# Spin-Orbital Transformation in a Tight Focus of an Optical Vortex with Circular Polarization
## 1 Introduction  
In modern optics, many optical effects are based on the interaction between the polarization of light and the vortex phase of light. A review on spin-orbital conversion (SOC) (interaction, coupling) in photonics can be found in [1]. By SOC, we mean the ability of light with circular polarization to create a rotation of the transverse energy flow. The spin-orbital transformation underlies the spin Hall effect in an inhomogeneous medium [2], upon reflection from the interface between two media [3; 4; 5], tight focusing [6; 7; 8; 9; 10], light scattering [11; 12; 13], and while light transmission through waveguides [14; 15; 16], anisotropic crystals [17; 18], metasurfaces [19; 20; 21], and in nanostructures [22; 23; 24]. Due to SOC, excitation of optical vortices was observed in the nanodisk laser when the disk was excited by light with circular polarization [25]. The spin Hall effect was experimentally observed as a submillimeter Imbert-Fedorov shift when light was reflected from the surface of a birefringent symmetric planar waveguide with a metal shell [26]. Light in free space demonstrates an intrinsic quantum spin Hall effect when surface modes with strong spin-momentum locking are formed [27]. A review of works on SOC in optics is presented in [28]. Due to SOC, when light with linear polarization is reflected from twisted few-layer hyperbolic metasurfaces, two shifted beams with circular polarization of different directions are formed [29]. In this case, when a beam with circular polarization propagates at an angle to the optical axis, then the transverse flow of the Poynting vector through a plane perpendicular to the optical axis will be displaced or separated. In [30], this phenomenonwas called the geometric spin Hall effect. If a paraxial vortex beam with cylindrical vector polarization is limited by a sector diaphragm, that is, the circular symmetry of the beam is broken, then regions with left and right circular polarization appear in the transmitted beam [31]. The spin Hall effect also arises if a radially polarized laser beam passes through a sector diaphragm [32]. Then, regions with left and right circular polarization will be formed in the beam after passing the aperture. In the works listed above, the question of what part of the spin angular momentum (SAM) is transferred to the orbital angular momentum (OAM) is not considered. SAM is the spin density in the cross-section of the light beam. Light with linear polarization has no spin, i.e., SAM is zero. However, light with circular polarization has a maximum spin denseness, that is, SAM is equal to the power of the beam. The normalized to beam power SAM of circularly polarized light is plus one for right-hand polarization and minus one for left-hand circular polarization. The normalized SAM of light with elliptical polarization is less than one. OAM is a value that indicates the presence of a vortex energy flow in the beam. The normalized to the beam power OAM of vortex laser beams is equal to the topological charge. Therefore, spin-orbit transformation occurs when circularly polarized light that has SAM is converted into light with a vortex energy flow that has OAM.  
SOC is a universal physical phenomenon and is present in many processes. For example, it is known that due to SOC, no collapse occurs during Bose-Einstein condensate [33]. That is, due to SOC, the initial Gaussian quantum state does not decrease to zero in a finite time, but transforms into a vortex state and increases in size with time.  
In our study, within the Richards-Wolf formalism, exact formulas are obtained for the case of sharp focusing of a circularly polarized optical vortex. Here, we proposed right-hand circular polarization. These formulas show what part of the total longitudinal SAM transforms into the total longitudinal part of the OAM in the focus. It is also shown that the maximum part of the SAM that can convert into the OAM is equal to half of the beam power.