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Abstract
By designing a liquid crystal cell with comb electrode structure, the alignment modulation of nematic liquid crystal in the cell can be realized after the electric field is applied. In different orientation regions, the incident laser beam can deflect at different angles. At the same time, by changing the incident angle of the laser beam, the reflection modulation of the laser beam on the interface of the liquid crystal molecular orientation change can be realized. Based on the above discussion, we then demonstrate the modulation of liquid crystal molecular orientation arrays on nematicon pairs. In different orientation regions of liquid crystal molecules, nematicon pairs can exhibit various combinations of deflections, and these deflection angles are modulable under external fields. Deflection and modulation of nematicon pairs have potential applications in optical routing and optical communication.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
It is well known that the self-focusing effect can balance the diffraction of the transmitted beam in some optical medium and the self-trapping beam can be formed [1]. Such self-trapping beams, also known as optical spatial solitons, can be used to develop many all-optical photonic devices, such as light-written waveguides [2–7], light mode conversion devices in few mode fibers [8], X or Y-directional couplers [9], all-optical logic gates [10–14], beaming random lasers [15,16], multifunctional integrated beam control device [17], etc. With these great practical values, spatial optical soliton has become a research hotspot in nonlinear optics. With the further research on the application of optical spatial solitons in optical communication devices, much effort has contributed to exploring new beam self-trapping mechanism [18–20] and forming low-power optical spatial solitons in low-cost optical materials. Among these materials, liquid crystals and other soft materials are becoming important materials to study the self-trapping behavior of light beams because of their special optical properties and their wide applications in the field of optical information processing [21,22]. In particular, the reorientation mechanism and non-local response of nematic liquid crystal make it become an excellent platform for studying solitons [21]. Moreover, the phenomenon of beam self-trapping in nematic liquid crystals has also obtained an interesting and vivid name – nematicons [23]. The properties and transmission behaviors of nematicons have been systematically studied [24–26]. For example, vortex solitons [27–29], Airy solitons [30] and a series of other exotic optical spatial solitons have been continuously explored. Regardless of the type of solitons, the control of their deflection and routing is an unavoidable and attractive research topic because of its potential applications in all-optical information processing and all-optical interconnection. Physically, the deflection of the nematicons comes from the disturbance or defects of the refractive index distribution in the medium. At present, Nematicons deflection can be realized via applying external light perturbation [11,30], separating top electrode and applying different voltages [31,32], using planar cell with purely transverse or longitudinal modulation [33,34], employing spherical defects such as air bubbles and glass spheres [35,36], and utilizing periodic arrays of vortices to induce the movement of the nematicon [37]. Among them, the electric field is a flexible and convenient way to control the liquid crystal directors. Customized electrode structures can tunable the orientation of liquid crystals in a desirable region, which opens new vistas for novel optical steering elements. For example, transformation of light beams from one-dimensional bulk diffraction to discrete spatial solitons can be realized via comb electrodes [38]; All-optical switching is achieved by combining nematic liquid crystal with topological configurations using periodic electrode pattern [39].
In this paper, we fabricate periodic electrode arrays to form the periodic orientation distribution of liquid crystal molecules in liquid crystal cell. We study the deflection behavior of the nematicons in different orientation regions and the reflection characteristics at different incident angles. In addition, we also use liquid crystal molecules periodic arrangement to achieve nematicon pairs different combinations of deflection, and this deflection angle is modulated under voltage.
2. Experimental materials and devices
In our experiments, we employ the nematic liquid crystal (NLC) E7, with refractive indices of ne = 1.71 and no = 1.52 at λ = 589 nm and room temperature, is filled in a sandwich structure consisting of two parallel slides at a separation of 100 µm. The inner surfaces of both slides are covered with Indium-Tin-Oxide (ITO) to apply low frequency voltage to the cell. In order to align the NLC molecules in the plane (y, z) (as shown in Fig. 1), photoalignment technology is used in our experiment. SD1 molecules are spin-coated on ITO films of both slides. Polarized UV light is used to drive the orientation of SD1 molecules and the planar alignment of the NLC directors parallel to z direction can be obtained. The cell is sealed at the input by an extra glass slide with SD1 along x to prevent both beam depolarization and the formation of a meniscus.
