## 3 Numerical Simulations and Results
### Results
As we did before, we add a minus sign to the component with topological charge \(-3\) in order to take into account the change in polarization due to propagation to the focal plane; the polarization state in the focal plane is described by Equations (22) and (23). \[\left[ \begin{array}{c}V_{x}\\ V_{y}\end{array} \right]_{afterHWP}=\left(\sin\ \varphi\left[ \begin{array}{c}1\\ \mathrm{i}\end{array} \right]+\cos\ \varphi\left[ \begin{array}{c}1\\ -\mathrm{i}\end{array} \right]\right)\mathrm{e}^{-\mathrm{i}\,\theta}. \tag{20}\]  
\[\left[ \begin{array}{c}V_{+}\\ V_{-}\end{array} \right]_{lens}=\mathrm{i}\sin\ \varphi\left[ \begin{array}{c} \mathrm{e}^{-\mathrm{i}\,\theta}\\ -\mathrm{e}^{\mathrm{i}\,\theta}\end{array} \right]+\cos\ \varphi\left[ \begin{array}{c} \mathrm{e}^{-\mathrm{3i}\,\theta}\\ \mathrm{e}^{-\mathrm{i}\,\theta}\end{array} \right]\propto\left[ \begin{array}[] {c}\mathrm{e}^{-\mathrm{2i}\,\theta}\\ -\mathrm{i}\,\mathrm{e}^{\mathrm{i}\,\theta}\sin\ \varphi+\mathrm{e}^{-\mathrm{i}\,\theta}\cos\ \varphi\end{array} \right]. \tag{21}\]  
\[\left[ \begin{array}{c}V_{+}\\ V_{-}\end{array} \right]_{focus}=\mathrm{i}\sin\ \varphi\left[ \begin{array}{c} \mathrm{e}^{-\mathrm{i}\,\theta}\\ -\mathrm{e}^{\mathrm{i}\,\theta}\end{array} \right]+\cos\ \varphi\left[ \begin{array}{c} -\mathrm{e}^{-\mathrm{3i}\,\theta}\\ \mathrm{e}^{-\mathrm{i}\,\theta}\end{array} \right]\propto\left(\mathrm{i}\, \mathrm{e}^{\mathrm{i}\,\theta}\sin\ \varphi-\mathrm{e}^{-\mathrm{i}\,\theta}\cos\ \varphi\right)\left[ \begin{array}{c} \mathrm{e}^{-\mathrm{2i}\,\theta}\\ -1\end{array} \right]. \tag{22}\]  
\[\left[ \begin{array}{c}V_{x}\\ V_{y}\end{array} \right]_{focus}=\left(\mathrm{i}\,\mathrm{e}^{\mathrm{i}\, \theta}\sin\ \varphi-\mathrm{e}^{-\mathrm{i}\,\theta}\cos\ \varphi\right)\left[ \begin{array}{c}-\sin\ \theta\\ \cos\theta\end{array} \right]. \tag{23}\]  
For the simulations presented in Figure 7, we consider the low-NA aspherical lens, with back focal length 46 mm and clear aperture 23 mm in diameter, as the focusing element. Figure 7 also indicates that the phenomenon that leads to the generation of the circularly non-symmetrical focal spot is not  
Figure 6: Polarization state after each optical component, for each variation of the system. due to the strong focusing (presence of \(E_{\mathrm{z}}\) component), since the results are in perfect accord with simulations using a focusing element with higher NA, shown in Figures 4 and 5. In Figure 7a-d,f-h,j-l, we represent the transition between variation 1A and 1C of the system resulting from the clockwise rotation of QWP by \(10^{\circ}\) and the clockwise rotation of HWP by \(5^{\circ}\), which means \(\varphi=2\gamma\); cases \(\varphi=70^{\circ}\), \(\gamma=45^{\circ}\) and \(\varphi=60^{\circ}\), \(\gamma=45^{\circ}\) are represented in Figure 7e,i, respectively. Rotation of HWP is necessary in order to preserve the orientation of the focal spot and can be used to further control the orientation, which can be observed from the fact that Figure 7e does not have the same orientation as Figure 7f, while Figure 7i does not have the same orientation as Figure 7j. We can see how the polarization state of the beam before MC (represented by black ellipses) affects the focal spot. Results from Figure 7 are well explained by Equation (23). [PERSON] and [PERSON] [26] used a similar optical system, using a radial analyzer instead of MC and a polarization rotator that rotated the polarization by \(90^{\circ}\). They produced two different types of radially polarized vector beams depending on the input polarization on the radial analyzer. One of the vector beams produced a focal spot similar to that presented in Figure 4a, and the other vector beam produced a focal spot similar to configuration 1C (Figure 5b). They experimentally proved that the radially polarized focal spot possesses non-zero intensity along the optical axis.