## 3 Density of the Longitudinal Component of the Spin Angular Momentum Vector at the Focus  
Using the projections of the electric field strength vectors at the focus (3), we can find the distribution of the longitudinal projection of the spin angular momentum (SAM) at the focus of the field (1). The SAM vector is determined by the relation [6]:  
\[\mathbf{S}=\frac{1}{16\pi\omega}\text{Im}(\mathbf{E}^{*}\times\mathbf{E}), \tag{5}\]  
where \(\omega\) is the angular frequency of the monochromatic light. The constant \(1/(16\pi\omega)\) is omitted below. Further, we obtain an expression only for the longitudinal projection of the SAM (5), since it coincides with the third component of the Stokes vector and reveals the presence of regions with elliptical polarization in the beam cross section. Taking into account (3), the longitudinal projection of the SAM at the focus of the field (1) can be written as follows:  
\[\begin{array}{l}S_{z,\alpha}=2\text{Im}\big{(}E_{x}^{*}E_{y}\big{)}\\ =\frac{1}{1+\alpha^{2}}\Big{[}(1-\alpha^{2})I_{0,n}^{2}+\alpha^{2}I_{2,n-2}^{2 }-I_{2,n+2}^{2}+2\alpha\cos(2\varphi)I_{0,n}(I_{2,n-2}-I_{2,n+2})\Big{]}.\end{array} \tag{6}\]  
Equation (6) shows that at \(\alpha=1\), field (1) becomes linearly polarized and, at the focus, the longitudinal component of the SAM (6) is equal to the SAM of an optical vortex with linear polarization obtained in [22]:  
\[S_{z,\alpha=1}=\frac{1}{2}\Big{[}I_{2,n-2}^{2}-I_{2,n+2}^{2}+2\cos(2\varphi)I_ {0,n}(I_{2,n-2}-I_{2,n+2})\Big{]}. \tag{7}\]  
It can be seen from (7) that there are four regions at the focus, in two of them the longitudinal SAM is positive at \(\varphi=0\) and \(\varphi=\pi\), since near the optical axis \(I_{2,n-2}^{2}>I_{2,n+2}^{2}\), and in the other two regions at \(\varphi=\pi/2\) and \(\varphi=3\pi/2\) the SAM is negative. On the other circle, where \(I_{2,n-2}^{2}<I_{2,n+2}^{2}\), on the contrary, at \(\varphi=\pi/2\) and \(\varphi=3\pi/2\) the SAM is positive, and at \(\varphi=0\) and \(\varphi=\pi\) the SAM is negative. The presence of regions with longitudinal projection of the SAM with different signs at the focus demonstrates the spin Hall effect. Where the SAM is positive, the elliptical polarization vector rotates counterclockwise, and in the regions with the negative longitudinal SAM there is a left-handed elliptical polarization, and the polarization vector rotates clockwise.  
There follows another extreme case from Equation (6), when the parameter \(\alpha=0\). In this case, the longitudinal SAM at the focus will be equal to the SAM of an optical vortex with right-handed circular polarization obtained in [23]:  
\[S_{z,\alpha=0}=I_{0,n}^{2}-I_{2,n+2}^{2}. \tag{8}\]  
It is seen from (8) that for an optical vortex with right-handed circular polarization at the focus, the SAM varies only along the radius and can have a different sign at different radii. This effect can be called the radial spin Hall effect at the focus. Thus, we have shown that, by varying the parameter \(\alpha\) of the initial light field, it is possible to control the characteristics of the spin Hall effect at the focus, i.e., obtain different types of spin distribution at the focus, (7) and (8).