Control of Light by Topological Solitons in Soft Chiral Birefringent Media

In practically all branches of physics, different types of solitons, with a number of them enjoying topological protection, are found. Here we explore how one- and two-dimensional topological solitons formed by spatially localized continuous orientational patterns of optical axis in uniaxial birefringent media interact with light. These solitons, in the forms of one-dimensional twist walls and two-dimensional skyrmions, are controllably generated in thin films of cholesteric liquid crystals to introduce spatially localized patterns of effective refractive index. Laser light interacts with these solitons as quasiparticles or extended interfaces of different effective refractive indices seen by ordinary and extraordinary waves propagating within the liquid-crystal medium. Despite our system’s complex nature, our findings can be paralleled with the familiar phenomena of total reflection and refraction at interfaces of optically distinct media, albeit these behaviors arise in a medium with homogeneous density and chemical composition but with spatial variations of molecular and optical-axis orientations. By exploiting the facile response of liquid crystals to external stimuli, we show that the twist walls and skyrmions can be used to steer laser beams and to act as lenses and other optical elements, which can be reconfigured by low-voltage fields and other means. Analytical and numerical modeling, with the latter based on free-energy-minimizing configurations of the topological solitons, closely reproduce our experimental findings. The fundamental insights provided by this work potentially can be extended also to three-dimensional solitons, such as Hopfions, and may lead to technological applications of optical-axis topological solitons in telecommunications, nanophotonics, electro-optics, and so on.

Popular Summary

In fluid mechanics and optics, solitons are self-reinforcing solitary wave packets or optical fields that remain nearly unchanged during propagation. These solitons emerge from a delicate balance of nonlinear and linear effects within the host medium, attracting the interest of mathematicians and physicists since the first documented observation of the soliton in 1834. Solitons of very different types, often called topological solitons and nonsingular topological defects, are widely studied in topology, physics, and cosmology. Here, for the first time, we demonstrate that topological solitons in a uniaxial liquid crystal can provide a new means for controlling light while behaving similarly to objects that have a refractive index different than that of a surrounding medium.

In our experiments, we demonstrate refraction, reflection, and lensing of laser beams that are all based on the particlelike nature of these solitons and can be reconfigured by weak external stimuli. We show that interactions of light with such topological optical-axis solitons are described using a generalized Snell’s law, and we discuss how our fundamental findings may lead to technological uses in telecommunications, beam steering, waveguiding, nanophotonics, and electro-optics.

Our findings bridge the fundamental studies of topological solitons with a vast spectrum of technological applications of liquid crystals that can host such solitons.

Figure 1

(a) Schematic representation of a 1D soliton with a $\pi$ twist in a nonpolar LC. The orientation of the LC director is represented by cylinders, and the far field is denoted by ${\mathit{n}}_0$. (b) 3D embedding of the wall shown in (a) between plates (in gray) with moderate homeotropic anchoring. (c) Same as (b) with strong homeotropic anchoring. Two defect lines (in red) are necessary to accommodate the boundary conditions. (d) Schematic representation of a 2D soliton dubbed elementary skyrmion with a vortex of $\pi$ twist from its center in all radial directions. (e) 3D embedding of the elementary skyrmion shown in (d) between two plates (in gray) with moderate homeotropic anchoring; no singular defects are present in this LC bulk. The director field lines resting on nested toroidal surfaces are represented in black. (f) Same as (e) with strong homeotropic anchoring; similar to (c), two defects (in red) are present due to the strong anchoring. The colors of the surfaces in (e),(f) are only there for visual enhancement and do not have special meaning.

Figure 2

Schematics of the (a) microscope, (b) liquid-crystal cell, and (c) optical setup used to probe interactions between Gaussian beams and topological solitons in frustrated cholesteric LCs. (Not to scale.)

Figure 3

Side (a) and top (b) view of the system, which consists of a LC layer confined between two plates in blue, with a topological soliton inside the LC layer along the $z$ axis. A light beam with wave vector ${\mathit{k}}_i$ is incident on the soliton in the $yz$ plane and generates reflected and transmitted beams. (c) Side view of the LC director field inside the simulated translationally invariant soliton. This soliton called a twist wall is invariant by translation along the $z$ axis and represents the simplest type of translationally invariant soliton in frustrated cholesterics, with a $\pi$ twist of the director field along the $y$ axis represented schematically in (d). In (a)–(c), ${\mathit{n}}_0$ represents the orientation of the background director field far from the soliton. The projection of the translationally invariant director configuration’s components along the axis of invariance ($z$ axis) is indicated by the color scale in (c).

