Chapter 6: Dynamics of Optical Solitons in Bias-Free Nematic Liquid Crystals

Yana V. Izdebskaya, Anton S. Desyatnikov and Yuri S. Kivshar

Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra, ACT, Australia

6.1 Summary

We present a review of our experimental work on spatial optical solitons in nematic liquid crystals (NLCs) in a planar cell without external fields. We investigate the nonlocal interaction of two mutually incoherent copropagating (CO) and counterpropagating (CP) bias-free nematicons; the analysis of the dynamics of CP solitons versus their relative separation and excitation shows the existence of stable vector solitons composed of two oppositely propagating nematicons. Efficient steering of nematicons is achieved by introducing defects in the liquid crystal layer: we demonstrate significant bending as well as large-angle total internal reflection of nematicons interacting with a curved interface between NLCs and air. Nematicons induce guiding channels capable of confining a weak signal: we identify experimentally the domains of nematicon power where higher order guided modes are supported. Finally, we present recent results on the astigmatic transformation of vortex beams in NLCs into spiraling dipole azimuthons accompanied by power-dependent charge-flipping of on-axis phase singularity.

6.2 Introduction

Spatial optical solitons have been studied extensively in diverse nonlinear media, in both one- and two-dimensional geometries [1]; they have significant potentials for signal processing, switching, and readdressing in the next generations of all-optical circuits. In this context, the giant optical nonlinearity arising from molecular reorientation in NLCs has attracted significant attention [2, 3]. The reorientational nonlinearity allows generation of spatial optical solitons at relatively low optical powers, in the milliWatt region or below [4], and provides direct access to the study of fundamental aspects and applications of light interaction with self-assembling nonlinear soft matter. Both theoretical [5] and experimental [6] results have been reported for spatial optical solitons in NLC, also called nematicons [7]. The reorientational optical nonlinearity can be modulated by prealigning the NLC molecules with external electric field. Such bias allows for a versatile control over the orientation of the organic molecules at rest and, therefore, provides fine tuning of the strengths of both nonlinearity and nonlocality [8]. However, nematicons can also exist in simpler settings with unbiased NLC cells, where the molecule pretilt in the bulk of the planar cell is achieved by the anchoring conditions at the boundaries [9–13]. The nonlinear effects, induced by small tilts of the molecules by laser light, are sufficiently strong to observe self-focusing effects at milliwatt input powers, provided the input beam is extraordinarily polarized, that is, with electric field and molecular director coplanar in the plane (x, z), parallel to the cell interfaces [9]. Because of the optical anisotropy of the liquid crystalline molecules and the resulting birefringence of the nematic medium, light beams propagating in NLC walk-off the direction of their wave vectors (see also Chapter 1).

In this chapter, we present an overview of our recent experimental results on the generation and dynamics of spatial optical solitons in short planar NLC cells without external bias [14–18]. We investigate in detail the power-dependent nonlocal interaction of two identical mutually incoherent CO nematicons (Section 6.3) and CP nematicons (Section 6.4), demonstrating the existence of a class of vector nematicons consisting of two self-trapped beams propagating in opposite directions. Depending on the parameters of the setup we observe either stable stationary states or dynamic instabilities of two CP beams in the form of splitting and spatial entanglement. In Section 6.5, the interaction of spatial optical solitons with curved dielectric surfaces is studied in an NLC layer with injected air bubbles of various sizes; this interaction allows large-angle bending of nematicons as well as total internal reflection preserving the nondiffracting character of the self-trapped beams. In Section 6.6, we show that the waveguides induced by spatial solitons in unbiased NLCs can support various copolarized higher order guided modes. Section 6.7 presents our experimental observations of the so-called dipole azimuthons, robust higher order spatial solitons forming in nematics from an input vortex beam. Finally, Section 6.8 concludes this chapter.