Fig. 1. (a) Optical path diagram for observation of solitons (L, the red(λ=633 nm) cw laser; λ/2, half-wave plates; BS, beam splitter; M1, M2, mirrors; P, polarizers; MO, 20${\times} $ microscope oective; NLC, sample; CCD, charge-coupled device camera) (b) The liquid crystal cell internal electrode pattern.
Nematicons are generated by the optical setup shown in Fig. 1(a), where 633 nm laser is used. The input beam obtains linear polarization (polarized in the x-direction) through a 1/2λ wave plate and a polarizer. Then the laser beam is focused on the NLC layer by the 20× micro-objective into a laser spot of several micrometers in the NLC cell. When we need multiple beams, beam splitters and mirrors in the optical path are used to split our beam into two beams incident on the liquid crystal cell. A charge-coupled device (CCD) is utilized to monitor the propagating beam inside the NLC cell by collecting the light scattered out of the top slide, as sketched in Fig. 1(a). The designed electrode pattern is shown in the Fig. 1(b). The stripe pattern with 910-µm width in the front part is used to generates the deflection change of the nematicons under different voltage. The width of the stripes is wide enough for us to observe the effect of different areas of the stripes on the nematicons. At the end of the stripe pattern there is another uniform region can be used to applied voltage.
3. Simulation of liquid crystal molecular orientation and beam propagation in liquid crystal cells
3.1 Simulation about the arrangement of the liquid crystal molecules at different voltages applied to customized stripe electrodes
We simulated the arrangement of the liquid crystal molecules at different voltages applied to customized stripe electrodes. As Fig. 2 shows, a voltage of 3 V has been chosen as it provides sufficient refractive index variation in the ITO region and etched region while avoiding nonlinear saturation of the nematic liquid crystal, since the nematic liquid crystal is an elastic continuum, a slow change in the tilt angle of liquid crystal molecules at the junction of the ITO region and the etched region can be seen in Fig. 2(b). For higher voltages, the liquid crystal molecules in the middle region of the cell also create a large inclination (Fig. 2(c)(d)), which is not only unfavorable for the formation of nematicon, but the weak refractive index difference makes it difficult to deflect the beam. In addition, we can see that at the boundary of the ITO region and etched region, the y-component of the electrostatic field also plays an important role in the alignment of the liquid crystal molecules, twisting the molecules out of the xz plane.
Fig. 2. The arrangement of the liquid crystal molecules and contour plot of the tilt distribution at different voltages. (a)U = 0 V; (b)U = 3 V; (c)U = 6 V; (d)U = 10 V.
3.2 Simulation of beam propagation in different regions of liquid crystal cells
According to the orientation of the liquid crystal molecules in Fig. 2, we simulated the propagation state of the light beam at different positions in the liquid crystal cell [40,41]. In the simulation, the light beam propagates along the z-direction, and the polarization direction of the light beam is in the x-direction. The electric field is applied to the liquid crystal cell in the x direction. In the simulation, we take the orientation distribution of liquid crystal molecules corresponding to an applied voltage of 3 V. In Fig. 3(a) the light beam traves along the ITO etched region. In this region, the orientation of liquid crystal molecules is perpendicular to the polarization direction of light, and the beam exhibits natural diffraction. When the light beams propagate in the ITO regions, due to the orientation of liquid crystal molecules parallel to the polarization direction of the light, the beams can be self-trapped and propagate in the nematicons, as shown in Fig. 3(c). When the incident position of the beam moves from the ITO etching region to the ITO region, due to the gradual orientation of the liquid crystal molecules between the two regions, the beam will deflect towards the region with a higher refractive index (ITO region) during transmission, as shown in Fig. 3(b). From the simulation results in Fig. 3, we can see that the customized stripe electrodes can form different orientation distributions of liquid crystal molecules after applying an electric field, and this distribution of liquid crystal molecule orientation can regulate the transmission state and deflection of incident light beams in different regions.
