## 2 Materials and Methods
### Vector Beams and Their Propagation Dynamics  
Vector beams can be described as the superposition of two LG modes with orthogonal circular polarization  
\[\mathbf{V}=\cos(\delta)LG_{0}^{\ell_{R}}\mathbf{e}_{\mathbf{R}}^{\wedge}+ \sin(\delta)e^{2\varphi_{L}}LG_{0}^{\ell_{L}}\mathbf{e}_{\mathbf{L}} \tag{7}\]  
where \(\ell_{R}\) and \(\ell_{L}\) are the topological charges encoded in the RCP/LCP states. The parameter \(\delta\in[0,\pi]\) controls the relative magnitude between the RCP and LCP components, while \(2\varphi\) is their relative phase and induces a \(\varphi\) counterclockwise rotation of the polarization pattern. It is well-known that pure vector beams (\(\ell_{R}=-\ell_{L}\)) retain their polarization pattern upon propagation. However, hybrid vector beams undergo a transformation caused mainly by the Gouyphase difference in the RCP/LCP components. For the superposition with equal weight of the two circular polarization components (\(\delta=\pi/4\)) Equation (7) can be rewritten as:  
\[\mathbf{V}(z)=\frac{1}{\sqrt{2}}\Big{[}u_{\ell_{R}}(z)e^{-i(\ell_{R}\theta+\psi_{ G}^{R}(z))}\overset{\Lambda}{\mathbf{e_{R}}}+u_{\ell_{L}}(z)e^{-i(\ell_{L} \theta+\psi_{G}^{L}(z)+2\varphi)}\overset{\Lambda}{\mathbf{e_{L}}}\Big{]} \tag{8}\]  
where \(\psi_{G}^{R/L}(z)\) denotes the RCP/LCP Gouy phases. In cases where the difference in charge \(|\ell_{R}|-|\ell_{L}|\) is small enough to maintain a superposition of the two modes, the amplitudes can be approximated to be the same \(u(z)\cong u_{\ell_{R}}(z)\cong u_{\ell_{L}}(z)\), and the following approximation can be considered [33]:  
\[\mathbf{V}(z)\cong\frac{1}{\sqrt{2}}u(z)e^{-i\overline{\ell}\theta}e^{-i \overline{\psi}_{G}(z)}e^{-i\varphi}\Big{[}e^{-i(m\theta+\frac{1}{2}\Delta \psi_{G}(z)-\varphi)}\overset{\Lambda}{\mathbf{e_{R}}}+e^{+i(m\theta+\frac{1} {2}\Delta\psi_{G}(z)-\varphi)}\overset{\Lambda}{\mathbf{e_{L}}}\Big{]} \tag{9}\]  
written in terms of the mean topological charge \(\overline{\ell}=\frac{1}{2}(\ell_{R}+\ell_{L})\) and the semi-difference charge \(m=\frac{1}{2}(\ell_{R}-\ell_{L})\). Here \(\overline{\psi}_{G}(z)=\frac{1}{2}\big{(}\psi_{G}^{R}(z)+\psi_{G}^{L}(z)\big{)}\) is the mean Gouy phase, and \(\Delta\psi_{G}\) is the relative Gouy phase shift between the RCP and LCP components, given by  
\[\Delta\psi_{G}(z)=\psi_{G}^{R}(z)-\psi_{G}^{L}(z)=(|\ell_{R}|-|\ell_{L}|)\xi(z) \tag{10}\]  
Noting that a phase shift between circular polarization components is equivalent to a polarization rotation, and using the Jones vectors \(\overset{\Lambda}{\mathbf{e_{L}}}\) and \(\overset{\Lambda}{\mathbf{e_{R}}}\), Equation (9) can be rewritten as  
\[\mathbf{V}(z)\cong u(z)\mathbf{R}\Big{(}\frac{1}{2}\Delta\psi_{G}(z)\Big{)} \cdot\left( \begin{array}{c}\cos\!\left(m\theta+\varphi+\frac{1}{2}\Delta \psi_{G}(z)\right)\\ \sin\!\left(m\theta+\varphi+\frac{1}{2}\Delta\psi_{G}(z)\right)\end{array} \right) \tag{11}\]  
where \(\mathbf{R}(\alpha)\) is the \(2\times 2\) Jones matrix of a polarization rotator [35] and where, for the sake of simplicity, we omit the global phase terms in Equation (9) that do not affect the polarization state. Equation (11) reveals useful aspects for our analysis. First, the Jones vector (\(\cos(m\theta+\varphi)\), \(\sin(m\theta+\varphi)\))\({}^{t}\) describes the linear cylindrically polarized pattern at the beam waist (\(z=0\)), which is defined only by the semi-difference charge \(m\) and the relative phase \(\varphi\). The propagated field, however, undergoes an effective polarization rotation \(\frac{1}{2}\Delta\psi_{G}(z)=\frac{1}{2}(|\ell_{R}|-|\ell_{L}|)\xi(z)\), which modifies the polarization map along the propagation. Since \(\xi(z)\) changes from being zero at the waist (\(z=0\)) to becoming \(\xi(z)=\pi/2\) at the far field, the total Gouy phase shift for this propagation reads \(\Delta\psi_{G}(z)=(|\ell_{R}|-|\ell_{L}|)\pi/2\).