**2. Higher-charged vortex solitons in defected radial lattices**

### **2.1. Introduction**

Defects and defect states exist in a variety of linear and nonlinear systems, including solid state physics, photonic crystals, and Bose-Einstein condensates. When lights propagate in an optical lattice with a local defect, the band-gap guidance results in the formation of linear or nonlinear defect modes [32, 33]. Recently, defect guiding phenomena of light in diverse settings, such as photonic crystals [13], fabricated waveguide arrays [34, 35], and optically induced photonic lattices [36–42], have been predicted theoretically and observed experimentally. Ye and his coworkers proposed that stable nonlinear modes can be trapped in a lower-index defect sandwiched between two optical lattices, or in the cylindrical core of a radial lattice [43]. The variation of defect scales, depths and shapes can be used to stabilize and reshape the fundamental, dipole and vortex solitons [20].

In this section, we reveal that the defocusing media with an imprinted radially symmetric lattice with a lower-index defect covering several lattice rings can support stable vortex solitons with higher charges under appropriate conditions. In contrast to the cases in competing media, vortex solitons can propagate stably at lower or moderate energy flow. In lattices with fixed depth and defect scale, vortex solitons are completely stable provided that the propagation constant exceeds a critical value. In particular, we illustrate that the variation of topological charges slightly influences the existence and stability domains of vortex solitons. This is in contrast to all previous studies and allows the experimental realization of vortex solitons with higher charges without changing the parameters of an optical setting.

#### **2.2. Theoretical model**

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cutoffs of propagation constant. Although higher-charged vortices at higher energy flow suffer oscillatory instability, they can survive very long distances without visible distortions. Vortex solitons at lower or moderate energy flow are completely stable under appropriate conditions. The variation of topological charges slightly influences the existence and stability domains of vortex solitons. This property provides an effective way for the experimental realization of vortex solitons with higher charges in an optical setting with fixed parameters. In section 3, the existence, stability and propagation dynamics of vortex solitons in a defocusing Kerr medium with an imprinted azimuthally modulated Bessel lattice are investigated. Since the special amplitude distribution of vortex soliton resembles to the azimuthons stated in [31] and the phase distribution is also a staircase function of the polar angle, such vortex solitons can also be termed as "azimuthons". The azimuthal refractive index modulation admits stable vortex solitons with lower or higher topological charges. The "stability rule" of azimuthons in defocusing cubic media is exactly opposite to that of vortex solitons in focusing media with the same transverse refractive index modulation. It is the first

In section 4, the stability of vortex solitons supported by a circular waveguide array with out-of-phase modulation of linear and nonlinear refractive indices is studied. The out-of-phase competition between two effects substantially modifies the stability properties of vortex solitons. Vortex solitons undergo remarkable power-dependent shape transformations. They expand or shrink radially with the propagation constant, depending on the phase difference between the neighboring lobes. In particular, we revealed that increasing waveguide number of circular array can stabilize vortex solitons with higher topological

Defects and defect states exist in a variety of linear and nonlinear systems, including solid state physics, photonic crystals, and Bose-Einstein condensates. When lights propagate in an optical lattice with a local defect, the band-gap guidance results in the formation of linear or nonlinear defect modes [32, 33]. Recently, defect guiding phenomena of light in diverse settings, such as photonic crystals [13], fabricated waveguide arrays [34, 35], and optically induced photonic lattices [36–42], have been predicted theoretically and observed experimentally. Ye and his coworkers proposed that stable nonlinear modes can be trapped in a lower-index defect sandwiched between two optical lattices, or in the cylindrical core of a radial lattice [43]. The variation of defect scales, depths and shapes can be used to stabilize

In this section, we reveal that the defocusing media with an imprinted radially symmetric lattice with a lower-index defect covering several lattice rings can support stable vortex solitons with higher charges under appropriate conditions. In contrast to the cases in competing media, vortex solitons can propagate stably at lower or moderate energy flow. In lattices with fixed depth and defect scale, vortex solitons are completely stable provided that the propagation constant exceeds a critical value. In particular, we illustrate that the variation of topological charges slightly influences the existence and stability domains of vortex solitons. This is in contrast to all previous studies and allows the experimental realization of vortex solitons with higher charges without changing the parameters of an

**2. Higher-charged vortex solitons in defected radial lattices**

and reshape the fundamental, dipole and vortex solitons [20].

example of stable azimuthons in local nonlinear media.

charges.