Fig. 3. Simulation of beam propagation in different regions of liquid crystal cells (U = 3.0 V, light beam propagates along the z-direction, and the polarization direction of the light beam is in the x-direction.)
4. Variable nematicons deflection
4.1 Position variable nematicons “Refraction”
When voltage is supplied, we can see intriguing events caused by nematicons at the junction of no electrodes and electrodes. Figure 4 shows the variation of the nematicons when incident at different positions of the strip electrode at a voltage of 3 V, and the etching boundary of the electrode is marked with white dotted lines in Fig. 4. This voltage is necessary to assist a few milliwatts of beam to produce nematicons because the rotation of liquid crystal molecules needs to overcome Freédericksz threshold. The beam is clearly diffracted in the electrode-free region because the power of the beam is not sufficient to induce a reorientation of the liquid crystal molecules [Fig. 4(c)]. On the contrary, the beam in the electrode region can induce the redirection of liquid crystal molecules assisted by voltage and thus forms nematicons. When the incident position is about 72 µm from the junction line between the electrode region and electrode-free region, nematicons tend to deflect toward the electrode region, with a deflection angle of about 16 degrees [Fig. 4(a), (e)]. A larger deflection angle of about 26 degrees can be observed when the incident position is about 155 µm from the dividing line as shown in the Fig. 4(b) and (d). In Fig. 4(g) we show the relationship between different incidence positions and deflection angles in the y-axis direction at 3 V, with input power of 1.05 mW. We can see that the beam deflection or refraction angle can be tuned within 26 degrees by changing the incident position of the beam in the electrode etching area.
Fig. 4. Variation of the beam at different incident positions at 3 V, with input power 1.05 mW. The distance between the incident position of the beam and the nearest dotted line is (a) 53.2 µm; (b) 154.7 µm; (c) 417 µm; (d) 164 µm; (e) 66 µm; (f) 230.2 µm, respectively. Between the dotted lines indicates the electrode-free area. The green arrow indicates the distance from the incident position of the beam to the ITO divided line. (g) Relationship between different incidence positions and deflection angles in the y-axis direction.
4.2 Angle variable nematicons “reflection”
Next, we make the beam collide with the dividing line of the electrode pattern, which makes nematicons have a total internal reflection at the interface. And $\alpha $ is the angle between the direction normal to the dividing line and the incident light wave vector $\overrightarrow {\textrm{k}}$, as shown in Fig. 5 (a). When the incident power is 1.2 mW and the electric field with a voltage of 3.5 V is applied to the cell, the total internal reflection (TIR) from nematicons can be observed at the electrode boundary. The beam relies on nonlinear reorientation to maintain its self-confinement when encountering such an interface. The total internal reflection produced by nematicons at different incident angles are shown in Fig. 5(a)-(f). Keeping the incident light power and applied voltage unchanged, we can find that with the increase of the incident angle α, the depth of penetration decreases, and the deflection angle of the beam also decreases. The relationship between α(°) and depth of penetration is shown in Fig. 5(g).
Fig. 5. Total internal reflection of nematicons at different angles of incidence (a) α=68.9°; (b) α=68.9°; (c) α=68.9°; (d) α=68.9°; (e) α=68.9°; (f) α=68.9°. The etched area and ITO area has been marked in the figure. (g) Relationship between $\alpha (^\circ )$ and depth of penetration
Then we keep the angle of incidence α=77°, and observe the reflection behavior of nematicons at the interface with the voltage gradually increasing. When the voltage is 2 V, the refractive index difference caused by the rotation of liquid crystal molecules under the electrode and electrode-free regions is small, making the nematicons insufficient for total internal reflection back to the ITO region [ Fig. 6(a)]. When the voltage rises to 3 V, a significant total internal reflection of the nematicons can be observed [Fig. 6(b)]. The total internal reflection of the nematicons is observed to decrease as the voltage is gradually increased to 5 V. [Figure 6(c)]. When the voltage is increased to 6 V, the tilt angle of the liquid crystal molecules is too large leading to reorientation saturation and thus it is difficult to maintain the soliton state in the ITO etched region [Fig. 6(d)].
