## 3 Numerical Simulations and Results
### Results  
In Figure 3a, we show the intensity profile of the beam right before (red) and after (blue) MC for variation 1 of the system, with green representing the conversion efficiency depending on the input coordinate. The beam is helical due to the presence of SPP, placed 1 m before MC. We can observe that the conversion efficiency is nearly 50%, except in the central region, where it is nearly zero. In Figure 3b, we have a ray-fan diagram (red) which represents the position of the ray in the focal plane depending on the position before the MC. We can observe small oscillations near zero that are caused by the inability of aspherical lens formula to perfectly represent the ideal corrected rear surface of MC. In Figure 3c (inset), we represent MC for variation 1 with the high-NA asphere in the longitudinal section, where the distance between components is shrunk to fit in the image. In Figure 3d (inset) is the ray spot diagram of the system, where we can observe that the geometrical spot is smaller than the wavelength and, therefore, the system is nearly diffraction-limited. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **MC\({}_{1}\)-Front** & **MC\({}_{1}\)-Rear** & **MC\({}_{2}\)-Front** & **MC\({}_{1}\)-Rear** & **A\({}_{1}\)-0.21 NA** & **A\({}_{2}\)-0.47 NA** \\ \hline R & 5 mm & 7.45 mm & 19.15 mm & 14.42 mm & 25.560 mm & 18.651 mm \\ \hline \(\kappa\) & –67.33 & –54.74 & –55.7 & –14.6 & –1.01 & –1.408 \\ \hline cen. & 9 mm & & 9 mm & 6 mm & 9 mm \\ \hline \(\alpha_{2}\) & 0.002076 & & 0.01293 & & \\ \(\alpha_{4}\) & –0.00003008 & & \(-0.00009489\) & \(3.270\times 10^{-6}\) & \(1.4755\times 10^{-5}\) \\ \(\alpha_{6}\) & \(4.857\times 10^{-7}\) & & \(9.713\times 10^{-7}\) & \(7.720\times 10^{-10}\) & \(-5.557\times 10^{-9}\) \\ \(\alpha_{8}\) & \(-5.705\times 10^{-9}\) & & \(-8.430\times 10^{-9}\) & \(1.630\times 10^{-13}\) & \(-3.511\times 10^{-12}\) \\ \(\alpha_{10}\) & \(4.446\times 10^{-11}\) & & \(5.312\times 10^{-11}\) & & \\ \(\alpha_{12}\) & \(-2.158\times 10^{-13}\) & & \(-2.207\times 10^{-13}\) & & \\ \(\alpha_{14}\) & \(5.879\times 10^{-16}\) & & \(5.343\times 10^{-16}\) & & \\ \(\alpha_{16}\) & \(-6.843\times 10^{-19}\) & & \(-5.668\times 10^{-19}\) & & \\ \hline \multicolumn{5}{l}{n@800 nm} & 1.7601 & 1.7601 & 1.5108 & 1.7761 \\ \hline \end{tabular} MC\({}_{1}\) – the mode converter for variation 1, MC\({}_{2}\) – the mode converter for variation 2, A\({}_{1}\) – the aspherical lens with low NA, A\({}_{2}\) – the aspherical lens with high NA.  
\end{table} 
Table 1: A detailed description of all surfaces employed in the simulation using the aspherical lens formula. Figure 3: The intensity profile before mode converter (MC) is shown in red, the intensity profile after MC is shown in blue, and the conversion efficiency is shown in green (**a**); the ray-fan diagram is shown in red (**b**); the MC and aspherical lens, with ray-tracing, in longitudinal section view (**c**); ray intersection with the focal plane (**d**). The focal spot for variation 1A, which is polarized azimuthally, is presented in Figure 4b, while a focal spot with radial polarization is presented in Figure 4a for comparison. The radially polarized focal spot can be generated using a polarization rotator that rotates the polarization anti-clockwise instead of clockwise. The high-NA aspherical lens (0.47 NA) was used for focusing, in both cases. For both cases, the polarization state is the same after PR and in the focal plane because the beam is formed by a superposition between two helical beams with topological charge +1 and \(-\)1 with orthogonal circular polarization states. For the azimuthally polarized beam (variation 1A), the intensity profile in the focal plane is typical for a beam used for STED microscopy, having zero intensity on the optical axis. The radially polarized focal spot has non-zero intensity on the optical axis, and this behavior is a marker for high NA focusing, as pointed out, for example, in Reference [22].