Fig. 1. Differential geometry of ray^[11]

Fig. 2. Ray tube model

Fig. 3. Univalent hyperboloid ray model of Gaussian beam. (a) Two rays launching from same point Q at beam waist; (b) relationship between wavefront and rays with equal length^[55]

Fig. 4. Two rays passing through arbitrary point S out of beam waist^[55]

Fig. 5. Distributions of Gaussian windowed Fourier transforms of optical wave field. (a1)-(c1) Correspond to Eq. (41); (a2)-(c2) correspond to Eq. (42); (a3)-(c3) correspond to Eq. (43)

Fig. 6. Ray distributions of two different beams. (a) Airy beam; (b) Hermit-Gaussian beam

Fig. 7. Physical insight of single ray in SAFE mothed

Fig. 8. Axicon-based generation of nondiffracting beam. (a) “X”-like rays which can be generated by refracting parallel rays through axicon lens; (b) ray presentation of self-repairing property of beam

Fig. 9. Intensity distributions. (a) Intensity distribution calculated by SAFE; (b) intensity distribution calculated by angular spectrum diffraction method; (c)(d) self-repairing situation of beam blocked at different regions, calculated by angular spectrum diffraction method

Fig. 10. Ray presentation of Airy beam. (a) Ray model of Airy beam built by choosing arbitrary point at parabolic caustic as reference point; (b) schematic of ray model

Fig. 11. Airy beam reconstructed by SAFE method. (a) Normalized intensity distribution of xoz plane; (b) comparison of normalized intensity distribution of z = 0 plane with Airy beam

Fig. 12. Schematic of self-repairing property of Airy beam. (a) Ray model of Airy beam; (b) (c) optical-wave-field distributions of Airy beam blocked at different regions

Fig. 13. Beam having spiral wavefront composed by uniform skew rays. (a) Schematic of skew rays; (b) specific analyse of single ray

Fig. 14. Phase distributions of optical wave field with different values of m. (a) m=0; (b) m=1; (c) m=3

Fig. 15. Corresponding relationship between point on Poincaré sphere and ellipse in real space^[78]. (a) Location of point on Poincaré sphere determined by longitude angle φ and latitude angle θ; (b) parameters of ellipse in real space determined by point on Poincaré sphere

Fig. 16. Corresponding relationship between circular curve on Poincaré sphere and different kinds of Hermit-Laguerre Gaussian beams. (a1)-(c1) Circular curves on Poincaré sphere with normals toward different latitude angles θ; (a2)-(c2) ellipse distributions of normalized sizes on beam-waist plane; (a3)-(c3) corresponding ray models

Fig. 17. Structured Gaussian beam with decagram inner caustic^[100]. (a) Curve on surface of Poincaré sphere; (b) ellipses and their outer and inner caustics; (c) normalized intensity distribution on beam-waist plane; (d) phase distribution on beam-waist plane