**3. Higher charged vortex solitons in azimuthally modulated Bessel lattice**

#### **3.1. Introduction**

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photons decreases with the increase of rotational energy. The conclusion may be generalized

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).

all vortices in radial lattices with defects can be observed in experiments.

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

**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.

to vortex solitons with continuous intensity distributions in other models.

Besides the harmonic lattice, there is another important optical lattice with unique symmetry, the Bessel lattice, which can be created by nondiffracting Bessel beams with cylindrical symmetry. Kartashov *et al.* systematically investigated the dynamics of various types of solitons supported by the Bessel lattices, including multipole-mode solitons [46], ring-profile vortex solitons [17], spatiotemporal solitons [27] etc. Necklace [47], broken ring solitons [48] can also be trapped stably in different order Bessel lattices. For a review of the early works, see [23].

Interestingly, Bessel lattices with azimuthal modulation are also possible [25, 49]. Such lattices resemble highly nonlinear micro-structured fibres [50] and may be realized in experiment by several incoherent Bessel beams with different intensities and orders [49]. The complex lattices can also be created in photorefractive crystals by the phase-imprinting technique [50, 51]. The azimuthally modulated lattices exhibit several discrete guiding channels of linear refractive index. Stable soliton complexes and azimuthal switching in focusing cubic media with modulated Bessel lattices were reported in [49]. Neighboring components in soliton complex are out-of-phase. Ring-shaped and single-site solitons were observed in azimuthally modulated lattices [50, 51]. Especially, by using group-theory techniques, Kartashov *et al.* derived a general "charge/stability rule" for vortex solitons supported by the azimuthal Bessel lattice [25].

In Ref. [31], Desyathikov and his coworkers introduced a novel class of spatially localized self-trapped ringlike singular optical beams in focusing cubic and saturable media, the so-called "azimuthons". The amplitude of such states is a spatially localized ring modulated azimuthally, and the phase of the azimuthon is a staircase function of the polar angle. This concept provided an important missing link between the radially symmetric vortices and rotating soliton clusters [52]. Following this work, stable azimuthons in nonlocal nonlinear media were found when the nonlocality parameter exceeds a certain threshold value [53, 54].

However, stable azimuthons are only found in media with nonlocal responses. Azimuthons in local nonlinear media unavoidably experience azimuthal modulation instability upon propagation. In this section, we elucidate the existence and stability properties of azimuthons (vortex solitons with special amplitude distribution) supported by the azimuthally modulated Bessel lattices. It is the combination of nontrivial phase and lattice confinement who affords the existence of azimuthons. Thus, the azimuthons we obtained provide a missing link between the radially symmetric vortices and nonrotating soliton clusters although they break the radial symmetry due to the potential we used. Similar to the "azimuthons" stated in [31], the nonlinear localized modes we discussed can also be attributed to the two contributions induced by the internal energy flow and the modulated beam respectively. In sharp contrast to the cases in focusing cubic media [25], we reveal that the "stability rule" in defocusing cubic media is quite the reverse. The result is in good agreement with the conclusion given by [55] where the stability of discrete vortex solitons supported by hexagonal photonic lattices in focusing media is opposite to the stability in the defocusing one, though the discussions were limited to the single-charged and double-charged vortex solitons.

### **3.2. Theoretical model**

We consider beam propagation along the *z* axis in defocusing cubic media with an imprinted transverse refractive index modulation. The dynamics of the nonlinear modes supported by such a scheme can be described by Eq. (1)

Here, the parameter *p* describes the lattice depth. The profile of the modulated lattice is given by *R*(*x*, *y*) = *J*<sup>2</sup> *nJ* [(2*blin*)1/2*r*] cos2(*nφ*), where *nJ* denotes the order of the Bessel function, *φ* is the azimuthal angle, *n* stands for the azimuthal index and *blin* defines the transverse lattice scale. Typical transverse linear refractive index modulation induced by the first order Bessel lattice with azimuthal index *n* = 2 is shown in Fig. 6(a). The local lattice maxima situated closer to the lattice center are more pronounced than others. The number of guiding channels in the main ring is given by 2*n*.

