Fig. 1. (a) a-type and (b)b-type period-gradually-changing amplitude gratings; (c) a -type and (d)b-type 0-π binary period-gradually-changing phase gratings

Fig. 2. (a) Annular amplitude grating; (b) annular phase grating

Fig. 3. Schematic of vortex beam diffraction by gratings. (a) Vortex beam; (b) grating; (c) diffraction patterns

Fig. 4. Phase profile diagram of phase-corrected optical element

Fig. 5. Fan-out phase images under different number of copies. (a) 7 times; (b) 9 times

Fig. 6. Simulation results for four period-gradually-changing gratings. (a)-(d) a-type period-gradually-changing amplitude gratings; (e)-(h) a-type 0-π binary period-gradually-changing phase gratings; (i)-(l) b-type period-gradually-changing amplitude gratings; (m)-(p) a-type 0-π binary period-gradually-changing phase gratings

Fig. 7. Simulation results for two annular gratings. (a)-(d) Annular amplitude gratings; (e)-(h) annular phase gratings

Fig. 8. Experimental setup

Fig. 9. Experimental results of detection and correction for four period-gradually-changing gratings. (a)-(d) a-type period-gradually-changing amplitude gratings; (e)-(h) a-type 0-π binary period-gradually-changing phase gratings; (i)-(l) b-type period-gradually-changing amplitude gratings; (m)-(p) b-type 0-π binary period-gradually-changing phase gratings

Fig. 10. Experimental results of detection and correction for two annular gratings. (a)-(d) Annular amplitude gratings; (e)-(h) annular phase gratings