# Transverse Spin Hall Effect and Twisted Polarization Ribbons at the Sharp Focus
## 7 Conclusions  
The following optical polarization effects have been theoretically and numerically shown to occur at the tight focus of a linearly polarized optical vortex with topological charge \(n\). When moving along a full circle around the optical axis in the focal plane, the major axis vectors of the polarization ellipse have been shown to make \(2n\) half-turns. That is, in a vector field composed of the major axis vectors of the focal-plane-centered polarization ellipses, the polarization singularity index equals the topological charge of the original optical vortex. Irrespective of the topological charge, four regions have been found to occur in the focal plane in which the longitudinal SAM projection has an alternating sign, demonstrating a longitudinal Hall effect. We have shown that on moving along a closed, axis-centered (circular or elliptic) contour, the major axis vector of the polarization ellipse generates, in a 3D space, a two-sided twisted ribbon with an even number of half-turns, by analogy with a twisted ribbon in Ref. [6]. In the focal plane, regions have been found to occur in which the transverse spin vector projections located in the focal plane have an alternating handedness relative to the circular motion handedness. This is a manifestation of an azimuthal transverse spin Hall effect. We have also demonstrated that upon moving on a circle, the transverse spin vector projections on the circle radii can be alternatively directed either from or to the center, which is a manifestation of a radial transverse spin Hall effect.  
Figure 3: An intensity pattern (marked white-yellow-red) at the focus of an azimuthally polarized field in (21) at topological charge \(n=1\) and a pattern of polarization ellipse distribution in the focal plane. The green arrows inside the ellipses mark the major axis vector. Red ellipses mark right-handed elliptic polarization (\(S_{z}>0\)), and blue ellipses mark left-handed elliptic polarization (\(S_{z}<0\)). The axes are plotted in microns.  
The results obtained in this work can be employed for nanostructuring polarization-sensitive materials [17; 18; 19; 20; 21], for magnetization and data recording based on the inverse Faraday effect [22; 23; 24; 25], for manipulating microparticles [26; 27; 28] as well as in optical microscopy [29; 30; 31].  
Conceptualization, V.V.K.; methodology, V.V.K., A.A.K. and Z.G.; software, A.A.K., A.K. and K.G.; validation, V.V.K.; formal analysis, V.V.K.; investigation, V.V.K. and A.A.K.; resources, V.V.K.; data curation, V.V.K.; writing--original draft preparation, V.V.K. and E.S.K.; writing--review and editing, V.V.K.; visualization, A.A.K., A.M.T. and S.S.S.; supervision, V.V.K.; project administration, V.V.K.; funding acquisition, V.V.K. and A.A.K. All authors have read and agreed to the published version of the manuscript.  
The work was partly funded by the Russian Science Foundation under grant # 22-12-00137 (Theoretical background) and from the government project of the NRC \"Kurchatov Institute\" (numerical modeling).  
Not applicable.  
Data Availability StatementData are contained within the article.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.