Tunable Optical Vortex Arrays from a Single Nematic Topological Defect

We report on the generation of tunable structured light fields endowed with various sets of phase singularities from a single topological defect in a nematic liquid crystal mesophase. The experimental demonstration relies on the use of electric field-induced nonsingular topological defects called “umbilics.”

Figure 1

Typical meridional [panel (a)] and equatorial [$z=L/2$, panel (b)] cross sections of the director field of a $s=+1$ umbilic of the splay type; $r_c$ is the core radius of the defect and $L$ is the nematic film thickness. The color scale refers to the amplitude of the transverse part of $\mathbf{n}$, i.e., $\sin \vartheta$, see Eq. (1), whereas the white segments represent the local orientation of the director. (c) Calculated reduced amplitude $a$ of the tilt angle $\vartheta$ nearby the defect located at $r=0$. (d) Calculated reduced core radius $r_c/L$ vs reduced voltage $\tilde{U}$.

Figure 2

(a) Experimental intensity patterns of the optical transmission between parallel linear polarizers (here the passing direction is set along the $x$ axis) when $\tilde{U}$ is chosen to fulfill the bright field condition for the optical vortex arrays $A_m$, $m=(1,2,3,4)$ (see text for details). (b) Corresponding intensity patterns calculated from Eq. (9).

Figure 3

Calculated phase profile of the output light field under monochromatic illumination and parallel polarizers imaging for bright field arrays $A_m$ with $m=1$ (a), 2 (b), and 3 (c). The location of individual single charge optical vortex is indicated by circles (topological charge $+1$) and stars (topological charge $-1$). The calculated region of interest corresponds to a $6r_c\times 6r_c$ area.

Figure 4

Upper row: transmission between crossed circular polarizers at $\lambda =532\mathrm{nm}$ associated to the bright field arrays $A_1$, $A_2$, and $A_3$. Scale bar is $20\mu \mathrm{m}$. Bottom row: confrontation between experimental data (solid line) and the model (dashed line) for the radial intensity profile under crossed circular polarizers. Diamonds markers refer to the condition $\Delta (r_n)=(2n+1)\pi$ where $r_n$ corresponds to the radius of the circle on which lies the four single charge optical vortices of $Q_n$.

Figure 5

Wavelength tunable bright field arrays illustrated in the case $m=2$ using red (R), green (G), and blue (B) set of illumination wavelengths ${\lambda }_{\mathrm{R}}=632.8\mathrm{nm}$ (a), ${\lambda }_{\mathrm{G}}=532\mathrm{nm}$ (b), and ${\lambda }_{\mathrm{B}}=488\mathrm{nm}$ (c).

Figure 6

Upper row: electrically-controlled radial position in the ($x$, $y$) plane of quadrupoles $Q_1$ (▾), $Q_2$ (▪), and $Q_3$ (♦) for a $x$-polarized incident illumination at $\lambda =532\mathrm{nm}$, $\tilde{U}$ being varied from 1 to 1.6. Bottom row: polarization-controlled azimuthal position of the $m$ quadrupoles embedded in the bright field array $A_m$ when the angle $\alpha$ between the direction of the pair of parallel linear polarizers and the $x$ axis is varied from 0 to $\pi$. Quadrupoles labeling is the same as above.