Fig. 1. (a) FPS, (b) HOPS, and (c) HyOPS. S1(N,M), S2(N,M), and S3(N,M) are the Stokes parameters of the beams, and the superscript (M,N) denotes the topological charges carried by the LCP (southern pole) and RCP (northern pole) beams. When M=−N, the LCP and RCP beams are vortex beams of the same topological charge with opposite signs, and the HyOPS (c) is degraded to the HOPS (b). Moreover, when M=N=0, the LCP and RCP beams are plane waves, and the HOPS is degraded to an FPS (a).

Fig. 2. (a) Design flow and (b) meta-atom’s library of the dielectric metasurface for generating HOPS and HyOPS beams.

Fig. 3. (a) Scheme of generating arbitrary VVBs on a target HOPS or HyOPS. (b) The circular point represents a scalar beam having a position angle (2ϑ,2ψ). (c) The solid red point is the transmitted light beam’s position (−2ϑ,−2ψ) on a HOPS/HyOPS.

Fig. 4. (a) Simulated far-field, near-field, and phase profile distributions of the X component in the transmitted light beams over the target fifth-order HOPS by manipulating the position angles (2ϑ,2ψ). (b) Measured near-field distribution of the X component. (c1) and (c2) The simulated and experimental results of the X component’s propagation evolution from the metasurface (Z=0mm) to Z=16mm. Scale bars: 130μm.

Fig. 5. Experimental results: controlled generation of tightly focused fifth-order HOPS beams (a1)–(c3) along its longitude line and (d1)–(f3) along its equator. The first and fourth columns are the X components in the total focal fields T (the third and sixth columns), and the second and fifth columns are the Y components.

Fig. 6. Simulation results: controlled generation of 0-2 order HyOPS beams (a1)–(c3) along its longitudinal line and (d1)–(f3) along its equator. The first and fourth columns are the X components in the total focal fields T (the third and sixth columns), and the second and fifth columns are the Y components.

Fig. 7. Experimental results: controlled generation of 0-2 order HyOPS beams (a1)–(c3) along its longitudinal line and (d1)–(f3) along its equator. The first and fourth columns are the X components in the total focal fields T (the third and sixth columns), and the second and fifth columns are the Y components.

Fig. 8. Simulation results: controlled generation of 0-1 order HyOPS beams (a1)–(c3) along its longitudinal line and (d1)–(f3) along its equator. The first and fourth columns are the X components in the total focal fields T (the third and sixth columns), and the second and fifth columns are the Y components.

Fig. 9. Experimental results: controlled generation of 0-1 order HyOPS beams (a1)–(c3) along its longitudinal line and (d1)–(f3) along its equator. The first and fourth columns are the X components in the total focal fields T (the third and sixth columns), and the second and fifth columns are the Y components.