# Spin Hall Effect before and after the Focus of a High-Order Cylindrical Vector Beam
## 5 Conclusions  
It is known that in the cross-section of a high-order cylindrical vector beam (CVB), polarization is locally linear. The number of full rotations of the locally linear polarization vector, when passing along a closed contour around the optical axis, is equal to the CVB order, for instance, \(n\). It is also known that both in the initial plane and in the focus, the CVB has neither the spin angular momentum (SAM), nor the orbital angular momentum (OAM). In this work, we demonstrated that near the focal plane of the CVB, for instance, at a distance of wavelength before and beyond the focus, 4(\(n\!-\!1\)) local subwavelength areas are generated, where the polarization vector is rotating in each point. These areas appear because, at \(n\
eq 1\), cylindrical polarization violates the rotational symmetry of the focused beam. The centers of these local areas reside on a certain-radius circle with the center on the optical axis. Furthermore, in the neighboring areas, the polarization vector is rotating in different directions so that the longitudinal component of the SAM vector in the neighboring areas has opposite signs. The full SAM, obtained by averaging the SAM density over the whole transverse section of the beam, is equal to zero. Such space separation of the left and right rotation of the polarization vectors manifests so that the optical spin Hall effect takes place. Such a phenomenon can be used in optical sensors for determining the CVB order. For example, metasurface-based optical sensors are known [26; 27; 28; 29], that detect left-handed and right-handed circular polarizations by separating them in space. In our case, the CVB order \(n\) is determined by counting the local areas with left and right circular  
Figure 4: Distributions of intensity (**a,e**), radial component of the Poynting vector (**b,f**), normalized-to-maximum longitudinal component of the SAM vector (**c,g**), and normalized-to-maximum longitudinal component of the OAM vector (**d,h**) of a sharply focused Gaussian beam with 4 th-order cylindrical polarization before the focus (**a–d**) and beyond the focus (**e–h**). In all the figures, light and black colors mean, respectively, maximum and minimum. Scale marks (in the bottom left corner) denote 1 \(\upmu\)m.  
polarization near the sharp focus. Among other applications of the optical spin Hall effect are spin-dependent beam splitting [30], surface sensing [31], and determining biomolecular concentration [32].  
Conceptualization, V.V.K.; methodology, S.S.S.; software, A.A.K. and V.D.Z.; validation, V.V.K. and A.A.K.; formal analysis, V.V.K.; investigation, V.D.Z.; resources, V.V.K.; data curation, V.V.K.; writing--original draft preparation, V.V.K.; writing--review and editing, V.V.K.; visualization, A.A.K. 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 funded by the Russian Science Foundation under grant no. 22-12-00137.  
Not applicable.  
The authors declare no conflict of interest.