# Transverse Spin Hall Effect and Twisted Polarization Ribbons at the Sharp Focus
## 2 Projections of Strength, Intensity and Spin Angular Momentum at the Tight Focus of a Light Field  
Assume an original light field whose Jones vector is given by  
\[\mathbf{E}(\varphi)=\exp(in\varphi)\left(\begin{array}{c}1\\ 0\end{array}\right), \tag{1}\]  
where (_r_, \(\varphi\)) are the polar coordinates at the beam cross-section, \(n\) is the integer topological charge, with the linear polarization vector being directed along the horizontal _x_-axis. Projections of the electric and magnetic field vectors at the focus of the original field in Equation (1) are given by [13]  
\[\begin{array}{l}E_{x}=\frac{i^{n-1}}{2}e^{in\varphi}\left(2I_{0,n}+e^{2i \varphi}I_{2,n+2}+e^{-2i\varphi}I_{2,n-2}\right),\\ E_{y}=\frac{i^{n}}{2}e^{in\varphi}\left(e^{-2i\varphi}I_{2,n-2}-e^{2i\varphi}I _{2,n+2}\right),\\ E_{z}=i^{n}e^{in\varphi}\left(e^{-i\varphi}I_{1,n-1}-e^{i\varphi}I_{1,n+1} \right),\end{array} \tag{2}\]  
Entering Equation (2) are the functions \(I_{\mu\
u}\) of a single radial variable _r_:  
\[I_{\
u,\mu}=2 kf\int\limits_{0}^{\alpha}\sin^{\
u+1}\left(\frac{\theta}{2} \right)\cos^{3-\
u}\left(\frac{\theta}{2}\right)\cos^{1/2}(\theta)A(\theta)e^ {ikz\cos\theta}J_{\mu}(kr\sin\theta)d\theta, \tag{3}\]  
where \(k=2\pi/\lambda\) is the wave number of monochromatic light of wavelength \(\lambda,f\) is the focal length of a focusing lens, \(\alpha\) is the maximum angle of the incident ray with the optical axis, which defines the numerical aperture of an aplanatic lens, NA = sin \(\alpha,J_{\
u}(\xi)\) is the Bessel function of the first kind and \(\
u\)-th order. \(A(\theta)\) is a real function that defines the radially symmetric amplitude of the original field, which depends on the incidence angle \(\theta\) of an outgoing ray from the original wavefront converging at the center of the focal plane.  
Next, using Equation (2), we can derive all three projections of the SAM vector [14]:  
\[\mathbf{S}=\frac{1}{8\pi\omega}\text{Im}(\mathbf{E}^{*}\times\mathbf{E}), \tag{4}\]  
where \(\omega\) is the angular frequency of light. Hereafter, the constant \(1/(8\pi\omega)\) is dropped.  
Substituting (2) into (4) yields relationships for projections of the spin density vector, or the spin angular momentum vector at the tight focus for the original field (1):\[\begin{array}{l}S_{x}=M(r)\sin\varphi+N(r)\sin 3\varphi,\\ S_{y}=O(r)\cos\varphi+M(r)\cos\varphi+N(r)\cos 3\varphi,\\ S_{z}=\frac{1}{2}\big{(}I_{2,n-2}-I_{2,n+2}\big{)}(I_{2,n-2}+I_{2,n+2}+2I_{0,n} \cos 2\varphi),\\ M(r)=I_{1,n-1}I_{2,n-2}-I_{1,n+1}I_{2,n+2},\\ N(r)=I_{1,n-1}I_{2,n+2}-I_{1,n+1}I_{2,n-2},\\ O(r)=2I_{0,n}(I_{1,n-1}-I_{1,n+1}).\end{array} \tag{5}\]  
By analogy, we can express the full intensity, \(I=I_{x}+I_{y}+I_{z}=\left|E_{x}\right|^{2}+\left|E_{y}\right|^{2}+\left|E_{z} \right|^{2}\) and its separate components at the focus:  
\[\begin{array}{l}I_{x}=I_{0,n}^{2}+\frac{1}{4}I_{2,n-2}^{2}+\frac{1}{4}I_{2,n +2}^{2}+\cos 2\varphi\big{(}I_{0,n}I_{2,n-2}+I_{0,n}I_{2,n+2}\big{)}+ \frac{1}{4}I_{2,n-2}I_{2,n+2}\cos 4\varphi,\\ I_{y}=\frac{1}{4}I_{2,n-2}^{2}+\frac{1}{4}I_{2,n+2}^{2}-\frac{1}{4}I_{2,n-2}I _{2,n+2}\cos 4\varphi,\\ I_{z}=I_{1,n-1}^{2}+I_{1,n+1}^{2}-2I_{1,n-1}I_{1,n+1}\cos 2\varphi,\\ I=I_{0,n}^{2}+\frac{1}{2}I_{2,n-2}^{2}+\frac{1}{2}I_{2,n+2}^{2}+I_{1,n-1}^{2} +I_{1,n+1}^{2}+\cos 2\varphi\big{(}I_{0,n}I_{2,n-2}+I_{0,n}I_{2,n+2}-2I_{1,n-1}I_{1,n +1}\big{)}.\end{array} \tag{6}\]