321004, China*

Huijun Li *Department of Physics, Zhejiang Normal University, Jinhua, 321004, China*
