## 2 Materials and Methods
### Q-Plates and Their Polarization Conversion Efficiency  
A q-plate is a phase-plate retarder with uniform retardance \(\phi\) whose principal axis follows \(q\) times the azimuthal angle \(\theta\) in the device plane. The \(q\) value is an integer or semi-integer, so that the principal axis has a singularity in the plate center, and its sign is positive (negative) for counterclockwise (clockwise) rotation. Q-plates of fixed retardance (static q-plates) are commercially available with \(q=1/2\) and \(q=1\), although other values can be ordered on demand [34]. The Jones matrix for a q-plate can be written as [23; 24]:  
\[\mathbf{M}_{q}(\phi)=\cos\!\left(\frac{\phi}{2}\right)+i\sin\!\left(\frac{\phi }{2}\right)\mathbf{Q}_{q} \tag{1}\]  
where \(\mathbf{I}\) is the identity matrix and \(\mathbf{Q}_{q}\) is the matrix for a q-plate tuned at \(\pi\) retardance, given by  
\[\mathbf{Q}_{q}=\left( \begin{array}{cc}\cos(2q\theta)&\sin(2q\theta)\\ \sin(2q\theta)&-\cos(2q\theta)\end{array} \right) \tag{2}\]  
When the q-plate is tuned to \(\pi\) radians (half-wave (HW) q-plate) a full polarization conversion is achieved. This means that a beam of uniform linear polarization entering the q-plate is fully converted into a pure vector beam. For retardance values other than half-wave, the q-plate is detuned and the conversion efficiency decreases. As discussed in [28], the case of quarter-wave (QW) q-plate is of special interest. In this situation, Equation (1) results in \(\mathbf{M}_{q}(\phi=\pm\pi/2)=\frac{1}{\sqrt{2}}\Big{(}\pm i\mathbf{Q}_{q}\Big{)}\) and, if illuminated with an input state \(\mathbf{V}_{in}\), the output beam is given by  
\[\mathbf{V}_{out}=\mathbf{M}_{q}(\phi=\pm\pi/2)\cdot\mathbf{V}_{in}=\frac{1}{ \sqrt{2}}\Big{(}\pm i\mathbf{Q}_{q}\Big{)}\cdot\mathbf{V}_{in} \tag{3}\]  
This relation shows that the output is the superposition of the input beam with the one generated by a HW q-plate, with a \(\pm\pi\)/2 phase shift in between. The situation where the HW q-plate is illuminated with circularly polarized light is used to generate vortex beams, since the device reverses the helicity of the light and transfers a spiral phase \(\exp(\pm i2q\theta)\)[23]. This effect can be described approximately as modifications of Laguerre-Gauss (LG) modes. In our work we follow [35; 36] for the definition of the LG modes and we only consider modes with radial index \(p=0\), which are vortex beams of azimuthal index \(\ell\) given by:  
\[LG_{0}^{\ell}=u_{\ell}(z)e^{-i\ell\theta}e^{-i\psi_{\mathrm{C}}(z)} \tag{4}\]  
where \(u_{\ell}(z)\) is the complex amplitude given by  
\[u_{\ell}(z)=\sqrt{\frac{2}{\pi|\ell|!}}\frac{1}{w(z)}\Big{(}\frac{\sqrt{2}r}{ w(z)}\Big{)}^{|\ell|}\exp\!\left(-\left(\frac{r}{w(z)}\right)^{2}-i\frac{kr^{2}}{2 R(z)}\right) \tag{5}\]  
\(w(z)=w_{0}\sqrt{1+\left(z/z_{R}\right)^{2}}\) is the beam width, \(w_{0}\) is the beam waist, \(z_{R}\) is the Rayleigh range and \(R(z)=z\big{[}1+\left(z/z_{R}\right)^{2}\big{]}\) is the radius of curvature. The term \(\psi_{\mathrm{C}}(z)=\left(|\ell|+1\right)\xi(z)\) is the Gouy phase, where \(\xi(z)=\tan^{-1}(z/z_{R})\) is zero at the beam waist (\(z=0\)) and \(\xi=\pm\pi/2\) at \(z\rightarrow\pm\infty\). When a circularly polarized \(LC_{0}^{\ell}\) beam traverses a q-plate of a given \(q\) value, the effect can be approximated as  
\[\mathbf{Q}_{q}\Big{[}LG_{0}^{\ell}\mathbf{e}_{\mathbf{R}}\Big{]}\cong LG_{0} ^{\ell-2q}\mathbf{e}_{\mathbf{L}}\text{ and }\mathbf{Q}_{q}\Big{[}LG_{0}^{\ell}\mathbf{e}_{ \mathbf{L}}\Big{]}\cong LG_{0}^{\ell+2q}\mathbf{e}_{\mathbf{R}} \tag{6}\]  
where \(\mathbf{e}_{\mathbf{L}}^{\wedge}=\frac{1}{\sqrt{2}}(1,-i)^{t}\) and \(\mathbf{e}_{\mathbf{R}}^{\wedge}=\frac{1}{\sqrt{2}}(1,+i)^{t}\) are the normalized Jones vectors for the LCP and RCP states, respectively. It is worth at this point to explain what we mean by the approximation in Equation (6). The q-plate imparts a helical phase \(\exp(\pm i2q\theta)\)[23].