**Figure 6.** (a) First-order Bessel lattice with azimuthal index *n* = 2. (b) Profile of the first linear modes with *n* = 2.

Experimentally, Eq. (1) can be realized by launching a modulated Bessel beam into a photorefractive crystal in the ordinary polarization direction and a soliton beam in the extraordinary polarization direction [46]. In the particular case of optical lattice induction in SBN crystal biased with dc electric field <sup>∼</sup> 105 V/m, for laser beams with 10*μ*m the propagation distance *z* ∼ 1 corresponds to 1mm of actual crystal length, while amplitude *<sup>q</sup>* <sup>∼</sup> 1 corresponds to peak intensity about 50 *mW*/*cm*<sup>2</sup> [25]. Note that Eq. (1) can also be treated as Gross-Pitaevskii equation for a 2D Bose-Einstein condensate with repulsive interatomic interactions trapped in an optical lattice created by an azimuthally modulated Bessel beam.

We search for stationary solutions of Eq. (1) in the form of *A*(*x*, *y*, *z*)=[*qr*(*x*, *y*) + *iqi*(*x*, *y*)] exp(*ibz*), where *qr* and *qi* are real and imaginary parts of the solution profiles and *b* is a nonlinear propagation constant. The twisted phase structure of the stationary solutions can be defined by *m* = arctan(*qi*/*qr*)/2*π*, where *m* is the so-called "topological charge" of vortex solitons. Substituting the expression into Eq. (1), we obtain:

$$\frac{1}{2} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) q\_{r,i} - bq\_{r,i} - (q\_r^2 + q\_i^2)q\_{r,i} + p\mathbb{R}q\_{r,i} = 0 \tag{4}$$

We fix *blin* ≡ 2 and vary *b*, *p* and *n* without loss of generality.

To elucidate the stability properties of solitons, we search for perturbed solutions of Eq. (1) in the form *A*(*x*, *y*, *z*)=[*qr* + *iqi* + (*ur* + *iui*) exp(*λz*)] exp(*ibz*), where *ur* and *ui* are the real and imaginary parts of the perturbations, respectively. Substituting the perturbed solution into Eq. (1) and linearizing *ur*,*<sup>i</sup>* around *qr*,*<sup>i</sup>* yield a system of coupled Schrödinger-type equations for perturbation components *ur*,*i*:

$$\pm \lambda u\_{i,r} = \frac{1}{2} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) u\_{r,i} - b u\_{r,i} + p \mathcal{R} u\_{r,i} - (3q\_{r,i}^2 + q\_{i,r}^2) u\_{r,i} - 2q\_r q\_l u\_{i,r} \tag{5}$$

where *ur*,*<sup>i</sup>* may grow with a complex rate *λ* during the propagation of solitons. The eigenfunctions *ur*,*<sup>i</sup>* and eigenvalues *λ* can be solved numerically. The solitons are stable only when all real parts of *λ* equal zero.

#### **3.3. Discussions**

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We consider beam propagation along the *z* axis in defocusing cubic media with an imprinted transverse refractive index modulation. The dynamics of the nonlinear modes supported by

Here, the parameter *p* describes the lattice depth. The profile of the modulated lattice is given

the azimuthal angle, *n* stands for the azimuthal index and *blin* defines the transverse lattice scale. Typical transverse linear refractive index modulation induced by the first order Bessel lattice with azimuthal index *n* = 2 is shown in Fig. 6(a). The local lattice maxima situated closer to the lattice center are more pronounced than others. The number of guiding channels

