# Spin Hall Effect before and after the Focus of a High-Order Cylindrical Vector Beam
## 2 Spin Angular Momentum before and beyond the Focus
If we substitute the electric field components from Equation (2) into Equation (5) and take into account that the integrals (3) are complex-valued near the focus (but not in the focus itself), we obtain the longitudinal component of the SAM vector:  
\[E_{y}(r,\varphi) =i^{n-1}[\sin(n\varphi)I_{0,n}-\sin((n-2)\varphi)I_{2,n-2}],\]0  
where the asterisks \"*\" mean the complex conjugation. Separating the real and imaginary parts in the integrals (3), and using the linear approximation \(\exp(ikz\cos\theta)\approx 1+ikz\cos\theta\) near the focus (\(kz<<1\)), instead of Equation (6) we get:  
\[E_{y}(r,\varphi) =i^{n-1}[\sin(n\varphi)I_{0,n}-\sin((n-2)\varphi)I_{2,n-2}],\]1  
where we use the following designations:  
\[ \begin{split} R_{0}&=I_{0,n}(z=0),\ I_{0}= \overline{I}_{0,n},\\ R_{2}&=I_{2,n-2}(z=0),\ I_{2}=\overline{I}_{2,n-2},\\ \overline{I}_{\
u,\mu}&=\Big{(}\frac{4\pi f}{\lambda }\Big{)}\int\limits_{0}^{\varphi_{0}}\sin^{\
u+1}\Big{(}\frac{\theta}{2}\Big{)} \cos^{3-\
u}\Big{(}\frac{\theta}{2}\Big{)}\cos^{3/2}(\theta)A(\theta)I_{\mu}( \xi)d\theta.\end{split}  \tag{8}\]  
As seen from Equation (7), in the focus itself (\(z=0\)), \(S_{3}=0\), and, therefore, in each point of the focal plane, polarization is linear. However, at small defocusing (\(kz<<1\)), \(S_{3}\
eq 0\), and areas with elliptic and circular polarization appear, if \(n\
eq 1\). The condition \(n\
eq 1\) is consistent with Figure 1, demonstrating that the order \(n=1\) does not break the rotational symmetry of the beam and the areas with nonzero SAM cannot appear. Thus, such areas can appear only due to the symmetry violation at \(n\
eq 1\). In the areas where before the focus (\(z<0\)) the SAM was negative (\(S_{3}<0\)), after the focus (\(z>0\)) it becomes positive (\(S_{3}>0\)), and vice versa. According to Equation (7), near the focal plane, on a certain radius circumference with the center on the propagation axis, centers of 4(\(n\)-1) local subwavelength areas reside, with elliptic and circular polarization. In such neighboring areas, the polarization vector is rotating in opposite directions (clockwise or counterclockwise). A similar result was obtained in [22], but for \(n=0\). Since, at \(n\
eq 1\), near the focus of the light field (1), areas with left and right circular polarization (areas with a different \"spin\") are spatially separated, we can conclude that near the focus (before and after it), the spin Hall effect occurs, although in the focal plane itself, this effect vanishes. This effect can be approximately explained by decomposing the cylindrical vector beam (1) into a superposition of two circularly polarized optical vortices:  
\[E_{n}(\varphi)=\left( \begin{array}{c}\cos n\varphi\\ \sin n\varphi\end{array} \right)=\frac{1}{2}e^{in\varphi}\left( \begin{array}[] {c}1\\ -i\end{array} \right)+\frac{1}{2}e^{-in\varphi}\left( \begin{array}{c}1\\ i\end{array} \right).\]Both optical vortices have the SAM and the OAM. If \(n=1\), the SAM and the OAM annihilate each other, and the angular momentum of each vortex is zero. Thus, both vortices do not rotate on propagation and focusing, and the areas with a positive and with a negative SAM are not separated in space. If, however, \(n\
eq 1\), the angular momentum of each vortex is nonzero and, therefore, both beams rotate on propagation (Figure 2, beam rotations), i.e., there is a transverse (azimuthal) energy flow in each vortex. Since the vortices are of opposite orders \(n\) and \(-n\) and their circular polarizations are also opposite, the azimuthal energy flows of both vortices are also opposite and, in the transverse plane, they interfere with each other, thus generating the areas with the opposite SAM (Figure 2, near-focus plane). In the focal plane itself, however, these transverse energy flows annihilate each other, and the SAM is zero in each point (Figure 2, focus plane).