## 1 Introduction  
Helical beams are optical beams characterized by a transversal phase distribution with an additional helical phase term \(\updownarrow\theta\). Integer parameter \(\updownarrow\) is called topological charge, and, if different from zero, it denotes that there is a discontinuity in the phase map. Polarization state may change with the azimuthal angle around the discontinuity, whereby the resulted beam is called a cylindrical vector beam [1]. Structured light and especially helical beams and cylindrical vector beams are deemed suitable for applications like optical trapping and particle manipulation [2; 3; 4], stimulated emission depletion (STED) microscopy [5; 6], laser micromachining [7], or charged particle acceleration in plasma [8]. Cylindrical vector beams can be produced by selection of radial polarization from a circularly polarized beam using a conical window cut at Brewster angle [9], using microstructured birefringent plates, also known as spatially variant polarization converters [10] or liquid crystal modulators [11]. [PERSON] et al. reported an interferometric optical system that allows full control of the polarization and the shape of the beam [12], with measured efficiency around 50%. [PERSON] et al. [13] demonstrated that propagation of circularly polarized non-collimated beams through c-cut uniaxial crystals results in the generation of helical beams of \(\pm 2\) topological charge. Transformation of Gaussian fs-laser beams to single- or double-charge vortex beams using a uniaxial crystal was experimentally proven [14], with efficiency limited to 50% for double-charge beams and 75% for single-charge beams. The dynamics of spin-to-orbital angular momentum conversion in uniaxial crystals was further studied by [PERSON] et al. [15]. The interaction of circularly polarized, non-collimated, ring-shaped light beams with a single piece of homogeneous uniaxial medium was demonstrated to shape scalar and vectorial beams with nearly 100% conversion efficiency [16]. It was also shown previously by [PERSON] et al. [17] that Bessel beams can achieve nearly 100% efficiency of conversion into a helical mode due to propagation along the crystal axis of a uniaxial crystal. Real approximations of Bessel beams are characterized by a conical wavefront and a ring-like intensity pattern in far-field [18]; since conversion efficiency depends only on the angular spectrum of the beam, it is natural to consider that a beam with conical wavefront is ideal for mode conversion in uniaxial crystals. Flat-top beams can be useful for laser-scribing surface micro-structures, to improve the quantity of energy effectively used for ablation, to avoid damaging the material, and to create a sharper ablated channel [19]. Typical optical systems for generation of flat-top beams are highly sensitive to the shape and width of the incident beam [20] and do not produce a collimated beam; therefore, the flat-top profile changes quickly near the designated plane. Recent studies showed that control of polarization is an important aspect for the improvement of machining efficiency. Results regarding various types of femtosecond cylindrical vector beams used for drilling and scribing in crystalline silicon and fused silica were presented in Reference [7]. The conclusion presented therein was that radial polarization is the best option for drilling, to improve the quality of the boreholes. Furthermore, we observed that circular polarization produces better results than linear polarization. For STED microscopy, one of the main requirements for the depletion beam is to have zero intensity on the optical axis [21]. Intensity on the optical axis could be different from zero due to the effect of the longitudinal electric field that arises at focusing with high-numerical-aperture (NA) lenses.