## 2 Optical System
In this case, the beam after SPP and MC is a combination of a helical mode \(-3\) with right circular polarization and a helical mode \(-1\) with left circular polarization. The focal spot produced by this beam is more extended due to the contribution of the helical beam with topological charge \(-3\), and the polarization state is slightly different, but the dark area from the center of the focal spot has a similar extension. This particular case is called variation \(1\)C. Intermediate between these two configurations is variation \(1\)B (linear incoming polarization on MC), which produces a two-lobed spot with a highly elongated dark area in the center. Intermediate A-B configurations produce a small focal spot with an elliptical dark region in the center, whereas intermediate B-C configurations produce a larger focal spot with an elliptical dark region in the center. Rotation of HWP controls the orientation of a circularly non-symmetric focal spot, while rotation of QWP controls the shape of the focal spot. A general Jones vector for the field in the plane of the focusing element is given in Equation (3) and is then particularized for variation \(1\)A (Equation (4) using vertical/horizontal polarization and Equation (5) using circular polarization) and variation \(1\)C (Equation (6) using vertical/horizontal polarization and Equation (7) using circular polarization) of the system. \[\mathbf{V}_{lens}\ =\ \mathrm{e}^{-\mathrm{i}\,\theta}\mathbf{R}(-45^{\circ}) \mathbf{B}\,\mathbf{U}(\delta)\mathbf{B}^{-1}\,\mathbf{J}_{HWP}(\gamma)\, \mathbf{J}_{QWP}(\varphi)\,\mathbf{V}_{in}\ =\ \mathbf{K}(\delta)\mathbf{J}_{HWP}(\gamma)\,\mathbf{J}_{QWP}( \varphi)\,\mathbf{V}_{in}. \tag{3}\]Angles \(\gamma\) and \(\varphi\) give the orientation of the fast axis of HWP and QWP with respect to the horizontal axis. Matrix **B** makes the conversion from circular to linear polarization. \[\textbf{K}(\delta)=\left[ \begin{array}{ccc}1&1\\ -1&1\end{array} \right]\left[ \begin{array}{ccc}1&1\\ \text{i}&-\text{i}\end{array} \right]\left[ \begin{array}{ccc}\cos\ \delta&-\text{i}\,\text{e}^{-2\text{i}\,\theta}\sin\ \delta\\ -\text{i}\,\text{e}^{2\,\theta}\sin\ \delta&\cos\ \delta\end{array} \right]\left[ \begin{array}{ccc}1&-\text{i}\\ 1&\text{i}\end{array} \right]\left[ \begin{array}{ccc}\text{e}^{-\text{i}\, \theta}&0\\ 0&\text{e}^{-\text{i}\,\theta}\end{array} \right],\]  
\[\left[ \begin{array}{c}V_{x}\\ V_{y}\end{array} \right]_{lens}=\textbf{K}(45^{\circ})\left[ \begin{array}{ ccc}1&0\\ 0&-1\end{array} \right]\left[ \begin{array}{ccc}1&0\\ 0&-\text{i}\end{array} \right]\left[ \begin{array}{ccc}1\\ 1\end{array} \right]=\textbf{K}(45^{\circ})\left[ \begin{array}{c}1\\ \text{i}\end{array} \right]\propto\left[ \begin{array}{c}-\sin\ \theta\\ \cos\ \theta\end{array} \right], \tag{4}\]  
\[\left[ \begin{array}{c}V_{+}\\ V_{-}\end{array} \right]_{lens}=\left[ \begin{array}{ccc}1&-\text{i}\\ 1&\text{i}\end{array} \right]\left[ \begin{array}{c}-\sin\ \theta\\ \cos\ \theta\end{array} \right]\propto\left[ \begin{array}{c}\text{e}^{-\text{i}\, \theta}\\ -\text{e}^{\text{i}\,\theta}\end{array} \right]. \tag{5}\]  
\[\left[ \begin{array}{c}V_{x}\\ V_{y}\end{array} \right]_{lens}=\textbf{K}(45^{\circ})\left[ \begin{array}{ ccc}1&0\\ 0&-1\end{array} \right]\left[ \begin{array}{ccc}1&0\\ 0&\text{i}\end{array} \right]\left[ \begin{array}{c}1\\ 1\end{array} \right]=\textbf{K}(45^{\circ})\left[ \begin{array}{c}1\\ -\text{i}\end{array} \right]\propto\left[ \begin{array}{c}\cos\ \theta\\ \sin\ \theta\end{array} \right]\text{e}^{-2\text{i}\,\theta}, \tag{6}\]  
\[\left[ \begin{array}{c}V_{+}\\ V_{-}\end{array} \right]_{lens}=\left[ \begin{array}{ccc}1&-\text{i}\\ 1&\text{i}\end{array} \right]\left[ \begin{array}{c}\cos\ \theta\\ \sin\ \theta\end{array} \right]\text{e}^{-2\text{i}\theta}\propto\left[ \begin{array} []{c}\text{e}^{-3\text{i}\,\theta}\\ \text{e}^{-\text{i}\,\theta}\end{array} \right]. \tag{7}\]  
The second variation of this system is designed to produce a nearly uniform circular spot.