Transverse spin Hall effect and polarization twist ribbon at the sharp focus

In this work, using the Richards-Wolf formalism, we found explicit analytical expressions for the coordinates of the major and minor axes of the polarization ellipse centered in the focal plane for a cylindrical vector beam of integer order n. For such a beam, the major axis of the polarization ellipse lies in the focal plane, whereas the minor axis is perpendicular to the focal plane. Therefore, the polarization ellipse is perpendicular to the focal plane, with the polarization vector rotating clockwise or counterclockwise in this plane (producing 'optical wheels'). Considering that the wave vector is also perpendicular to the focal plane, the polarization ellipse and the wave vector turn out to lie in the same plane, so that at some point of time the polarization vector can coincide with the wave vector, which is not usual for transverse electromagnetic oscillations. For a cylindrical vector beam, the spin angular momentum vector lies in the focal plane. So, when going around a certain circle with the center on the optical axis, the spin vector is directed counterclockwise in some sections of the circle, and clockwise in other sections. This effect can be called the transverse (azimuthal) optical spin Hall effect, in contrast to the well-known longitudinal optical spin Hall effect at the sharp focus. The longitudinal spin Hall effect is understood as the separation in the focal plane of regions with different signs of the longitudinal projection of the spin angular momentum vector. This work shows that there is always an even number of such regions and that when going around a circle, the vector of the major axis of the polarization ellipse forms a two-sided twist surface with an even number of turns.