Fig. 1. Normalized light intensity and phase that the peak control parameter m and the curvature control parameter n are different. (a1)-(a3) are the light intensity distribution diagram when m=2 and n=0.25-0.75 respectively; (b1)-(b3) is the corresponding phase distribution diagram; (c1)-(c3) is the light intensity distribution when m=3 and n=0-0.75; (d1)-(d3) is the corresponding phase distribution diagram; (e1)-(e3) is the light intensity distribution when m=4-6 and n=0.5; (f1)-(f3) is the corresponding phase distribution diagram

Fig. 2. Normalized light intensity, gradient force and OAM when the peak control parameter m=2 and the curvature control parameter n=0-0.75. (a1)-(a4) are the light intensity distribution diagram when m=2 and n=0-0.75 respectively; (b1)-(b4) is an enlarged version of the gradient force distribution map corresponding to the white dashed frame region in (a1)-(a4); (c1)-(c4) enlarged view of the corresponding OAM density map in the area with the white dotted line

Fig. 3. Schematic of the experimental setup

Fig. 4. When the peak control parameter m=2 and the curvature control parameter n=0.02, the cycloid beam manipulates the particle to rotate

Fig. 5. When the peak control parameter m=2 and the curvature control parameter n=0.75, the cycloid beam simultaneously manipulates three yeast cells to rotate by adjusting the phase difference