Fig. 1. Schematic of the experimental setup for unconventional optical tweezers utilizing an RI stratified medium, coupled with SPV beams. (a) Illustration of the RI stratified medium composed of four layers, with the third layer (sample chamber) where particles are trapped at on-axis and off-axis positions, showing particles rotating in opposite directions relative to each other. (b) Depiction of the doughnut intensity pattern and spiral polarization distribution of the input ACW-SPV and CW-SPV beams. (c) Ray diagram of the experimental setup, showing how the ACW/CW SPV light is coupled into the optical tweezers for probing the spatially resolved LSAM.

Fig. 2. Time-lapse frames displaying the (a) ACW rotation and (b) CW rotation of the LC particles trapped at the beam center for the incident ACW-SPV (Erad+Eazi) and CW-SPV (Erad−Eazi) vector beams, respectively. The particles, being relatively larger (diameters of ∼7 to 8μm), follow the inherent spiral field orientations of the respective beams. White arrows serve as the reference (drawn at the onset of rotation), and the red arrows depict the instantaneous orientations (Video 1, MP4, 523 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s1]; Video 2, MP4, 648 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s2]).

Fig. 3. Time-lapse frames displaying the (a) CW rotation and (b) ACW rotation of the LC particles trapped at off-center locations for the incident ACW-SPV (Erad+Eazi) and CW-SPV (Erad−Eazi) vector beams, respectively. Both particles (diameter of ∼2μm) spin in a direction opposite to the inherent spiral field orientations of the respective beams. White arrows serve as the reference (at the onset of rotation), and the red arrows depict the instantaneous orientations (Video 3, GIF, 960 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s3]; Video 4, MP4, 261 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s4]; and Video 5, MP4, 181 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s5).

Fig. 4. Schematic of the geometric representation of the coordinate transformation from cylindrical to spherical coordinates through an aplanatic lens, illustrating the gradual bending of k vectors toward the focus, which leads to the emergence of longitudinal field components and the acquisition of a geometric phase.

Fig. 5. Numerical simulation of the intensity distribution at different positions: (a) at the focus, (b) at z=1μm, (c) at z=2μm, and (d) at z=3μm away from the focus for the input ACW-SPV (or CW-SPV) beam. At the focus, the intensity is high with a small spatial extent. As we move away from the focus, off-axis intensity annular ring forms, which are essential for trapping particles off-axis. However, beyond z=2μm, the intensity at the beam center reduces significantly, as shown in (d). (e) and (f) display the intensity and polarization (black arrows) distributions of the incident ACW-SPV (Erad+Eazi) and CW-SPV (Erad−Eazi) vector beams across the beam cross sections, respectively. (g) A line plot of the intensity distribution across the x axis at z=2μm and z=3μm away from the focus is presented for better visualization. (h) Comparison of the maximum intensity values at the beam center (solid red circles) and at the off-axis annular rings (solid black squares) as a function of the RI at z=2μm away from the focus.

Fig. 6. Simulated longitudinal spin distribution (LSAM) for the incident (a) ACW-SPV (Erad+Eazi) and (b) CW-SPV (Erad−Eazi) vector beams, showcasing the aggregation of LSAM components in concentric circular rings. (c) The line plot of spatially resolved positive/negative (σ=±1) and negative/positive (σ=∓1) helicities of the ACW-SPV/CW-SPV beams is shown by the red/black line, respectively, with opposite helicities accumulating at different radii. (d) The distributions of the transverse SAM (TSAM) density also show similar aggregation in circular regions. (e) Line plots of LSAM density (light blue) and intensity (orange) at z=2μm from the focal point, showing the overlap of spatially resolved LSAM and intensity peaks at both on-axis and off-axis positions. (f) The net spatially resolved LSAM (obtained by integrating the local spin density over the area covered by the particle) experienced by a particle as a function of its radius, when placed at the beam center or in the off-axis high-intensity ring, shows a critical radius of ∼1μm (or diameter of ∼2μm). Below this radius, the particle will experience negative/positive helicity and rotate CW/counterclockwise for the incident ACW-SPV/CW-SPV beam at off-axis positions. Beyond this radius, particles will always experience positive/negative helicity and rotate counterclockwise/CW due to the net positive/negative integration value. These distributions are computed in the transverse plane between z=1μm and z=2.5μm beyond the focus, for a coverslip with an RI of 1.814. However, z=2μm is the optimized distance from the focus for probing the off-axis distribution of LSAM (Video 6, MP4, 5326 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s6]; Video 7, MP4, 5187 kB [URL: https://doi.org/10.1117/1.APN.4.2.026006.s7]).