Fig. 1. Schematics of（a）physical structure and（b）diffraction properties ^［19］ for liquid crystal polarization grating；Schematics of（c）physical structure^［16］ and（d）diffraction properties^［20］ for liquid crystal polarization volume grating.

Fig. 2. （a）Composition of the AR waveguide display system^［21］；（b）Geometric relationship between field of view，eye relief and eye box.

Fig. 3. （a）Dual-channel waveguide model and（b）k-space diagram of dual-channel waveguide^［23］；（c）Triple-channel waveguide model and（d）k-space diagram of triple-channel waveguide^［24］.

Fig. 4. （a）Director distribution of liquid crystals in LCOM-PVG and schematic diagram of FOV stitching；（b）Simulated angular and spectral responses of LCOM-PVG and PVG；（c）Actual display effect pictures in a bright indoor environment based on LCOM-PVG and PVG^［25］.

Fig. 5. （a）LCs’ director distribution of PVG and CPVG^［26］；（b）Simulated angular and spectral responses of the CPVG^［26］；（c）Schematic diagram of a single unit with sixteen gradient regions and four different regions in PVGs and the director distribution at the bottom^［27］；（d）Angular and spectral responses of the PVG of the 3D periodic gradient structure^［27］.

Fig. 6. Basic configuration and light propagation path of the one-dimensional（1D）and two-dimensional（2D）exit pupil extension.（a）Schematic representation of the 1D exit pupil extension in the x-y plane（top）and x-z plane（bottom）；（b）k-space diagram of the 1D exit pupil expansion^［33］；（c）Schematic representation of the 2D exit pupil extension in the x-y plane（top），the y-z plane（left）and the x-z plane（bottom；（d）k-space diagram of the 2D exit pupil extension^［34］.

Fig. 7. 2D exit pupil extension based on three PVGs.（a）Schematic configuration of the PVG-based 2D exit pupil extension^［36］；（b）Variation of the efficiency of the exit light at different angles ρ^［36］；（c）Simulated diagram of 2D pupil extension effect under white light^［36］；（d）Schematic configuration of dual-channel 2D exit pupil expander，the upper red grating is the in-coupling grating，the upper left and right sides are the left and right deflection gratings，the lower part is the out-coupled grating^［33］；k-space diagrams for（e）no separation and（f）separation cases，the red rectangles refer to the FOV in k-space^［33］.

Fig. 8. Cross-PVG-based 2D exit pupil extension system.（a）Design diagram of the waveguide structure for 2D exit pupil extension，where “Left” and “Right” refer to the left- and right-spin chirality of the PVGs；（b）Shape of the designed waveguide and grating vector of the PVG coupler in the waveguide；（c）k-space diagram of the proposed 2D exit pupil extension system. The arrows with colors refer to the projections of the out-coupled grating and in-coupled grating vectors in the waveguide plane^［39］.

Fig. 9. Full-color display based on the three-layer waveguide structure.（a）Schematic diagram of the three-layer waveguide；（b）FOV as a function of the refractive index at different maximum diffraction angles^［21］.

Fig. 10. Full-color display based on two-layer waveguide structures.（a）Schematic diagram of the first-type two-layer waveguide structure；（b）Reflection of light by two PVGs at different incident angles^［32］；（c）Experimentally demonstrated full-color display effect^［32］；（d）Schematic diagram of the second-type two-layer waveguide structure；（e）Schematic of propagation of RGB three-color light beams inside the optical waveguide^［40］；（f）Experimentally demonstrated full-color effect^［40］.

Fig. 11. Full-color display based on single-layer waveguide structure.（a）Schematic diagram of single-layer waveguide structure；（b）Schematic diagram of the three-layer PVG；（c）Experimental full-color display effect；（d）FOV as a function of the refractive index at different maximum diffraction angles^［21］.

Fig. 12. Passively tunable PVGs.（a）Schematic diagram of mechanically tunable PVG based on flexible substrates；（b）Diffraction spectra at different shape deformations；（c）Change of the diffraction angle as the function of the strain at the incident wavelength of 532 nm^［41］.

Fig. 13. Actively tunable PVGs.（a）Electrically-switchable beam deflection effect^［47］；Transmission spectra at different voltages with the frequency of（b）1 kHz and（c）50 kHz，respectively^［48］；（d）Temperature-controlled transmission spectra of PVGs^［48］；（e）Optically-controlled transmission spectra of PVGs based on the azobenzene chiral dopant with different UV exposure times^［48］；（f）Schematic diagram of cholesteric liquid crystal PVG with photo-controlled chiral reversibility and modulation of beam deflection^［49］.

Fig. 14. Electrically tunable PVGs in optical waveguide display systems.（a）Electrically switchable PVGs；（b）Elimination of the rainbow effect by electrically switchable PVGs；（c）Sketches of the proposed waveguide configuration with a dynamic eye-box enabled by a pixelated PVG when the pupil is located at different positions；（d）Schematic diagram of the tunable PVG-based display system for expanding the FOV based on ultra-fast time-sequence switching^［50］.