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Abstract
We provide experimental evidence that stable vortex-solitons in nematic liquid crystals, termed vortex nematicons, can be generated in planar cells without any external biases, neither electric nor magnetic. We report on nonlinear vortices with extraordinary-wave beams in various undoped samples, pin-pointing how material nonlocality and birefringence aid their stable propagation. Finally, we also demonstrate confinement and waveguiding of an incoherent co-polarized probe signal by the nonlinear vortex.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light transverse localization by beam propagation in uniaxial nonlocal nematic liquid crystals (NLCs) has been widely investigated over the past two decades, with particular reference to extraordinarily-polarized spatial solitons in planar geometries [1–3]. Such stable two-dimensional (2D) self-confined wavepackets, referred to as nematicons [2], have been shown to exhibit numerous properties of relevance for photonics applications, including, for instance, the ability to mold permanent waveguides [4] or the beaming and control of random lasers in NLCs [5–6]. Nematicons rely on the graded refractive index increase produced in bulk NLCs by molecular reorientation due to extraordinarily-polarized light beams; such nonlinear response turns out to be highly nonlocal as it is in thermal media [7–9], with the resulting benefits of added stability and robustness of 2D solitary waves as well as, e.g., cavityless beam bistability and long-range interactions [10].
The generation and propagation of structured nonlinear beams with orbital angular momentum in self-focusing media, such as doughnut-shaped optical vortices encompassing a phase singularity on axis, has been hampered by an inherent azimuthal instability, with a tendency to break-up into two or more bright solitons [11]. Such modulationally unstable behavior of vortex-solitons was confirmed in early experimental reports using planar samples of reorientational NLCs, in spite of their highly nonlocal nonlinearity [12,13]. The lack of stable vortex-solitons represented an obstacle to their applications; among the theoretical/numerical studies indicating possible routes to circumvent vortex breaking, of particular interest were those addressing the high nonlocality of nematicons and their role in stabilizing collinear co-propagating and co-polarized vortex beams, supporting vector vortex-solitons [14,15] able to survive refractive perturbations [16–18]. Employing two extraordinary-wave beams of different wavelengths, Izdebeskaya et alia demonstrated the latter approach and generated stable vector vortex-nematicons, consisting of a bell-shaped nematicon collinear with a doughnut-shaped vortex of charge one [19,20]. A different route towards nonlinear vortex stabilization in NLCs relied on the taming role of a competing defocusing (thermal) contribution in the presence of absorption, as analyzed by Ramaniuk et alia in [21]. A major contribution to azimuthal instability was attributed to the proximity of boundaries in typically used NLC planar cells. To this extent, in a more recent experiment Izdebeskaya et alia have observed the formation of stable (one-color single beam) vortex-solitons using cells without lateral interfaces but with an external magnetic field able to induce and control molecular orientation in the nematic phase [22].
In this Paper we report on the formation of single-beam self-confined vortices in planar NLC cells, demonstrating vortex-nematicons which counteract their inherent instability by means of the high nonlocality, despite the presence of the transverse boundaries of a (standard) cell, particularly in low-birefringence NLCs. Our findings confirm that nonlocality in conjunction with an all-optical index increase is able to ensure stable propagation of vortex solitary waves [23,24], provided the beam does not interact with the cell interfaces [25] and the astigmatism linked to birefringence is moderate [26]. At variance with previous preliminary reports, these results open up wider perspectives for vortex-soliton applications in a whole class of nonlocal nonlinear media exhibiting self-focusing.
2. Samples and setup
We prepared planar cells consisting of two glass slides parallel to one another and coated with polyimide layers, subsequently rubbed in order to anchor the molecular director n at θ=π/4 with respect to z in the plane yz (see Fig. 1). Upper and lower interfaces were separated by either 30 or 100μm and the whole cells were sealed by orthogonally-mounted cover glasses to prevent NLC menisci (and depolarization effects) at input and/or output along z. The sealing interfaces were covered with polyimide, as well, and rubbed for molecular alignment along y. Various nematic liquid crystals were infiltrated by capillarity, including the standard 6CHBT ($n_ \bot ^{}$=1.49 and $n_{/{/}}^{}$=1.63 at 1.064µm, $n_ \bot ^{}$=1.52 and $n_{/{/}}^{}$=1.68 at 0.532µm) and the low-birefringence mixtures 903 ($n_ \bot ^{}$=1.47 and $n_{/{/}}^{}$=1.54 at 1.064µm, $n_ \bot ^{}$=1.48 and $n_{/{/}}^{}$=1.56 at 0.532µm) and 1110 ($n_ \bot ^{}$=1.452 and $n_{/{/}}^{}$=1.498 at 1.064µm).
Fig. 1. Configuration of NLC sample and beam excitation with relevant coordinates and quantities. The red arrows indicate the input wave-vectors of both the vortex and the probe beams (if present). The input is linearly polarized as an extraordinary wave with electric field along y. (a) Three-dimensional sketch, (b) Bottom view in the observation plane yz. Here θ=θ0=π/4, as suggested by the green ellipsoids oriented along n.
