Harnessing Optical Vortex Lattices in Nematic Liquid Crystals

By creating self-induced vortexlike defects in the nematic liquid crystal layer of a light valve, we demonstrate the realization of programable lattices of optical vortices with arbitrary distribution in space. On each lattice site, every matter vortex acts as a photonic spin-to-orbital momentum coupler and an array of circularly polarized input beams is converted into an output array of vortex beams with topological charges consistent with the matter lattice. The vortex arrangements are explained on the basis of light-induced matter defects of both signs and consistent topological rules.

Figure 1

Schematic representation of (a) optical vortex self-induction and (b) spin-to-orbital angular momentum transfer through the induced defect: An RHP (LHP) beam is converted into an LHP (RHP) beam with a phase singularity of charge $-2$ ($+2$); experimental interferograms are shown beside the respective panels. (c) A matter defect observed under crossed polarizers (left); spatially resolved polarimetry (middle); reconstructed molecular organization (right). (d) Spin-to-orbital angular momentum conversion efficiency measured versus voltage bias and for various input intensities.

Figure 2

Experimental setup. Obj, objective; BS, beam splitter; M, mirror; SLM, spatial light modulator; NDF, neutral density filter; F, filter transmitting the red components of the white light source for background illumination of the sample; POL, polarizer; HWP, half-wave plate; QWP, quarter-wave plate; PH, pinhole; L, lens; and CCD, charge-coupled device camera. Bottom inset: Example of square modulation mask as input to the SLM. Upper inset: Enlarged view of the sample observed under crossed polarizers when illuminated by the square grid (bright spots from the DPSS green laser) and white light background; on the left is an enlarged view of a single vortex.

Figure 3

Hexagonal vortex lattices: (a) Laser intensity distribution $V=19\mathrm{V}$; (b),(c) white light images under crossed polarizers, (b) independent vortices $V=18\mathrm{V}$ and (c) fully coupled lattice $V=22\mathrm{V}$; (d) interferogram $V=12\mathrm{V}$; and (e),(f) spatially resolved polarimetry, (e) $V=22\mathrm{V}$ and (f) $V=18\mathrm{V}$. The dashed lines mark the lattice structure, and the circles indicate the positions of the addressing light spots; the input intensity is $I=250\mu \mathrm{W}/{\mathrm{cm}}^2$.

Figure 4

Vortex lattices with various spatial distributions; the images were taken through crossed polarizers; input intensity $I=250\mu \mathrm{W}/{\mathrm{cm}}^2$. Squares for (a)–(c) $V=14$, 18, 22 V, hexagons with a defect in the center for (d)–(f) $V=14$, 18, 22 V, and Penrose lattice for (g)–(i) $V=14$, 18, 22 V.

Figure 5

Numerical distribution of (a) the molecular director and (b) energy density around the defect; lengths are normalized to the cell thickness. (c) Simulated vortex structure and spatial arrangement; two $+1$ vortices are initially addressed, with a $-1$ spontaneously appearing in between them. (d) Experimental demonstration of spin-to-orbital angular momentum transfer. Left panels: A $q=+1$ defect is created in the center of an illuminated area; a $q=-1$ defect is generated in between two spots. Central (right) panels: For an input LHP (RHP), the $q=+1$ defect yields an optical vortex with charge $-2$ ($+2$), and the $q=-1$ defect produces an optical vortex with charge $+2$ ($-2$).