**Figure 6.** (a) First-order Bessel lattice with azimuthal index *n* = 2. (b) Profile of the first linear modes

Experimentally, Eq. (1) can be realized by launching a modulated Bessel beam into a photorefractive crystal in the ordinary polarization direction and a soliton beam in the extraordinary polarization direction [46]. In the particular case of optical lattice induction in SBN crystal biased with dc electric field <sup>∼</sup> 105 V/m, for laser beams with 10*μ*m the propagation distance *z* ∼ 1 corresponds to 1mm of actual crystal length, while amplitude *<sup>q</sup>* <sup>∼</sup> 1 corresponds to peak intensity about 50 *mW*/*cm*<sup>2</sup> [25]. Note that Eq. (1) can also be treated as Gross-Pitaevskii equation for a 2D Bose-Einstein condensate with repulsive interatomic interactions trapped in an optical lattice created by an azimuthally modulated

We search for stationary solutions of Eq. (1) in the form of *A*(*x*, *y*, *z*)=[*qr*(*x*, *y*) + *iqi*(*x*, *y*)] exp(*ibz*), where *qr* and *qi* are real and imaginary parts of the solution profiles and *b* is a nonlinear propagation constant. The twisted phase structure of the stationary solutions can be defined by *m* = arctan(*qi*/*qr*)/2*π*, where *m* is the so-called "topological charge" of

*qr*,*<sup>i</sup>* <sup>−</sup> *bqr*,*<sup>i</sup>* <sup>−</sup> (*q*<sup>2</sup>

To elucidate the stability properties of solitons, we search for perturbed solutions of Eq. (1) in the form *A*(*x*, *y*, *z*)=[*qr* + *iqi* + (*ur* + *iui*) exp(*λz*)] exp(*ibz*), where *ur* and *ui* are the real and

*<sup>r</sup>* + *<sup>q</sup>*<sup>2</sup>

*<sup>i</sup>* )*qr*,*<sup>i</sup>* + *pRqr*,*<sup>i</sup>* = 0 (4)

vortex solitons. Substituting the expression into Eq. (1), we obtain:

*∂*2 *∂y*<sup>2</sup> 

We fix *blin* ≡ 2 and vary *b*, *p* and *n* without loss of generality.

1 2  *∂*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

[(2*blin*)1/2*r*] cos2(*nφ*), where *nJ* denotes the order of the Bessel function, *φ* is

**3.2. Theoretical model**

*nJ*

in the main ring is given by 2*n*.

by *R*(*x*, *y*) = *J*<sup>2</sup>

with *n* = 2.

Bessel beam.

such a scheme can be described by Eq. (1)

Before we discuss the dynamics of vortex solitons (azimuthons), it is important to understand the origin of such nonlinear modes. After removing the nonlinear term in Eq. (4), the linear equation has infinite eigenvalues and the corresponding linear eigen-modes. Nonlinear modes bifurcate from these linear modes while the nonlinearity cannot be ignored. Fundamental solitons always bifurcate from the first linear modes and higher order solitons associate with the other linear modes. Corresponding to the azimuthal Bessel lattice shown in Fig. 6(a), we plot the first eigen-mode of the linearized equation of Eq. (4) in Fig. 6(b). The profile of linear mode possesses several amplitude peaks covering on a constitutive ring-like substrate, thus it looks like azimuthons. Such linear modes intuitively indicate the possible profiles of nonlinear modes in the nonlinear system.

Mathematically, the refractive index modulation contributed by the modulated Bessel lattice increases linearly with the growth of lattice depth. However, this relationship cannot hold for practical crystal when the lattice is modulated very deep. Thus, the practical realization of stable vortex solitons with higher topological charges becomes infeasible by solely increasing the lattice depth of the first-order lattice to a very large value. Fortunately, the higher-order Bessel lattice can suppress the azimuthal instability of vortex solitons effectively [26]. To study the properties of vortex solitons with higher charges, one must consider the higher order modulated Bessel lattices.

