Fig. 1. (a) Poincare sphere and (b) the distribution of polarization on the sphere[30]

Fig. 2. Comparison of hybridly polarized and radially polarized vector optical fields[30]. (a) Hybridly polarized vector optical field; (b) radially polarized vector optical field

Fig. 3. Intensity distribution of hybridly polarized vector optical field and radially polarized vector optical field[30]

Fig. 4. Distributions of (a)(b) polarization and (c)(d) intensity pattern of the azimuthally varying hybridly polarized vector optical fields with 45° and 0° quarter wave plate, respectively[31]

Fig. 5. Simulated collapsing patterns of the azimuthally varying hybridly polarized vector optical fields[55]

Fig. 6. Distribution of intensity and polarization of ellipticity-variant vector optical fields[56]

Fig. 7. Simulated distribution of polarization, measured intensity patterns and Stokes parameters of radially variant vector optical field with hybrid state of polarization[70]

Fig. 8. Snapshots of the motion of trapped particles around the ring focus generated by radially variant vector optical fields with hybrid state of polarization[70]

Fig. 9. Full Poincare sphere vector optical fields[32]

Fig. 10. Hybrid Poincare sphere[75]

Fig. 11. Second harmonic generation with full Poincare sphere beams[77]. (a) Intensity and polarization distributions of fundamental full Poincare sphere beams; (b) numerical simulations of the intensity patterns of the second harmonic light field; (c) experimental results for the intensity patterns of the second harmonic light field

Fig. 12. Higher-order Poincare sphere representation for the ±1 topological charges[34]

Fig. 13. Polarization and intensity distributions of the vector optical fields on the higher-order Poincare sphere[80]

Fig. 14. Hybrid-order Poincare sphere[35]. (a)-(c) Phases for optical fields at points A, B, and C, respectively; (d)-(f) corresponding intensity and polarization distributions

Fig. 15. Generalized Poincare sphere[36]

Fig. 16. Vector optical fields on generalized Poincare sphere carrying OAM[84]

Fig. 17. Array vector optical fields with seven vector optical field bases and the weakly focused status[37]. (a)(b) Array vector optical fields with radially and azimuthally polarized vector optical fields as bases, respectively; (c)-(e) weakly focused fields of the base, lattice, and array vector optical field; (f) SEM image of the silicon surface ablated by the weakly focused fields

Fig. 18. Array vector optical fields with azimuthally polarized vector optical fields as bases and SEM images of micromachined silicon with the tightly focused fields[38]. (a)-(e) Total intensity patterns of the array vector optical fields; (f)-(j) x-component intensity patterns of the array vector optical fields; (k)-(o) SEM images of the silicon surfaces ablated by the tightly focused array vector optical fields

Fig. 19. Fractal optical fields with the space-variant parameters[39]. (a) Amplitude-only; (b) phase-only; (c) polarization-only; (d) amplitude-phase; (e) phase-polarization; (f) amplitude-polarization; (g) amplitude-phase-polarization

Fig. 20. Polarization and intensity distributions of two types of fractal vector optical fields based on the Sierpinski structure[39]. (a)(b) Polarization states; (c)(d) experimental total intensity patterns; (e)(f) experimental x-component intensity patterns

Fig. 21. Experimental intensity patterns of the generated fractal vector optical fields with radially polarized vector optical fields as bases and different lattices[40]

Fig. 22. Array vector fields at the focal planes of the type-B fractal vector optical fields[39]. (a) Simulation results of focused field intensity for dimension of 0.6 mm×0.6 mm; (b) simulation results of focused field intensity for dimension of 1.8 mm×1.8 mm; (c) simulation results of focused field intensity for dimension of 5.4 mm×5.4 mm; (d) experimental results of focused field intensity

Fig. 23. Multi-zone sector plates and corresponding array vectorial focused fields[86-87]. (a) Multi-zone plate divided in radial direction; (b) multi-zone plate divided in azimuthal direction; (c) vectorial focused fields corresponding to (a); (d) vectorial focused fields corresponding to (b)

Fig. 24. Two-dimensional vectorial multifocal array[88]. (a) Multifocal spots with controllable intensities and positions; (b) multifocal spots with controllable states of polarization

Fig. 25. Three-dimensional vectorial multifocal array with controllable parameters[88]. (a) Three-dimensional vectorial multifocal array; (b)-(e) corresponding intensity patterns of beams passing a polarizer with transmission direction marked by a red double arrow

Fig. 26. Dynamically controlled array vector optical fields and the applications. (a) Rotation of the array vector optical field and the corresponding simulated focal traces[89]; (b) shear transformation of the array fractal vector optical field and the corresponding trapping experiment results[40]

Fig. 27. Two kinds of full Poincare beams and C-point and L-line in these fields

Fig. 28. Intensity and polarization distributions of the output field (the C-points are marked by circles, and the L-line is represented by a yellow line)[93]

Fig. 29. Arrayed polarization singularity vector optical fields generated by interference of three polarized waves with different amplitude ratios (the green ellipses show the polarization states, the yellow lines indicate L-lines, and the blue circles and red squares represent the C-points)[41]. (a) 1∶1∶1; (b) 5∶5∶7; (c) 5∶5∶3

Fig. 30. Multiple polarization singularity vector optical fields with spatial state of polarization (SoP) structures similar to the electric field lines[42]

Fig. 31. Polarization distributions of six kinds of multiple polarization singularity vector optical fields[43]

Fig. 32. Two-dimensional orthogonal coordinates systems. (a) Parabolic coordinates system; (b) elliptic coordinates system; (c) bipolar coordinates system; (d) hyperbolic coordinates system

Fig. 33. Parabolic-symmetry vector optical fields with different topological charges and n=0[44]

Fig. 34. Elliptic-symmetry vector optical fields with different topological charges and n=0[45]

Fig. 35. Bipolar-symmetry vector optical fields with different topological charges and n=0[46]

Fig. 36. Hyperbolic-symmetry vector optical fields with different topological charges and n=0[47]

Fig. 37. Polarization structure in the focal plane depicted by Archimedean spiral pattern[48]

Fig. 38. Arbitrarily designed focal fields[49]. (a) Poincare sphere; (b)-(b3) simulated Stokes parameters of the vectorial focal field; (c)-(c3) corresponding experimental results

Fig. 39. Experimentally generated vectorial focal fields consisting of 2D ring curve and 3D Achimedean curve with continuously varying state of polariza-tion. The 2D ring is located at the focal plane[105]

Fig. 40. Two vector optical fields with elliptical symmetry[51]. (a) With elliptical radial symmetry; (b) with elliptical azimuthal symmetry

Fig. 41. Vector optical field with Taiji pattern[106]. (a) Total intensity pattern; (b) x-component intensity pattern; (c) y-component intensity pattern; (d) state of polarization distribution

Fig. 42. Comprehensive adjustment of optical field amplitude, state of polarization, and phase[107]. (a) Total intensity, (b) state of polarization, and (c) phase of the incident vector optical field; (d) triangle focal spot with uniform intensity