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Topological steering of light by nematic vortices and analogy to cosmic strings

Abstract

Liquid crystals are widely known for their technological uses in displays, electro-optics, photonics and nonlinear optics, but these applications typically rely on defining and switching non-topological spatial patterns of the optical axis. Here, we demonstrate how a liquid crystal’s optical axis patterns with singular vortex lines can robustly steer beams of light. External stimuli, including an electric field and light itself, allow us to reconfigure these unusual light–matter interactions. Periodic arrays of vortices obtained by photo-patterning enable the vortex-mediated fission of optical solitons, yielding their lightning-like propagation patterns. Predesigned patterns and spatial trajectories of vortex lines in high-birefringence liquid crystals can steer light into closed loops or even knots. Our vortex lattices might find technological uses in beam steering, telecommunications, virtual reality implementations and anticounterfeiting, as well as possibly offering a model system for probing the interaction of light with defects, including the theoretically predicted, imagination-capturing light-steering action of cosmic strings, elusive defects in cosmology.

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Fig. 1: Topological steering of light by LC vortices.
Fig. 2: Launching of optical solitons into a cell with LC vortices.
Fig. 3: Interactions of optical solitons with arrays of vortices.
Fig. 4: Vortex-mediated fission of optical solitons.
Fig. 5: Electrical and optical control of light deflection.
Fig. 6: Cosmology analogy, light deflection by LC vortices and effective geometry.

Data availability

All data generated or analysed during this study are included in the published Article, and the material and geometric parameters used in Landau–de Gennes modelling as well as the details of sample preparation and nematicon excitation are provided in the Supplementary Information. Source data are provided with this paper.

Code availability

The codes used for Landau–de Gennes modelling and simulation of polarized optical micrographs are published in ref. 73. The custom Lagrangian and Hamiltonion ray-tracing codes are provided in the Supplementary Information.

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Acknowledgements

We thank A. Hess, B. Li, H.-S. Kwok and T. Lee for discussions, as well as for providing technical assistance and materials. I.I.S. thanks the International Institute for Sustainability with Knotted Chiral Meta Matter with headquarters at Hiroshima University for hospitality during his stay in Japan, during which the initial version of this paper was written. Regarding funding, this research was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-SC0019293 with the University of Colorado at Boulder (C.M., J.-S.W. and I.I.S.). I.I.S. and C.M. also acknowledge the support of the US National Science Foundation (DMR-1810513) in the initial stages of this project.

Author information

Authors and Affiliations

Authors

Contributions

C.M. prepared the photo-patterned samples and performed the experiments. J.-S.W. performed the theoretical analysis and numerical simulations of ray tracing and polarizing optical micrographs, with input from I.I.S.; C.M. and I.I.S. analysed data. I.I.S. conceived, designed and directed the research, provided funding and wrote the manuscript, with input from all authors.

Corresponding author

Correspondence to Ivan I. Smalyukh.

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Nature Materials thanks Ruth Durrer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Principles of photo-patterning of LC vortices.

a, Optical response of azobenzene dye molecules to linearly polarized blue light, with the dye molecular structure shown above. When illuminated by the blue light, randomly oriented rodlike dye molecules collectively re-align to point orthogonally to the excitation light’s polarization direction. b, Schematic setup of a custom-built photo-patterning system. The photo-patterning is based on a microdisplay used to project a computer-controlled blue-light pattern that is relayed by three lenses, linearly polarized by an inserted half-wave plate and a polarizer, and then projected to the back aperture of an objective, which focuses the pattern on the azobenzene dye layers. The pattern size is defined by the magnification of the objective. c, Vortices and their arrays are generated via sequential illuminations of patterns corresponding to narrow angular sectors of azimuthal orientations of dye molecules, synchronized with the corresponding control of the linear polarization (double white arrows) and its azimuthal orientation angle (Φp) to define dye molecule orientations at surfaces and, thus, define boundary conditions for n(r).

Extended Data Fig. 2 Light steering in LCs with different Δn.

