Fig. 1. Phase and intensity profiles of different modes. (a1)(b1) LG₀₁mode; (a2)(b2) LG₁₁ mode; (a3)(b3) LG₂₁ mode

Fig. 2. Preparation methods of radial mode beam. (a) Holographic preparation method^[40]; (b) spiral phase plate preparation method^[42]; (c) metal metasurface preparation method^[43]

Fig. 3. Projection measurement of radial modes. (a) Experimental setting of intensity-flattening method^[50]; (b) measurement accuracy of radial modes^[50]; (c) experimental setting of phase-retrieval method^[51]; (d) high-precision and high-efficiency measurement results^[51]

Fig. 4. OAM mode sorter^[62]. (a) Experimental setting; (b) experimental result

Fig. 5. Experimental setting of radial mode sorter based on fractional Gouy phase classification method^[69]

Fig. 6. Experimental results of radial mode sorter^[69]

Fig. 7. High-order LG_pl mode sorter^[71]

Fig. 8. Experimental results of high-order LG_pl mode sorter^[71]

Fig. 9. Mode division multiplexing using radial and azimuthal modes^[44]. (a) Intensity profiles of 35 LG_pl modes; (b) experimental setting; (c) encoding ofLG_pl modes in SLM-1; (d) modal filter composed of SLM-2 and CCD camera; (e) identification of 105 modes in far field by using CCD camera

Fig. 10. RGB encoding colour images^[44]

Fig. 11. Experimental setting for radial mode entanglement of two photons^[54]

Fig. 12. Quantum correlation of radial modes^[53]

Fig. 13. Quantum correlation of radial modes based on radial pump modes^[48]

Fig. 14. Quantum correlation of radial modes based on high-order LG_pl pump modes^[48]

Fig. 15. Definition of radial position and radial momentum^[95]. (a)(b) Radial position; (c)(d) radial momentum

Fig. 16. Experimental setting of EPR correlation for radial degree of freedom^[95]

Fig. 17. EPR correlation of radial degree of freedom for two photons^[95]

Fig. 18. Experimental results of radial diffraction^[98]