Fig. 1. Simulation results of number of radial zero points and intensity distribution of LG beam. (a) Relationship between the number of radial zero points of Laguerre function and the radial index p when topological charge is 15; (b) radial intensity of LG beam with radial index of 0-5 when the topological charge is 15

Fig. 2. Schematic of relative position of LG beam and rotating object,O is the rotation center of the object,O1 is the center of the LG beam, and d is the distance between OandO1. (a) Schematic of relative position of LG beam and rotating object when d=0, i.e. O1 and O coincide; (b) schematic of relative position of LG beam and rotating object when the optical axis does not coincide with the rotating axis

Fig. 3. Experiment setup and phase distribution diagrams. (a) Schematic of RDE experiment setup of LG beam with different radial indices; (b) phase distribution diagrams of LG beam with radial index from 0 to 5 when the topological charge is ±20

Fig. 4. Experimental results. (a1)(a2) Intensity of LG beam with different radial indices; (b1)(b2) signal intensity of the reflected light in the time domain, dotted line indicates the mean of time domain signal intensity; (c1)(c2) rotational Doppler spectra in the frequency domain after transformation, dotted frame is the frequency domain signal broadening area, the straight solid line represents the average signal power value of the spectrum broadening range in the dotted frame, and the dot is the central frequency shift point

Fig. 5. Curves of measured rotational speed changed with radial index when the different radial index LG beam whose topological charge is ±20 is used as incident light and there is lateral misalignment between the center of rotating object and the center of LG beam

Fig. 6. Average RDE signal amplitude varied with radial index under different topological charges when the rotating speed is 50 Hz