# Transverse Spin Hall Effect and Twisted Polarization Ribbons at the Sharp Focus
## 3 Expressing the Field Strength Vector via Vectors of the Axes of the Polarization Ellipse  
Formulae for calculating the polarization singularity indices of vector fields have previously been reported [11]. Using those formulae, the \(\mathbf{E}\)-field vector can be expressed via vectors \(\mathbf{A}\) and \(\mathbf{B}\), which are oriented along the major and minor axes of a polarization ellipse centered on a given space point:  
\[ \begin{array}{l}\mathbf{E}=e^{i\gamma}(\mathbf{A}+i\mathbf{B}),\;\;I=| \mathbf{A}|^{2}+|\mathbf{B}|^{2},\\ \mathbf{A}\bullet\mathbf{B}=0,\;\;|\mathbf{A}|\geq|\mathbf{B}|,\\ \gamma=\frac{1}{2}\arg(\mathbf{E}\bullet\mathbf{E}),\\ \mathbf{A}=\frac{1}{\sqrt{\mathbf{E}\bullet\mathbf{E}}}\mathrm{Re}\Big{(} \mathbf{E}\sqrt{\mathbf{E}^{*}\bullet\mathbf{E}^{*}}\Big{)},\\ \mathbf{B}=\frac{1}{\left|\sqrt{\mathbf{E}\bullet\mathbf{E}}\right|}\mathrm{ Im}\Big{(}\mathbf{E}\sqrt{\mathbf{E}^{*}\bullet\mathbf{E}^{*}}\Big{)},\\ \mathbf{S}=\mathrm{Im}(\mathbf{E}^{*}\times\mathbf{E})=2(\mathbf{A}\times \mathbf{B}).\end{array}  \tag{7}\]  
In Equation (7), '\(\bullet\)' denotes the dot product of vectors, \(\times\) denotes the cross product, Re and Im are real and imaginary parts of the number, \(\mathbf{S}\) simultaneously describes the SAM vector and the normal vector to the polarization ellipse plane formed by the vectors \(\mathbf{A}\) and \(\mathbf{B}\), \(I\) is the intensity expressed through projections of the vectors of the polarization ellipse axes. We note that the decomposition of the complex field \(\mathbf{E}\) into the real vectors \(\mathbf{A}\) and \(\mathbf{B}\) is unambiguous, meaning that while not for any decomposition, the vectors \(\mathbf{A}\) and \(\mathbf{B}\) are mutually orthogonal, but for any decomposition, they lie in a perpendicular plane to the SAM vector, which is the polarization ellipse plane. Remarkably, representation (7) is also applicable to the light field at the focus defined by Equation (2), making it possible to derive explicit relationships for the two vectors \(\mathbf{A}\) and \(\mathbf{B}\) at the tight focus of field (1). Actually, the E-field vector in (2) can be represented in a similar way to (7) by merely separating the real and imaginary parts:  
\[ \begin{array}{l}\mathbf{E}=e^{i\gamma}(\mathbf{A}+i\mathbf{B}),\\ A_{x}=I_{0,n}+P(r)\cos 2\varphi,\\ A_{y}=P(r)\sin 2\varphi,\\ A_{z}=F(r)\sin\varphi,\\ B_{x}=Q(r)\sin 2\varphi,\\ B_{y}=-Q(r)\cos 2\varphi,\\ B_{z}=L(r)\cos\varphi,\\ P(r)=(I_{2,n-2}+I_{2,n+2})/2,\;\;F(r)=I_{1,n-1}+I_{1,n+1},\\ Q(r)=(I_{2,n+2}-I_{2,n-2})/2,\;\;L(r)=I_{1,n-1}-I_{1,n+1},\\ \gamma=n\left(\varphi+\frac{\pi}{4}\right)-\frac{\pi}{4}.\end{array}  \tag{8}\]  
Of course, the vectors \(\mathbf{A}\) and \(\mathbf{B}\) in (8) cannot be guaranteed to form two axes of the polarization ellipse at point (\(x,y,z=0\)); however, it can be checked whether the derived projections of vectors in (8) enable obtaining the SAM projections in (5) and (6), because  
\[ \begin{array}{l}\mathbf{S}=\mathrm{Im}(\mathbf{E}^{*}\times\mathbf{E})=2( \mathbf{A}\times\mathbf{B}),\\ I=|\mathbf{A}|^{2}+|\mathbf{B}|^{2}.\end{array}  \tag{9}\]  
The SAM vector in (9) is perpendicular to the vectors \(\mathbf{A}\) and \(\mathbf{B}\), with its absolute value being equal to the product of the modules of \(\mathbf{A}\) and \(\mathbf{B}\), multiplied by the sine of an unknown angle \(\eta\) between the vectors:  
\[|\mathbf{S}|=|\mathbf{A}||\mathbf{B}|\sin\eta. \tag{10}\]  
From (10) it follows that linear polarization of field (1) at the focus corresponds to the zero modulus of either vector \(\mathbf{A}\) or vector \(\mathbf{B}\): \(|\mathbf{S}|=|\mathbf{A}|=|\mathbf{B}|=0\). Equation (8) suggests that for any topological charge \(n\), there exists an axis-centered radius \(r_{0}\) of a circle in the focal plane (\(z=0\)), such that the following equality holds:  
\[2Q(r_{0})=I_{2,n+2}-I_{2,n-2}=0. \tag{11}\]  
Along this circle in the focal plane, the vector \(\mathbf{B}\) of the polarization ellipse has only a longitudinal projection:  
\[