Fig. 1. Optical vortices and its three common generating elements. (a) Phase and wavefront distributions for optical vortices with topological charges from -3 to +3; (b) schematic diagram of optical vortex generated by SPP; (c) schematic diagram of optical vortex generated by projecting diffraction fork grating on SLM; (d) schematic diagram of optical vortex with topological charges of ±1 generated by Q-plate with q=1/2

Fig. 2. OAM manipulation by Dove prism. (a) Angle between bottom surface of Dove prism and horizontal plane is 0°; (b) angle between bottom surface of Dove prism and horizontal plane is α

Fig. 3. Schematic diagram of realization of single-photon OAM qubit encoding with SPP^[37,42]. (a) Inserting SPP in Sagnac interference loop to produce various polarization-OAM composite states; (b) inserting two SPPs in two spatial modes of single photon to expand encoding in three degrees of freedom of polarization, path and OAM

Fig. 4. Discrete-time quantum random walk experiments based on OAM. (a) Realization of quantum random walk based on OAM by several groups of Q-plate^[56]; (b) realization of periodic lattice quantum random walk with two degrees of freedom by combining OAM with path mode^[58]

Fig. 5. Preparation of hyper-entangled sources. (a)(b) Two different methods for preparation of polarization and path hyper-entangled sources^[73-74]; (c) polarization and OAM hyper-entangled sources^[70]

Fig. 6. OAM spectrum distribution generated by spontaneous parametric down-conversion^[83]

Fig. 7. High-dimensional maximum entangled states with OAM prepared by adaptive modulation. (a) three-dimensional maximum entangled state with OAM^[86]; (b) four-dimensional maximum entangled state with OAM^[88]

Fig. 8. OAM beam splitting by using interferometer. (a) Utilizing two BSs for beam splitting and combining^[35]; (b) rotating angle α between two different Dove prisms to sort various OAM modes^[35]; (c) utilizing two PBSs for beam splitting and combining^[93]

Fig. 9. Utilizing coordinate transformation to sort various OAM modes. (a) Experimental scheme for realizing transformation from Cartesian coordinate system to logarithmic-polar coordinate system^[94]; (b) using spiral transformation instead of logarithmic-polar coordinate transformation to improve sorting efficiency^[96]

Fig. 10. 18-bit GHZ states with six photons' degrees of freedom of polarization, path and OAM^[42]

Fig. 11. Experimental schemes for high-dimensional multi-photon quantum gates^[119]. (a) Experimental scheme of three-dimensional CNOT. PS1 and PS2 are used in control part, and PS₂ is only needed in controlled part. Introduced auxiliary entangled state is two-dimensional 5-photon entangled state; (b) experimental scheme of three-dimensional controlled phase gate, control part and controlled part are symmetrical; (c) experimental scheme of high-dimensional Toffoli gate, in which two control photons are two-dimensional quantum states and controlled photon has 4-dimensional space