6.3 From One to Two Nematicons

In the experiments, we use a short planar cell of length L = 1.1 mm (Fig. 6.1a) consisting of two polycarbonate slides spaced by 100 μm and filled with 6CHBT [19, 20], an NLC with relatively low absorption. The inner surfaces of the slides are unidirectionally rubbed in order to align the NLC molecular director n in the plane (x,z) at 45° with respect to the z axis. Such boundary conditions in (x,z) are analogous to the bulk prealignment in (y,z) by an external bias voltage [6]. Two additional glass slides are attached perpendicularly to the propagation axis z at two opposite sides of the cell in order to define air–NLC input/output interfaces, avoiding lensing and depolarization due to an NLC meniscus [6]. We use microscope objectives and CCD cameras to collect the light at the sample output and the light scattered above the cell along propagation.

First of all, we study the propagation of a single beam in such a cell for various excitations. Figure 6.1 shows typical top-view photos of (b,d) the ordinarily polarized o beam, with electric field parallel to y, and (c,e) the extraordinarily polarized e beam, with electric field parallel to x, launched with input powers P = 0.4 mW in (b,c) and P = 9 mW in (d,e). Clearly, the o beams in Figure 6.1b and 6.1d and the low power e beam in Figure 6.1c diffract, whereas the e beam in Figure 6.1e experiences self-focusing at power P ≥ 0.9 mW and gives rise to a self-localized state, that is, a nematicon. Owing to the high birefringence Δϵ ≈ 0.15 (refractive indices n_e = 1.6718 and n_o = 1.5225 at room temperature of 23°C at wavelength λ = 532 nm) in our geometry, the nematicon travels in the NLC along the direction of the Poynting vector with a significant spatial walk-off angle, α ≈ 4.2^°, with respect to the z direction [15, 21]. The reduction of walk-off α with input power is plotted in Figure 6.1f.

Noteworthy, the nematicons are generated in a liquid uniaxial medium subject to slow dynamics [22] and instabilities [15, 23], where the birefringence-induced walk-off is power sensitive [15, 21] and nonlocality leads to beam breathing [24, 25]. In the absence of external bias, as in our configuration, these effects result in a bent nonuniform waveguide with fluctuations on a time scale smaller than the inverse maximum frame-rate of our camera (25 fps). Therefore, to quantitatively characterize the nematicon profile, we define the nematicon half-width at half-maximum (HWHM), w_{x, y}, as two half-sizes of a rectangle enclosing the contour line of the averaged intensity profile at the half-peak level. Figure 6.1g shows the experimental results at the cell output for a nematicon excited with the green laser at 532 nm. At low input power (P< 0.9 mW) the gradual reduction of beam size in Figure 6.1g shows its self-focusing, whereas at higher power P > 0.9 mW a nematicon is formed.

Next, we study the dynamics and interaction of two mutually incoherent—initially parallel—copropagating beams in NLC at various excitation levels. In order to generate two CO beams, we used a standard Mach–Zehnder arrangement. As discussed earlier [26], owing to the long-range character of the nonlocality the nematicons attract each other, more and more for increasing input powers. In most cases, this attraction is independent of the relative phase of the solitons, even when they are mutually coherent [26]. A sequence of images in Figure 6.2 shows the stationary trajectories of two initially parallel beams for different input powers. At low input power (0.5 mW) self-focusing is too weak to overcome diffraction, so the beams keep spreading as they propagate. By increasing the power to 2 mW we achieve stable propagation of solitons and their weak attraction. For higher excitations, the attraction is sufficiently strong to induce one (at P ≥ 3 mW) or multiple (P = 13.5 mW) intersections in the nematicon trajectories.

6.4 Counter-Propagating Nematicons

One of the simplest processes that leads, in nonlinear optics, to a variety of complex phenomena is the mutual interaction of two CP optical beams. Numerous concepts, such as phase conjugation, Bragg reflection by volume gratings, and wave-mixing, are based on this geometry. Nevertheless, a simple CP geometry can give rise to an extremely complicated and sometimes counterintuitive dynamic behavior, including mutual beam self-trapping and the formation of stationary states, as well as complex spatiotemporal instabilities [27].

Fundamental concepts motivated earlier studies of optical solitons created by two beams propagating in opposite directions, and it was shown that mutual self-trapping of two CP beams can lead to the formation of a novel type of vector soliton [28, 29], for both coherent and incoherent interactions. More detailed analyses [30, 31] revealed that CP solitons may display a variety of instabilities, accompanied by nontrivial temporal and spatial dynamics.

