## 2 Optical System  
We present two different layouts. Variation 1 is designed to produce a beam that achieves azimuthal or nearly azimuthal polarization in the focal plane, with the advantage that the shape of the focal spot can be modified, and an extended or circularly non-symmetric focal spot can be generated. Variation 2 is designed to produce an intensity profile that is nearly uniform in the focal plane, while the polarization is circular near the optical axis and radial near the edge. The principal component in our optical system is a uniaxial crystal with the crystal axis (CA) aligned parallel with the optical axis, called a mode converter (MC). The longitudinal section view of MC with conical shaped surfaces is shown in Figure 1. Characteristics of MC, aperture diameter \(D\), crystal thickness \(L\), physical angle \(\alpha\), and ray tilt angle inside the crystal \(\alpha_{0}\), which is connected with the refractive index, are represented. MC is considered to be made of sapphire (\(n_{o}=1.7601\), \(n_{e}=1.7522\)). Transparent interfaces of the mode converter are conically shaped in an ideal case, while the description of a more realistic MC is presented in the next section; however, for now, let us assume an ideal case. The first surface modifies a collimated incident beam of light in such a way to produce the wavefront desirable for mode conversion inside the component (conical wavefront). When the beam of light passes through the second surface, it becomes collimated. Due to the specific shape of the interfaces and the birefringent nature of the material used, light propagation through the component induces an exchange between the orbital angular momentum and spin angular momentum of the beam. The efficiency of mode conversion is the fraction \(\eta\) of light energy that is converted into a helical mode. Characteristics of MC are chosen in such a way that the efficiency of the mode conversion is \(\eta=50\%\) for variation 1 of the system and around \(\eta\approx 80\%\) for variation 2 of the system. Fraction \(\eta\) of the energy of the incident beam is transferred to a helical mode having a polarization state that is orthogonal on the input polarization while \(1-\eta\) of the energy remains in a non-helical mode with the same polarization state as the incoming beam, thereby preserving the total angular momentum. The optical system for variation 1 is shown in Figure 2a, while the optical system for variation 2 is shown in Figure 2b. We consider a linearly polarized helical beam with topological charge \(\updownarrow=-1\) and Gaussian intensity profile, at wavelength \(\lambda=800\) nm, having 16 mm in diameter, which is supposedly generated using a spiral phase plate (SPP). For variation 2, we consider a simple Gaussian beam with the same characteristics, which did not pass through an SPP. The optical beam encounters a quarter-wave plate (QWP) and becomes elliptically polarized, then encounters a half-wave plate (HWP) that controls the orientation of the polarization ellipse. MC is considered to be after the HWP on the light path 1 m away from the starting point. The thickness of the crystal is 9 mm, and the physical angle of the conical surface is \(7^{\circ}\) for variation 1 of the system and \(7.7^{\circ}\) for variation 2. The focusing element is placed 150 mm away from MC. A polarization rotator (PR) that rotates the plane of polarization \(45^{\circ}\) clockwise is inserted between the MC and focusing element. Polarization optical components QWP, HWP, MC, and PR can actually be placed at any \(z\)-coordinate in the system; however, their order on the light path should not be changed. The position of SPP with respect to MC may have a slight influence on the focal spot. An optical system similar to variation 1 was used by [PERSON] and [PERSON] to experimentally evaluate the longitudinal component of the electric field in the focal plane of a high-numerical-aperture (NA) microscope objective [26].