Fig. 1. Typical degrees of freedom in optics and typical structured light fields

Fig. 2. Principle of mode sorter using log-polar mapping. (a) Intensity and phase distribution during the geometric coordinate transformation of the beam^[46]; (b) Beam evolution of mode sorter by log-polar mapping （From left to right: multiplexing; From right to left: demultiplexing)^[49]; (c) Log-polar transformation to achieve two-dimensional spatial separation for the OAM beams with radial varying phase^[50]

Fig. 3. Compact integrated geometric coordinate transformation device. (a) Design and working principle of the compact OAM mode decomposition multiplexing diffracted optical elements based on optical transformation, with the expander and rectifier elements integrated on one glass substrate^[52]; (b) Non-paraxial OAM mode sorter^[53]; (c) SEM picture of the combined mode sorter fabricated by 3D laser printing^[54]

Fig. 4. Influence of spot broadening on mode overlap^[31]

Fig. 5. Schematic diagram of high-resolution mode sorter

Fig. 6. Multi-plane light conversion. (a) LG mode classifier based on MPLC. Gaussian beam at a certain position in Cartesian coordinate system is converted to LG mode of corresponding order after passing through MPLC^[62]; (b) Fabricated transmissive MPLC device, which generates Gaussian spots at different positions from incident LG modes of different topological charge^[64]; (c) Optical vortex converter based on optical diffraction neural network^[67]

Fig. 7. Geometric transformations for modal density measurement

Fig. 8. Optical OAM multiplier and divider. (a) OAM divider^[76]; (b) OAM multiplier^[77]; (c) Integrated OAM multiplier and divider^[73]; (d) OAM multiplier and divider based on spiral transformation^[75]

Fig. 8. [in Chinese]

Fig. 9. Geometric transformation for classical and quantum optical communication. (a) Schematic diagram of classical OAM-MDM-WDM optical communication system and spiral coordinate transformation principle^[79]; (b) High dimensional entanglement transfer from path to OAM degree of freedom^[80]