In order to investigate the propagation dynamics and interaction of CP solitons, we launched in our cell beams from opposite sides. When the two inputs propagate toward each other with a finite impact parameter r (the offset, see dashed lines in Figure 6.3A), we observe the formation of two nematicons and their bending in opposite transverse directions owing to walk-off. Because walk-off is power dependent (see also Chapter 11), the interaction of CP nematicons is parameterized by both the offset r and the input power P.

Typical experimental results on the interaction of CP solitons are presented in Figure 6.3. As expected from the single beam case in Figure 6.1, an increase in excitation P > 0.9 mW leads to self-focusing and the formation of nematicons. For a given offset r and small powers the beams do not interact, as shown in Figure 6.3A. Here and later, we distinguish the absence or the presence of interaction by blocking one of the beams and observing the other one that relaxes (in time) to its independent (unperturbed) trajectory. With decrease in the offset r (or increase in the input power P), the two CP beams still remain spatially separated but the attraction between them leads to additional bending, clearly visible in Figure 6.3B for r = 130 μm. The bound state of two CP nematicons is obtained for specific parameters r and P, as seen in Figure 6.3C. On the other hand, the sequence of Figure 6.3C and 6.3D for r = 105 μm shows how such bound state can be destroyed by an increase in power, which leads in Figure 6.3D to the development of a dynamic transverse instability of the two time-averaged CP beams in the form of spatial splitting and spatial entanglement.

The results of numerous experiments are summarized in Figure 6.4a with differently shaded regions for qualitatively different interaction scenarios, as described earlier. Importantly, in all our experiments, the beams are launched parallel to the optic axis z with equal powers P and, because the interaction is phase independent, the key parameters governing the dynamics are offset r and power P.

For powers below a threshold, no interaction is observed (blank area) and each nematicon trajectory does not bend when the CP soliton is launched (Fig. 6.3A). The threshold is offset dependent and it can be seen as the border of the light gray domain of attraction marked with black triangles in Figure 6.4a. By increasing the power we enter this domain of mutual attraction and observe the two trajectories bending toward each other while propagation remains stationary in time (Fig. 6.3B). Vector nematicons (Fig. 6.3C) are formed in the dark area with square markers, and the instability (Fig. 6.3D) is observed in a narrow region with circles in Figure 6.4a.

The domains in Figure 6.4a point out a counterintuitive feature, namely, an asymmetry with respect to the (dashed) line in r = 105 μm. This offset value corresponds to a “head-on” collision of the two CP solitons at small powers. Above and below this value a qualitative difference appears between the two configurations sketched in Figure 6.4b and 6.4c. For an offset r > 105 μm in Figure 6.4b, the walk-off of the two nematicons, indicated by curved arrows, acts in the same transverse direction of the mutual nonlinear attraction. Conversely, for r < 105 μm in Figure 6.4c, the two nematicons bend away from each other due to walk-off, counteracting mutual attraction. As a result, the bound states of CP nematicons with relatively straight trajectories appear at low powers only for offsets r≃105 μm (see the dark area in Fig. 6.4a), because they require a lesser amount of curvature in order to merge into a joint waveguide. For r ≤ 105 μm (such as that shown in Fig. 6.3C and Fig. 6.4a) vector CP nematicons require larger excitation powers.

6.5 Interaction of Nematicons with Curved Surfaces

Nematicons are excellent candidates for all-optical soliton control and signal routing. Efficient and tunable routing of nematicons can be achieved by reflection or refraction at the interface between two nonlinear media, each controlled by an independent external bias [32, 33] or by anchoring at the boundaries [34]. Similarly, localized nonlinear defects induced by an additional “control” light beam can be repulsive [13, 35] or attractive [13, 36], depending on the induced change (negative or positive) of the refractive index. However, if the boundary conditions are linear, for example, fixed by the anchoring of NLC molecules, the interface is repulsive [37, 38].

In this section, we report on the experimental demonstration of the strong interaction of nematicons with curved dielectric surfaces. To induce a curved linear/nonlinear interface between air and NLC, we injected microscale air bubbles of different sizes into the layer. From the side of the cell, such extended defects have cylindrical shapes with generatrices orthogonal to the planar cell boundaries. In our experiments, we kept the input nematicon power P = 3.5 mW constant while varying the relative distance between soliton and defect.

