# Transverse Spin Hall Effect and Twisted Polarization Ribbons at the Sharp Focus
## 6 Numerical Modeling  
In this section, based on Richards-Wolf formalism [12], we derive projections of the E-vector for some original light fields and intensity patterns at the tight focus. Next, using Berry formulae (7) for the same light fields, we calculate the coordinates of the major and minor axis vectors of the polarization ellipses centered in the focal plane. The wavelength is 532 nm, and the numerical aperture is NA = 0.95. In Figure 1, depicted as a vertically elongated, inhomogeneous yellow-and-red ring, there is an intensity pattern at the tight focus for the original field (1) at \(n=3\). The white color marks the maximum intensity, yellow marks medium intensity, red marks low intensity and black marks zero intensity. In compliance with Equation (6), the on-axis intensity in Figure 1 is seen to be zero. Equation (6) also suggests that the intensity pattern should have two maxima on a near-circular ring located on the vertical or horizontal axis, depending on the sign of the expression in the round braces, which is multiplied by \(\cos(2\varphi)\). The maximum intensity values (marked white) in Figure 1 are seen to be located on the vertical axis; hence, we infer that the expression in (6) that is multiplied by \(\cos(2\varphi)\) is negative. Although the on-axis intensity is zero, the near-axis transverse amplitude of field (2) can effectively be expressed as  
\[\left( \begin{array}{c}E_{x}\\ E_{y}\end{array} \right)\approx-e^{i\varphi}I_{2,1}\bigg{(} \begin{array}{c}1\\ i\end{array} \bigg{)}. \tag{26}\]  
Relationship (26) suggests that in the optical axis vicinity, the light is near right-handed circularly polarized. In Figure 1, inside the light-colored intensity ring, the polarization ellipses are near circular and marked red, indicating the right-handed circular polarization. From Figure 1, on a full circle around the optical axis, the major axis of the ellipses within  
Figure 1: Intensity distribution (a slightly elliptical white-yellow-red ring) and projections of polarization ellipses onto the focal plane for the original field (1), describing a horizontally linearly polarized optical vortex (\(n=3\)). In-ellipse arrows mark the direction of the major axis of the polarization ellipse. Blue ellipses correspond to the left-handed circular polarization and negative longitudinal SAM projection (\(S_{z}<0\)), with red ellipses corresponding to right-handed circular polarization and positive longitudinal spin (\(S_{z}>0\)). the intensity ring is seen to rotate by \(2\pi\), implying that \(lc=1\). The same conclusion can be drawn from Equation (26), as near the optical axis in the focal plane, the topological charge equals \(1\) (at \(n=3\)). In Figure 1, the yellow ellipse-shaped intensity pattern is seen to be fringed with inner and outer red ellipses. A closer look at the inner red intensity ellipse reveals that the polarization vectors are marked blue at the top and bottom and red on either side. So, we infer that the longitudinal SAM component, \(S_{z}\), within the intensity ring is negative at the top and bottom and positive on either side. Therefore, there are four regions with opposite-sign spin, which is a manifestation of the longitudinal spin Hall effect. Vice versa, the outer red intensity ellipse is fringed with red arrows of polarization at the top and bottom and blue arrows on either side. So, in the outer red intensity ellipse, the longitudinal SAM component, \(S_{z}\) is positive at the top and bottom and negative on either side, also showing that there are four regions with the opposite-sign spin. Moving along the edge of the yellow intensity ellipse, we find that at the boundary where the red arrows change to the oppositely directed blue ones, the major axis vector of the polarization ellipse experiences an abrupt flip-over in 3D space. When making a full circle around the optical axis, the major axis vector forms a 3D two-sided strip with an even number of 'lip-overs', equal to the number of regions with the opposite-sign longitudinal spin.