1. Introduction

Recent interest in the study of nonlocal optical solitons is explained by a number of experimental observations of self-trapping effects and spatial solitons in different types of nonlocal nonlinear media [1]. It has already been shown that a diversity of nonlinear media such as atomic vapors [2], nematic liquid crystals [3], and optical media with thermal self-action such as lead glasses [4] can satisfactorily be modeled as nonlocal nonlinear media.

The study of solitons in nonlocal media is a rich field of research in nonlinear optics, and it has been already shown that the spatially nonlocal nonlinear response may bring interesting novel physics also introducing new effects such as strong modification of modulational instability [5], suppression of beam collapse [6], dramatic change of the soliton interaction [7, 8], formation of multi-soliton bound states [9], as well as stabilization of spatially localized vortex solitons [10] against symmetry breaking azimuthal instability [11, 12]. Many of the predicted and demonstrated properties of nonlocal nonlinear systems suggest that in nonlinear optical media we should expect stabilization of different multihump nonlinear structures such as ringlike clusters of many solitons [13] and modulated localized vortex beams or azimuthons [14].

Recently, we studied the simplest example of multihump solitons predicted in nonlinear optics, namely dipole-like structures composed of two interacting fundamental beams with opposite phases that undergo angular rotation during propagation [15]. We revealed that nonlocality can provide an effective physical mechanism for stabilizing these dipole solitons which are known to be unstable in all types of realistic nonlinear media with a local response.

Because the dipole soliton can be viewed as a special type of azimuthon [14], we wonder if other types of multihump solitons can be stabilized by nonlocal nonlinear response. In this paper, we study the higher-order azimuthons and demonstrate that spatial nonlocality allows to stabilize azimuthons against symmetry-breaking instability, as well as to increase their family with novel types of azimuthons which can only exist in nonlocal media. We find that stable azimuthons with N peaks may exist when the nonlocality parameter exceeds a certain threshold value and, in a sharp contrast to local media [14], their general condition for existence, N ≥ 2, becomes independent on the beam topological charge. The topological structure of azimuthons with N < 2m is presented by a circular array of N single-charge vortices with one additional dislocation at the origin with the charge given by the rule m-N, so that the total topological charge of azimuthon m is calculated as an algebraic sum of charges of all vortices.

2. Variational approach

We consider propagation of paraxial optical beams in a nonlinear medium described by the generalized nonlinear Schrödinger equation [1] for the scalar electric field envelope E,

where z and $\overrightarrow{r}$ = (x,y) stand for the propagation and transverse coordinates, respectively, and ∆_⊥ is the transverse Laplacian The kernel K of the nonlinear response is defined by a physical process responsible for the medium nonlinearity. Here we assume the Gaussian response

 

Fig. 1. (a) Profiles R(r) of single-charge azimuthons (m = 1) with n = 0 (vortex soliton), and with n = 0.5 for N = 2,3. Contours n = const for azimuthons with m = 1 for N = 2 (b) and N = 3 (c). Existence domains for azimuthons with topological charges m = 1 (d), 2 (e), and 3 (f), for different values of N. The domain for N = 2 is shaded as an example.

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where σ is the nonlocality parameter (when σ → 0, we recover the local Kerr model). We introduce the scaled variables, $\overrightarrow{r}$ = $\overrightarrow{r}$ ′σ, z = z′σ², and omit primes. Following Ref. [14], we look for stationary solutions in the rotating frame $\overrightarrow{r}$ = {r, θ = φ - ωz}, in the form E($\overrightarrow{r}$ ,z) = (√π/σ)V(r, θ)exp(ikz), and employ variational method [16] with the following ansatz [14],

where ω ₀,n, and R(r) are the variational parameters. Solving variational equation for ω ₀ we find ω ₀ = 2 ${\int }_0^{\infty }$ R ² r ^-1 dr/ (${\int }_0^{\infty }$ R ² rdr), and we define the normalized rotational velocity Ω = ω/ω ₀. The parameter n ∊ [0,1] is the contrast of azimuthal modulation. The structure of azimuthon is defined by two integer indices, the number of peaks N and topological charge m.