The formation of a “free” nematicon, without air bubble, is accompanied by walk-off, which effectively tilts the straight-line trajectory by the angle α with respect to the axis z (see dashed arrows in Fig. 6.5a and b). In sharp contrast with the previous case, when the soliton approaches the bubble, its trajectory bends toward the surface, as is seen from gray-shaded intensity profiles in Figure 6.5a and b. Such bending indicates attraction by the surface. In good qualitative agreement, the numerically calculated changes of molecular orientation and refractive index induced by the bubble in Figure 6.5c show a significant increase of the latter (dark regions), or the induction of an attractive potential.

At the same time, the bending is quantitatively different for nematicons approaching the bubble from the top and the bottom (Fig. 6.5a and b). This apparent difference can be ascribed to the counteraction of two “forces,” the walk-off and the attraction to the surface. Qualitatively, whereas attraction is in the direction of walk-off in Figure 6.5a and increases the nematicon inclination δ, it is opposite to and compensates the walk-off in Figure 6.5b.

Finally, when the target soliton trajectory crosses the bubble surface, as shown in Figure 6.5d, the soliton experiences total internal reflection, observed earlier in different settings [32]. Unlike the cases studied earlier, in our geometry, the nematicon steering may exceed 90°. Surprisingly, despite the very different propagation direction after reflection, the solitons propagate similar to rays of light, with hardly any visible differences in widths and profiles before and after collision with the curved surface.

6.6 Multimode Nematicon-Induced Waveguides

An attractive property of soliton-induced waveguides is their ability to guide weak signals of different wavelengths [6, 39, 40] as well as the possibility to reconfigure such waveguides by spatial steering. Another dimension in soliton-based optical switching is offered by the multimodal character of the self-induced waveguides, observed earlier with photorefractive solitons [39, 41]. In this regard, the nematicon waveguides, that is, those induced by spatial optical solitons in NLCs [6, 42], are of particular interest because of the long-range or highly nonlocal character of the reorientational nonlinearity. The transverse size of the nematicon-induced index perturbation can be up to one order of magnitude wider than the beam [24, 43, 44] and, therefore, such a waveguide is expected to be multimodal. Additional evidence of such multimodality is the existence of higher order nonlocal solitons [45–47] and incoherent solitons in photorefractive crystals [48] and nematics [49].

Here, we describe experimental observations of higher order modes guided by bias-free nematicons [17]. First, we study the dynamics of a weak extraordinary polarized signal (red, wavelength 632.8 nm) fundamental H₀₀¹ mode launched collinearly with the green nematicon (λ = 532 nm). Figure 6.6a and 6.6b shows top-view images of the extraordinary signal beam, resulting in either diffraction without nematicon (Fig. 6.6a) or guided-wave propagation by the presence of a nematicon (Fig. 6.6b). The power of the input signal (red) beam was 82 μW in all experiments.

Figure 6.6c–e shows experimental results at the cell output for a H₀₀ mode. At low nematicon power (P<0.9 mW), self-focusing reduces the waist in the green region and the corresponding size of the mode in the red region, until a nematicon is generated for P > 0.9 mW (Fig. 6.1g) and the red signal becomes guided (Fig. 6.6c). The shaded region in Figure 6.6c marks the existence of a stable H₀₀ guided mode, with a typical averaged output transverse profile displayed in Figure 6.6d. Further increases in nematicon power P lead to a soliton waveguide supporting a larger number of guided modes, with a stronger mixing and an output exhibiting a multi–hump profile, as in Figure 6.6e. The latter transition for P > 3 mW is accompanied by an increase in HWHM for the red signal, as apparent in Figure 6.6c, whereas the nematicon maintains its robust structure (Fig. 6.1g). Clearly, the nonlinear refractive index potential gets reshaped with nematicon excitation, inducing a multimoded channel waveguide at the output of which a complex pattern appears owing to superposition and mixing of guided modes [50].