**2.1. Introduction**

optical setting.

We consider light propagation along the *z* axis of a defocusing Kerr medium with an imprinted transverse modulation of the refractive index. Dynamics of the beam can be described by the nonlinear Schrödinger equation for the dimensionless complex field amplitude *A*:

$$i\frac{\partial A}{\partial z} = -\frac{1}{2} \left( \frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2} \right) + |A|^2 A - p\mathbb{R}(x, y)A. \tag{1}$$

Here, the longitudinal *z* and transverse *x*, *y* coordinates are scaled in the terms of diffraction length and beam width, respectively; *p* denotes the lattice depth; the refractive-index profile is given by *<sup>R</sup>*(*x*, *<sup>y</sup>*) = cos2(Ω*r*) for *<sup>r</sup>* <sup>≥</sup> (2*<sup>N</sup>* <sup>−</sup> <sup>1</sup>)*π*/(2Ω) and *<sup>R</sup>*(*x*, *<sup>y</sup>*) = 0 otherwise, where *r* = (*x*<sup>2</sup> + *y*2)1/2 is the radial distance, Ω is the frequency, and *N* = 1, 2... is the number of rings removed from the lattice and characterizes the defect scale. Thus, the transverse modulation of refractive index features a lower-index guiding core. By comparing the defocusing bulk media without external potentials, the radial lattices with defects can confine the beams in a local region. An example of such refractive-index landscapes is shown in Fig. 1.

**Figure 1.** Radial lattice with a defect.

Although there are defects in radial lattices, the wings of nonlinear modes still penetrate into the bulk of lattices. Thus, the existence of nonlinear modes strongly depends on the transverse lattices. Since the term 1/*rd*/*dr* in Laplacian can be neglected at *r* → ∞, the band-gap structure of a radially symmetric lattice is slightly different from that of 1D periodic lattice [43]. Thus, it is convenient to use the band-gap structure of 1D periodic lattice to approximately analyze the existence of solitons. Due to the fact that nonlinear modes in defocusing Kerr media can only be found in the finite gaps, we are interested in the solitons residing in the first finite gap. Equation (1) conserves several quantities, including the energy

flow *U* = <sup>∞</sup> −∞ <sup>∞</sup> <sup>−</sup><sup>∞</sup> <sup>|</sup>*A*(*x*, *<sup>y</sup>*)<sup>|</sup> <sup>2</sup>*dxdy* and the Hamiltonian *H* = <sup>1</sup> 2 <sup>∞</sup> −∞ <sup>∞</sup> −∞( *∂A ∂x* 2 + *∂A ∂y* 2 − 2*pR*|*A*| <sup>2</sup> <sup>+</sup> <sup>|</sup>*A*<sup>|</sup> <sup>4</sup>)*dxdy*:

We search for stationary solutions of Eq. (1) by assuming *A*(*x*, *y*, *z*) = *q*(*r*) exp(*ibz* + *imφ*), where *q* is a *r*-dependent real function depicting the profile of stationary solution, *b* is a propagation constant associating with the energy flow, and *m* is an integer known as the topological charge of vortex soliton. The nonlinear mode degenerates to a fundamental radially symmetric mode when *m* = 0. The substitution of the light field into Eq. (1) yields:

$$\frac{d^2q}{dr^2} + \frac{1}{r}\frac{dq}{dr} - \frac{m^2}{r^2}q - 2bq - 2q^3 + 2pRq = 0,\tag{2}$$

which can be solved numerically by means of a Newton iterative method. Mathematically, various families of stationary solutions are determined by the propagation constant *b*, lattice depth *p*, modulation frequency Ω and defect scale *N*. We vary *b*, *p*, *N* and fix Ω ≡ 2 in this section.

