1. Introduction

A link between fundamental optical spatial solitons [1] and doughnut-shaped vortices [2–5] is provided by the existence of dynamic bound states in the form of rotating soliton clusters [6] and azimuthally modulated vortex solitons, or azimuthons [7]. Since a nonlocal response was predicted to suppress azimuthal instability of vortex beams [8, 9], there is a growing interest towards azimuthons in various nonlocal models [10–16] and geometries [17–21]. Although the theoretical results are plentiful [10–23], the experimental realizations are scarce, being limited so far to lead glass [24,25] with a boundary-dependent thermo-optic nonlinearity, and rubidium vapor [26] with a nonlocality too weak to avoid vortex breakup.

Spatial solitons in nematic liquid crystals, or nematicons [27–29], provide exciting opportunities to test theoretical predictions on the stability of various nonlocal vortex solitons. While spiraling of two nematicons was demonstrated earlier [30–32], the dynamics of azimuthons carrying phase singularities and optical vortices remains unexplored. Particularly interesting among them are the most robust lowest-order single-charged vortices [8,9] and spiraling dipole azimuthons [10–16]. An important open question is whether radially symmetric or azimuthally modulated vortex nematicons can be supported and stabilized by a reorientational nonlocal nonlinearity.

Here we report on the generation of dipole azimuthons in nematic liquid crystals. We observe experimentally and describe theoretically, for the first time to our knowledge, the formation of dipole azimuthons with nontrivial charge-flipping of on-axis phase dislocations. In these topological reactions the central phase dislocation (vortex line) splits into three lines with alternating topological charges, resembling a pitchfork bifurcation. We argue that these transformations are due to self-induced astigmatic deformation of the vortex beam [22, 23], which do not rely on the external anisotropy stemming from the boundaries [17–21].

2. Experimental results

Our experiments are carried out in a planar cell filled with the nematic liquid crystal 6CHBT, with two parallel polycarbonate slides spaced by 110μm and uniformly rubbed at an angle π/4 with respect to the direction z of the input beam wavevector, see Figs. 1(a,b) and Refs. [33–35]. Glass plates at the input and output interfaces seal the cell to prevent lens-like effects and avoid light depolarization [see Fig. 1(b)]. A single-charged vortex beam is generated with a fork-type amplitude diffraction hologram using an extraordinarily polarized cw laser beam of wavelength λ = 800 nm and power P. The beam is coupled into the cell with a 10× microscope objective, resulting in an input ring of radius w ≈ 4μm at peak intensity, see Fig. 1(c). The beam dynamics in the medium is monitored by collecting both the light out-scattered through the top plate of the cell (top view, longitudinal dynamics) and the light transmitted through the output interface (output, transverse dynamics), using objectives and high-resolution CCD cameras.

 

Fig. 1 (a) Perspective and (b) top views of the unbiased planar cell; the ellipses indicate the oriented molecules. MO – microscope objectives; CCD – cameras. (c) Input vortex profile and (d–f) output intensity distributions. (d) Diffracted e-polarized single-charged (m = +1) vortex beam with P = 0.7 mW. (e,f) Vortex breakup into two filaments for P = 2.1 mW, tilted in the opposite direction with respect to the dark stripe, for input topological charges m = +1 in (e) and m = −1 in (f). Dashed lines in (d–f) indicate the edges of the cell.

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Figures 1(c–f) present experimental results for vortex beams propagating in the sample. At low input power, P < 0.9 mW, the vortex beam uniformly diffracts without any noticeable self-action, see Fig. 1(d). As the power is increased, 1 < P[mW]< 2.2, the beam experiences self-focusing and the output spot visibly reduces, see Figs. 1(e,f). In this regime, the initial radially symmetric vortex undergoes a drastic transformation: the vortex “doughnut” breaks up into two bell-shaped beams and the dark core becomes a tilted stripe. Such symmetry-breaking can be explained in the context of vortex astigmatic transformations [36, 37] due to the anisotropic structure of the planar cell and the anisotropic character of nematic liquid crystals. Specifically, as the intensity of a vortex beam grows, the (extraordinary) refractive index increases in the middle of the cell but remains constant along x at the cell boundaries where the molecules are anchored, see, e.g. Fig. 1(c) in Ref. [38]. The index distribution thus becomes cylindrical and the sample behaves as a cylindrical lens. In full agreement with Ref. [37], we observe different transverse tilts of the elongated beam with respect to the dark stripe and the cell boundaries, depending on whether the charge of the input vortex is positive as in Fig. 1(e), or negative as in Fig. 1(f).