The following discussion will focus on azimuthons (vortex solitons) carrying different topological charges supported by azimuthally modulated Bessel lattices imprinted in defocusing cubic media. For the convenience of comparing with the results of Ref. [25], we assume *R*(*x*, *y*) = *J*<sup>2</sup> *<sup>n</sup>*[(2*blin*)1/2*r*] cos2(*nφ*), where the order of lattice equals to the azimuthal index. We also search for stationary solutions of azimuthons by a relaxation method. A Gauss beam multiplying a phase dislocation with charge *m* was selected as an initial iterative guess solution. Figure 7 displays some instances of azimuthons supported by the azimuthally modulated lattices with *n* = 4 and 6. The azimuthons exhibit spatially modulated patterns which are in contrast to the vortices in unmodulated Bessel lattices [17], where the vortices are ring-shaped. The number of amplitude peaks is determined by the azimuthal index *n*. Like the vortices in focusing media [25], azimuthons with similar amplitude distributions allow different topological charges. In the fourth-order Bessel lattices with azimuthal index *n* = 4, azimuthons can be found only for *m* = 1, 2 and 3. In sufficiently deep lattices with fixed *b* and *p*, the discreteness of azimuthons increases with the growth of the topological charge *m*, while the "radii" of the azimuthons are almost the same. Such properties are similar to the vortex solitons in focusing media [25]. For fixed *p* and *n*, the azimuthons will expand to the outer

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lattice rings at small *b* and shrink to the main guiding lattice ring at larger *b*. The local minima of azimuthons around the lattice ring approaches to zero when *b* → *bco*.

We also find azimuthons supported by lattices with different azimuthal indices. Numerical study reveals that azimuthon solutions can be found only when the relation 0 < *m* < *n* is satisfied. The relation also holds for the vortex solitons in focusing cubic media [25]. The phase difference between the neighboring components is *mπ*/*n*, which differs from the vortex solitons in harmonic lattices [56] or necklace solitons in Bessel lattices [47].

The properties of azimuthons supported by azimuthally modulated Bessel lattice are summarized in Fig. 8. The power of azimuthons is a descending curve due to the defocusing nonlinearity [Fig. 8(a)]. Azimuthon solutions cannot be found when the propagation constant exceeds a certain value which corresponds to an eigen-value of the linearized equation of Eq. (4). The upper propagation constant cutoffs of vortex solutions with *m* = 1 and 3 are displayed in Fig. 8(b) and 8(c). The existence areas expand with the growth of lattice depth for fixed topological charge and shrink with the growth of topological charges for fixed lattice depth. There is a lower threshold lattice depth for supporting azimuthons. Comparing the points of *b* → 0 in Fig. 8(b) and 8(c), we find that the threshold lattice depth grows with the increase of topological charge *m*.

To comprehensively understand the stability properties of azimuthons supported by lattices with different depths and azimuthal indices, we performed the linear stability analysis on azimuthons in lattices with order (azimuthal index) *n* up to 10 and lattice depths *p* ≤ 80. We numerically derived an important "stability rule" for azimuthons in the azimuthally modulated Bessel lattice imprinted in defocusing media. That is, azimuthons might be stable only when the topological charge satisfies the condition:

$$0 < m \le n/2\tag{6}$$

**Figure 7.** Amplitude distributions of azimuthons with *m* = 1 (a), 2 (b) and 3 (c, d). Parameters *n* = 4, *p* = 30, *b* = 0.5 in (a-c) and *n* = 6, *p* = 45, *b* = 0.5 in (d). Bottom row: the corresponding phase structures.


**Table 1.** The stability status of azimuthons for different lattice orders.

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lattice rings at small *b* and shrink to the main guiding lattice ring at larger *b*. The local minima

We also find azimuthons supported by lattices with different azimuthal indices. Numerical study reveals that azimuthon solutions can be found only when the relation 0 < *m* < *n* is satisfied. The relation also holds for the vortex solitons in focusing cubic media [25]. The phase difference between the neighboring components is *mπ*/*n*, which differs from the vortex

