## \(E\left(r,\rho ,z\right)=\frac{k}{z}{\int }_{0}^{{\rho }_{0}}\text{e}\text{x}\text{p}\left(ik\sqrt{{r}^{2}+{z}^{2}}-\sqrt{{f}^{2}+{\left(r-{r}_{0}\right)}^{2}}\right)\bullet {J}_{l}\left(\frac{k\rho r}{z}\right)rdr\), \(\left(3\right)\)

where *ρ* is the integral radius of the aperture, *ρ**0* is the lens size, and *J**l*(***) is the *l*th Bessel function, which can be considered as the combination of the circular aperture diffraction of the PB phase metasurface and the *l*th OV light.

## The metasurface design.

We used an all-dielectric PB phase metasurface consisting of elliptic nanoposts [36] array on a SiO2 substrate, as shown in figure 1(a). A right-handed circularly polarized (RCP) Gaussian light normally illuminated the SiO2 substrate from the bottom up, passing through the nanoposts array and form a Q-PV with LCP polarization state [37]. The beam’s working wavelength was set with 532 *nm* and the waist radius was twice the radius of the metasurface. To uniformly produce a Q-PV, the nanoposts on the SiO2 substrate were arranged in ring shape. According to PB phase principle, the rotation angle of each nanopost should be \(\alpha =\phi ⁄2\). Extract the phase term of equation 2 and accept the phase-angle conversion relationship, we obtain the following expressions for the phase and rotation angle distributions, respectively:

\(\phi \left(r,l,\theta \right)=k\left(\left(f-\sqrt{{f}^{2}+{\left(r-{r}_{0}\right)}^{2}}\right)\right)+l\bullet \theta\) , \(\left(4\right)\)

\(\alpha \left(r,l,\theta \right)=\frac{k}{2}\left(\left(f-\sqrt{{f}^{2}+{\left(r-{r}_{0}\right)}^{2}}\right)\right)+\frac{l\bullet \theta }{2}\) . \(\left(5\right)\)

figure 1(b) shows the local arrangement of the metasurface. The distance between the adjacent ring is 330 *nm*, and the distance between each nanopost in the ring is 330 *nm*, too. The toroidal arrangement of the metasurface could make the light field of Q-PV more even. Figure 1(c) shows the structure of a single nanopost with a height of *H* = 600 *nm*, the long axis is *L* = 250 *nm*, and the short axis is *W* = 95 *nm*. The TiO2 elliptic nanopost is an artificial birefringent material with high refractive index at visible light wavelength. It can accurately control the emission phase of the partially transmitted light within 2*π* range [34]. For high dielectric index metasurfaces, due to a strong local waveguide effect, the interaction among adjacent nanoposts can be ignored. [34]. At the same time, the TiO2 nanopost has a transmittance of up to 90% in the visible light band.

## The generated Q-PVs by metasurfaces.

To generate Q-PV, we firstly chose metasurfaces with aperture diameter *D* of 20 *µm*, *l* of 1, 5, 10, *f* of 15 *µm*, *r*0 of 5 *µm* to perform numerical simulations and made a comparison with the light field calculated by Fresnel integral diffraction theory. figures 2(a), (e), and (i) show the simulation results of the designed phase distributions of different TCs when *l* equals 1, 5, and 10, respectively. Phase values were obtained by extracting the transmission electrical field components and calculating with \(arctan\left[Im\right(Ex)/Re(Ex\left)\right]\). The simulated phase distributions agree with the theoretical ones. This proves that the designed metasurface can well accomplish phase control. figures 2(b), (f), and (j) respectively show the light field distributions of the Q-PV field in the *xOz* plane when *l* equals 1, 5, and 10. We can see that as the TC increases, the actual focal ring gradually deviates from the position of the designed focal length of *f* = 15 *µm*, which can be attributed to the Poynting vector’s oblique nature of the light field. However, the ring-shaped distribution in the designed focal plane remains unchanged. figures 2(c), (g), and (k) show the simulated light field in the focal plane with different *l*, their ring-shape distribution hardly changed with TC. The insets of figures 2(c), (g), and (k) are the phase distribution of the corresponding focal planes. The number of the mutation in the phase pattern corresponds to the number of TC. figures 2 (d), (h), and (i) show the theoretical light fields of Q-PVs calculated with equation 3, we can see that the simulation results were consistent with theoretical results. Therefore, the designed PB phase metasurface can successfully generate Q-PVs whose ring-shaped light field distributions hardly changed with TC.

