Fig. 1. Principle model of rotational Doppler effect. (a) Velocity projection model based on Poynting vector^[74]; (b) scattering model of interference fringe^[34]; (c) lateral phase gradient model^[32]; (d) OAM mode conversion model^[33]; (e) phase analysis of rough surface^[33]

Fig. 2. Rotation detection technique based on rotational Doppler effect. (a) Heterodyne detection method using singlet OV^[79]; (b) stripe detection method using superimposed OV^[79]; (c) scattering light OAM detection method^[38]

Fig. 3. Rotational velocity measurement based on misaligned RDE. (a) Detection of OV deviation from rotation center^[55]; (b) detection of OV oblique incidence^[56]; (c) misaligned OV and corresponding OAM modal spectra^[57]; (d) detection under arbitrary incidence conditions of OV^[40]

Fig. 4. Rotation detection based on RDE. (a) Detection of angular acceleration^[43]; (b) rotation direction measurement^[41]; (c) rotation direction measurement based on vector beam^[87]; (d) locating rotating center^[44]; (e) rotating axis measurement based on spliced OV^[46]; (f) rotating axis detection based on noncoaxial rotational Doppler effect of multi-mode OV^[90]

Fig. 5. Composite motion detection based on OV. (a) Measurement of particle velocity in spiral motion using beams with different spatial modes^[91]; (b) simultaneous measurement of both the longitudinal and angular speed based on composite polarized beams^[92]; (c) composite motion detection based on OAM interferometry^[93]; (d) simultaneous detection of procession and rotation using OV^[47]

Fig. 6. Digital spiral imaging and quantum digital spiral imaging. (a) Effect of π phase jumps at different distances and different widths on spiral spectrum^[105]; (b) quantum digital spiral imaging scheme^[106]; (c) results of quantum digital spiral imaging^[106]

Fig. 7. Azimuth measurement and symmetry detection based on optical vortices. (a1)‒(a6) Distribution of target azimuth angles^[107]; (b1)‒(b6) detecting results of azimuth^[107]; (c) result of symmetry detection^[73]

Fig. 8. Experimental setup of common-path interferometry and results of single-pixel complex-amplitude imaging using LG modes^[108]. (a) Experimental setup; (b) chessboard encoding method; (c1)-(f2) imaging results

Fig. 9. Complex amplitude imaging based on LG beams^[109]。 (a) Experimental setup; (b) phase of superimposed LG beams; (c1)‒(c4) intensity distribution of superimposed LG beams with different phase shifts; (d1)‒(d2) patterns to be imaged; (e1)‒(g3) imaging results of different mode numbers

Fig. 10. Generating asymmetric perfect vortex using a 2D HOBBIT system and the imaging result^[110]. (a) HOBBIT setup; (b)imaging principle; (c) comparison of beam generated by simulation and experiment; (d) imaging result

Fig. 11. Experimental results of complex amplitude object reconstruction by SOC algorithm and compressed sensing algorithm^[111]

Fig. 12. Image decomposition and noise reduction based on LG mode^[112]. (a) Amplitude spectrum of the image; (b) phase spectrum of the image; (c) reconstructed image; (d) difference between reconstructed and original image; (e) images with noise; (f) azimuthal coefficient of noisy image; (g) image after noise removal

Fig. 13. Setup and results of rotation target imaging based on LG spectrum^[113]. (a) Experimental setup; (b) LG amplitude spectrum; (c) LG phase spectrum; (d)‒(g) LG amplitude spectra at four different sampling rates; (h)‒(o) imaging results at four different sampling rates