Figure 1

Experimental setup. (a) A moderately focused circularly polarized Gaussian beam is normally incident onto a homeotropic NLC film. The output beam is collimated and analyzed using two electrically controlled retarders $R_{1,2}$ and a polarizer $P$. (b) Orientation of the polarizer and of the slow axes of the retarders.Reuse & Permissions

Figure 2

Self-induced optical vortex precession for left-handed circularly polarized incident beam. (a) Transient vortex dynamics observed by recording $I_R(x,y)$, the input beam being turned on at $t=0\mathrm{s}$. Illustrative representations of the projection of the director field on the ($x$, $y$) plane, $\Vert \mathbf{n}-(\mathbf{n}·\mathbf{z})\mathbf{z}\Vert$, are shown at $t=0$, 10 and 110 s. Arrows on images indicate the motion of the optical vortex. (b) Dynamics of the optical vortex trajectory over 500 s with time step $dt=1\mathrm{s}$. The stationary trajectory is fitted by an ellipse. (c) Steady state three-dimensional trajectory of the precessing optical vortex core, $T$ being the precession period. The experimental conditions are $z=350\mu \mathrm{m}$, ${\theta }_0=8.5°$ and here $P=780\mathrm{mW}$.Reuse & Permissions

Figure 3

(a) Snapshot of the intensity profile of optical vortex field during steady state precession under the experimental conditions of Fig. 2. The circle marker refers to the initial location of the vortex at $t=0$ whereas the star marker refers to the actual location of the optical vortex. (b) Local phase spatial distribution nearby of the optical phase singularity. (c) Azimuthal dependence of the phase along the dashed circle shown in panel (b).Reuse & Permissions

Figure 4

Power dependence of (a) $\rho$ and (b) $T$ (diamond markers) and $e$ (square markers), see text for definitions. The shadowed area refer to two distinct regimes, below and above the threshold power $P=P_{\mathrm{th}}$. The experimental conditions are those of Fig. 2, for which $P_{\mathrm{th}}=705\mathrm{mW}$.Reuse & Permissions

Figure 5

Optical vortex precession characteristics at the onset of the transition versus $\delta =d/L$. (a) Threshold power $P_{\mathrm{th}}$. (b) Reduced off-axis distance ${\rho }_{\mathrm{th}}$. (c) Precession period $T_{\mathrm{th}}$. (d) Aspect ratio of the steady state elliptical trajectory $e_{\mathrm{th}}$. The experimental conditions correspond to ${\theta }_0=10.6°$. The solid curve in panel (a) is the optical Fréedericksz transition best fit, see text for details.Reuse & Permissions

Figure 6

Maps of the reduced third Stokes parameter $s_3$ for an incident power just below (a) and above (b) the threshold power $P_{\mathrm{th}}$. The ellipse in panel (b) is the steady state vortex trajectory, the star marker being the location of the charge two phase singularity at given time whereas up and down triangle markers refer to the corresponding spin up and down centers of gravity. (c) Enlargement of panel (b) that shows the steady state trajectories of S, $G_{+}$ and $G_{-}$. (d) Dynamics of the angle $\Delta \alpha$ over of few precession periods. The experimental conditions correspond to a left-handed circularly polarized incident beam with ${\theta }_0=9.5°$ and $z=310\mu \mathrm{m}$.Reuse & Permissions

Figure 7

(a) Power dependence of the time-averaged angle $⟨\Delta \alpha {⟩}_t$ in the steady state optical vortex precession regime. Error bars refer to the standard deviation of the well-defined oscillation of $\Delta \alpha (t)$, as illustrated in Fig. 6d. (b) Power dependence of ${\varrho }_{\parallel ,\perp }$. The experimental conditions are those of Fig. 6.Reuse & Permissions