1. Introduction

In local isotropic self-focusing media the spatial optical solitons [1] are, in general, unstable. If the collapse instability is removed by, e.g., saturation of nonlinearity, the fundamental bell-shaped solitons become stable while the higher-order solitons still break up because of modulational instability or phase-sensitive interactions [2]. The nonlocality of media response, such as in liquid crystals [3] and lead glases [4], allows to suppress collapse [5] as well as to stabilize vortex solitons [6–8], mulipoles [9], and azimuthons [10–13].

Nonlocal media supports great variety of spatial solitons and affects dramatically their stability. Most intriguing are periodic solitons [14], including multi-modal beams [15–17] and two-dimensional solitons transforming their symmetry during propagation [10, 18–20]. Quasi-periodic dynamics can be triggered by the internal modes of solitons when two different states, such as necklaces and matrices of solitons [20], can couple without violating conservation of power and Hamiltonian. In thermal media the stability of solitons is strongly affected by the boundaries [8, 9, 21, 22] and the transformation dynamics can be effectively controlled, e.g., by changing the aspect ratio of a rectangular sample [23].

The process of the dynamic reshaping of nonlocal solitons [20] resembles astigmatic transformations in mode converters. For linear Hermite-Laguerre-Gaussian beams [24] it can be described with the help of a single parameter, 0≤α≤π/4, transforming between Hermite (α=0) and Laguerre (α=π/4) profiles. The transformations of solitons is essentially more complex because the conservation of orbital angular momentum (OAM) leads to their spiralling in nonlinear media [25] and restricts conversion to the states with the same OAM. As a result, the lowest-order dipole [10] and tripole [25] spiralling solitons do not transform.

In this paper we analyze robust long-lived quasi-periodic transformations of nonlocal solitons, including transformations accompanied by spiralling. First we show that the field oscillations of solitons with zero OAM [20] exhibit equidistant spectrum characteristic for discrete breathers [26]. Second representative example is the 2×3 soliton matrix whose family includes the pure Hermite beam and the double-ring single-charge vortex soliton with non-zero OAM.

 

Fig. 1. Top row: intensities of generalized quadrupole GN₁₁ for several values of α. (a) Power vs. propagation constant and (b) Hamiltonian vs. power for three different families GN_mn with indices (m,n) indicated next to the curves; for each family five curves are shown with α=π (0,1,2,3,4)/16. (c) Stability domain of the generalized quadrupole.

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The oscillation spectra of such spiralling soliton include discrete set of spatial frequencies produced by algebraic sum of soliton frequency, rotation rate, and transformation frequency.

We consider the system described by the nonlinear Schrödinger (NLS) equation [1]

where the medium response function is assumed to be Gaussian, R(r)=exp(-r ²). To find approximate soliton profiles we use variational principle with the generic ansatz, E(x,y, z)=U(x,y)exp(ikz), introducing soliton propagation constant k. Corresponding Lagrangian, 𝓛=-kP-𝓗, is defined through intergrals of motion, the power P=∫|U|² d r, and Hamiltonian

An important for the following integral of motion is the OAM, M=Im∫U*(x∂_yU-y∂_xU)d r.

We construct different families of generalized nonlocal solitons, U=GN_mn, using linear optical generalized Hermite-Laguerre-Gaussian modes HLG_mn(x,y;α) [24]

where H_j are Hermite polynomials and the coefficients C^j_mn(α) are expressed in terms of Jacobi polynomials P ^(µ,ν) _j as follows:C^j_mn(α)=i^j cos^n-jα sin^m-jαP_j ^(n-j,m-j) (-cos2α). Variational parameters are the amplitude A and soliton widths a and b.

2. Quasi-periodic transformations: quadrupole soliton

Quadrupole-type family of solutions GN₁₁ can be described by the following ansatz:

and the variational solutions are derived numerically in the parameter domain (k,α), see Fig. 1. The family includes pure quadrupole beam for α=0, or 2×2 matrix of out-of-phase solitons, as well as the radial mode of a fundamental soliton for α=π/4. Importantly, the OAM is zero for the whole family. For comparison, the families with nonzero OAM of the tripole soliton GN₂₀ [25] and the 2×3 soliton matrix GN21 are also shown in the diagrams in Figs. 1(a)–1(b).

 

Fig. 2. Quasi-periodic breather-like dynamics of the GN₁₁ soliton with P=200 and α=π/8 is shown at small (a, Media 1) and large (b, Media 2) propagation distances. At one spatial location in the transverse plane we show (c) the intensity and (d) the spectrum of oscillations of the field. (e) Transformation frequency ν_t vs. power P for α=π (1,2,3)/16.

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In direct simulations of the beam propagation we use the split-step FFT algorithm and the variational solutions Eq. (4) as input profiles. Varying α and P we investigate the stability properties of the quadrupole beams, the results are summarized in Fig. 1(c). The scaling of our model is such that the degree of nonlocality increases with power P [11, 20], so that at low power, closer to the local limit, we observe the solitons breaking up. At higher power this scenario changes dramatically and the beams evolve in distinctive ways depending on α.

