## 1 Introduction
Azimuthally polarized fields are known to be purely transverse (for example, see Figure 5 in Reference [22] and explanations therein) and, therefore, are often employed for fluorescence microscopy beyond the diffraction limit [6; 23] (not necessarily STED). A more common approach for STED depletion beams is to make use of circularly polarized helical beams with non-zero total angular momentum, for which the longitudinal electric field undergoes destructive interference along the optical axis and, therefore, the intensity is zero. In this work, we present the design and analysis of an optical system, capable of producing cylindrical vector beams with adjustable characteristics in the focal plane. Specifically, the focal spot can be changed from a typical doughnut spot with azimuthal polarization to an extended doughnut spot or a bi-lobed focal spot without circular symmetry, via simple rotation of certain optical elements around the optical axis; moreover, if the focal spot is circularly non-symmetric, then it could be rotated around the optical axis by rotation of certain components. The intensity on the optical axis is always zero; therefore, this system has great potential for STED microscopy. The main (first) optical layout proposed herein is composed of a spiral phase plate (SPP), a quarter-wave plate (QWP), a half-wave plate (HWP), a component capable of converting a Gaussian beam into a cylindrical vector beam, called a mode converter (MC), a polarization rotator (PR), and an objective lens (L), situated on the light path in the order given by the enumeration. MC is a uniaxial crystal with conical-like shaped surfaces aligned with the crystal axis along the symmetry axis of the system. The QWP is used to achieve elliptical polarization, while the HWP is used to rotate the polarization. The mode converter transforms a circularly polarized beam into a cylindrical vector beam. As explained in the previously mentioned paper [16], the uniaxial crystal works similarly with a wave plate having the fast axis oriented radially (or azimuthally, depending on the properties of the material). If a certain length of the crystal is met, the mode converter will act as a quarter-wave retarder. In this situation, the linear (locally linear) polarization resulting from right circular polarization has the direction oriented at 45\({}^{\circ}\) clockwise from the radial direction. Circularly polarized modes of the incoming field on the MC are partially converted to helical modes in such way that the total angular momentum is preserved; thus, after the mode converter and SPP, there will be four distinct beams/modes in the system with different topological charges and polarization states, and each will acquire a different amount of Gouy phase shift [24] due to the propagation to the focal plane. As a result, the diffraction pattern in the focal plane can be very different in comparison with the beam after MC and could even be circularly non-symmetric. The diffraction pattern could be modified by a change in weight of or phase difference between the component modes, which is controlled by the rotation of polarization components QWP and HWP placed before MC. As discussed later, such an optical system could generate a beam with some important features to present (a) a desired polarization state in the focal plane, namely azimuthal or partially azimuthal polarization, (b) zero intensity on the optical axis in the focal plane, and (c) changeable intensity pattern in the focal plane between different configurations. A second layout of the optical system is also presented, which is capable of producing cylindrical vector beams with nearly top-hat intensity profile in the focal plane based on an idea similar to that presented in Reference [25], i.e., to superimpose two scalar beams with a different profile and orthogonal polarization states. In the second section, the optical system is presented, as well as the function of the component MC in an ideal case. Jones matrix description of the optical system is also presented in this section.