Fig. 1. Geometrical representations of the first- and second-order singularity states. The poles of the Poincaré sphere denote the screw singularity states with opposite topological charges, while the equator describes the edge states. (a), (b) The screw singularity states with positive topological charges. (g), (h) The screw singularity states with negative topological charges. (c)–(f) The first- and second-order edge states located at α=π [(c), (e)] and α=0 [(d), (f)].

Fig. 2. Theoretical demonstrations of the edge-to-screw conversions. (a) The designed structured light state comprising a higher-order edge state (blue curves) and an off-center-packed circular vortex array (red dots), for three different cases of topological orders. (b)–(e) Geometrical mappings of the first- and second-order edge-to-screw conversion onto the first [(b), (c)] and second [(d), (e)] order orbital Poincaré sphere. The state evolutions induced by the circular vortex V(u) [(b), (d)] and antivortex V(u*) [(c), (e)] arrays are considered. The propagation distance is set to 400 mm. Three different values of r0 are examined, as indicated in the panels.

Fig. 3. (a) Experimental setup. BE, beam expander; P, polarizer; BS, beam splitter; SLM, spatial light modulator; CCD, charge-coupled device; M, mirror. The laser was working at the wavelength of λ=632.8  nm. The iris was used to isolate the unwanted orders of diffraction from the SLM. (b), (c) The computer-generated phase-only holograms for observing the first-order (l=1) (b) and second-order (l=2) (c) edge-to-screw state conversions, based on Eqs. (9) and (10).

Fig. 4. Observations of the first- [(a), (b)] and second-order [(c), (d)] edge-to-screw singularity state conversions. Panels (a1)–(d1) denote the petal-like structured light fields measured at the focal plane z=0. Panels (a2)–(d2) present measurements of the propagating light fields recorded at z=400  mm. Panels (a3)–(d3) show the plane-wave interference patterns of the singularity states at z=400  mm. (a), (c) The theoretical results; (b), (d) the corresponding experimental measurements. Panels in the first and second columns share the same scale bar: 0.5 mm. Scale bars in (a3), (b3) and (c3), (d3) are 0.1 mm and 0.13 mm.

Fig. 5. Observations of the higher-order edge-to-screw state conversions. (a)–(c) The experimentally generated higher-order edge states at z=0 with imbedded circular vortex array for three different orders: (a) l=3, (b) l=5, and (c) l=10. The scale bars in (a), (b) and (c) are 0.47 mm and 0.63 mm, respectively. (d)–(f) The theoretical results corresponding to (a)–(c). (g)–(i) The plane-wave interference patterns experimentally measured at z=400  mm. The scale bars: (g) 0.2 mm, (h) 0.25 mm, and (i) 0.35 mm. (j)–(l) The theoretical outcomes corresponding to (g)–(i).

Fig. 6. The linear relationship between topological orders of higher-order edge state and the topological charge of the resultant screw singularity state. The black-rectangle scatters and red line represent experimental and theoretical results, respectively. |l| denotes the order of the edge state while lnew is the new-born topological charge.