Fig. 1. Numerical simulations of eight-ring vector perfect vortex beams with l = ±5 generated by two methods under varying ring half-width, along with their corresponding intensity crosslines along the horizontal direction. (a)–(c) Generated using the direct SLM modulation method; (d)–(f) produced using the Bessel beam kinoform method. (a), (d) ω_0nx = 10 µm, (b), (e) ω_0nx = 6 µm, (c), (f) ω_0nx = 2 µm.

Fig. 2. Experimental setup utilizing Bessel beam kinoform for the generation of multi-ring vector perfect vortex beams with conjugate superposition and its Doppler shift detection. HWP, half-wave plate; BE, beam expander; SLM, transmissive spatial light modulator; L, lenses with focal lengths of 250 mm (L₁, L₂) and 100 mm (L₃, L₄); M, mirror; X/Y-LP, linear polarizers; RG, Ronchi grating; BS, beam splitter; DMD, digital micromirror device; PD, photodetector; OSC, oscilloscope. (a1) Hologram loaded onto the SLM (scale bar: 1.2 mm). (a2) The +1-order and (a3) −1-order diffraction patterns through the SLM (scale bar: 1.6 mm). (b) The multi-ring Bessel–Gaussian vector beam with conjugate superposition multiplexed by the RG (scale bar: 1.6 mm). (c) A multi-ring vector perfect vortex beam was generated and projected onto the rotating object loaded by the DMD (scale bar: 1.2 mm). The rotating object is a rectangular object (with a length of 9.72 mm and a width of 0.1 mm) rotating at 50 r/s.

Fig. 3. Doppler velocity measurements using multi-ring vector perfect vortex beams with varying beam radii generated by two methods. The beam radius decreases progressively from left to right, and below the intensity plots are the corresponding Doppler frequency spectra. (a1)–(d1) Beams generated by the Bessel beam kinoform method, with corresponding Doppler frequency spectra presented in (a2)–(d2). (e1)–(h1) Beams generated by the direct SLM modulation method, accompanied by their respective Doppler frequency spectra in (e2)–(h2). In this experiment, the employed beam is an eight-ring perfect vortex beam formed with TCs of ±5, with the ring half-width ω₀ fixed at 0.006 mm. However, the inter-ring spacing of these beams varies, specifically (a), (e) 0.07 mm, (b), (f) 0.06 mm, (c), (g) 0.05 mm, and (d), (h) 0.04 mm.

Fig. 4. Measurement of a microscopic object with a radius of 16 µm rotating at 50 r/s. A beam is generated via the direct SLM modulation method with (a) inter-ring spacing of 0.2 mm and ring half-width of 0.03 mm, (c) inter-ring spacing of 0.04 mm and ring half-width of 0.001 mm. (e) A beam generated via the Bessel beam kinoform method, with inter-ring spacing of 0.04 mm and ring half-width of 0.001 mm. All objects are outlined with white circles in each figure.

Fig. 5. Experimental results of velocity gradient detection. (a) Schematic diagram of an eight-ring vector perfect vortex beam incident on multiple rotating objects. (b) Experimental Doppler frequency spectrum. (c) The relationship between rotational speed and radius, corresponding to (b). Additionally, the following pairs of subfigures correspond to each other: (d) to (g); (e) to (h); and (f) to (i). For all Doppler signals, eight-ring vector perfect vortex beams were employed as the incident light. The beam parameters are as follows: beam radius of 1.3 mm, ring half-width of 0.03 mm, and inter-ring spacing of 0.15 mm. The TCs are arranged from the innermost to the outermost as 7, 8, 9, 6, 5, 4, 3, and 2, respectively.