To excite the first-order mode in the soliton-induced waveguide, we insert a thin glass plate in front of half of the signal beam and tilt it, introducing a π-phase jump and reproducing the phase profiles of H₁₀ and H₀₁ modes, respectively. The HWHM of both modes versus nematicon excitation (Fig. 6.7a–c) exhibits (shaded) regions of stable guidance, with averaged profiles shown in Figure 6.7b–d. We temporally average the output images of higher order modes as recorded during data acquisition (as discussed in Section 6.3; Fig. 6.1g). At low excitation powers, the strong diffraction of the signal beam is limited only in the y direction by the cell boundaries, thus w_y < w_x. At variance with soliton waveguides in photorefractive crystals, subject to a directional bias [41], both dipole-like modes coexist in the same interval of excitations, 1.2 < P[mW] < 2, suggesting that, in this case, the role of the cell boundaries and the related index anisotropy [51] are minimal.

Using two glass plates, we generate the red signal mode H₁₁; see, for example, Figure 3e and f in Reference 17. The HWHM of this mode in both x and y corresponds well to the sizes of the H₁₀ and H₀₁ modes above, ∼ 12 μm, approximately twice wider than the nematicon. As expected, the stable domain for H₁₁ is shifted toward higher powers with 1.4 < P[mW] < 2.3 [17]. Finally, Figure 6.7 displays experimental results for the H₂₀ and H₀₂ modes, generated with two glass plates, as well. The H₂₀ mode displays a relatively broad domain of stable existence, 1.8 < P[mW] < 3. Its HWHM w_x≃15 μm is nearly three times larger than the nematicon size. In sharp contrast, the vertically oriented H₀₂ mode (Fig. 6.7h) is only quasi-stable in a narrow domain, 1.7 < P[mW] < 2.2, due to the (anisotropic) role of the cell boundaries at y = 0, d (where d is the cell thickness) with fixed director orientation and despite the fact that its total width, 2w_y < 24 μm, is much smaller than d = 100 μm. It follows that the order of the supported guided modes is limited from above not only by the nematicon power and mode mixing process but also by the boundary-induced anisotropy [51]. Noteworthy, for H_mn with (m, n) ≤ 1, the nematicon waveguides can be considered isotropic and circularly symmetric.

6.7 Dipole Azimuthons and Charge-Flipping

A link between fundamental optical spatial solitons [1] and doughnut-shaped vortices [52, 53] is provided by the existence of dynamic bound states of solitons in the form of rotating soliton clusters [54] and azimuthally modulated vortex solitons, or azimuthons [55, 56]. In this section, we report on the formation of dipole azimuthons in unbiased NLC, with nontrivial charge-flipping of on-axis phase dislocation [18].

A single-charged vortex beam is generated with a fork-type amplitude diffraction hologram from an extraordinarily polarized cw (continuous wave) laser beam of wavelength 800 nm. At low input power, P<0.9 mW, the vortex beam uniformly diffracts without any noticeable self-action. As the power increases, 1 < P[mW] < 2.2, the beam experiences self-focusing and its output size visibly reduces. In this regime, the initial radially symmetric vortex undergoes a drastic transformation, see for example, Figure 1 in Reference 18: the vortex “doughnut” breaks up into two beams with a dark core transformed into a tilted stripe, similar to vortex astigmatic transformations [57].

A further increase in input vortex power (P > 2.3 mW) leads to stronger narrowing of the output beam and to the formation of a self-trapped dipole-like beam. Figure 6.8 shows the longitudinal dynamics and the transverse output profiles of the dipole beam for three different input powers. The whole beam is tilted and bent due to walk-off and diffraction is suppressed. Note the dark line along the beam, splitting it into two lobes in Figure 6.8a–c. The corresponding output transverse patterns in Figure 6.8d–f contain two pronounced bright spots, clearly coupled together in a structure similar to a numerically simulated dipole azimuthon [58]. For different input powers the output dipole has a strongly varying elliptic shape tilted at different angles with respect to the boundaries. Importantly, this spatial twist of the beam inside the cell is defined by the topological charge of the input vortex beam, in excellent agreement with theoretical predictions [58].