The stability of solitons can be analyzed by considering the perturbed solution in the form of: *A*(*x*, *y*, *z*)=[*q*(*r*) + *u*(*r*) exp(*λz* + *inφ*) + *v*∗(*r*) exp(*λ*∗*z* − *inφ*)] exp(*ibz* + *imφ*), here the perturbation components *u*, *v* could grow with a complex rate *λ* during propagation, and *n* is an integer representing the angle dependence of the perturbation and is termed as an azimuthal index. The substitution of the perturbed solution into Eq. (1) results in a system of eigenvalue equations:

$$i\lambda u = -\frac{1}{2}(\frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr} - \frac{(m+n)^2}{r^2})u + bu + (v+2u)q^2 - pRv$$

$$-i\lambda v = -\frac{1}{2}(\frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr} - \frac{(m-n)^2}{r^2})v + bv + (u+2v)q^2 - pRv. \tag{3}$$

The coupled equations can be solved by a finite-difference method. In Cartesian coordinates, the square of the above linearization operator is self-adjoint if the stationary solutions are angle independent (fundamental solitons). Thus, the discrete eigenvalue is either purely real or purely imaginary. The instability growth rates with purely real parts correspond to the Vakhitov-Kolokolov (V-K) instability [44]. When the stationary solutions are angle dependent (vortex solitons), the eigenvalues may have both real and imaginary parts associating with an oscillatory instability. Stationary solutions are completely stable provided that all real parts of eigenvalues equal zero.

#### **2.3. Discussions**

First, we consider vortex solitons with unit charge supported by the defocusing Kerr media with an imprinted radial lattice with a defect. Without loss of generality, we set the defect scale *N* = 10 in the following discussions. In contrast to the fundamental solitons, the energy flow *U* of vortices is always a monotonically decreasing function of propagation constant *b* [Fig. 2(a)]. We stress that the vortex solitons here are bright or ring-profile ones whose amplitudes decay to zero at infinity. Such vortices cannot be bifurcated from the dark vortices supported by the bulk defocusing media in the vanishing lattice. In other words, vortex soliton only exists when the lattice depth exceeds a critical value. For example, as shown in Fig. 2(b), the threshold value of lattice depth for the appearances of vortices with unit charge is *pth* ≈ 1.18, below which no vortex solutions can be found. Therefore, the vortex in the present lattice system is not a continuum of dark vortex in the vanishing lattice case, and it belongs to a different soliton family. For *p* ≤ 2.57, the existence domain expands with the lattice depth. It shrinks with the growth of lattice depth if *p* ∈ (2.57, 7.05], due to the restriction of the ascending lower edge of the first band gap. It is the restriction of the first gap which accounts for the hoofed existence domain [Fig. 2(b)].

Figures 2(c) and 2(d) display two typical profiles of vortex solitons with unit charge at different energy flow. The vortex at higher energy flow looks like a flat-topped beam embedded with a dark core at which the amplitude is zero and the phase is undefined. Vortices become more localized with the growth of propagation constant. Vortices at higher energy flow penetrate deeply into the bulk of lattice, which leads to the multi-ring structures of beam intensity distributions.

To examine the stability of vortex solitons with unit charge, we conduct linear-stability analysis on the stationary solutions according to Eqs. (3). Typical spectra of the linearization operator for vortex solitons at *b* = −0.8 and *b* = −0.1 in lattice with defect scale *N* = 10 at *p* = 4.2 are shown in Figs. 3(a) and 3(b), respectively. Vortices at higher energy flow suffer an oscillatory instability with complex growth rates [Re(*λ*) �Im(*λ*)], while vortices at lower or moderate energy flow are completely stable [Re(*λ*) = 0]. To confirm the stability analysis results, we numerically integrate Eq. (1) with a standard beam propagation method code, using the stationary solutions as the initial inputs. Representative unstable and stable propagation examples are illustrated in Figs. 3(c) and 3(d). Obviously, unstable vortex solitons can survive large distances (hundreds of diffraction lengths), greatly exceeding the present experimentally feasible sample lengths.

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which can be solved numerically by means of a Newton iterative method. Mathematically, various families of stationary solutions are determined by the propagation constant *b*, lattice depth *p*, modulation frequency Ω and defect scale *N*. We vary *b*, *p*, *N* and fix Ω ≡ 2 in this

The stability of solitons can be analyzed by considering the perturbed solution in the form of: *A*(*x*, *y*, *z*)=[*q*(*r*) + *u*(*r*) exp(*λz* + *inφ*) + *v*∗(*r*) exp(*λ*∗*z* − *inφ*)] exp(*ibz* + *imφ*), here the perturbation components *u*, *v* could grow with a complex rate *λ* during propagation, and *n* is an integer representing the angle dependence of the perturbation and is termed as an azimuthal index. The substitution of the perturbed solution into Eq. (1) results in a system of