The experiments were carried out with a continuous-wave Nd:YAG laser operating at 1.064µm. The beam, suitably polarized with the electric field along the y axis in order to excite extraordinary-waves in the NLCs (Fig. 1), was shaped by a fork-type holographic diffraction mask (Thorlabs) into a first-order azimuthally-symmetric vortex of topological charge one, i.e., a beam with transverse field distribution
Since NLCs are positive uniaxial materials with $n_{/{/}}^{} > n_ \bot ^{}$ [27], all-optical reorientation due to extraordinary waves with electric field coplanar to the beam wave-vector k and the optic axis n (molecular director) yields an increase in orientation angle θ (with respect to k) and, therefore, an increase in the extraordinary refractive index ${n_e}$, with
Fig. 2. Sketch of the experimental setup: BS – beam splitter, λ/2 - half-wave plates and P –polarizers; VM vortex mask, M – mirrors, MO – microscope objectives; F – pass-band filters; CCD – charge coupled device camera; NLC – planar cell with sample; Inset (a) and inset (b) typical intensity profile and phase interferogram of the input vortex beam at 1.064µm, respectively; inset (c) intensity profile of the Gaussian probe beam at 0.532µm.
3. Results and discussion
The aim of the experiments on single-beam extraordinary-wave vortices is to show that reorientation, in conjunction with the nonlocality stemming from the NLCs’ elastic properties, suffice to provide stable self-confinement of wavepackets as in (1). Reorientational self-focusing determines a potential ruled by $\Delta n_e^2(\theta )$ [28,29], with a nonlocal range linked to the cell thickness across x and an effective nonlinear figure
First, we present some experimental results on vortex-nematicons generated in planar cells of thickness 30µm and filled with the NLC mixture 1110, a low-birefringence liquid crystal with an elastic constant K2${\approx}$8.4pN [32]. The high value of the Frank constant and the low-anisotropy ($n_{/{/}}^2 - n_ \bot ^2 \cong$0.1354) of this material determine a relatively small Kerr-like figure ${\kappa _{NL}} \approx$10−3 V-2. As a result, self-confinement is expected at input powers higher than for typical nematicons [2]. Figure 3 displays the output intensity profile of the obtained nonlinear vortex after propagation for 2mm in a 30µm-thick sample. Despite some diffraction and divergence over the extended propagation length in the presence of scattering losses, the typical doughnut shape of a low-order vortex is preserved in (a) through (c), with the fork phase-dislocation appearing in the corresponding output interference pattern and confinement also being appreciable versus propagation in the observation plane yz. Fringe visibility around the fork bifurcation and beam localization in yz appear to improve slightly with excitation, at power levels nearly one order of magnitude higher than for nematicons in standard mixtures such as E7 or compounds as 6CHBT [1–2].
Fig. 3. Experimental results on observed vortex-nematicon in sample 1110 of thickness 30µm. Top panels: output intensity distribution (left) and corresponding interferogram showing the fork dislocation in the transverse phase distribution (right). Bottom panels: evolution of the vortex-soliton in the plane yz for various input beam powers: (a) 27mW; (b) 32mW; (c) 37mW, respectively.
Similar results were gathered in the NLC mixture 903, a nematic liquid crystal with an elastic constant K2${\approx}$7.4pN, dielectric anisotropy $n_{/{/}}^2 - n_ \bot ^2 \cong$0.21866 and ${\kappa _{NL}} \approx$2.10−3 V-2. Figure 4 shows confinement results on vortex propagation and preservation of singularity versus input power, although the used power levels were halved as compared to the previous examples in Fig. 3. In both such materials with moderate dielectric anisotropy (and birefringence), the generation of self-confined vortex-nematicons in planar cells of limited thickness confirms that nonlocality, with range well exceeding the transverse beam width, suffices in providing vortex-nematicon stability over an extended propagation length.
Fig. 4. Measured vortex-soliton at λ=1.064µm in 903 with thickness 30µm. Top panels: output intensity distribution (left) and corresponding interferogram showing the fork dislocation in the transverse phase distribution (right). Bottom panels: evolution of the vortex-soliton in the plane yz for various input beam powers: (a) 11mW, (b) 14mW and (c) 16mW, respectively.
Noteworthy, both in Fig. 3 (1110) and Fig. 4 (903), even though with sharper visibility in the latter case, the vortex-nematicon evolution in the propagation plane yz exhibits a breathing character, with a minimum beam width occurring closer and closer to the input as the power is increased. Such oscillatory behavior, observed over half-period in Fig. 4(a) and nearly one period in Fig. 4(c) is consistent with nonlinear propagation in highly nonlocal NLCs, as discussed earlier, e.g., in [2].
