Fig. 1. (a) FPS, (b) HOPS, and (c) HyOPS. $S_1^{(N,M)}$, $S_2^{(N,M)}$, and $S_3^{(N,M)}$ 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 $(2\vartheta ,2\psi )$. (c) The solid red point is the transmitted light beam’s position $(-2\vartheta ,-2\psi )$ 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 $(2\vartheta ,2\psi )$. (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 ($Z=0\mathrm{mm}$) to $Z=16\mathrm{mm}$. Scale bars: $130\mu \mathrm{m}$.

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.

Table 1. Parameters and target of metasurface samples.