A further increase of the input vortex power (P > 2.3 mW) leads to more pronounced narrowing of the output beam and to the formation of self-trapped dipole azimuthons [10], as demonstrated by the results in Fig. 2. First, we discuss the longitudinal dynamics (Figs. 2(a–c)) for three input powers. The whole beam appears tilted and bent owing to walk-off [33–35], but its transverse size varies only slightly during propagation (in comparison with the diffraction pattern in Fig. 1(d)) as diffraction is suppressed. Nevertheless, the variation in waist indicates the breathing of the self-trapped beam [28, 39, 40], and the excitation of oscillatory modes of a stable dipole soliton [5], particularly robust and long-lived in nonlocal dielectrics [8,9].

 

Fig. 2 Generation and propagation of a dipole azimuthon, with power-driven beam twist and breathing. (a–c) Evolution in the plane (x,z) of a beam with input charge m = +1 for three different input powers. The walk-off angle of about 4.5° is defined by the molecular alignment at the cell boundaries. (d–f) Transverse intensity patterns at the cell output for beams with m = +1; (g–i) as in (d–f) but for m = −1. (j) Beam HWHM versus input power P (see text).

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The vortex develops an astigmatic deformation in propagation, splitting into two bright spots as it travels in the first half of the sample (i.e. small z). The longitudinal dark line visible in Figs. 2(a–c) disappears when the two dipole lobes are aligned vertically (i.e. along y), by the output in Fig. 2(b) and well inside the cell in Fig. 2(c). The corresponding output transverse patterns in Figs. 2(d–f) display two well pronounced bright spots, clearly coupled together in a structure similar to that of simulated dipole azimuthons [10]. By varying the power excitation, the output dipole exhibits a strongly varying elliptic shape with tilt at different angles with respect to the boundaries.

The dipole tilt at the output indicates the spatial twist of the beam inside the finite-length sample. Such twist is in opposite directions for opposite input topological charges, see Figs. 2(g–i), at variance with the observed behavior of composite dipole solitons in a photorefractive medium [41–43] where the inherent anisotropy suppresses the rotation and pins the dipole in a preferred direction. Conversely, in our reorientational dielectric the direction of the twist is defined by the topological charge and the rate of twist depends on the excitation level, in excellent agreement with theoretical predictions [10] and previous experimental results [31,32].

Since we observe nematicon deformations in real time, for a better quantitative estimate of soliton parameters we time-average the output images recorded during data acquisition. Figure 2(j) presents our measurements of the output beam half-width at half-maximum (HWHM) w_x,y, i.e. the half-size of a rectangle enclosing the contour of the averaged transverse intensity profile at the half-peak. The graph shows that, by increasing the input power from low values up to P ∼ 2.5 mW, the beam width gradually decreases, indicating a transition from self-focusing to solitary regime. The latter corresponds to the interval 2.5 < P[mW]< 6.5 with the formation of stable azimuthons. In this domain the beam can get more elongated in the vertical (y) direction, w_y > w_x, as emphasized by the shaded regions in Fig. 2(j) and confirming the power-dependent twist. Higher powers, P > 6.8 mW, lead to destabilization of the dipole, in contrast with theoretical predictions [10]. At high excitations, in fact, the temporal dynamics is amplified while the beams develop multi-humped and irregular structures.

Next, we investigate the singular phase structure of dipole azimuthons by employing an interferometric technique: a tilted broad Gaussian beam at an angle with the dipole interferes with it at the output. The interferograms in Fig. 3 allow detecting phase dislocations by the characteristic presence of fork dislocations. Two opposite topological charges, m = ±1, are distinguished by the fork orientation, either down or up.