Applying the variational technique [16], we obtain the equation for the radial envelope,

where I_N is the modified Bessel function of the first kind of the order N. The parameter k is expressed as k = k ₀ + ωS, where S = mδ + Ω(1 - δ) and $\delta =2\sqrt{\frac{1-n}{\left(2-n\right)}}$. Soliton “spin” S is a ratio of the orbital angular momentum, M = Im ∫ V ^* V _θ d $\overrightarrow{r}$ , and the beam power, P = ∫ |V|² d $\overrightarrow{r}$ , i.e. S = M/P. Parameter g determines the radial profile of the azimuthon through the asymptotic relation R → r^g at r → 0. It is given by g ² = δm ² + (Ω² + N ²/4)(1 - δ), the latter expression indicates that larger values of the topological charge m, rotational velocity Ω, or the number of peaks N correspond to a larger radius of the ring. Finally, from Eq. (4) and a variational equation for n we obtain a condition that should be satisfied simultaneously with Eq. (4), and it defines the rotation frequency,

 

Fig. 2. Propagation of the azimuthons with m = 1, N = 3, n = 0.5 and different k. (a,b) Breakup for (a) k = 2.3 and ω = 1.2 and (b) k = 0.7 and ω = -0.25. (c) (2 Mb) Unstable azimuthon with ω = 40 that remains trapped by a strong nonlocal potential at k = 143.4. (d) (1.4 Mb) Stable azimuthon with ω = -9 and k = 91.7. Top row - variational solutions at z = 0, and bottom row - after propagation of (a,b) z = 8.9, (c) z = 7.4, and (d) z = 120.

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To obtain radial profiles R(r), we solve Eqs. (4) and (5) using a numerical shooting method within an iterative procedure. We find that there always exist two branches of solutions with ω _± = ω ₀Ω_±, these results are summarized in Fig. 1.

3. Existence conditions

In the limit n → 0, the ansatz (3) generates a radially symmetric vortex soliton with θ = 1, S = g = m, and k = k ₀ + mω. Thus, solving Eq. (4) for the vortex solitons with n = 0, from Eq. (5) we can obtain two cutoff branches ${\omega }_{\pm }^{c.o.}$ = ω ₀${\mathrm{\Omega }}_{\pm }^{c.o.}$ for different N. Cutoff frequencies determine the existence domains for azimuthon solutions, shown on the plane (ω,k) for azimuthons with topological charges m = 1,2,3 [see Figs. 1(d–f)].

In contrast to the local Kerr media [14], two cutoff frequencies do not always have opposite signs in nonlocal media, and we find solutions when the condition N < 2m holds; these azimuthons always rotate with a positive angular velocity. In particular, for even N > 2, e.g., N = 4 in Fig. 1(d–f), azimuthons with charges m = N/2 + 1 were found to exist with positive angular velocities only [see Fig. 1(f)], while for m = N/2 we obtain the azimuthons also with zero (or almost zero) angular velocity [see Fig. 1(e)], and for m = N/2 - 1 there exist azimuthons with either positive, zero, or negative angular velocity [see Fig. 1(d)].

Since the function F_N in Eq. (5) vanishes as the beam power (and k) grows, the azimuthon angular velocity approaches the value Ω_± = m ± N/2 for k → ∞. From this relation we find that, for N = 2m, one of the cutoff branches that bound the azimuthon existence domain always tends to zero, ω_^c.o. → 0. In the case of dipole solitons [15] with m = 1, the angular velocity does not vanish but becomes small. For larger number of peaks N, our method predicts the existence of stationary nonrotating azimuthons with four peaks and a double charge, six-peak azimuthons with a triple charge, etc. We note that the non-rotating azimuthons appear due to a balance between two contributions to nonzero angular momentum, the energy flow produced by a nontrivial singular phase, and the contribution due to the beam spiraling.

 

Fig. 3. Dynamics of (a) (1.8 Mb) unstable azimuthon with m = 1, N = 4, k = 0.5, n = 0.5, and ω = -0.5, and (b) (2 Mb) stable azimuthon with n = 0.5,k = 115.3, and ω = -17.3. (c) (2 Mb) Propagation of a double-charge azimuthon with n = 0.5, k = 200, and ω = 0.0025. (d) (2.2 Mb) Propagation of a triple-charge azimuthon with n = 0.5, k = 173.9 and ω = 9. Top row: variational solutions; bottom row: after propagation of (a) z = 8.1, (b) z = 110, and (c,d) z = 10.

