Fig. 1. Schematic of 3D position detection with a Dove prism. (a) Light propagating along the longitudinal axis is inverted by 180° and the angle of rotation of the image is twice that of the rotation angle of the prism. (b) Schematic of the experiment setup. The green spot in the inset is a trapped particle that is illuminated by another 532 nm laser.

Fig. 2. PSD of the motion in the X-Y-Z axis. The resonance frequencies are $\mathrm{\Omega }=52.7\mathrm{kHz}$, 128.0 kHz, and 164.8 kHz, respectively. The black lines are fitting curves according to Eq. (8). The air pressure in the vacuum chamber is 8.1 mbar. The diameter of the trapped nanosphere is $154\pm 1\mathrm{nm}$.

Fig. 3. Damping rate decreases with the air pressure. The dashed line is a fit according to Eq. (9).

Fig. 4. PSD of the position signal at different rotation angles of the Dove prism.

Fig. 5. Peak height of the X and Y frequency parts in the PSD according to the rotation angle of the Dove prism. (a) The PSD when the Dove prism rotation angle is 25°. The black line is the fitting curve according to Eq. (15). (b) The scale factors of the X and Y parts according to Eq. (15) vary with the rotation angle of the Dove prism. The blue and red lines demonstrate ${\cos }^2(2\phi )$ and ${\sin }^2(2\phi )$, respectively. $\phi$ is the rotation angle of the Dove prism.