## 8 Conclusions  
In this paper, we study a spin-orbit conversion in a spherical lens with a high numerical aperture. Utilizing the Richards-Wolf theory [34], analytical formulas are obtained for the density of the longitudinal SAM and OAM projections at the focus of the lens for the initial light field (1). The amplitude of the field depends on the real parameter \(\alpha\). The initial light field (1) is an axial superposition of two optical vortices with the same topological charge \(n\) and right- and left-handed circular polarizations. Amplitudes of both beams in the superposition are different. The parameter \(\alpha\) is the ratio of the amplitudes of the beams. In this work, analytical formulas are obtained for the total longitudinal projections of the SAM and OAM at the focus of the field (1). It follows from the Formulas (10), (11) and (19), (21) that when passing through a spherical lens (or a diffractive lens), the total SAM decreases due to a spin-orbit conversion, while the OAM increases by the same amount. In this case, the sum of the SAM and OAM remains unchanged during focusing (Equation (20)). Simulations are also carried out using the [PERSON] and Richards-Wolf integrals [34]. When the light field (1) propagates before a spherical lens, the total SAM and OAM are shown to remain unchanged separately. After passing through the lens, the total SAM decreases, and the total OAM increases by the same value. When propagating after the lens, the total SAM and OAM remain unchanged separately too. The sum of the SAM and OAM before the lens and after it does not change (Figure 1). When the parameter \(\alpha\) increases from 0 to 1, the spin-orbit conversion in a spherical lens is shown numerically to decrease from its maximum value to zero. With a numerical aperture of 0.95 and with \(\alpha=0\), the initial normalized total longitudinal SAM is equal to 1 before the spherical lens, and equal to 0.76 after it. That is, only a quarter of the SAM transferred to the OAM (Figure 2). The maximum spin-orbit effect in a spherical lens is shown numerically to occur at a numerical aperture close to 1 (Figure 3). For smaller aperture values, the effect decreases. It is shown that, at a numerical aperture less than 0.7, the spin-orbit conversion can be neglected, since the total SAM remains almost unchanged when light passes through a spherical lens. Theoretically (Equation (7)) and numerically (Figure 4) it is shown that there is a spin Hall effect at the focus of the light field (1). This effect leads to appearance of regions with elliptical polarizations with different signs in the sharp focus. That is, at the focus, light with left- and right-handed elliptical polarizations is separated in space. In this case, in the initial plane only linearly polarized light is present (Figure 4). By changing the parameter \(\alpha\) the spin Hall effect at the focus can be controlled. When \(\alpha=0\), the initial field is circularly polarized, and the Hall effect is minimal at the focus. When \(\alpha=1\), there is a linearly polarized initial field, and the Hall effect achieve its maximum at the focus. Measuring the magnitude of the spin-orbital conversion effect in the sharp focus of a microobjective with a high numerical aperture requires measuring the third component of the Stokes vector averaged over the beam transverse section. The value of this quantity gives the full longitudinal SAM component. In addition, the third component of the Stokes vector can be measured immediately in the microobjective's exit pupil with a diameter of several millimeters, rather than in the focus itself whose size can be less than a micron. The effects of spin-orbit conversion in a spherical lens and the spin Hall effect at the focus considered in this work can be used to control the rotation of microparticles [38; 39]. **Author Contributions:** Conceptualization, V.V.K. ; methodology, V.V.K. ; software, A.G.N. and A.A.K. ; validation, V.V.K. ; formal analysis, V.V.K. ; investigation, V.V.K., A.A.K. and A.G.N. ; resources, V.V.K.