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More importantly, we also find azimuthons that only exist in nonlocal media, as shown in Fig. 1. For example, azimuthons with m = 2 and two peaks do not exist in local Kerr media [14]; the case of a local medium corresponds to the limit k → 0 in our model. In contrast, here we find that the azimuthon with N = 2 and m = 2 is indeed possible for k > 16.6, see Fig. 1(e). For the azimuthon with the indices N = 3 and m = 2, we also find a region k > k _offset where it exists, this region is separated by a small gap k _offset from the local limit k = 0. Similar situation occurs for the case m = 3 [Fig. 1(f)]: as the number of peaks N grows, the gap k _offset decreases as the general existence condition N > 2 $\sqrt{F_N}$ should be satisfied at n → 0 in Eq. (5).

An important result we obtain is that the main condition for the azimuthon existence in nonlocal media becomes N ≥ 2, and it does not depend on the value of its charge m; this is in a sharp contrast to the results obtained for the cases of local Kerr and saturable media [14]. Thus, we find that nonlocality stabilizes the azimuthally modulated singular beams with almost any combination of the parameters (N,m). However, for a given m, the existence domain for azimuthons quickly shrinks with N → 2, as seen in Fig. 1.

4. Propagation dynamics and stability

To test the stability of azimuthons, we add up to 10% of noise to our stationary profiles and numerically propagate azimuthons using a pseudo-spectral split-step beam propagation algorithm. We observe two different scenarios of the modulational instability of azimuthons. When the power (and the value of k) is below the threshold, the azimuthons break up and split into the fundamental solitons which move away from each other; this is similar to what happens in local media [see Figs. 2(a,b) and 3(a)]. For larger degree of nonlocality the induced effective potential is strong enough to trap the beams even when the azimuthal instability develops. For example, the strong nonlocal potential can force the splinters generated by azimuthal instability to fuse into a single soliton [15]. We also observe that in some band of values of k, azimuthons can remain localized as rings for very long distances, even when the number of peaks along the ring is not recognizable already after a short propagation, see Fig. 2(c). Nevertheless, similar to the dipole solitons [15], we observe that azimuthons with different combinations of the parameters (N, m) become stable for large enough k, see Figs. 2(d) and 3(b).

We also study numerically the dynamics of the novel states, specific to nonlocal media, such as shown in Fig. 3 azimuthons with N = 4, m = 2 (c) and m = 3 (d). Most interesting higher-order azimuthons are presented in Fig. 4. In (a) we show the rotating beam generated by the variational solution with N = 2 and m = 3. While initially it carries a single triple-charged vortex, the dislocation quickly splits to three elementary vortices, similar to the higher-charge dark vortex solitons in defocusing local media [10]. However, instead of simple repulsion, three spatially separated vortices form a bound state within an intensity ring with two peaks. The dynamics, visible in movie, shows that the ratios of spiraling of phase structure and surrounding modulated ring are different, so that the rotation is double-periodic.

 

Fig. 4. (a) (2.2 Mb) Propagation of the azimuthon with m = 3, N = 2, n = 0.5, k = 257.5, and ω = 19.5. Examples of higher-order azimuthons with m = 6 and n = 0.5, for the cases: (b) (2.3 Mb) N = 5, k = 645.7, ω = 24.9; (c) (2.2 Mb) N = 6, k = 624.3, ω = 21.3; and (d) (2.2 Mb) N = 7, k = 603, and ω = 17.8. Top: variational solutions; middle: after propagation of (a) z = 10 and (b-d) z = 2; bottom: corresponding phases.

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The interplay of two indices m and N leads to an interesting topological structures of azimuthon’s phase front for N < 2m. In Figs. 4(b–d) we show the dynamics of azimuthons with the charge m = 6 for N = 5, 6, and 7, correspondingly. We observe that an initial charge-six vortex evolves into topologically different states with different numbers of single-charge (“elementary”) vortices, depending on the symmetry order N of intensity ring modulation. We conclude that the phase topology follows two constrains: first is the superimposed rotational symmetry of the order N, surprisingly robust, and second is the total topological charge, conserved in all cases. Actually, the main configuration of the observed “vortex cluster” contain a ring of N single-charged vortices (thus building a charge of m ₁ = N), and an additional vortex of the charge m ₂ = m - N in the origin, keeping the total charge m ₁ + m ₂ = m. It is clearly seen in Figs. 4(b–d) that the central dislocation is of the order m ₂ = 1 (b), m ₂ = 0 (c), and m ₂ = -1 (d).

5. Conclusions

We have found that nonlocal nonlinear response of optical media allows for the existence of novel classes of stable azimuthally modulated singular optics beams-azimuthons, characterized by two integer indices N and m. Some of those azimuthons can only exist in nonlocal media, and the stabilization is achieved when the nonlocality parameter exceeds a threshold value.