Fig. 1. Distributions of polarization states of four polarization singularities. (a) Lemon C-point with m=0.5 and τ=r, (b) star C-point with m=−0.5 and τ=r, (c) lemon V-point with m=0.5 and τ=1, and (d) star V-point with m=−0.5 and τ=1. (e)–(h) Corresponding distributions of the orientation angles of the major axes of polarization states to (a)–(d), respectively. (i)–(l) Corresponding distributions of the angles of ellipticity of the polarization states to (a)–(d), respectively.

Fig. 2. Schematic of experimental setup. SLM, spatial light modulator; L1, plano-concave lens; L2, plano-convex lens; L3 and L4, bi-convex lenses; SF, spatial filter; QWP, quarter-wave plate; G, Ronchi grating; CCD, charge coupled device.

Fig. 3. Measured intensity distributions around a lemon V-point (m=0.5) at different propagation distances as z=0.1, 0.4, 0.7, and 1.0 m.

Fig. 4. Distributions of the intensity and polarization states around a V-point at a propagation distance z=0.5  m. First, second, and third rows show the intensity, orientation angles of the major axes of polarization states, and angle of ellipticity, respectively. First (third) column shows the simulated results with m=0.5 (2.5), and second (fourth) column shows the corresponding measured results.

Fig. 5. Distributions of the intensity and polarization states around a C-point at a propagation distance z=0.5  m. First, second, and third rows show the intensity, orientation angles of the major axes of polarization states, and angle of ellipticity, respectively. First (third) column shows the simulated results with m=0.5 (2.5), and second (fourth) column shows the corresponding measured results.

Fig. 6. Any polarization state can be seen as a superposition of right- and left-handed circularly polarized components carrying the phases. (a) Polarization states at all points surrounding the V-point are linearly polarized but different in orientation, which are located on the equator of the Poincaré sphere. (b) Polarization states at all points surrounding the C-point are elliptically polarized but different in orientation, which are located on a certain latitude in the northern (or southern) hemisphere of the Poincaré sphere.

Fig. 7. Comparison of the polarization topological index T and algebraic sum of the singularities of vector optical fields with different topological charge m.

Fig. 8. Distributions of the orientation angles 2ϕ of the major axes of the polarization states for different singular lines. (a) Single straight singular line case, (b) triple straight singular case (n=3 and η=0), and (c) triple spiral singular case (n=3 and η=r).

Fig. 9. Illustration of the experimentally measured intensity and polarization distribution of the ternary case [(a)–(c)] and the ternary spiral case [(d)–(f)] at a propagation distance z=0.5  m. (a), (d) Intensity; (b), (e) orientation angles of the major axes of the polarization states; and (c), (f) angle of ellipticity.