For the sake of completeness, we carried out additional experiments in the standard compound 6CHBT, previously investigated with reference to nonlinear vortex propagation in conjunction with a collinear nematicon [19,20] or in the presence of an external magnetic field [22]. 6CHBT exhibits an elastic constant $K_{2}\approx 3.6{\rm{pN}}$ [32], dielectric anisotropy $n_{//}^2 - n_ \bot ^2 \cong 0.42821$ and ${\kappa _{NL}} \approx 12\cdot10^{-3}\ {\rm{V}}^{-2}$. Figure 5 displays selected results in both thinner and thicker planar samples, with interface separation of 30µm [Figs. 5(a) and 5(b)] and 100µm [Figs. 5(c) and 5(d)], respectively. While in either cells we managed to observe vortex-nematicons as characterized by a fork phase dislocation in the output fringe-pattern [see Figs. 5(a) and 5(c)], the vortex profile was significantly altered, more so in thinner samples, as expected. Such results are consistent with the azimuthal instability observed and reported earlier [12,13].
In order to assess this apparent discrepancy, we resorted to acquiring temporal sequences of both intensity and phase output distributions at fixed input powers and beam profiles. Figures 5(b) and 5(d) illustrate the measured patterns: it can be clearly seen that the vortex-nematicon in 6CHBT undergoes a modulationally unstable evolution with time, with both intensity and phase alterations. The fork dislocation, in particular, is lost and subsequently recovered versus time, supporting the noted ambiguity present in literature.
Fig. 5. Experimental results obtained at 1.064µm in 6CHBT planar samples of thicknesses (a-b) 30µm and (c-d) 100µm, respectively. (a) Vortex-nematicon after time-averaging over 2s: left, evolution in the observation plane yz of a P = 4.5mW vortex; middle, output intensity profile; right, corresponding interferogram with phase dislocation. (b) Time sequence of the output vortex: (top panels) intensity profile and (lower panels) corresponding interferogram. (c) As in (a) but in a 100µm-thick sample and an input power P = 3.25mW; (d) as in (b) but in a sample of thickness 100µm and P = 3.25mW.
Similar time sequences were then collected in 1110 and 903, and typical results are presented in Figs. 6(a) and 6(b), respectively, in a thin planar cell. Noteworthy, in both these NLCs with lower anisotropy and correspondingly smaller figure ${\kappa _{NL}}$, the structure of the vortex-nematicon is maintained over time, despite some temporal variations partly due to the liquid nature of the samples and their molecular agitation at room temperature [33,34]. Furthermore, the intensity profiles appear to remain more symmetric in low-birefringence 1110 than in 903, such trend being consistent with the observations reported above in 6CHBT.
Fig. 6. Time sequence of a vortex-nematicon output propagating in a 30µm-thick cell. (a) Input power P = 32mW in 1110 and (b) P = 14mW in 903. In both (a) and (b) the intensity distributions are shown above the corresponding fringe patterns.
Finally, using the NLC with moderate birefringence in the thinner planar cells of thickness 30µm, we investigated the refractive index waveguide formed by the vortex-nematicon via nonlinear reorientation. To such extent, we co-launched a collinear co-propagating weak Gaussian beam at 0.532µm, as sketched in Fig. 2, observing its evolution in the observation plane yz [ Fig. 7(c)] and its output profiles in the presence of either a weak input vortex (not shown) or a vortex-nematicon [Fig. 7(d)]. It is apparent that the reorientational response induces self-focusing and confinement of the green probe, which acquires an intensity distribution resulting from the vortex profile in the nonlinear regime.
Fig. 7. Pump-probe experiment in 903 of thickness 30µm. (a) 1.064µm vortex propagating (left) at low power P < 1mW or (right) high power P = 12mW; (b) corresponding output intensity profile and interferogram for P = 12mW. (c) Low power (P = 1mW) Gaussian probe (0.532µm) evolution in yz when co-launched with (left) a low-power or (right) a high-power vortex, respectively. (d) Corresponding output intensity profile of the probe propagating within a 12mW vortex-nematicon.
4. Conclusions
Vortex-nematicons, i.e. spatial optical solitary waves with orbital momentum and topologic charge in nematic liquid crystals, can be generated and observed as self-confined (propagating rather than stationary as in, e.g., [35]) wavepackets in planar cells, over propagation distances of 2mm even in the absence of externally applied fields such as electric or magnetic biases. Vortex-nematicons tend to be comparatively more stable at moderate powers and in low-birefringence nematic mixtures -such as 1110 and 903- than in the standard 6CHBT. We have also analyzed vortex instability versus time, as well as signal confinement and waveguiding using vortex-solitons in NLCs with moderate dielectric anisotropy. Our findings support earlier theoretical reports on the paramount role of nonlocality, contribute to a better comprehension of vortex–solitons in birefringent nonlinear media and are an important step forward in vortex-nematicon generation and potential applications.
Funding
Narodowe Centrum Nauki (UMO-2016/22/M/ST2/00261).
Acknowledgments
GA acknowledges the gracious hospitality from the Faculty of Physics, University of Warsaw, Poland.
Disclosures
The authors declare no conflicts of interest.
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