# 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 the initial light field is an \(n\)th-order cylindrical vector beam, the Jones vectors of the electric and magnetic fields are given by:  
\[E_{n}(\varphi)=\left( \begin{array}{c}\cos n\varphi\\ \sin n\varphi\end{array} \right),\ \ \ \ H_{n}(\varphi)=\left( \begin{array}{c}-\sin n \varphi\\ \cos n\varphi\end{array} \right), \tag{1}\]  
with (\(r\), \(\varphi\)) being the polar coordinates in the transverse initial plane of the beam. Figure 1 depicts the intensity distribution of a Gaussian beam and the electric field strength vectors, when the beam has cylindrical polarization (1) with \(n=1\) and \(n=3\). It is seen in Figure 1 that on a closed contour around the field, the polarization vector makes one full rotation in Figure 1a and three full rotations in Figure 1b. Thus, on space propagation and in the focus, the field in Figure 1a (\(n=1\)) should retain its rotational symmetry, whereas 3 rd-order cylindrical polarization (\(n=3\), Figure 1b) violates this rotational symmetry. Adopting the [PERSON] integrals [19], we derive all the Cartesian components of the strength vectors of the electric and magnetic fields in the sharp focus of the light field with polarization (1):  
\[E_{x}(r,\varphi) =i^{n-1}[\cos(n\varphi)I_{0,n}+\cos((n-2)\varphi)I_{2,n-2}], \tag{2}\] \[E_{y}(r,\varphi) =i^{n-1}[\sin(n\varphi)I_{0,n}-\sin((n-2)\varphi)I_{2,n-2}],\] \[E_{z}(r,\varphi) =2i^{n}\cos((n-1)\varphi)I_{1,n-1},\] \[H_{x}(r,\varphi) =-i^{n-1}[\sin(n\varphi)I_{0,n}+\sin((n-2)\varphi)I_{2,n-2}],\] \[H_{y}(r,\varphi) =-i^{n-1}[-\cos(n\varphi)I_{0,n}+\cos((n-2)\varphi)I_{2,n-2}],\] \[H_{z}(r,\varphi) =-2i^{n}\sin((n-1)\varphi)I_{1,n-1}.\]  
In Equation (2), the functions \(I_{\
u,\mu}\) depend only on the radial and longitudinal coordinates \(r\) and \(z\) and equal to:  
\[I_{\
u,\mu}=\left(\frac{4\pi f}{\lambda}\right)\int\limits_{0}^{\theta_{0}} \sin^{\
u+1}\left(\frac{\theta}{2}\right)\cos^{3-\
u}\left(\frac{\theta}{2} \right)\cos^{1/2}(\theta)A(\theta)e^{i\mathbb{R}z\cos\theta}J_{\mu}(\xi)d\theta, \tag{3}\]  
where \(k=2\pi/\lambda\) stands for the wavenumber of monochromatic light with the wavelength \(\lambda,f\) is the focal distance of an aberration-free spherical lens that focuses the beam, \(z\) is the propagation axis (\(z=0\) is the focal plane), \(\xi=kr\) sin \(\theta\) is a dimensionless argument of the \(\mu\)th-order Bessel function of the first kind \(J_{\mu}(\xi)\), \(NA=\sin\theta_{0}\) is the numerical aperture of an aplanatic focusing system, whereas an arbitrary real-valued function \(A(\theta)\) describes the amplitude of an input field with a cylindrical symmetry (plane wave, Gaussian beam, Bessel-Gaussian beam). In the functions \(I_{\
u,\mu}\) in Equation (3), the first index \(\
u=0\), \(1\), \(2\)  
Figure 1: Intensity distributions of a Gaussian beam and the electric field strength vectors, when the beam has cylindrical polarization (1) with \(n=1\) (**a**) and \(n=3\) (**b**). describes the first part of the integrand, whereas the second index \(\mu=0\), \(1\), \(2\),..., \(n\) defines the order of the Bessel function. To derive the focal distribution of the spin density or the spin angular momentum (SAM) for the beam with polarization (1), we use the general expression for the spin angular momentum vector given in [20]:  
\[\mathbf{S}=\frac{1}{16\pi\omega}\text{Im}(\mathbf{E}^{*}\times\mathbf{E}), \tag{4}\]  
where \(\omega\) is the angular frequency of light. From now on, the constant factor \(1/(16\pi\omega)\) will be omitted. It is seen in Equation (4) that the longitudinal SAM component (without the constant factor) is coinciding with the non-normalized third Stokes parameter \(S_{3}\):  
\[S_{3}=S_{z}=2\text{Im}\big{(}E_{x}^{*}E_{y}\big{)}. \tag{5}\]  
It is known that the third Stokes component indicates circular or elliptic polarization of a light field [21]. If \(S_{3}=0\), then the field has only linear polarization.