To analyze the singular phase structure of dipole azimuthons, we employ an interferometric technique: a tilted broad Gaussian beam interferes at an angle with the dipole at the cell output. The obtained interferograms in Figure 6.9 allow detecting the position of the phase dislocations by the characteristic presence of fork dislocations. Two opposite topological charges, , are distinguished by the fork orientation, either down or up. Remarkably, we observe nontrivial topological reactions of the on-axis phase dislocation of nonlocal dipole azimuthons with the sudden appearance of a triplet of vortices. As a representative example, Figure 6.9 shows the output interferograms for the input vortex with topological charge m = +1. At small powers, the input phase dislocation is preserved (Fig. 6.9a, left). However, increasing power above a critical value, such as P = 2.6 mW in Figure 6.9b, we observe three spatially separated phase singularities in Figure 6.9a, middle. The central (on-axis) vortex has a topological charge m₀ = − 1 opposite to the one in Figure 6.9a, left; two satellite vortices with m_{1, 2} = + 1 compensate this difference keeping the total charge unchanged, m = m₀ + m₁ + m₂ = + 1. Increasing the input power results in spatial separation of the triplet and then attraction of vortices, followed by a second flipping of the central vortex. This time, the charge flips from m₀ = − 1 to m₀ = + 1 above P ∼ 3.3 mW (see Fig. 6.9a, right). With further increase in power up to 3.7 mW, the charge-flipping process repeats again.

It is noteworthy that analogous experiments with an input vortex beam carrying charge m = −1 lead to an opposite direction for the spatial twist of the dipole (Fig. 6.8) accompanied by topological reactions of three vortices similar to Figure 6.9. This observation allows us to conclude that charge-flipping is a robust and reproducible effect. The sudden splitting of a vortex line into three spatially separated lines or, conversely, the reconnection of three vortex lines into one, is similar to the diagrams of pitchfork bifurcations (supercritical and subcritical, respectively) with transition driven by optical power. The numerical simulations of the initially deformed vortex beam in isotropic nonlocal media reveal a similar dynamic behavior of vortex lines, that is, the formation of pitchfork topological reactions [59] during propagation, for example, Figure 4 in Reference 18. The apparent reason for such complex dynamics is the anisotropic deformation of the nematicon-induced waveguide, similar to astigmatic mode conversion [57], which also facilities the quasi-periodic transformations of nonlocal solitons [47, 60, 61].

6.8 Conclusions

We have presented an overview of experimental studies of a rich variety of phenomena with spatial optical solitons in an NLC-filled planar cell without external bias. The reorientational nonlinearity is strong enough to support stable nematicons as well as their robust interactions. We have investigated the nonlocal interactions of CO and CP nematicons and observed the formation of vector nematicons with two components propagating in opposite directions. We have experimentally determined the domains of two key parameters, transverse separation of CP beams r and their excitation power P, where the vector nematicon remains stable or exhibits dynamic instabilities. Furthermore, we have experimentally studied the interaction of nematicons with curved dielectric surfaces and demonstrated that, by varying the impact parameter, the soliton beam can be efficiently steered by exploiting its attractive interaction with the surface. We have experimentally observed total internal reflection of soliton beams from an interface between air and liquid crystals, with nematicon deviations exceeding 90°. Each nematicon induces an actual waveguide: we reported on the experimental observation of higher order modes guided by soliton-induced waveguides. We found that the nematicon waveguides operate in a bounded power region specific to each guided mode. Below this region, the guided beams diffract; above this region, mode mixing and coupling give rise to unstable outputs. Finally, we have generated self-trapped vortices in NLCs and observed their transformations into spiraling dipole azimuthons. We have observed nonlinearity-induced charge-flipping of the central phase dislocation as splitting of the on-axis phase singularity into three vortex lines, similar to pitchfork bifurcations. Our findings contribute toward a better understanding of nonlinear optical processes in anisotropic media and the dynamics of self-trapped laser beams in nonlocal media, and they can be potentially useful for all-optical processing of information in futuristic soliton-based optical switches and interconnects.

Acknowledgments

The authors acknowledge financial support from the Australian Research Council, and they are grateful for essential contributions of their collaborators, G. Assanto, V. Shvedov, W. Krolikowski, and M. Beli.

Notes

¹ We use the notation H_mn for Hermite-like modes with indices (m,n) because the proper distinction between hybrid EH and HE modes is not possible [50] in birefringent liquid crystals.

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