 

Fig. 3 (a–h) Acquired intensity profiles of nonlocal dipole azimuthons (left) and corresponding interferograms (right); the circles indicate the positions of vortices with charges +1 (green) and −1 (blue). (i) Topological charge m₀ of the on-axis phase singularity; the dashed vertical lines mark the input powers in (a–h).

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Figure 3 shows results for an input vortex with m = +1. At small powers, i.e. P = 2.3 mW in Fig. 3(a), the input phase dislocation is preserved, even if the intensity changes substantially, consistently with theoretical predictions [10, 11]. However, as the power is increased further, alterations in the interferograms are significant enough that the topological structure can no longer be clearly identified, as in Figs. 3(b,g). This regime corresponds to the flipping of a topological charge in Fig. 3(i), at P ∼ 2.45 mW and P ∼ 3.3 mW, respectively.

Excitations above the critical value corresponding to charge flipping result in stable output interferograms, see Fig. 3(c) for P = 2.6 mW. Remarkably, we observe a triplet of vortices, with the central (on-axis) vortex of charge m₀ = −1 and two satellite vortices with an opposite m_1,2 = +1, so that the total charge remains unchanged, m = m₀ + m₁ + m₂ = +1. By monitoring the output versus increasing power P [see Figs. 3(c–e)], we observe first the spatial separation of the triplet and then the attraction of vortices in Fig. 3(f), followed by flipping of the central vortex charge from m₀ = −1 to m₀ = +1 above P ∼ 3.3 mW, see Figs. 3(g–i). With a further increase of power the charge-flipping process repeats again at P ∼ 3.7 mW before a vortex triplet with m₀ = −1 remains in the field.

Noteworthy, similar experiments with input vortex beams carrying charge m = −1 lead to an opposite direction of the spatial twist and different profiles, see Fig. 2(g–i). Nevertheless, we reliably measure the same critical power values for charge-flipping as in Fig. 3(i) and observe similar topological reactions involving triplets of vortices. We can therefore conclude that the reported charge-flipping is robust and reproducible.

3. Numerical simulations and discussion

In order to better understand and interpret the surprising complexity of the observed topological structure of the field of nonlocal dipole azimuthons we resort to numerical simulations. Our scope is twofold: verifying the charge-flipping phenomenon and studying its origin while identifying the symmetry-breaking mechanism. In particular, we aim at confirming the hypothesis that neither the boundaries or the medium anisotropy are significant in triggering the topological reactions; hence, the nonlocal self-action of a vortex nematicon itself suffices. To this end, we consider a generic isotropic and unbounded medium governed by the nonlinear Schrödinger equation with a nonlocal nonlinearity [10,11],

We approximate the experimental conditions by numerically simulating the dynamics of a vortex soliton [8,9] with p = 1, astigmatically deformed at the input by stretch in one direction, y → y/1.2, see Fig. 4. This initial deformation leads to a corresponding elliptic distortion of the optically induced refractive index profile, similar to stressed optical fibers [46–48] supporting mode-transformations and vortex charge-flipping. However, in our case the trapping potential is nonlinear, i.e. self-induced, and it is nontrivial to aseess whether its initial deformation can propagate and sustain astigmatic mode transformations [36,49].

 

Fig. 4 Numerical dynamics of an initially stretched stable nonlocal vortex soliton. (a) Intensity profiles. (b) Volume half-intensity surface (green) and trajectories of vortices: red for m = +1 and blue for m = −1. (c,d) Magnified fragments with pitchfork topological reactions in (b), including their sequence at z ≃ 29.5. (e) Evolution of the zero-level contours ReE = 0 (red) and ImE = 0 (blue) at the super-critical pitchfork reaction in (c); the circles indicate vortex locations.