Fig. 6. Total internal reflection of nematicons at (a) U = 2 V; (b) U = 3 V; (c) U = 4 V; (d) U = 5 V; ($\alpha = {77^\circ }$.). The area above the dotted line indicates the electrode-free region.
5. Modulation of beam/nematicon pair deflection
Based on the deflection behavior of the light beam in different liquid crystal molecular orientation regions, exploring the propagation behavior of the light beam/nematicons pair will help us to use this array structure to guide the light beams. Because of the artificially introduced refractive index difference due to the etched electrode pattern, the region of low refractive index will repel nematicons. In order to realize the interaction of soliton pairs more clearly, we have changed the width of the etching electrode. The new width of stripes is 193 µm. The power of the two beams is 1.9 mW (the upper one in Fig. 7) and 1.4 mW (the lower one in Fig. 7) respectively. When U = 4 V, such power is sufficient to form nematicons. Different combinations of double nematicons deflections are achieved by changing the incidence position of the two nematicons (see Fig. 7).
In Fig. 7(a), (b), (d), (e), one nematicons propagates in a straight line, the other nematicons deflects upward or downward. In Fig. 7(c), two nematicons deflect outward to two striped electrode regions and in Fig. 7(f) two nematicons deflect inward to a striped electrode region. Between the adjacent white dashed lines in the Fig. 7 is the ITO electrode area.
The LCs in the region between the ITO and the etched ITO region form a refractive index gradient for the e light polarization state when the voltage is applied. However, as the voltage changes, the refractive index gradient caused by this part of the region also changes, and the final result is that the soliton pairs produce different deflection angles at different voltages. We tried the experiment of deflection regulation of soliton pair under different voltages (see Fig. 8). The beams can produce soliton pairs with the help of voltage when the voltage is 2 V, and it can be seen that the soliton pairs at this time have greatly crossed due to the influence of the ITO and etched region's refractive index gradient, as shown in Fig. 8(a). And as the voltage increases, the refractive index gradient between the incident position of the beam and the ITO region is gradually decreasing, resulting in two beams gradually moving away from each other and becoming difficult to maintain the nematicons, as shown in Fig. 8(b), (c), (d), (e), and (f).
Fig. 8. Nematicons routing direction via voltage tuning. The upper beam power is 1.9 mw, the lower beam power is 1.4 mw. (a) U = 2 V; (b) U = 2.5 V; (c) U = 3 V; (d) U = 3.5 V; (e) U = 4 V; (f) U = 5 V. Between the adjacent white dashed lines in the figure is the ITO electrode area.
Moreover, in order to try more possibilities of different periodic electrode stripes on the regulation of nematicons pairs, we also tried the modulation effect under 400 µm stripes (see Supplement 1). The experimental results show that the refractive index gradient formed by the liquid crystal molecules induced by the electric field can well modulate the angles of the nematicon pairs at a suitable electrode width.
6. Conclusions
In this paper, we constructed a kind of liquid crystal cell structure with strip electrode pattern, and then realized the periodic alignment array of liquid crystal molecules. In different orientation regions of liquid crystal molecules, the incident laser beam can deflect at different angles. In our experiment, with electric field of 3 V and input power 1.05 mW, the beam deflection or refraction angle can be tuned within ±26 degrees by changing the incident position of the beam in the electrode etching area. At the same time, the effects of incident angle and applied voltage on beam deflection are also studied. At the electrode interface, different deflection angles and penetration depths can be realized with tunable incident angle and electric field. Finally, based on the above results, we study the transport behavior of nematicon pairs in liquid crystal molecular alignment arrays and the deflection and modulation of the nematicons pairs can be realized. These results can potentially be used to realize the light-induced guided-wave circuits by tuning beam position and electric field.
Funding
National Key Research and Development Program of China (2022YFA1203700); National Natural Science Foundation of China (51873060, 61822504, 62035008, 62275081); Scientific Committee of Shanghai (2021-01-07-00-02-E00107); Shanghai Education Development Foundation and Shanghai Municipal Education Commission (21SG29).
