Fig. 1. Spatial distribution of instantaneous electric vector field for several conventional modes and CV modes^[37]. (a) x-polarized fundamental Gaussian mode; (b) x-polarized HG₁₀ mode; (c) x-polarized HG₀₁ mode; (d) y-polarized HG₀₁ mode; (e) y-polarized HG₀₁ mode; (f) x-polarized LG₀₁ mode; (g) Radially polarized mode; (h) Azimuthally polarized mode; (i) Generalized CV beams

Fig. 2. Geometric representation of polarization^[39]. (a) Polarization ellipse; (b) Representation of the polarization ellipse on the the Poincaré sphere

Fig. 3. Higher-order PS representation for l=+1 and |l|≠|σ|^[50]

Fig. 4. Generalized PS representation for m = +1 ^[62]

Fig. 5. Conformal mapping simulation of a spatiotemporal vortex tube transforming into a vortex ring^[76]. The spatial-temporal vortex tube phase Ф₁(x, y) propagates and evolves in free space to become a vortex ring, after which a second phase mask Ф₂(u, v) can be applied for collimation, with color coding representing the magnitude of the expanded phase

Fig. 6. System configuration for vector beam generation based on a slit-structured metasurface. He-Ne, He-Ne laser: Helium-neon laser; DF: Intensity filter; QWP: Quarter-wave plate; P1, P2: Polarizers; S: Sample; MO: Microscope objective lens; CCD: Charge-coupled device; GLP1, GLP2: Glan laser polarizers; MS: Metasurface; Lens: Lenses. (a) Apparatus for the generation of different orders of cylindrical vector fields^[78]; (b) Schematic diagram of the optical needle field generation system configuration using a metasurface (the optical needle field is represented by purple arrows)^[79]; (c) Setup for generating vector vortex beams utilizing a metasurface^[80]

Fig. 7. Cylindrical nanostuctured metasurfaces and cascaded metasurfaces for vector optical field generation. (a) Side view (left) and top view (right) of a metasurface composed of hexagonal units^[81]; (b) Elliptical amorphous silicon pillar structure within the hexagonal unit cell^[81]; (c-e) Schematic illustrations of near-axis distributions for cascaded metasurfaces with topological charges q = 0.5, 1.0, and 1.5, respectively^[83]; (f-h) Cross-polarized microscopy images of metasurfaces with q values of 0.5, 1.0, and 1.5, where q is a constant determined by the positional variation and slow axis orientation of the metasurface unit structures^[83]; (i) Nanoscale structure of the metasurface for generating three-dimensional cylindrical vector optical fields accompanied by SEM images^[87]

Fig. 8. Concept illustration of the streamlined metalens^[89]. The time-reversed electric fields (blue arrows) are obtained by the radiation of a circularly polarized point source and could be generated by a half-wave plate with spatially variant anisotropic axes. The red streamline is obtained by the trajectory of the vectorial field (orange arrows) formed by spatially variant anisotropic axes mentioned above

Fig. 9. Optical hologram based on asymmetric PSOI. (a, b) Fei Zhang et al’ work^[93] ; (c, d) Concurrent work of Harvard university^[94]

Fig. 10. MIM metasurface structure^[102]. (a) The metasurface configuration, where yellow rings denote double-nanorod structures and light brown rings represent single-nanorod structures. Inset: Magnified view of the structure; (b) A sector within the first ring; (c) Detailed illustration of the double-nanorod structure; (d) Single-nanorod structure depicted explicitly

Fig. 11. Orbital angular momentum detection with a metasurface. (a) Structure of the holographic metasurface^[105]; (b) Simulated interference pattern generated^[105]; (c) Binary representation of the simulated results^[105]; (d) Scanning electron microscope (SEM) image of the holographic surface, showing grooves at phase-matched positions^[105]; (e) The OAM detector upon left-handed circularly polarized (LCP) incidence^[106]; (f-h) Simulated intensity distributions of the OAM detector when illuminated by vortex beams carrying different topological charges: (f) l = 0, (g) l = −1, and (h) l = −2^[106]; (i) Optical micrograph of an eight-segment silicon cutoff-line spiral phase plate with π/4 phase steps on the left, and on the right, the SEM image of this structure alongside its corresponding vortex beam intensity distribution map^[106]

Fig. 12. Nanoparticle localization^[112]. (a) A microscope objective tightly focuses a radially polarized light beam onto silicon antennas on a glass substrate, where an oil-immersion objective collects and focuses light with numerical aperture (NA) ranging between 0.95 to 1.3, and a CCD is positioned at the rear focal plane; (b) Far-field intensity distribution diagram of the antenna along the optical axis; (c) Far-field intensity distribution diagram for a lateral displacement of 40 nm

Fig. 13. The design approach for an OAM-multiplexing hologram^[123]