## 3 Results and Discussion  
### Focused Optical Vortex Generation  
A succinct view of the designed transmission-type focused optical vortex (FOV) generator is presented in Figure 3a, the device combines the functions of a lens and a spiral phase plate into a single metasurface. The phase distribution profile of the FOV generator can be divided into the optical vortex profile \(\varphi_{OV}\) and lens profile \(\varphi_{lens}\). Thus, the expression for the phase profile can be written as follows:  
\[ \begin{array}{ll}\phi_{m}(x,y)&=\varphi_{OV}+\varphi_{lens}\\ &=l\star arctan\Big{(}\frac{y}{x}\Big{)}-\frac{2\pi}{\lambda}\Big{(}\sqrt{x^{ 2}+y^{2}+f^{2}}-f\Big{)}\end{array}  \tag{3}\]  
where \(\lambda\) is the design wavelength (here \(\lambda\) = 355 nm), f is the focal length, \(l\) is the topological charge, which defines the degree of helicity at the focal point, while \(x\) and \(y\) are the transverse coordinates on the metasurface plane. Then we can obtain the spatial distribution of the phase profile compensated by the metasurface through adjusting the rotation angle \(\theta x,y\) of the optimized Nb\({}_{2}\)O\({}_{5}\) nanobars discussed above, where \(\theta(x,y)~{}=~{}\phi_{m}(x,y)/2\). \[NA~{}=~{}sin(\phi)~{}=~{}\frac{D}{\sqrt{4f^{2}+D^{2}}}. \tag{4}\]  
Figure 2: Excitation of anti-parallel magnetic dipoles (AMDs) in a Nb\({}_{2}\)O\({}_{5}\) nanobar. (**a**,**d**) Normalized total electric (\(|\)E\(|^{2}\)) fields and (**b**,**e**) normalized total magnetic (\(|\)H\(|^{2}\)) field existing in the optimized nanobar under 355 nm \(x\)- and \(y\)-polarized wave incidence, respectively. The white arrows denote the induced corresponding field’s vector profiles in and near the nanobar, and the black lines distinguish the boundaries of the nanobar. Simplified sketches of exited antiferromagnetic modes with even and odd dipoles are shown in (**c**) and (**f**), respectively. The calculation for numerical aperture (NA) is displayed in Figure 3b and Equation (4). Here, \(D\) is the diameter of the metasurface and \(s\) is the hypotenuse of the right triangle. The diameter of the metasurface is kept at 12 \(\upmu\)m to achieve a NA of 0.514. A full-wave numerical simulation of all the designed FOV generators is calculated through FDTD methods. PML boundary conditions are applied along all the \(x\), \(y\), and \(z\) axes, and the incidence is the RCP plane wave. A perfect electric conductor (PEC) ring is placed between the RCP source and the metasurface to limit the area of incidence to the circular metasurface region. Figure 4a-c are the intensity profiles of \(x\)-\(z\) cross sections for \(l\) = 1,2,3, respectively, and the white dashed line marks the position of the focal-plane (z = 10.0 \(\upmu\)m). There is a small deviation (\(<\)1%) between the simulated focal length and the designed focal length. The deviation can be attributed to the discrete phase of each meta-atom for an approximation of the continuous phase distribution. Figure 4d-f demonstrate the donut-shaped focal spot on the focal plane. It can be observed that the minimum intensity of the vortex ring is at the center of the focal plane. The bottom-left insets of Figure 4d-f show corresponding phase patterns at the focal plane. It illustrates that, as the topological charge \(l\) increases, the number of spirals in the phase pattern also increases. Figure 4g-i demonstrate the corresponding horizontal cuts of focal spots. The full widths at half maximum (FWHM) of horizontal cuts of the optical vortex ring are 311 nm, 335 nm, and 355 nm, respectively. The efficiencies of the FOV are as high as 75.1% (for \(l=1\)), 73.0% (for \(l=2\)), and 71.0% (for \(l=3\)). Here, the efficiency of the FOV is defined as the ratio of the transmitted power passing through the focal spot in a circle whose radius is three times the FWHM. The definition of the FOV's efficiency is the same as in [36]. The focusing efficiency decreases as the topological charge increases, mainly due to the increase in the vortex ring's diameter.