## 3 Results on the Generation of Standard Hybrid Vector Beams
### Case with Input Circularly Polarized Gaussian Beam  
We first consider \(\ell=0\), i.e., the input beam is an RCP \(LG_{0}^{\ell}\) Gaussian beam. This simple case was analyzed in the far field in [30] and we include it here to compare with the next cases. According to Equation (12) the output is \(\mathbf{V}_{out}=\frac{1}{\sqrt{2}}\Big{(}LG_{0}^{\alpha}\mathbf{e_{R}}-iLG_{ 0}^{-1}\mathbf{e_{L}}\Big{)}\). Therefore, this hybrid vector beam is a Poincare beam with fractional order \(m=\) 1/2. Figure 3 shows numerical and experimental results in the NF (Figure 3a) and in the FF (Figure 3b). For each case we show Matlab numerical calculations using the exact relation in Equation (8) (Simul) and the corresponding experimental results (Exp). For the NF calculations we simply illustrate the polarization map, since the experiments only show the central area of the beam, cropped with a diaphragm. For the FF, the image shows the polarization map and the intensity distribution. Right and left-handed states are depicted in magenta and cyan ellipses respectively. Linear polarization is depicted in green. The experimental captures, both for NF and FF, are the images in the two columns to the right of the polarization maps. In each case we show captures without analyzer (NA) and with analyzers transmitting the typical polarization states (linear vertical and horizontal, linear at \(\pm\)45, RCP and LCP). The experimental polarization maps are obtained by deriving the Stokes parameters from the experimental images. Note that the FF images here displayed were saturated in order to better visualize the intensity pattern, but equivalent non-saturated images were used to calculate the polarization maps. The FF experimental intensity distribution corresponds to the averaged \(S_{0}\) Stokes parameter. The cases analyzed in the rest of the paper will present the figures in the same format as Figure 3.  
Figure 2: Optical setups for the generation of vector beams. (**a**) Setup to capture the near field. (**b**) Setup to capture the far field. P: linear polarizer; QWP: quarter-wave plate; qP: static q-plate; L: converging lens. Let us analyze the NF captures (Figure 3a). First, we note that illuminating the QW q-plate with a circular state provides an output that is a spatially-variant linearly polarized beam. Since \(m=1/2\), the orientation of the linear polarization shows an azimuthal rotation of half a cycle in the complete plane. The polarization pattern is rotated \(45^{\circ}\) clockwise with respect to the vertical direction due to the angle \(\varphi=-\pi/4\). This pattern is experimentally verified in the captures with a linear analyzer, which show a dark lobe opposed to a bright lobe, which rotate as the analyzer does. The captures with circular analyzers show a dark central dot for the LCP component, verifying the phase singularity in this component, while the RCP shows no singularity. The polarization map in the FF shows different polarization states that change along the azimuthal and radial coordinate, thus manifesting the Poincare beam nature. The changes along the azimuthal angle can be easily understood by the polarization rotation described in Equation (11). Like in the NF, the captures for linear analyzers show a bright spot that rotates as the analyzer does. However, note the \(\pi/2\) rotation of the FF intensity compared to the NF, for each orientation of the analyzer. This can be viewed as a consequence of the Gouy phase polarization rotation \(\frac{1}{2}\Delta\psi_{G}=-\frac{\pi}{4}\), that causes a \(90^{\circ}\) counterclockwise rotation of the intensity pattern transmitted by the linear analyzers. Polarization changes along the radial coordinate are a consequence of the different amplitude of the modes encoded on each circular component. The analysis with circular analyzers confirms that there is a phase singularity only for the LCP component, while the RCP component is bright in the center, denoting the absence of singularity. Consequently, the corresponding polarization map shows RCP at the center.