## 2 Optical System
[PERSON] and [PERSON] used a commercially available radial analyzer, instead of MC, which blocks the radial component and allows the azimuthal component of the circularly polarized incoming field to pass; they also used a combination of two HWPs to produce a polarization rotator. We compare our results to those in Reference [26] in the next section. Let us consider an incoming field with plane wavefront, which means that it can be represented by a bundle of parallel rays, incident on MC. In this case, each of the refracted rays is situated in a plane common with the optical axis and moves away from the optical axis as the \(z\)-coordinate increases. As the ordinary axis is parallel with the optical axis, the extraordinary axis must be radially oriented  
Figure 1: Longitudinal section through the mode converter. Figure 2: Schematic representation of the optical system corresponding to variation 1 (**a**); schematic representation of the optical system corresponding to variation 2 (**b**). since it has to perpendicular on the ordinary axis and included in the plane formed by the ordinary axis and ray direction. Therefore, MC could be described by Jones matrix \(\mathbf{R}(\theta)\)\(\mathbf{J}(\delta)\)\(\mathbf{R}(-\theta)\), where \(\mathbf{R}(\theta)\) is a rotation matrix, \(\theta\) is the rotation angle around the optical axis, and \(\delta\) is the phase shift between extraordinary and ordinary polarizations, which should be the same for all rays in the case of a perfect conical wavefront. The effect of MC on an arbitrary field represented using circular polarization states is described by a Jones matrix according to Equation (1), where \(\delta\) is connected with the length of the crystal \(L\), birefringence of the material, and ray tilt inside the crystal, as described by Equation (2), where \(k_{z}{}^{(\theta)}(\phi)\) and \(k_{z}{}^{(\phi)}(\phi)\) are respectively the longitudinal part of the extraordinary and ordinary plane-waves tilted at angle \(\phi\) with respect to the optical axis, \(S\) is the value of the length along \(z\) traveled by the plane-wave indexed by \(\phi\) through the material. The angle \(\phi\) is characteristic for a ray; therefore, it is easy to determine the matrix \(\mathbf{U}\) for each coordinate in the transversal plane of the beam using the points of intersection between rays and MC. The matrix \(\mathbf{U}\) was also noted by [PERSON] et al. (Equation (3) in Reference [15]), although parameter \(\delta\) was different because the wavefront considered was spherical. \[\mathbf{U}(\delta)=\left[ \begin{array}{cc}\cos\ \delta&-\mathrm{i}\,\mathrm{e}^{- 2\mathrm{i}\,\theta}\,\sin\ \delta\\ -\mathrm{i}\,\mathrm{e}^{2\mathrm{i}\,\theta}\,\sin\ \delta&\cos\ \delta\end{array} \right], \tag{1}\]  
\[\delta=\frac{k_{z}^{(e)}(\phi)-k_{z}^{(o)}(\phi)}{2}S(L,\alpha,\phi). \tag{2}\]  
In the case of right circular polarization of the incoming optical Gaussian beam on MC, the transmitted beam would be a combination of a helical mode with topological charge \(+2\) and left circular polarization and a non-helical beam with right circular polarization, but since the incoming beam is characterized by topological charge \(\updownarrow=-1\), the resulting beam is actually a combination of a mode with topological charge \(+1\) and left circular polarization and a mode with topological charge \(-1\) and right circular polarization. Superposition of the two modes generates a cylindrical vector beam characterized by the fact that the direction of polarization is rotated \(45^{\circ}\) clockwise from the radial direction. The polarization rotator does rotate the plane of polarization \(45^{\circ}\) clockwise, regardless of the input orientation, thereby producing the azimuthally polarized beam. This particular configuration of variation \(1\) is called variation \(1\)A throughout the paper. Via rotation of the QWP by \(90^{\circ}\), a different focal spot can be obtained.