The properties of azimuthons supported by azimuthally modulated Bessel lattice are summarized in Fig. 8. The power of azimuthons is a descending curve due to the defocusing nonlinearity [Fig. 8(a)]. Azimuthon solutions cannot be found when the propagation constant exceeds a certain value which corresponds to an eigen-value of the linearized equation of Eq. (4). The upper propagation constant cutoffs of vortex solutions with *m* = 1 and 3 are displayed in Fig. 8(b) and 8(c). The existence areas expand with the growth of lattice depth for fixed topological charge and shrink with the growth of topological charges for fixed lattice depth. There is a lower threshold lattice depth for supporting azimuthons. Comparing the points of *b* → 0 in Fig. 8(b) and 8(c), we find that the threshold lattice depth grows with the

To comprehensively understand the stability properties of azimuthons supported by lattices with different depths and azimuthal indices, we performed the linear stability analysis on azimuthons in lattices with order (azimuthal index) *n* up to 10 and lattice depths *p* ≤ 80. We numerically derived an important "stability rule" for azimuthons in the azimuthally modulated Bessel lattice imprinted in defocusing media. That is, azimuthons might be stable

**Figure 7.** Amplitude distributions of azimuthons with *m* = 1 (a), 2 (b) and 3 (c, d). Parameters *n* = 4, *p* = 30, *b* = 0.5 in (a-c) and *n* = 6, *p* = 45, *b* = 0.5 in (d). Bottom row: the corresponding phase

0 < *m* ≤ *n*/2 (6)

of azimuthons around the lattice ring approaches to zero when *b* → *bco*.

solitons in harmonic lattices [56] or necklace solitons in Bessel lattices [47].

increase of topological charge *m*.

structures.

only when the topological charge satisfies the condition:

where *n* > 2. Azimuthons with *m* = *n*/2 for even *n* may be stable or unstable depending on the lattice parameters. There exists a narrow instability area near *b* → 0 when the lattice is modulated shallow (near its lower threshold value). For deeper lattices, completely stable azimuthons are possible. A summary of "stability rule" is presented in Table 1. The table shows the stability status of azimuthons for different lattice orders. It is exactly opposite to the Table I in [25], which was derived by the group-theory and is valid in the focusing cubic media. This finding also verifies the very recent reports [55, 57] in which the stability of discrete vortex solitons supported by hexagonal photonic lattices in focusing media is proved to be opposite to the stability in the defocusing ones. We note that our conclusion is more general since the above two studies are restricted to the single- and double-charge discrete vortex solitons.

Linear instability analysis results of some unstable azimuthons supported by the fourth-, fifth-, and sixth-order lattices are shown in Fig. 8(d)-8(f). Note the relation between the azimuthal index and topological charge does not satisfy the condition Eq. (6). The instability domain vanishes only when propagation constant approaches to the upper cutoff. Direct propagation simulation examples of stable and unstable propagation of azimuthons are presented in Fig. 9.

**Figure 8.** (a) Power of azimuthons with *m* = 1, 2 and 3 vs propagation constant, *n* = 4, *p* = 30. Propagation constant cutoff *bco* vs lattice depth *p* for azimuthons with *m* = 1 (b) and 3 (c). Real parts of instability growth rate *λ* vs propagation constant for azimuthons supported by the fourth- (d), fifth- (e), and sixth- (f) order lattices with *p* = 30, *m* = 3, *p* = 35, *m* = 3, *p* = 35, *m* = 5, respectively. Azimuthal index *n* = 4 in (a-d), 5 in (e) and 6 in (f).

The azimuthons aforementioned are restricted to the particular cases of *n* = *nJ*. In fact, azimuthon solutions can also be found when *n* � *nJ*. Contrary to the intuition and the cases in nonlocal media [53], the charge *m* of available azimuthon solutions is independent of the lattice order *nJ* but less than the azimuthal index *n*. The initial input guess solutions with *m* ≥ *n* may converge to the nonlinear modes of the following three different categories: 1. an azimuthon with charge *m*� < *n*; 2. a multipole mode or necklace soliton with neighboring components out-of phase; 3. a multipole mode or necklace soliton embedded into a global skew phase whose charge *m*�� = *m* − *n*. Thus, we conclude that azimuthon solutions can only be found for *m* < *n*. The reason may be attributed to the Kerr media with a local nonlinear response in our model [31].