*dr* <sup>−</sup> (*<sup>m</sup>* <sup>+</sup> *<sup>n</sup>*)<sup>2</sup>

*dr* <sup>−</sup> (*<sup>m</sup>* <sup>−</sup> *<sup>n</sup>*)<sup>2</sup>

The coupled equations can be solved by a finite-difference method. In Cartesian coordinates, the square of the above linearization operator is self-adjoint if the stationary solutions are angle independent (fundamental solitons). Thus, the discrete eigenvalue is either purely real or purely imaginary. The instability growth rates with purely real parts correspond to the Vakhitov-Kolokolov (V-K) instability [44]. When the stationary solutions are angle dependent (vortex solitons), the eigenvalues may have both real and imaginary parts associating with an oscillatory instability. Stationary solutions are completely stable provided that all real parts of

First, we consider vortex solitons with unit charge supported by the defocusing Kerr media with an imprinted radial lattice with a defect. Without loss of generality, we set the defect scale *N* = 10 in the following discussions. In contrast to the fundamental solitons, the energy flow *U* of vortices is always a monotonically decreasing function of propagation constant *b* [Fig. 2(a)]. We stress that the vortex solitons here are bright or ring-profile ones whose amplitudes decay to zero at infinity. Such vortices cannot be bifurcated from the dark vortices supported by the bulk defocusing media in the vanishing lattice. In other words, vortex soliton only exists when the lattice depth exceeds a critical value. For example, as shown in Fig. 2(b), the threshold value of lattice depth for the appearances of vortices with unit charge is *pth* ≈ 1.18, below which no vortex solutions can be found. Therefore, the vortex in the present lattice system is not a continuum of dark vortex in the vanishing lattice case, and it belongs to a different soliton family. For *p* ≤ 2.57, the existence domain expands with the lattice depth. It shrinks with the growth of lattice depth if *p* ∈ (2.57, 7.05], due to the restriction of the ascending lower edge of the first band gap. It is the restriction of the first gap which accounts

Figures 2(c) and 2(d) display two typical profiles of vortex solitons with unit charge at different energy flow. The vortex at higher energy flow looks like a flat-topped beam embedded with a dark core at which the amplitude is zero and the phase is undefined. Vortices become more localized with the growth of propagation constant. Vortices at higher energy flow penetrate deeply into the bulk of lattice, which leads to the multi-ring structures of beam intensity

*<sup>r</sup>*<sup>2</sup> )*<sup>u</sup>* <sup>+</sup> *bu* + (*<sup>v</sup>* <sup>+</sup> <sup>2</sup>*u*)*q*<sup>2</sup> <sup>−</sup> *pRu*

*<sup>r</sup>*<sup>2</sup> )*<sup>v</sup>* <sup>+</sup> *bv* + (*<sup>u</sup>* <sup>+</sup> <sup>2</sup>*v*)*q*<sup>2</sup> <sup>−</sup> *pRv*. (3)

section.

eigenvalue equations:

eigenvalues equal zero.

**2.3. Discussions**

distributions.

*<sup>i</sup>λ<sup>u</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup>

<sup>−</sup>*iλ<sup>v</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup>

for the hoofed existence domain [Fig. 2(b)].

2 ( *d*2 *dr*<sup>2</sup> <sup>+</sup>

2 ( *d*2 *dr*<sup>2</sup> <sup>+</sup> 1 *r d*

1 *r d* We summarize the linear-stability analysis results in Fig. 2(b). We show the critical value of propagation constant *bn*=<sup>1</sup> *cr* above which no perturbations with the azimuthal index *n* and nonzero real part of growth rate were found. Vortex solitons are dynamically stable in a broad region near the upper cutoffs of propagation constant. It is the combination of defocusing nonlinearity and confining potential who affords the stability of vortex solitons. The precise structure of instability regions (patched) is rather complicated. There may exist multiple narrow stability windows.