An example of quasi-periodic behavior of the GN₁₁(x,y;π/8) soliton is shown in Fig. 2. The rows of intensity snapshots in Fig. 2(a, Media 1) and 2(b, Media 2) are taken from the same simulation at different propagation distances. In the case shown (P=200) the soliton constant is k=88.6 so that soliton period π/k=0.036, thus the row Fig. 2(b) is taken at the distances of about 1800 soliton periods. Although the fine details of the soliton structures are slightly different between Fig. 2(a, Media 1) and (b, Media 2) because the oscillations are not strictly periodic, the revival of the same states is remarkable. The period of oscillations can be roughly estimated from Figs. 2(a)-2(b) to be T=2.84 and thus there were about 24 such oscillations.

For a better characterization of the transformation dynamics we record during propagation the sequences of the values of complex field at randomly chosen points in the transverse plane. An example of the intensity at one such point is shown in Fig. 2(c), small variations of the amplitude are visible during 25 oscillations while the period seems to be constant. The logarithm of the Fourier transform of the recorded complex field, plotted in Fig. 2(d), reveals the fine structure of oscillations. The main peak at the frequency ν=14.04 corresponds to the soliton propagation constant, k/2π. The side peaks are equidistant from the main peak with frequency comb defined by the quasiperiodic transformations, ν_t=0.357; note that this value is in good agreement with directly observed inverse period of oscillations, ν_t⋍1/T=0.352. The clear “ladder”-like structure of the spectrum in Fig. 2(d) resembles an equidistant spectra of breather solutions in discrete nonlinear systems [26].

 

Fig. 3. Family of the 2×3 soliton matrix GN₂₁; top row: intensities for P=200. (a) Normalized OAM, (b) stability diagram, and (c) the spiralling velocity vs. k.

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We examine the dependence of ν_t for the family of solitons, parameterized by α and P, by performing extensive numerical simulations of the beam propagation also applying initial random noise to confirm the robustness of results summarized in Fig. 2(e). We find that the transformation frequency ν_t increases monotonically with power, before saturating at high powers. In our model this observation points out that for larger scale of nonlocality the transformations are faster. In the limit of infinite nonlocality the linear Snyder-Mitchell model [27] approximates our system Eq. (1), thus beating (interference) of different linear modes of harmonic potential can be used to find related spatial frequencies [28].

3. Transformations and spiralling: 2×3 soliton matrix

We construct the family of the generalized 2×3 soliton matrix using the GN₂₁ mode,

Varying α from 0 to π/4 in Eq. (5) produces the family ranging from the 2×3 matrix to a double-ring single-charge vortex soliton, see Fig. 3. The OAM is largely defined by the modulation parameterα and practically does not change with power, as seen in Fig. 3(a). The stability diagram in Fig. 3(b) is qualitatively similar to the one for generalized quadrupole in Fig. 1(c), namely at low power the beams break up while for powers above a certain threshold (red dashed line in (b)) they transform and rotate with propagation. The rate of spiralling Ω [25] strongly changes with power in Fig. 3(c), in contrast to the OAM in Fig. 3(a), so that for larger scale of nonlocality the rotation is faster.

The simulations of Eq. (1) reveal complex dynamics of GN₂₁ soliton with three simultaneous processes on different scales: fast breathing oscillations, spiralling, and transformations (see Fig. 4 and Media 3). It is impossible to distinguish different frequencies from direct observations and we resort to the analysis of oscillation spectra. Because of rotation, at any given point the field oscillates strongly without smooth structure as in Fig. 2(c). Therefore, we record sequences at ten different points, E_n(z)=E(x_n,y_n, z), n=1,2..10, take the Fourier transform of each field, 𝓕_n=𝓕[E_n], and average them, F=0.1∑¹⁰ _n=1|𝓕_n|², see bottom frame of Fig. 4.

 

Fig. 4. Quasi-periodic transformations and spiralling of the GN₂₁(x, y;π/8) soliton with power P=300 (Media 3). Bottom: the averaged spectrum of oscillations with vertical lines showing composite spatial frequencies ν_ij.

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Analysis of the spectrum in Fig. 4 allows to draw several conclusions. The dominating frequency ν=19.2 is given by the soliton constant k=120.3. Other side peaks are produced by the linear combination of two distinct frequencies (indices i and j are integer),

namely the transformation frequency ν_t=0.607 (period T=1.647), and the spiralling rate ν_r=0.179 (period of rotation T_r=5.587 corresponds to angular velocity Ω=2π/T_r=1.125, close to variational parameter Ω=0.982). We indicate in Fig. 4 with vertical lines the frequencies νij obtained for −3≤(i, j)≤3. Remarkably, all major peaks in the spectrum belong to the ν_ij set while some combinations (i, j) are clearly missing. We conclude that, despite distortions of the profile during propagation, the major contributions to the complex dynamics of the GN₂₁ soliton can be distinguished as spiralling and mode transformations.

4. Conclusions

We demonstrated, using generalized nonlocal solitons GN₁₁ and GN₂₁ as representative examples, that nonlocality of the nonlinear medium response supports quasi-periodic transformations between different symmetries of self-trapped optical beams. Despite fast breathing oscillations, typical for perturbed solitons, the oscillation spectra contain discrete set of spatial frequencies of transformation and spatial rotation. The missing composite frequencies ν_{i j} in Fig. 4 indicate that only specific combinations (i, j) in Eq. (6) contribute to the final spectrum, reflecting specific (unknown) structure of the soliton. Note that transformations can not be described in linearized perturbation analysis because they are not small. On the other hand, the periods of rotation and transformation are at least one order of magnitude lower than soliton period. Therefore, the ratio ν_t/k can be used as a small adiabatic parameter to address the question of whether truly periodic two-dimensional solitons exist.