Fig. 1. The extra-cavity manipulation of VVBs. (a) The generation of optical skyrmions by OAM manipulation on opposite SAM states^[64]; (b) The generation of perfect VVBs using cascaded LC-SLMs^[67]

Fig. 2. The intra-cavity manipulation of VVBs. (a) The generation of high-order Poincare sphere beams from a laser using Q-plate ^[75]; (b) The non-linear generation of wave-tunable CVBs in OPO cavity^[76]; (c) The scheme of single-frequency CVBs laser^[78]; (d) The generation of hybrid Poincare sphere beams using meta-surface^[79]

Fig. 3. The representation and manipulation of SU (2) coherent states. (a) SU (2) Poincare sphere beams^[80]; (b) The SU(2) coherent states decided by the frequency degeneracy and the coherent phase^[83]; (c) The manipulation principle of 3-DoFs 8-dimensional nonseparable states^[85]; (d) The digital modulation of SU(2) coherent states^[87]; (e) The intra-cavity manipulation of 3-DoF nonseparable states^[90]

Fig. 4. The complex vortex array coupled by multi-DoFs. (a) The vector vortices array manipulated by 2D grating^[99]; (b) The five DoFs manipulation on vector vortices array using phase-only grating^[98]; (c) The higher dimensional vector vortices array manipulated by 3-DoFs^[100]

Fig. 5. The novel structured beams manipulated by multi-DoFs in the space-time domain. (a) Spatiotemporal beams with two OAMs^[106]; (b) The schematic diagram of scalar spatiotemporal vortices^[107]; (c) The experimental scheme and the mode conversion of vector spatiotemporal vortices^[109]; (d) Vortex rings of light^[110]; (e) “Photonic conchs” ^[111]; (f) Flying electromagnetic doughnuts manipulated by metasurface^[112]; (g) The photonic hopfions with 3D topological structure^[113]