Fig. 1. Schematic diagram of the generation of ultra-dense vector perfect vortex beam and rotational Doppler frequency shift measurement. The setup comprises a spatial light modulator (SLM); two lenses (L1 and L2) with focal length 250 mm used to construct a 4-f system; polarizers (X-LP and Y-LP) with polarization along the x-axis and y-axis, respectively; a Ronchi grating (RG) used to synthesize x- and y-polarization components; a rotating object for detection (Rotor); and a photodetector (PD). Simulated beam intensities: (b) l=±6, (c) l=±(5,8,12) (from inner to outer), (d) l=±(3:3:36), (e) l=±(6,9,12,15) at multiple axes, (f) l=±(2:2:12) for detecting rotation radius, and (g) l=±(2:2:12) for detecting object size.

Fig. 2. Experimentally captured intensities and corresponding crosslines with l=6 and N=2, 4, 6, 8, 10, and 12: (a1)–(f2) scalar multi-ring perfect vortex beams, and (a3)–(f4) ultra-dense vector perfect vortex beams.

Fig. 3. Simulated and experimental intensities and corresponding crosslines of ultra-dense vector perfect vortex beams with l=6 and ω=0.15  mm, 0.10 mm, and 0.05 mm. (a1)–(c2) Simulation results. (d1)–(f2) Corresponding experimental results.

Fig. 4. Frequency spectra of rotational Doppler frequency shift of ultra-dense vector perfect vortex beams: (a) N=2, (b) N=4, (c) N=6, (d) N=8, (e) N=12. Insets show intensities. (f) Signal-to-noise ratio (SNR) of 20 datasets with error bar in red.

Fig. 5. Experimental generation of frequency combs. (a1)–(c1) Intensity distributions; (a2)–(c2) time-domain intensity fluctuations; and (a3)–(c3) frequency-domain signals.

Fig. 6. Simulation and experimental results of object size and location perception. Intensity used for detection: (a) simulated; (b) experimental. The TCs used to generate petal-like intensities are l=±(2:2:12) with radius interval Δr=0.25  mm. Object location sensing of ball object: (c) simulated; (d) experimental. Size detection of rectangular object: (e) simulated; (f) experimental.

Fig. 7. Simulation results of misalignment issues. Frequency combs of coaxial nested ultra-dense vector perfect vortex beam: (a) aligned and (b) misaligned. TCs of each ring are ±(6,9,12,15), located at radius 1.2, 1.8, 2.4, and 3.0 mm, respectively. Frequency combs of non-coaxial nested beam for misalignment issues. From the inner to the outer rings, shift is 0.1 mm in positive x-, positive y-, negative x-, and negative y-directions, and object rotation axis shifts 0.052  mm diagonally to (c) first quadrant, (d) second quadrant, (e) third quadrant, and (f) fourth quadrant, respectively. The red dashed line is the fitted envelope.

Fig. 8. Experimental results of error margin. (a) Frequency domain results with 100 s sampling time. (b) Mean peak value and error bar with different sampling times.