Now, we focus on the vortex solitons with higher topological charges in radially lattices with defects imprinted in a defocusing Kerr medium. Figure 4(a) shows the hoofed existence domain of vortex solitons with *m* = 3. Vortex solitons can be found in lattices with *p* ∈ [1.18, 6.97]. By comparing the existence domains of vortex solitons with *m* = 1 [Fig. 2(b)]

**Figure 2.** (a) Energy flow of vortex solitons with *m* = 1 vs *b* for different *p*. (b) Areas of existence and instability (patched) on the (*p*, *b*) plane. Solid lines denote the edges of the first gap of 1D periodic lattice. (c, d) Contour and amplitude profiles of vortices at *p* = 4.2, *b* = −0.8, *U* = 556.2 and *p* = 4.2, *b* = −0.1, *U* = 37.9, respectively. In all cases *N* = 10.

#### 6 Will-be-set-by-IN-TECH 372 Optical Communication

and *m* = 3, one can find that the upper cutoff of propagation constant drops from ∼ −0.04 to ∼ −0.08, which leads to the decrease of the upper threshold value of lattice depth *p* (from 7.05 to 6.97) and thus the shrinkage of the existence domain. Yet, the existence area of vortex solitons with *m* = 3 still occupies almost the whole of the first gap of undefected lattice (for *b* < 0).

The energy flow of vortex solitons with *m* = 3 also decreases with the propagation constant. The pronounced decaying oscillations of such modes in the bulk of lattice become stronger with the decrease of propagation constant [Fig. 4(b)]. The maxima of intensity distribution around the phase dislocation move towards the center of the defect core with the growth of propagation constant. Figure 4(c) displays an example of screw-type phase distribution of vortex soliton with *m* = 3.

The instability of vortex solitons with higher charges usually depends on the azimuthal index *n* [17, 26, 45]. Linear-stability analysis results reveal that for vortices with *m* = 3, the instability area associating with *n* = 2 is always dominant. For vortex solitons with *m* = 3 in a lattice with *p* = 5, the widths of instability windows associating with azimuthal indices *n* = 1, 2 and 3 occupy ≈ 26.77%, ≈ 38.69% and ≈ 16.93% of the width of the whole existence domain, respectively. An example of instability growth rate corresponding to azimuthal index *n* = 2 versus propagation constant is illustrated in Fig. 4(d). It indicates that vortex solitons suffer weak azimuthal instability, which allows them to propagate without obvious shape distortion over large propagation distances. Vortex solitons will be completely stable provided that the propagation constant exceeds a critical value.

By comparing the stability areas of vortex solitons with *m* = 1 [Fig. 2(b)] and *m* = 3 [Fig. 4(a)], one finds that the stability area of vortices with *m* = 3 is slightly narrower than that of

**Figure 3.** Spectra of the linearization operator (a, b) and unstable and stable propagations (c, d) of vortex solitons shown in Figs. 2(c) and 2(d).

vortices with *m* = 1, which constitutes one of our central results. That is to say, the stability area is slightly affected by the growth of topological charge, which allows one to realize stable vortex solitons with even higher charges. Since vortices with different charges share a collective stability area, one can input beams with different charges to excite vortex solitons with corresponding charges in certain parameter windows without changing the lattice depth, defect scale, modulation frequency, etc. It should be noted again that vortex solitons with different charges can propagate stably at lower or moderate energy flow, which is in sharp contrast to the cases in competing media, where very high energy flow is needed to stabilize the vortices [8, 10]. Thus, in addition to the Bessel lattice [17], the radial lattice with a defect is another effective alternative for the realization of stable vortex solitons at lower or moderate energy flow, especially for vortices with higher charges.

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and *m* = 3, one can find that the upper cutoff of propagation constant drops from ∼ −0.04 to ∼ −0.08, which leads to the decrease of the upper threshold value of lattice depth *p* (from 7.05 to 6.97) and thus the shrinkage of the existence domain. Yet, the existence area of vortex solitons with *m* = 3 still occupies almost the whole of the first gap of undefected lattice (for

The energy flow of vortex solitons with *m* = 3 also decreases with the propagation constant. The pronounced decaying oscillations of such modes in the bulk of lattice become stronger with the decrease of propagation constant [Fig. 4(b)]. The maxima of intensity distribution around the phase dislocation move towards the center of the defect core with the growth of propagation constant. Figure 4(c) displays an example of screw-type phase distribution of

