## 3 Numerical Simulations and Results
### Results
The presence of longitudinal electric field on the optical axis for the radially polarized helical beams was also experimentally demonstrated in Reference [26]. This is proof that, by using a lens with 0.47 NA, we are in the strong focusing regime. We should observe a non-zero intensity on the optical axis for variation 1B and 1C if this effect is present, and we should be able to determine if the focal spot generated by variation 1B or by variation 1C is proper for STED microscopy. Focal spots produced by variation 1B and variation 1C using the same focusing element are shown in Figure 5a,b respectively, where the polarization state is also shown. We can observe in Figure 5 that, around 2 mm from the optical axis, polarization becomes azimuthal. This may seem unexpected since the polarization for variation 1C is radial after the polarization rotator (Equations (6) and (7)). This change in polarization state suggests that the phase difference between the two circularly polarized modes that are superimposed increases with \(\pi\). This phenomenon is attributed to the Gouy phase shift [24], which is different for the mode with topological charge \(-\)3. For both cases, the intensity is zero on the optical axis, and, for both cases, the polarization is partially azimuthal. The main difference between the two configurations is that variation 1B produces a focal spot with axial symmetry, while, in the case of variation 1C, the focal spot is circularly symmetric. The focal spot generated by variation 1A is also circularly symmetric but less extended than in the case of variation 1C. Figure 4: Radially polarized focal spot (a); azimuthally polarized focal spot, generated by variation 1A (b). The expression of the complex electric field generated by a paraxial Gauss-Laguerre mode is given in Equation (14), where the only phase term which depends on the \(z\)-coordinate and is not common for modes with different indexes is the Gouy phase term \(\psi_{p\uparrow}(z)=(2\;p+[\downarrow]+1)\;\mathrm{arctan}(z\;\lambda\!/(\pi\;w^ {2}{}_{0}))\). Considering the focal plane as the origin, we can evaluate the Gouy phase in the plane of the lens, with the expression given in Equation (15). Other notations are as follows: \(w(z)\) is the \(z\)-dependent half-diameter of the beam, \(q(z)\) is the complex beam parameter, and \(L_{p}{}^{[2]}\) is the generalized Laguerre polynomial. \[E(\mathbf{r},z)=\frac{w_{0}}  
mode with topological charge \(-3\) is equal to the amplitude of the mode with topological charge \(-1\). This affirmation is correct also for the spot produced by variation 1C, represented in Figure 5b. We summarize the polarization states of the beam after each optical component and for each variation of the system 1A, 1B, 1C, and 2 in Figure 6. Variation 2 is the configuration of the system for the generation of flat-top beams. Note that the first polarization state represented is the polarization state of the beam after QWP; note that MC for variation 2 is different from MC for variation 1A, 1B, and 1C, and also note that HWP produces a relevant effect only in the case of variation 1B for which it can be used to rotate the diffraction pattern. For variation 2, the polarization state along the optical axis is maintained through the system. We also consider the case with QWP and HWP rotated simultaneously in the same direction, with the rotation angle of QWP twice the rotation angle of HWP (\(\varphi=2\gamma\)). The Jones vector for the beam after the HWP is given in Equation (20) as a superposition between right and left circularly polarized modes, and we can see that the weight of each mode is controlled by the rotation angle \(\varphi\) of the QWP. As a consequence, the beam in the plane of the lens (expression given in Equation (21)) is described as the summation of one azimuthally polarized beam with zero angular momentum (expression given in Equation (5)) and a radially polarized beam with non-zero angular momentum (with expression given Equation (7)).