Acknowledgments
The authors acknowledge the support from the National Key Research and Development Program of China (2022YFA1203700), National Science Foundation of China (grantnos. 61822504, 51873060, 62035008, 52272109 and 62275081), Innovation Program of Shanghai Municipal Education Commission, Scientific Committee of Shanghai (2021-01-07-00-02-E00107), and “Shuguang Program” of Shanghai Education Development Foundation and Shanghai Municipal Education Commission (21SG29).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003). [CrossRef]
2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286(5444), 1518–1523 (1999). [CrossRef]
3. A. Bezryadina, T. Hansson, R. Gautam, B. Wetzel, G. Siggins, A. Kalmbach, J. Lamstein, D. Gallardo, E. J. Carpenter, and A. Ichimura, “Nonlinear Self-Action of Light through Biological Suspensions,” Phys. Rev. Lett. 119(5), 058101 (2017). [CrossRef]
4. A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93(17), 171104 (2008). [CrossRef]
5. A. Piccardi, A. Alberucci, N. Kravets, O. Buchnev, and G. Assanto, “Power-controlled transition from standard to negative refraction in reorientational soft matter,” Nat. Commun. 5(1), 5533 (2014). [CrossRef]
6. N. Karimi, M. Virkki, A. Alberucci, O. Buchnev, M. Kauranen, A. Priimagi, and G. Assanto, “Molding Optical Waveguides with Nematicons,” Adv. Opt. Mater. 5, 1700199 (2017). [CrossRef]
7. A. d’Alessandro and R. Asquini, “Light Propagation in Confined Nematic Liquid Crystals and Device Applications,” Appl. Sci. 11(18), 8713 (2021). [CrossRef]
8. H. Y. Zhu, W. L. Zhang, J. C. Zhang, X. X. Chen, R. Ma, and Y. J. Rao, “Nematicon-Assisted Mode Coupling to Optical Fibers,” IEEE J. Sel. Top. Quantum Electron. 27(2), 6 (2021). [CrossRef]
9. M. Peccianti, K. Brzdąkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27(16), 1460–1462 (2002). [CrossRef]
10. M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81(18), 3335–3337 (2002). [CrossRef]
11. S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photonics Technol. Lett. 18(12), 1287–1289 (2006). [CrossRef]
12. A. A. Armando Piccardi, Umberto Bortolozzo, and Stefania Residori, “Soliton gating and switching in liquid crystal light valve,” Appl. Phys. Lett. 96(7), 071104 (2010). [CrossRef]
13. E. G. Anagha and R. K. Jeyachitra, “Review on all-optical logic gates: design techniques and classifications - heading toward high-speed optical integrated circuits,” Opt. Eng. 61(06), 31 (2022). [CrossRef]
14. A. Javed, T. Uthayakumar, and U. Al Khawaja, “Simulating an all-optical quantum controlled-NOT gate using soliton scattering by a reflectionless potential well,” Phys. Lett. A 429, 127949 (2022). [CrossRef]
15. S. Perumbilavil, A. Piccardi, O. Buchnev, G. Strangi, M. Kauranen, and G. Assanto, “Spatial solitons to mold random lasers in nematic liquid crystals Invited,” Opt. Mater. Express 8(12), 3864–3878 (2018). [CrossRef]
16. S. Perumbilavil, A. Piccardi, R. Barboza, O. Buchnev, M. Kauranen, G. Strangi, and G. Assanto, “Beaming random lasers with soliton control,” Nat. Commun. 9(1), 3863 (2018). [CrossRef]
17. U. Mur, M. Ravnik, and D. Se, “Controllable shifting, steering, and expanding of light beam based on multi-layer liquid-crystal cells,” Sci. Rep. 12(1), 352 (2022). [CrossRef]
18. S. Slussarenko, A. Alberucci, C. P. Jisha, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Photonics 10(9), 571–575 (2016). [CrossRef]
19. C. P. Jisha, S. Nolte, and A. Alberucci, “Geometric Phase in Optics: From Wavefront Manipulation to Waveguiding,” Laser Photonics Rev. 15, 2100003 (2021). [CrossRef]