The instability of vortex solitons with higher charges usually depends on the azimuthal index *n* [17, 26, 45]. Linear-stability analysis results reveal that for vortices with *m* = 3, the instability area associating with *n* = 2 is always dominant. For vortex solitons with *m* = 3 in a lattice with *p* = 5, the widths of instability windows associating with azimuthal indices *n* = 1, 2 and 3 occupy ≈ 26.77%, ≈ 38.69% and ≈ 16.93% of the width of the whole existence domain, respectively. An example of instability growth rate corresponding to azimuthal index *n* = 2 versus propagation constant is illustrated in Fig. 4(d). It indicates that vortex solitons suffer weak azimuthal instability, which allows them to propagate without obvious shape distortion over large propagation distances. Vortex solitons will be completely stable provided that the

By comparing the stability areas of vortex solitons with *m* = 1 [Fig. 2(b)] and *m* = 3 [Fig. 4(a)], one finds that the stability area of vortices with *m* = 3 is slightly narrower than that of

**Figure 3.** Spectra of the linearization operator (a, b) and unstable and stable propagations (c, d) of

*b* < 0).

vortex soliton with *m* = 3.

propagation constant exceeds a critical value.

vortex solitons shown in Figs. 2(c) and 2(d).

To confirm the above conclusions, we investigate the dynamics of vortices with *m* = 4, 5...10. The existence and stability domains shrink slowly with the topological charge due to the slow decrease of the upper cutoff of propagation constant. The energy flow decreases with the growth of topological charge when the lattice parameters are fixed [Fig. 5(a)]. This can be explained by the law of conservation of energy. For linearly polarized vortex beam, the total energy includes two parts. The first part is the energy carried by the photons, and the second part is the rotational energy associating with the orbital angular momentum. From the definition of the energy flow of vortex solitons, one finds that the expression of energy flow only defines the energy carried by the photons. Yet, the rotational energy of vortex solitons is proportion to the square of topological charge and effectively rotational radius. Given that the allowed energy of different modes in a fixed system is a constant, the energy flow carried by

**Figure 4.** (a) Areas of existence and instability (patched) of vortex solitons with *m* = 3 on the (*p*, *b*) plane. Solid lines denote the edges of the first gap of 1D periodic lattice. (b) Profiles of vortices at *p* = 4.2. (c) Phase distribution at *b* = −0.5, *p* = 4.2. (d) Real part of instability growth rate associating with *n* = 2 vs *b* at *p* = 5. In all cases *N* = 10.

photons decreases with the increase of rotational energy. The conclusion may be generalized to vortex solitons with continuous intensity distributions in other models.

The above discussions can also explain the decrease of the thickness of vortex solitons shown in Fig. 5(b). With the growth of topological charge, the decrease of effective mass of the beam is in companion with the increase of effectively rotational radius and angular velocity. For fixed propagation constant, the delocalization of vortex soliton weakens with the growth of topological charge. A representative propagation example of unstable vortex solitons is illustrated in Fig. 5(c). The vortex can propagate without visible shape distortion over hundreds of diffraction lengths. Figures 5(d) and 5(e) show two instances of stable propagations of vortex solitons at *b* = −0.4 with topological charges *m* = 6 and 9, respectively. The phase structure of vortex soliton at *z* = 1024 with *m* = 9 is displayed in Fig. 5(f).

Finally, we briefly discuss the influence of lattice parameters on the existence of vortex solitons. Vortex solutions cannot be found in radial lattices without defects. The existence domain expands with the growth of defect scale and approaches an ultimate at *N* = 4. It shrinks with the increase of modulation frequency Ω. The existence domain shrinks slowly with the topological charge if other parameters are fixed. No matter what topological charge or lattice parameters are, the stable area always occupies a region near the upper cutoffs of propagation constant. We stress that although the vortex solitons residing in the patched areas shown in Figs. 2(b) and 4(a) are unstable, they can survive large propagation distances. Unstable vortex solitons with higher charges exhibit a similar behavior. Thus, we expect that all vortices in radial lattices with defects can be observed in experiments.

**Figure 5.** (a) Energy flow of vortex solitons with different *m* vs *b*. (b) Profiles of vortex solitons with different *m* at *b* = −0.4. (c, d) Unstable and stable propagations of vortex solitons with *m* = 6 at *b* = −0.67 and *b* = −0.4, respectively. (e, f) Field modulus and phase distribution of vortex soliton with *m* = 9 at *b* = −0.4, *z* = 1024. In all panels, *p* = 5.
