Fig. 1. Design principle of generating a linearly polarized beams with a rotating polarization angle based on optical heterodyne interference.(a) Linearly polarized beam with angle θ to the x-axis can be generated by superposition of two circularly polarized beams with opposite handedness and the phase difference of 2θ. (b) Linearly polarized beam with a rotating polarization angle by superposition of two circularly polarized beams with opposite handedness. The polarization angle rotated at frequency of Δf/2, where Δf is the heterodyne frequency of two laser fields.

Fig. 2. Schematic diagram of the experiment.The trapping beam is produced by a 532 nm laser and a Mach-Zehnder interferometer based on heterodyne interference. Devices in the green dashed box: a polarization detector made of a liquid crystal vortex half wave-plate and a polarizer which determines the polarization angle of the output trapping field. WP: wave plate; DM: 561 nm long pass dichroic mirror mounted on a flippable frame; PD: photodiode; PBS 1-4: polarized beam splitter; LC-VP: Liquid crystal based vortex half-wave plate; Obj 1: 10x objective; Obj 2: 100x objective with NA of 1.4; Filter 1-2: 635 nm long-pass filter.

Fig. 3. Verifying the polarization orientation distribution after passing through a LC-VP device.(a) Polarization angle distributions every T/4, where T is beat period that equals to 1/∆f. (b) The simulated light-field intensity distributions of the output trapping field recorded after passing the polarization detector consisting of a liquid crystal vortex half wave-plate and a polarizer when heterodyne frequency was set to 125 Hz. (c) The experimental light-field intensity distributions recorded by a camera confirming that the polarization angle of the output trapping light rotated counterclockwise corresponding to (b) (Supplementary Video 1).

Fig. 4. Video snapshots of disc-shaped vaterite particles recorded by optical tweezer apparatus.(a) Top view of a vaterite particles before being trapped. Note that the optical axis is perpendicular to the top facet. (b) Top view of the vaterite particles after being trapped by the linear polarized beam with the polarization angle indicated by the yellow arrow. (c) Video images recorded every 1.4 ms for a 1.5 μm diameter and 1 μm thickness vaterite micro-disc being rotated clockwise in the optical trap (Supplementary Video 2)

Fig. 5. The scattered li ght signals o f one trapped vaterite particle.(a) The signals of the probe laser scattered by rotating vaterite particle at heterodyne frequency tuned with the step of 100 Hz. (b) The frequency spectra obtained by Fourier transform to the signals in (a).