20. G. Poy, A. J. Hess, A. J. Seracuse, M. Paul, S. Zumer, and II Smalyukh, “Interaction and co-assembly of optical and topological solitons,” Nat. Photonics 16(6), 454–461 (2022). [CrossRef]
21. G. Assanto, “Nematicons: reorientational solitons from optics to photonics,” Liq. Cryst. Rev. 6(2), 170–194 (2018). [CrossRef]
22. C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005). [CrossRef]
23. M. Peccianti and G. Assanto, “Nematicons,” Phys. Rep. 516(4-5), 147–208 (2012). [CrossRef]
24. G. Liang, F. J. Shu, W. J. Cheng, and L. B. Jiao, “Nonlinearity-mediated collimation of optical beams,” Opt. Express 30(7), 10770–10778 (2022). [CrossRef]
25. G. Assanto and N. F. Smyth, “Self-confined light waves in nematic liquid crystals,” Phys. D 402, 132182 (2020). [CrossRef]
26. S. Aya and F. Araoka, “Kinetics of motile solitons in nematic liquid crystals,” Nat. Commun. 11(1), 3248 (2020). [CrossRef]
27. U. A. Laudyn, M. Kwasny, M. A. Karpierz, and G. Assanto, “Vortex nematicons in planar cells,” Opt. Express 28(6), 8282–8290 (2020). [CrossRef]
28. P. S. Jung, Y. V. Izdebskaya, V. G. Shvedov, D. N. Christodoulides, and W. Krolikowski, “Formation and stability of vortex solitons in nematic liquid crystals,” Opt. Lett. 46(1), 62–65 (2021). [CrossRef]
29. H. C. Zhang, T. Zhou, and C. Q. Dai, “Stabilization of higher-order vortex solitons by means of nonlocal nonlinearity,” Phys. Rev. A 105(1), 013520 (2022). [CrossRef]
30. Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017). [CrossRef]
31. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2(11), 737–742 (2006). [CrossRef]
32. R. Barboza, A. Alberucci, and G. Assanto, “Large electro-optic beam steering with nematicons,” Opt. Lett. 36(14), 2725–2727 (2011). [CrossRef]
33. U. A. Laudyn, M. Kwaśny, F. A. Sala, M. A. Karpierz, N. F. Smyth, and G. Assanto, “Curved optical solitons subject to transverse acceleration in reorientational soft matter,” Sci. Rep. 7(1), 12385 (2017). [CrossRef]
34. U. A. Laudyn, M. Kwasny, M. A. Karpierz, N. F. Smyth, and G. Assanto, “Accelerated optical solitons in reorientational media with transverse invariance and longitudinally modulated birefringence,” Phys. Rev. A 98(2), 023810 (2018). [CrossRef]
35. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Deflection of nematicons through interaction with dielectric particles,” J. Opt. Soc. Am. B 30(6), 1432–1437 (2013). [CrossRef]
36. Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Soliton bending and routing induced by interaction with curved surfaces in nematic liquid crystals,” Opt. Lett. 35(10), 1692–1694 (2010). [CrossRef]
37. C. Meng, J.-S. Wu, and I. I. Smalyukh, “Topological steering of light by nematic vortices and analogy to cosmic strings,” Nat. Mater. 22(1), 64–72 (2023). [CrossRef]
38. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29(13), 1530–1532 (2004). [CrossRef]
39. P. S. Jung, M. Parto, G. G. Pyrialakos, H. Nasari, K. Rutkowska, M. Trippenbach, M. Khajavikhan, W. Krolikowski, and D. N. Christodoulides, “Optical Thouless pumping transport and nonlinear switching in a topological low-dimensional discrete nematic liquid crystal array,” Phys. Rev. A 105(1), 013513 (2022). [CrossRef]
40. A. Alberucci, A. Piccardi, R. Barboza, O. Buchnev, M. Kaczmarek, and G. Assanto, “Interactions of accessible solitons with interfaces in anisotropic media: the case of uniaxial nematic liquid crystals,” New J. Phys. 15(4), 043011 (2013). [CrossRef]
41. J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A: Pure Appl. Opt. 8(2), 214–220 (2006). [CrossRef]
