## 4 Results on the Generation of Hybrid Petal-Type Vector Beams
### Case with Input Higher-Order Radial Polarization  
In this section we study the resulting beam when illuminating the QW q-plate with a pure radial beam of order \(\ell\). These beams are described by the following relation:  
\[\mathbf{V}_{out}=\frac{1}{\sqrt{2}}\Big{[}LG_{0}^{\ell}\mathbf{e_{R}^{\star}}+ LG_{0}^{-\ell}\mathbf{e_{L}^{\star}}\Big{]} \tag{14}\]  
Figure 5: Results for a QW q-plate illuminated with linear vertical polarization and \(\ell=0\). (**a**) Results in the near field; (**b**) results in the far field. Simulated and experimental polarization maps, with equivalent color representation as in Figure 3. Experimental images captured for different orientations of the analyzer, indicated as white arrows (NA indicates no analyzer). The output vector beam, after applying Equations (3) and (6), adopts the form of a more complex non-separable structure of wavefront and local polarization which reads  
\[\mathbf{V}_{out}=\frac{1}{2}\left[\left(LG_{0}^{\ell}-iLG_{0}^{-(\ell-1)} \right)\mathbf{e}_{\mathbf{R}}^{\alpha}+\left(LG_{0}^{-\ell}-iLG_{0}^{\ell-1} \right)\mathbf{e}_{\mathbf{L}}\right] \tag{15}\]  
Like in the previous case, there is a superposition of two LG modes in each circular polarization. This superposition generates sidelobes along the azimuthal coordinate for each circular component due to the azimuthal phase difference between each pair of LG modes. This situation is similar to the superposition that generates scalar beams with angular lobes, in what has been named as petal beams [40]. In order to gain physical insight of these angular lobes, we consider an approximation where we ignore the difference in the modes' amplitude, i.e., we assume \(u_{\ell-1}(z)\cong u_{\ell}(z)\). Within this approximation we can consider that  
\[\frac{1}{2}\left(LG_{0}^{\ell}-iLG_{0}^{-(\ell-1)}\right)\cong u_{\ell}(z)We ^{i\theta/2}\cos\left(\frac{1}{2}\left[-(2\ell-1)\theta-\Delta(z)+\frac{\pi}{ 2}\right]\right) \tag{16a}\] \[\frac{1}{2}\left(LG_{0}^{-\ell}-iLG_{0}^{\ell-1}\right)\cong u_{\ell}(z)We ^{i\theta/2}\cos\left(\frac{1}{2}\left[(2\ell-1)\theta-\Delta(z)+\frac{\pi}{ 2}\right]\right) \tag{16b}\]  
where \(W=e^{-i(2+|\ell|+|\ell-1)\xi(z)/2}e^{in/4}\) is a global phase and \(\Delta(z)=\left(|\ell|-|\ell-1|\right)\xi(z)\) denotes the Gouy phase difference between the modes \(LG_{0}^{\mp\ell}\) and \(LG_{0}^{\pm(\ell-1)}\). Note that this Gouy phase shift is different to that in Equation (10), which is the phase difference between the RCP and LCP components. Instead, \(\Delta(z)\) is the phase shift between the two modes now encoded in each circular polarization. The intensity of the RCP and of the LCP components, within this approximation, is given by  
\[i_{R}(z)\cong|u_{\ell}|^{2}(1+\sin[(2\ell-1)\theta+\Delta(z)]) \tag{17a}\] \[i_{L}(z)\cong|u_{\ell}|^{2}(1-\sin[(2\ell-1)\theta-\Delta(z)]) \tag{17b}\]  
These relations indicate that the intensity of each circular polarization component show \((2\ell-1)\) lobes along the polar coordinate. Since \(\Delta(z)=0\) in the NF the patterns \(i_{R}\) and \(i_{L}\) are complementary. However, propagation produces an opposite rotation of the lobes in each polarization component. Noting that \(|\ell|-|\ell-1|=\pm 1\) (positive for \(\ell\geq 0\) and negative for \(\ell<0\)), then \(\Delta(z)=\pm\pi/2\) in the FF limit, and the two sidelobes patterns perfectly match since the two relations in Equations (17a) and (17b) become the same:  
\[i_{R}(z\rightarrow\infty)=i_{L}(z\rightarrow\infty)\cong|u_{\ell}|^{2}(1\pm \cos[(2\ell-1)\theta] \tag{18}\]  
Figure 6 shows the corresponding results for a 2 nd-order radial beam as input state, i.e., the situation for the value \(\ell=2\). This input vector beam incident on the QW q-plate was obtained by adding a static q-plate of \(q=1\) to the PSG. The numerically calculated NF polarization map (Figure 6a) shows three L-lines that separate angular sectors of opposite handedness. The experimentally retrieved polarization map matches very well this calculation.