Figure 1

Scanning-mirror laser tweezers setup used to generate 2D patterns of torons. The CW collimated ytterbium fiber laser is controlled with a pair of scanning mirrors. Lenses L1 and L2 in a 4f configuration direct the beam into the back aperture of an objective. The scanning-mirror angles determine the 2D positions of the focused beam within the sample. The dichroic mirror (DM) allows for simultaneous laser scanning for generation of torons and POM studies of the optically controlled samples. The crossed polarizer and analyzer are used to obtain POM textures.Reuse & Permissions

Figure 2

Stokes polarimetry phase characterization setup used to map the phase of diffracted laser light. A Mach-Zehnder-type interferometer is implemented by dividing a He-Ne laser beam into two paths and then recombining these two beams by beam splitters BS1 and BS2, respectively. The block and iris allow for selecting the diffraction order of interest. M1–M3 are silver mirrors used for redirecting the beam. The polarization of the beam is controlled by polarizers P1–P3 and the λ/4 plate. The beam is collimated by the use of lens pairs L1-L2 and L3-L4.Reuse & Permissions

Figure 3

Experimental and computer-simulated POM textures of torons. (a),(c) Experimental and (b),(d) corresponding computer-simulated POM textures of laser-generated torons of different size in a (a),(b) 9-μm-thick cell with CNLC pitch of about 10 μm and (c),(d) 3.5-μm-thick sample with pitch of about 5 μm. The used CNLC materials were mixtures of the nematic host E7 and chiral dopant CB15 at concentrations corresponding to pitch values of 5 and 10 μm. In computer simulations, we utilized values of ordinary refractive index $n_o$ = 1.5, extraordinary refractive index $n_e$ = 1.7, and optical anisotropy Δ$n$ = 0.2, closely matching that of the used LC mixtures. The crossed polarizer and analyzer are shown by white double arrows and marked by “P” and “A”, respectively.Reuse & Permissions

Figure 4

Laser-generated torons in a confined CNLC. (a) Schematic of a toron n(r) configuration consisting of two hyperbolic point defects (blue dots on the top and bottom of the schematic) and twist-escaped disclination loop (red line in the schematic midplane) that conserve the total topological charge of structures embedded into a cell with a uniform vertical far-field director. (b) Vertical $xz$ computer-simulated cross section of n(r) with two blue dots (top and bottom) marking the hyperbolic point defects and two red dots (on the left and right sides of the schematic) marking the intersection of the disclination loop with the cross section. (c),(d) Computer-simulated (c) and experimental (d) 3PEF-PM textures of a laser-generated toron in a 5-μm-thick cell visualized using the paraview software for sample volumes of high-intensity signal representing azimuthal deviations of the director away from vertical and towards the direction of probing 3PEF-PM polarization [48]. The intensity threshold was adjusted to show a continuous 3D representation of the director's in-plane orientation for which four experimental data sets (corresponding to four different polarizations of 3PEF-PM excitation light) were used. The different colors (grayscale levels) visualize sample regions yielding high intensity of 3PEF-PM signal when imaged using different azimuthal orientations of linear polarization of the 3PEF-PM laser excitation light (as depicted in the inset for four angles that the linear 3PEF-PM polarization makes with the $x$ axis). The lateral size of the toron shown in (d) is about 5 μm. (e) Vector-field representation of the axially symmetric toron structure that yields the two point defects having opposite hedgehog charges and the rest of the 3D texture having zero hedgehog charge. For the selected direction of vector field (arrows) at the global point of reference (chosen to be the pointReuse & Permissions

Figure 4-a

defect at the bottom part of the structure) in (e), the hyperbolic hedgehog of “$+$1” charge is marked by a yellow solid circle (bottom part of the schematic) while the defect of “$-$1” charge is marked by a blue solid circle (top part of the schematic). The white solid circles indicate the location of the axis of the looped double-twist cylinder intersecting the schematic (e) in its midplane. Note that (due to nonpolar symmetry of the director) the choice of the vector directions is arbitrary and that the signs of both of the hyperbolic point defects can be reversed by changing the direction of arrows in the vector field.Reuse & Permissions

Figure 5

Periodic arrays of torons as voltage-tunable phase gratings. POM textures of (a) square and (b) hexagonal periodic arrays of laser-generated torons with the insets showing the corresponding diffraction patterns obtained using a 633-nm HeNe laser. Thin lines in-between the beams of the same diffraction orders shown in (b) are guides for the eye and have colors (grayscale levels) matched to the colors (grayscale levels) of corresponding data points presented in (d). (c) Horizontal intensity profile of the square-periodic diffraction pattern along the C-C line marked on (a). (d) Intensity of light in the hexagonal diffraction pattern vs applied voltage across the cell with the phase grating of torons for different diffraction orders. The crossed polarizer and analyzer are shown by blue double arrows and marked by “P” and “A”, respectively.Reuse & Permissions