Numerically simulated optical trajectories (red/cyan lines) of light passing near the (a) k = 1, Ψ0 = π/2 and (b) k = 1, Ψ0 = 0 vortices formed within LCs with Δn of 0.015, 0.2, 0.8 and 1.7. The beams incident from the left side are initially parallel to the horizontal edges of the structures. White cylinders depict the spatial orientation of n(r). c, Numerically simulated deflection angle θ for such systems, with the value of θ being positive (negative) for deflection towards (away from) the center, as defined in the insets. Vertical dashed lines mark the material parameters for LCs including CCN-47 (Δn = 0.015), E7 (Δn = 0.24) and SHB-6 (Δn = 0.79). no = 1.53 for all simulations.

Source data

Extended Data Fig. 3 Guiding of light by various topological vortices.

a-d, (left) Computer-simulated light trajectories (red lines) overlaid atop of corresponding n(r) (white cylinders) and (right) experimental light propagation trajectories of 1064 nm laser beams emerging from different entry points (cyan arrows with yellow frame). The studied vortices from (a) to (d) are defects with: (a) k = -1, Ψ0 = 0; (b) k = -1, Ψ0 = π/2; (c) k = -1/2, Ψ0 = 0 and (d) k = 1/2, Ψ0 = π/8, respectively. The simulated beam trajectories are obtained using the Hamiltonian approach of ray tracing (Methods). All scale bars are 100 μm.

Extended Data Fig. 4 Optical soliton steering by high-winding-number vortices.

a-d, (left) Polarizing optical micrographs of vortices with winding numbers of k = 3, 4, 6 and 9 (marked on images) obtained between crossed polarizers (double yellow arrows). (3 middle panels) Experimental optical soliton trajectories for 650 nm laser beam launched at sites marked by red arrows with yellow frame, which closely match (right) computer-simulated trajectories plotted atop of associated n(r) (white cylinders). All scale bars are 100 μm.

Extended Data Fig. 5 Optical soliton steering by topological vortex dimers.

a-b, (left) Polarizing optical micrographs of vortex dimers formed by vortices with (a) k = -2 and k = 2 and (b) k = 3 and k = -3, obtained under crossed polarizers (double yellow arrows). (2 middle panels) Optical soliton trajectories of 650 nm laser beam closely match (right) computer-simulated trajectories shown atop of the associated n(r) (white cylinders). All scale bars are 200 μm.

Extended Data Fig. 6 Optical characterization of a vortex array.

a-b, Polarizing (a) and brightfield (b) optical micrographs of a 3×2 array containing a vortex with k = 3, Ψ0 = 0 surrounded by 12 bright and 12 dark brushes corresponding to 6π azimuthal n(r)-rotation in (a). The dark spots in (b) are half-integer defects to which cores of k = 3 (hexagons of dark spots) and k = -1 (pairs of dark spots) vortices split. Double white arrows mark orientations of crossed polarizers. Both scale bars are 200 μm.

Extended Data Fig. 7 Optical soliton propagation within arrays with k = 1 vortices.

a, Lightning-like optical soliton propagations after slightly shifting 1064 nm laser beam with respect to the array. b-e, Meandering solitonic light trajectories generated by 650 nm (b,d) and 1064 nm (c,e) laser beams. The trajectories are captured (upper) with and (lower) without illumination backlight of the microscope. Double yellow arrows mark orientations of crossed polarizers. All scale bars are 200 μm.

Extended Data Fig. 8 Controlled fission of optical solitons.

a-d, Interaction of light beams with arrays containing (a) k = 3, Ψ0 = π/2 vortices and (b-d) k = 3, Ψ0 = 0 vortices: (left panels) polarizing optical micrographs and (other panels) the corresponding experimental beam trajectories. Rightmost panel in (a) shows superposition of the individual light trajectories distinguished by false colours. All different soliton trajectories and fission events are obtained by slight shifting of the launching site of the beam, as marked by colored arrows with yellow frame. All scale bars are 200 μm, showing dimensions of the studied vortex arrays. The angular step of photo-patterning of vortex arrays is 11.25° in (a) 45° in (b,d) and 22.5° in (c).