Figure 4

(a) BPM simulation of a beam reflected by a twist wall. The green color corresponds to a 3D volumetric rendering of the optical field intensity. The colored surfaces correspond to isosurfaces of the $z$ component of the LC director field. The beam profile is initially Gaussian but then spreads out along the $x$ axis after interacting with the twist wall while confined between the two glass plates represented in gray. (b) Simulated microscope images of the system represented in (a), with the axis of observation along $x$. The POM image of the twist wall is simulated assuming crossed polarizers, and the green beam corresponds to the 3D volumetric data of (a) averaged along $x$. (c) Same as (a) with a different incident angle. Instead of a reflected mode, two transmitted modes associated with different polarizations are observed. (d) Same as (b) with the system of (c). The intensity of the $o$ mode is enhanced 10 times in order to be seen with the naked eye. In (a),(c), the polarization state of light is represented with double arrows. In (b),(d), the white thick line represents $50\mu \mathrm{m}$; the white dashed lines represent the directions of propagation of transmitted or reflected beams predicted by Eq. (12) assuming $n^{(\alpha ,m)}\approx n_{\alpha }$. The angle of beam incidence is ${\theta }_i=70°$ for (a),(b) and ${\theta }_i=55°$ for (c),(d).

Figure 5

Comparison of our modification of Snell’s law to experiments with an ordinary (left) or extraordinary (right) polarized beam impinging on a CF3. Angles of transmission and reflection are shown for an $o$-mode incident beam in (a) or an $e$-mode incident beam in (h). Predicted reflection ($R_{o,e}$ modes) and transmission ($T_{o,e}$ modes) modes are indicated by the colored solid lines. Relative mode power as a function of the incidence angle is depicted by the color scale as shown. Experimental data, at incidence angles with nonzero mode powers, verify the trajectories predicted by our generalization of Snell’s law. POM images of an incident $o$-mode (b)–(g) or $e$-mode (i)–(n) beam interacting with a CF3 are shown below (a),(h). The beam propagates from right to left in each micrograph. In (b)–(d) and (i)–(k), the sample is illuminated with a backlight to view the birefringent medium’s soliton. In (e)–(g) and (l)–(n), the backlight is extinguished to capture the beam’s optical interaction with the soliton. Each interaction, as indicated by the incidence angles as shown, is portrayed as a pair of POM images viewed with and without a backlight. In general, as the incidence angle increases, different mode powers become nonzero as shown with a transition from total transmission to a regime with weak transmission and dominant reflection in (h). The scale bar’s length is $100\mu \mathrm{m}$.

Figure 6

(a) Simulated microscope images of an extraordinary beam reflected by a twist wall in a thin sample of $10\mu \mathrm{m}$. The incidence angle is ${\theta }_i=75°$. Similar to Fig. 4, the simulated birefringent-medium twist wall is observed under crossed polarizers, and the green part of this image is obtained by averaging along $x$ the three-dimensional optical field intensity simulated with BPM. The white bar represents $10\mu \mathrm{m}$. (b) Calculated power spectrum at points $\mathcal{P}1$ and $\mathcal{P}2$ in (a) as a function of the waveguide mode number. The dominant waveguide mode numbers for $\mathcal{P}1$ and $\mathcal{P}2$ are used to predict the main directions of propagation of reflected modes [dashed white lines in (a)] using Eq. (12). (c) Experimental images in a thick sample of $40\mu \mathrm{m}$ showing a splitting of reflected modes consistent with those in (a). The white bar is $100\mu \mathrm{m}$.

Figure 7

Top (a) and side (b) view of the director field of a baby skyrmion. We use the same axis convention as in Fig. 3. The color scale is the same for (a) and (b). (c) General trajectories of extraordinary rays deflected by the baby skyrmion, as predicted by the simple ray-tracing model of the main text. (d) Variations of the quantity $g$ introduced in the main text along a line parallel to $y$ and centered on the baby skyrmion. $h$ is the sample thickness. The sign of $g$ allows us to predict the sign of deflection of the rays.

