Fig. 1. (a) Three examples of q-plate patterns with α₀ being the initial optical axis orientation, reprinted with permission from Ref. [32], Copyright (2021) by the American Physical Society. (b) Illustration of the optical action of a q-plate with q = 0.5 on left circularly polarized light beam^[33].

Fig. 2. Experimental scheme to generate and analyze VBBs^[45]. The left inset shows the optical axis orientation of one q-plate and the phase acquired by the wavefront in the transverse plane. The right inset shows the intensity distribution of the generated VVB under different polarization. The sample preparation and analysis process are for studying the transmission of VVBs in dispersive media in the original work. PBS, polarized beam splitter; QWP, quarter-wave plate; HWP, half-wave plate; H, horizontal polarization; V, vertical polarization; D, diagonal polarization; A, antidiagonal polarization; L, left circular polarization; and R, right circular polarization.

Fig. 3. Different schemes to generate vector beams using CLC. (a) Generic mirror-backed Bragg–Berry optical element, reprinted with permission from Ref. [62]. Copyright (2021) by the American Physical Society. (b) Stacking two opposite-handed CLCs, reprinted with permission from Ref. [63]. Copyright (2019) by The Optical Society.

Fig. 4. Experimental scheme to generate self-engineered LC q-plates. Reprinted with permission from Ref. [72]. Copyright (2021) by the American Physical Society.

Fig. 5. Experimental setups for generating vector beams using SLMs. (a) Experimental setup for generating arbitrary vector beams via a triangular common-path interferometer^[80]; (b) generation of arbitrary vector beams by interferometric methods using a single SLM, reprinted with permission from Ref. [81]. Copyright (2019) by The Optical Society. (c) Schematic representation to generate multiple vector beams by the use of an SLM, reprinted with permission from Ref. [82]. Copyright (2019) by The Optical Society.

Fig. 6. Sketch of the biphoton hyperentangled state^[98].

Fig. 7. Schematic representation of the investigated field with a z-dependent degree of entanglement^[102]. (a) Basic concept with a z-dependent degree of entanglement. (b) Experimental scheme.

Fig. 8. Tunable two-photon quantum interference by using a q-plate^[112]. Different transformation orbits for different angles α₀ when the input photon states lie (a) on the poles and (c) on the equator; (b) and (d) represent the two photon coincidence probabilities for the corresponding orbits by changing the q-plate phase. Reprinted with permission from Ref. [112]. Copyright (2021) by the American Physical Society.

Fig. 9. Application of SLMs in high-dimensional two-photon interference^[113]. (a) Two-photon interference in a 3D mode splitter. (b) Experimental setup. Reprinted with permission from Ref. [114]. Copyright (2021) by the American Physical Society.