# Transverse Spin Hall Effect and Twisted Polarization Ribbons at the Sharp Focus
## 4 Polarization-Twisted Ribbons in the Focal Plane  
Assuming initial field (1), let us now choose in the focal plane a different circle of radius \(r_{1}\), on which the longitudinal projection \(B_{z}\) equals zero:  
\[L(r_{1})=I_{1,n-1}-I_{1,n+1}=0. \tag{14}\]  
In this case, the vector \(\mathbf{B}\) lies in the focal plane, while the vector \(\mathbf{A}\) 'almost' lies in the focal plane and is oriented along the horizontal axis, Equation (13). From (5), it is seen that at \(0<\varphi<\pi/4\) the longitudinal SAM projection is positive, \(S_{z}=(1/2)(I_{2,n-2}-I_{2,n+2})\)\((I_{2,n-2}+I_{2,n+2}+2I_{0,n}\cos 2\varphi)\)\(>0\) if \(I_{2,n-2}-I_{2,n+2}>0\) on the circle of radius \(r_{1}\), whereas at \(\varphi=\pi/2\) the longitudinal SAM projection is negative because \(2I_{0,n}>I_{2,n-2}+I_{2,n+2}\). Hence, at some angle \(\varphi\) from the interval \(\pi/4<\varphi<\pi/2\), the longitudinal SAM projection becomes zero:  
\[S_{z}=0,\ I_{2,n-2}+I_{2,n+2}=2I_{0,n}\cos 2\varphi. \tag{15}\]  
In view of Equation (14), it follows from (5) that on the circle of radius \(r_{1}\), the transverse SAM projections take a simpler form:  
\[\begin{array}{l}S_{x}=2M(r_{1})\sin\varphi\cos 2\varphi,\\ S_{y}=-2M(r_{1})\cos\varphi\sin 2\varphi.\end{array} \tag{16}\]  
The change of sign of the longitudinal SAM projection (1) means that at the focus, we observe a longitudinal spin Hall effect. When making a full circle of radius \(r_{1}\), the longitudinal SAM component changes sign four times at any number \(n\). We conclude that at small values of the angle \(\varphi\), surfaces of the polarization ellipses centered on the circle of radius \(r_{1}\) in the focal plane almost lie in the focal plane, \"looking\" along the optical axis. At an angle at which \(S_{z}=0\) (15), the polarization ellipse surfaces are perpendicular to the focal plane. At an angle somewhat larger than the angle in (15), the polarization ellipse surfaces get tilted oppositely, \"looking\" in the opposite direction to the optical axis. Hence,on a full circle centered on the optical axis lying in the focal plane, a surface constructed from the polarization ellipses \"looks\" twice along the optical axis (with the major axis directed almost along the \(x\)-axis), \"looking\" twice in the opposite direction to the optical axis (then, the ellipse major axis is directed oppositely to the \(x\)-axis). At \"twis\" points, which occur four times on the circle, defined by condition (15), the polarization plane \"look\" perpendicularly to the optical axis. Thus, we have shown that the longitudinal spin Hall effect is associated with the presence of two-sided polarization-twisted ribbons at the focus [4], unlike a one-sided polarization Mobius strip [1,2]. The number of twists of the twisted ribbon should coincide with the number of regions of alternating-sign longitudinal spin. Below, this conclusion is validated by the numerical simulation. We note that the number of areas with the spin of a different sign and the number of turns of the polarization ribbon is the same (in total, 4) and do not depend in this case on the topological charge of the linearly polarized optical vortex (1).  
Other Types of Light Fields for Which Projections of Polarization Ellipses at the Focus Can Be Analytically Derived