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At low powers the vortex breaks into two uncoupled filaments [8–10], while at large powers there is no charge-flipping because of the transition to the quasi-linear limit of high nonlocality [39]. Closer to the stability border, the dipole azimuthons show strong spatial oscillations in Fig. 4(a,b), similar to the experimental results in Fig. 2. Persistent oscillations indicate the excitation of internal vibrational modes of the dipole azimuthon, which lead to the appearance of additional vortex lines as shown in Fig. 4(b). Several vortices are visible and the central phase dislocation changes its vorticity several times. The two magnified plots in Fig. 4(c,d) confirm the unexpected character of the charge-flipping, with the sudden appearance of a vortex triplet. These topological reactions are reminiscent of super-critical [Fig. 4(c)] and sub-critical [Fig. 4(d) at z ≃ 31.5] pitchfork bifurcations in the theory of nonlinear dynamical systems.

The detailed mechanism of the super-critical pitchfork topological reaction in Fig. 4(c) can be followed in Fig. 4(e) by observing the zero-level contour lines of both real and imaginary parts of the field E = 0, whose crossings define positions and signs of the phase dislocations. As seen in Fig. 4(e) at z = 9.1, there is only a single crossing (vortex with charge +1) (see Fig. 3(a) for P = 2.3mW) but the ReE = 0 line (red) is noticeably S-shaped, so that at the next propagation step z = 9.2 there are already three crossings and the triplet of vortices, with the central vortex of opposite charge, two side vortices moving away with propagation, and the conserved total topological charge, see Fig. 3(c) for P =2.6 mW.

Similar topological reactions were previously observed in interference patterns of two Laguerre-Gaussian vortex beams [50]. In general, perturbations of a perfect pitchfork, as in Figs. 4(c,d) and in Fig. 1(a,b) of Ref. [50], lead to triplet splitting into a perturbed but well defined original vortex line and a vortex-antivortex pair, as in Fig. 1(c,d) of Ref. [50]. Such reaction was previously resported in experiments with photonic lattices [51]. The main difference with our experimental data in Fig. 4 is that the spatial separation between phase singularities at the bifurcation points in Figs. 4(b,g) is so small that cannot be resolved. As a result, we cannot trace the original vortex line of charge m₀ = +1 in Fig. 4(a) to one of the two side-vortices in Fig. 4(c). Otherwise stated, the experimentally observed topological reactions are indeed very close to the ideal pitchfork and in perfect agreement with numerics in Figs. 4(b,g).

It should be underlined that the analysis is further complicated by the fact that images acquired at the cell output (Fig. 3) cannot be directly compared with propagation dynamics at fixed power (Fig. 4). Nevertheless, since the rotation rate and breathing frequency depend on power in a continuous fashion, the sequence of output images in Fig. 3 can be qualitatively mapped onto the propagation dynamics with pitchfork reactions in Figs. 4(c,d). Furthermore, considering the dynamics of vortex lines and topological reactions versus a control parameter, e.g. either the power in experiments or the propagation variable z in numerics, then the experimental results in Fig. 3 pinpoint the occurrence of pitchfork topological reactions.

4. Conclusions

In conclusion, we generate self-trapped vortices in nematic liquid crystals and observe their transformations into spiraling dipole azimuthons. We observe nonlinearity-induced charge-flipping of the central phase dislocation in the form of splitting of the on-axis phase singularity into three vortex lines [50], analogous to pitchfork bifurcations. Our simulations suggest that astigmatic deformations of the vortex soliton and excitation of its internal modes are sufficient to trigger the azimuthon transformations and the charge flipping. Seemingly, such complex dynamics stems from the anisotropic deformation of a soliton-induced waveguide, similar to astigmatic mode-converters [36], which facilitates the quasi-periodic evolution of nonlocal solitons [22] and can lead to nontrivial topological reactions of vortex modes [46–49]. We believe our results provide the first experimental evidence that nonlocal nonlinear media are able to support efficient modal self-conversion [22].

Finally, we note the relevance of our results in other fields, such as dissipative nonlinear optics, with rich and somewhat unexpected vortex interactions [52, 53]. Very recently, the nucleation of vortex-antivortex pairs has attracted attention in dissipative optical solitons [54], as well as an evidence of superfluidity in exciton-polariton Bose-Einstein condensates in semiconductor microcavities [55]. It would be rather interesting to extend our observations of complex pitchfork topological reactions of quantized vortices in nematic liquid crystals to dissipative systems, as well.