# Transverse Spin Hall Effect and Twisted Polarization Ribbons at the Sharp Focus
## 3 Expressing the Field Strength Vector via Vectors of the Axes of the Polarization Ellipse
\begin{array}{l}B_{x}=Q(r_{0})\sin 2\varphi=0,\\ B_{y}=-Q(r_{0})\cos 2\varphi=0,\\ B_{z}=L(r_{0})\cos\varphi.\end{array}  \tag{12}\]  
Considering that the SAM vector in (9) is perpendicular to the vector \(\mathbf{B}\), the longitudinal projection of the SAM vector equals zero, and the SAM vector \(\mathbf{S}\) lies in the focal plane. The vector \(\mathbf{B}\) is of zero length at two points of a circle of radius \(r_{0}\), namely, at \(\varphi=\pi/2\) and \(\varphi=3\pi/2\), where \(B_{z}=0\). Let us analyze in more detail how the SAM vector varies in the neighborhood of those points. For instance, at angles slightly smaller than \(\varphi=\pi/2\), the longitudinal projection of the \(\mathbf{B}\) vector in (12) is positive (\(L(r_{0})\)cos \(\varphi>0\)) at \(L(r_{0})>0\), whereas at angles slightly larger than \(\varphi=\pi/2\), the longitudinal projection of the vector \(\mathbf{B}\) in (12) is negative (\(L(r_{0})\)cos \(\varphi<0\)). The \(\mathbf{A}\) vector is not found in the focal plane at points lying on the circle of radius \(r_{0}\) and hardly changes its direction at angles smaller and larger than \(\varphi=\pi/2\), being predominantly directed along the \(x\)-axis, as  
\[ \begin{array}{l}A_{x}=I_{0,n}(r_{0})+P(r_{0})\cos 2\varphi\approx I_{0,n}(r_ {0})>0,\\ A_{y}=P(r_{0})\sin 2\varphi\approx 0,\\ A_{z}=F(r_{0})\sin\varphi\approx F(r_{0}),\\ I_{0,n}(r_{0})>>P(r_{0}),I_{0,n}(r_{0})>>F(r_{0}).\end{array}  \tag{13}\]  
The inequalities in (13) are based on the results reported in Ref. [8]. Then, varying along the circle of radius \(r_{0}\), the SAM vector will be near opposite to the vertical \(y\)-axisat angles slightly smaller than \(\varphi=\pi/2\), being almost coincident with the vertical \(y\)-axis at angles slightly larger than \(\varphi=\pi/2\). Thus, when moving along a circle of radius \(r_{0}\), the direction of the SAM vector changes to the opposite upon reaching the point at angle \(\varphi=\pi/2\). We note that at the circle point corresponding to \(\varphi=\pi/2\), the SAM vector equals zero. The change of direction of the SAM vector lying in the focal plane upon passing a linear polarization point (\(\mathbf{B}=0\)) may be termed the transverse spin Hall effect. It should be differentiated from the longitudinal spin Hall effect [8; 9]. If the small axis of the polarization ellipse is zero, i.e., \(\mathbf{B}=0\), then the polarization ellipse reduces to a segment, i.e., polarization becomes linear. There is no polarization singularity at this point. When passing through such a point (\(\mathbf{B}=0\)), the SAM vector, which is parallel to the focus plane in this case, changes its sign. Before the point with liner polarization (\(\mathbf{B}=0\)), the SAM vector was directed nearly vertically downwards, i.e., the plane of the polarization ellipse is nearly perpendicular to the focus plane, and the polarization vector is rotating counterclockwise. On the contrary, after the point with linear polarization (\(\mathbf{B}=0\)), the SAM is directed almost vertically upwards in the focus plane, the plane of the polarization ellipse is perpendicular to the focus plane, and the polarization vector is rotating clockwise. Both of these polarization ellipses form two \"optical wheels\", where the polarization ellipses are rotated like spokes in a wheel. In (5), being \(\cos(2\varphi)\)-dependent, the longitudinal SAM projection changes sign four times on making a full circle. Hence, in the focal plane, there are four regions, in two of which the SAM projection is positive and in the other two negative. The separation of focal plane regions with different-sign longitudinal SAM components may be called the longitudinal spin Hall effect [8; 9]. Summing up, in this section we demonstrated that given the initial light field (1), in the focal plane there are circles centered on the optical axis where the SAM vector lies in the focal plane and changes the direction to the opposite an even number of times. We have given this effect the name transverse spin Hall effect.