Extended Data Fig. 9 Electric control of topological steering by an array with k = 3 vortices.

a, Evolution of brightfield micrographs when applied voltage increases from 0 to 5.0 V, as marked on images. b-d, Light deflection trajectories at voltages corresponding to (a) when beams are launched from positions y1, y2 and y3 marked in (a). Blue (a) and white (b-d) dashed lines mark the edge of indium tin oxide electrodes, with the right-side of the sample having the electrode and being responsive to applied voltage. All scale bars are 200 μm.

Extended Data Fig. 10 Engineering knots of light beams with the help of knotted vortices.

a, Waveguiding of a light beam (red line) in a high-index knotted region defined by a pair of trefoil-knot-shaped loops of vortices (black lines). b, A zoom-in view of the tube-like high-index region forming the trefoil knot shown in (a). c, A detailed view of the director structure and refractive index distribution within a cross-section of the knotted tubelike region, where the right-side insets show details of director rotations around the vortex lines and the bottom inset provides the colour scheme for the refractive index varying between the ordinary and extraordinary values. d, The three-dimensional distribution of the refractive index variation and e, its corresponding variation in a cross-sectional plane depicted in (d), with the high-index regions corresponding to intersections of the knotlike high-index structure with the plane, as clearly seen. Once coupled to such a knot-shaped topological waveguide, light would be confined to propagate within the knot.

Supplementary information

Supplementary Information

Supplementary Notes 1–5, Figs. 1–3 and Codes 1 and 2.

Supplementary Video 1

Polarizing optical microscopy textures of LC vortices with increasing winding number as marked in the lower-left corners. The video was obtained by rotating the sample between fixed crossed polarizers (double white arrows). Scale bars are 50 μm.

Supplementary Video 2

Optical soliton trajectories on top of polarizing micrographs for 650 nm (left) and 1,064 nm (right) laser beams while shifting the beam relative to the vortex with k = 1, Ψ0 = 0. Double white arrows mark orientations of crossed polarizers. Scale bars are 100 μm.

Supplementary Video 3

Polarizing optical microscopy texture of an array formed by k = 1 and k = –1/2 vortices arranged into quasi-hexagonal periodic lattices. The video was obtained by rotating the sample between fixed crossed polarizers (double white arrows). Scale bar is 100 μm. The video is shown in real time.

Supplementary Video 4

Polarizing optical microscopy textures of arrays with k = 3, Ψ0 = 0 vortices (left) and k = 3, Ψ0 = π/2 vortices (right). The videos were obtained by rotating crossed polarizers (initial orientations marked by double yellow arrows) relative to the sample with photo-patterned arrays. Scale bars are 200 μm. Polarizing optical microscopy texture of an array formed by k = 1 and k = –1/2 vortices arranged into quasi-hexagonal periodic lattices. The video was obtained by rotating the sample between fixed crossed polarizers (double white arrows). Scale bar is 100 μm. The video is shown in real time.

Supplementary Video 5

Soliton trajectories for a 1,064 nm beam launched from different sites of an array with k = 3, Ψ0 = 0 vortices. Fine translation of sample in the vicinity of these two sites yields 180° (left) and 90° (right) light deflection, respectively. Double white arrows mark orientations of crossed polarizers. Scale bars are 200 μm.

Supplementary Video 6

Optical soliton trajectories on top of polarizing optical micrographs. The solitonic trajectories are generated by a 1,064 nm laser beam shifted upward (left) and downward (right) relative to the array with k = 1, Ψ0 = 0 vortices. Double white arrows mark orientations of crossed polarizers. Scale bars are 200 μm.

Supplementary Video 7

Reconfiguration of split-core defects using laser tweezers. The ON/OFF status of the laser tweezers is marked in the lower-left corner. Scale bar is 100 μm.

Source data

Source Data Fig. 2

Source data for Fig. 2e,h.

Source Data Fig. 5

Source data for Fig. 5c,e.

Source Data Fig. 6

Source data for Fig. 6d.

Source Data Ext. Data Fig. 2

Source data for Extended Data Fig. 2c.

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Meng, C., Wu, JS. & Smalyukh, I.I. Topological steering of light by nematic vortices and analogy to cosmic strings. Nat. Mater. 22, 64–72 (2023). https://doi.org/10.1038/s41563-022-01414-y

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