As we know, the radius of the focal OV ring is positively correlated with TC. In order to study the divergence of the Q-PV, we quantitatively calculated the relationship between the measured radius (*r*) and TC by replacing the lens phase with the annular focusing phase. When designed radii (*r*0) were different, as shown in figure 3(a). We took focal OV as a comparison, as shown by the black line, the slope of the fitted line is 0.904. When *r*0 increases to 5 *µm*, we found that *r* still increased with TC, but the trend of the increase significantly reduced, appearing as the slope of the fitted straight line decreases to 0.299. When *r*0 was set to 10 *µm*, the increasing slope of the *r* was further limited to 0.221. This kind of divergence was also found in reference [35] as the Q-PV has a lower slope for the relationship between TC and *r* when *r*0 was larger. Meanwhile, when the focal length was set from 2 to 20 *µm*, we can see that *r* hardly changes with different focal lengths, as shown in figure 3(b). Therefore, the divergence characteristics of the Q-PV are not affected by the designed focal length.

## Optical spanners with Q-PVs.

Particles can be trapped and rotated in the focal OV. Since Q-PV has fixed radius of the focal ring, we will explore the new possibility of Q-PV being used in the applications of optical tweezers and spanners. We firstly simulated the Q-PV to trap SiO2 Mie-particles by choosing metasurfaces with aperture *D* of 20 *µm*, *l* of 1, 3, 5, *f* of 15 *µm*, *r*0 of 5 *µm*. Meanwhile, the amplitude of the incident light was set as 300 *hν* (i.e., the power of the light source was set to 0.00093 *mW*). The position of the SiO2 dielectric particle was set to the maximum intensity point of the focal ring (*z* = 14 *µm*), and along the *x*-axis from −10 *µm* to 10 *µm*, every 0.2 *µm* a step. We calculated the horizontal component of optical force exerted on the SiO2 particles, as shown in figure 4(a). Since the force distribution has symmetry at *x* = 5 *µm* and −5 *µm*, *x* = −5 *µm* was taken as an example. The force at *x* = −6 *µm* shows that when the dielectric particle was close to the outer side of the focal ring, it firstly received an increasing optical force pointing to the center of the focal ring, and then the optical force gradually decreased to 0, which was just at the center of the focal ring of Q-PV (i.e., *x* = −5 *µm*). As the particle continue to move towards the inner side of the focal ring, it was pulled by increasing and decreasing optical pulling forces in turn, suggesting that the dielectric particle would be trapped in the center of the Q-PV focal ring. By using equation 11, we calculated the trapping potential for SiO2 particle that integrated from *x* = −10 *µm*, as shown in figure 4(b). The trapping potential under each TC reached the lowest point of potential energy when the radius equals 5, and the depth of the potential wells reach below −200 *k*B*T*. In general, when the depth of trapping potential reaches −1 *k*B*T*, the particles can be stably trapped in the light field. Therefore, the dielectric particles can be stably trapped in the center of the orbit of the Q-PV.

Then we detected the OAM of Q-PV by detecting the angular optical force of metallic particles. The using of metallic particles is because they have high absorption and scattering that maximizes the transfer of OAM from the Q-PV [38]. Using Ag particles with a radius of 0.2 *µm*, we tested the angular force of the micro-sized particles every 30*°* on the focal ring, and the obtained results are shown in figure 4(c). As TC increases, the angular optical force exerts on the metallic particles gradually increases. The fluctuation of the force distribution was due to the uneven intensity of the light field, which was caused by the discrete distribution of nanoposts of the metasurface. We concluded that Q-PV not only can stably trap particles on fixed orbits, but also it can provide optical angular force that increases with TC for optical spanner use.

## Q-PV arrays.

In order to design a single metasurface with multi-channel OAM states, coaxial and non-coaxial Q-PV arrays are generated by stacking the complex amplitudes of Q-PV that carrying different TCs. By superimposing complex amplitudes of different single Q-PVs, the summation complex function is calculated, and the phase profile of the Q-PV array is then determined.

By introducing a linear phase gradient along the *θ* direction onto the phase of Q-PV: \(\text{e}\text{x}\text{p}\left[ikrS\left(\text{s}\text{i}\text{n}\theta \bullet \text{s}\text{i}\text{n}{\theta }_{n}+\text{c}\text{o}\text{s}\theta \bullet \text{c}\text{o}\text{s}{\theta }_{n}\right)/f\right]\) [35], where (\(r,\theta\)) is the polar coordinate parameter of each sampling point on the metasurfaces, (*S, θ**n*) is the designed position of the Q-PV in polar coordinates in the focal plane, the Q-PV will propagate away from the \(z\)-direction at a certain diffraction angle. The single phase of Q-PV in the array could be expressed as