Figure 6

Hexagonal periodic arrays of torons as voltage-tunable phase gratings. (a)–(f) POM textures of square-periodic arrays of laser-generated torons with the insets showing corresponding diffraction patterns obtained using a circularly polarized 633 nm HeNe laser; the applied voltages are marked in the bottom left corners of the images. The crossed polarizer and analyzer are shown by white double arrows and marked by “P” and “A”, respectively.Reuse & Permissions

Figure 7

Square-periodic arrays of torons as voltage-tunable phase diffraction gratings. (a)–(f) POM textures of square-periodic arrays of laser-generated torons with the insets showing corresponding diffraction patterns obtained with a 633 nm circularly polarized HeNe laser. The orientation of crossed polarizer and analyzer at all voltages is shown in (d) by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions

Figure 8

Computer-simulated diffraction patterns obtained for (a) square-periodic and (b) hexagonal arrays of torons.Reuse & Permissions

Figure 9

Control of phase singularities by defects in toron gratings. (a) POM texture of a square-periodic array of torons with an edge dislocation labeled by a red arrow (located near the center of the image) and (b) a corresponding diffraction pattern with diffraction orders marked on the image and a large-area view containing higher diffraction order beams shown in the inset. (c) An enlarged image of the first-order diffracted beam and (d) an interference pattern of the first-order diffraction and reference Gaussian beams obtained for their oblique incidence. (e) Phase profile of the first-order beam shown in (c). (f) An enlarged image of a second-order diffracted beam and (g) interference pattern of this beam and a reference Gaussian beam obtained for their oblique incidence. (h) Phase profile of the second-order beam shown in (f). Three-dimensional plots of the phase vs coordinates in the lateral plane of the diffracted beams of first (i) and second (j) orders, respectively, with green vertical lines showing locations of the phase singularities. The orientation of crossed polarizer and analyzer in (a) is shown by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions

Figure 10

Hexagonal arrays of torons with defects and the corresponding diffraction patterns with phase singularities. (a),(b) POM textures of hexagonal periodic array of torons with dislocations at applied voltages of (a) $U$ = 0 and (b) $U$ = 0.8 V${}_{\mathrm{p}-\mathrm{p}}$. The inset of (b) shows the location of 5–7 defects in the hexagonal toron array by means of a Voronoi diagram. (c) Diffraction pattern obtained using a grating shown in (a). (d),(e) Phase profiles of the first-order diffraction beams marked in (c). The orientation of crossed polarizer and analyzer in POM textures is shown in (a) by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions

Figure 11

Square-periodic array of torons with defects and the corresponding diffraction pattern. (a) POM texture of the array and the corresponding diffraction pattern shown in the inset obtained for a 633-nm HeNe laser; the two diffracted beams in different diffraction orders are marked in the inset as $m_x$ = 1, $m_y$ = 0 and $m_x$ = 1, $m_y$ = 1. (b) Phase profile (top) and an interference pattern of the $m_x$ = 1, $m_y$ = 0 diffraction beam and a reference Gaussian beam obtained for their oblique incidence (bottom). (c) A 3D representation of the phase vs coordinates in the lateral plane of the diffracted beam of the $m_x$ = 1, $m_y$ = 0 diffraction order. (d) Phase profile (top) and an interference pattern (bottom) of the $m_x$ = 1, $m_y$ = 1 diffraction order beam and a reference Gaussian beam obtained for their oblique incidence (bottom). (e) A 3D representation of the phase vs coordinates pattern in the lateral plane of the diffracted beams of the $m_x$ = 1, $m_y$ = 1 diffraction order beam. (f)–(h) Computer simulation of intensity (f) and phase distribution (g) of the diffraction pattern plotted separately [(f),(g)] and then overlaid on top of each other (h). The orientation of crossed polarizer and analyzer in (a) is shown by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions

Figure 12

Computer-simulated (a),(d) intensity and (b),(e) phase distributions for diffraction gratings that correspond to the experimental counterparts shown in Figs. 10 and 9, respectively. (c),(f) Overlaid plots of lateral intensity and phase distributions of the diffraction patterns obtained by superimposing parts (a) with (b) and (d) with (e), respectively.Reuse & Permissions

Figure 13

Square-periodic arrays of torons with defects as voltage-tunable phase gratings. (a)–(d) POM textures of square-periodic arrays of laser-generated torons at different applied voltages (marked on the images) with the insets showing corresponding diffraction patterns obtained with a 633-nm HeNe laser. The orientation of crossed polarizer and analyzer in the POM textures is shown in (c) by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions

Figure 14

POM images of (a) a honeycomb lattice of torons and (b) a square-periodic array of torons with a bifurcated homeotropic channel. The orientation of crossed polarizer and analyzer in these POM textures is shown by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions

Figure 15

(a)–(c) POM images of laser-generated quasi-crystal-like structures of torons obtained by mimicking quasicrystal structures described in Ref. 50. The lateral size of torons is about 10 μm in all POM images. The orientation of crossed polarizer and analyzer in the POM textures is shown by use of blue double arrows marked by “P” and “A”, respectively.Reuse & Permissions