Figure 8

A demonstration of simulated (a)–(c) and experimental (d)–(f) beam deflection and lensing from a toron at no electric field. In (a)–(c), the white lines correspond to calculated ray trajectories, while the color images are simulated as in Fig. 4 with the beam propagation method. As depicted in Figs. 7 and 7, the angle of deflection is attributable to the sign and magnitude of the deflection parameter $g$. Small, negative values of $g$ produce a slightly deflected beam whose simulation is shown in (a) with experimental confirmation in (d). When $g$ switches sharply from a very negative value to 0, defocusing and enhanced deflection occurs, as shown theoretically in (b) and experimentally in (e). By virtue of a skyrmion’s rotational invariance, a beam injected at the toron’s center experiences symmetric broadening behavior as shown theoretically in (c) and experimentally in (f). The inset in (c) shows an enlarged view of the ray trajectories near the toron’s center. Differences in appearance between simulated (a)–(c) and experimental (d),(e) torons mostly result from director perturbation, only near the glass plates, used to pin a toron against diffusive motion within its sample plane and from experimental illumination with a coherent white-light source. The scale bar’s length represents $100\mu \mathrm{m}$.

Figure 9

Experimental measurements of incident beam deflection and lensing with a theoretical ray-tracing fit for a toron at no electric field. (a) A schematic representation of the coordinate system and parameters used for data measurement. The incident beam ${\mathit{k}}_i$ is deflected by an angle $\theta$ from the incidence trajectory, which is depicted as a white dashed line segment. The deflected ray ${\mathit{k}}_t$ may represent the center of a deflected beam profile or its half-intensity-maximum envelope rays. (b) Deflection angles for the beam envelope and beam center rays plotted against the normalized distance $\delta =y_i/R$. The incident beam has an extraordinary polarization. The theoretical ray-tracing fit for center angles is plotted as the black curve. Negative angles in (b) correspond to a clockwise deflection of ${\mathit{k}}_t$ as indicated in (a). The scale bar’s length represents $100\mu \mathrm{m}$.

Figure 10

Modulation of beam-center deflection by the application of an electric field along the far-field uniform director ${\mathit{n}}_0$, which is orthogonal to the beam-propagation plane. (a) Incident extraordinary-mode ($e$-mode) beam-center deflection plotted against the normalized distance $\delta$. Three datasets are collected for three different electric potentials as indicated. The solid curve is the behavior predicted by our simple ray-tracing model. For the beam-center deflection peaks at $\delta =\pm 0.5$, larger beam-center deflection is attributable to an asymmetric broadening and lensing of the beamwidth for different toron diameters, which map to applied potentials as indicated in (b). (b) Measured toron diameter plotted as a function of the electric potential for the experimental system. (c)–(e) Incident $e$-mode light deflected by three instances of a toron experiencing different electric potentials. The normalized distance of interaction $\delta =0.5$ is constant as indicated by the star in (a). The shift in beam-center deflection is attributable to the contracting toron’s distortion profile despite a constant incident beamwidth. Negative angles in (a) correspond to the clockwise deflection of the beam in (c)–(e), as defined in Fig. 9. The scale bar’s length represents $100\mu \mathrm{m}$.

Figure 11

Transmitted (a) and reflected (b) powers from the midplane permittivity profile of a CF3, assuming an incident ordinary mode. (c),(d) Same as (a),(b) with an incident extraordinary mode. We use the same convention as Fig. 5 for the notation $T_{e,o}$ and $R_{e,o}$.

Figure 12

(a) Side view of the director field of a CF3 in a $40\text{−}\mu \mathrm{m}$-thick sample. The director field is obtained from a numerical minimization of the total free energy of the system, assuming strong homeotropic anchoring on the confining plates. CF3s are generally associated with two defect lines parallel to the soliton invariance axis (here, $z$). These topological defects are represented by two black spots near the sample plate in this figure. (b) Same simulated microscope image as in Fig. 4 but with a CF3 instead of a fully nonsingular twist wall. Contrary to Fig. 4, the incident extraordinary beam is not totally reflected, and some part of the beam is transmitted through the soliton, as also observed in experiments. (c) Side view of the beam intensity along the dashed black line (b), showing the intensity profile of the incident and transmitted beam along $x$ (sample normal) and ${\mathit{k}}_i$ (incident wave vector). The defect lines of the CF3 are represented with two red ellipses, and the isolines $n_z=-0.7$ are represented with dashed white lines.