# Vector Beams with Only Transverse Intensity at Focus
## 3 Results  
### Azimuthally Polarized Light  
The polarization at which the electric field is directed strictly azimuthally is well known. The absence of the longitudinal component of the electric field, in this case, follows directly from [PERSON]'s vector equations written for a cylindrical coordinate system: when the derivative with respect to the azimuthal angle is set to zero, the system of six equations is divided into two systems of equations, each containing three equations (an analogy can be seen with TE- and TM-polarization in a Cartesian coordinate system). One of the systems is:  
\[\left\{ \begin{array}{l}\frac{\partial H_{\theta}}{\partial z}-\frac{\partial H _{z}}{\partial\varphi}=\varepsilon\epsilon_{0}\frac{\partial E_{\varphi}}{ \partial t}+\sigma E_{\varphi},\\ -\frac{\partial E_{\varphi}}{\partial z}=-\mu\mu_{0}\frac{\partial H_{\theta}} {\partial t},\\ \frac{1}{\rho}\frac{\partial(\rho E_{\varphi})}{\partial\rho}=-\mu\mu_{0} \frac{\partial H_{z}}{\partial t},\end{array} \right. \tag{3}\]  
where \(\mathbf{H}\) is the magnetic field, \(\varepsilon\) and \(\mu\) are the dielectric and magnetic permeability, and \(\varepsilon_{0}\) and \(\mu_{0}\) are the electric and magnetic constants. The field described by (3) contains only one electrical field component \(E_{\phi}\). For a classical azimuthally polarized beam, the electric field component of the initial beam in (1) has the form:  
\[E(\theta,\varphi)=\left( \begin{array}{c}a(\varphi)\\ b(\varphi)\end{array} \right)=\left( \begin{array}{c}-\sin\varphi\\ \cos\varphi\end{array} \right). \tag{4}\]  
Substituting Equation (4) into (2), we can see that  
\[P_{z}(\theta,\varphi)=\sin\theta\cos\varphi\sin\varphi-\sin\theta\sin\varphi \cos\varphi=0. \tag{5}\]  
Previously, only the case of azimuthal polarization was studied (without singularity lines). Azimuthal polarization is a special case of a beam with a V-point singularity [26] located at the center of the beam. The result of focusing an azimuthally polarized beam (4), calculated by (1), is shown in Figure 3. The distribution of intensity \(I=1\,|E_{x}\,|^{2}+1\,|E_{y}\,|^{2}+1\,|E_{z}\,|^{2}\) (Figure 3a) and its individual components are shown. From Figure 3d, it can be seen that the longitudinal component is equal to zero. It should be noted that the absence of the longitudinal component of the electric field leads to the fact that the spin angular momentum (SAM) vector will contain the longitudinal component only (or \(S_{z}\equiv 0\)):  
\[\mathbf{S}=\frac{1}{8\pi\omega}\text{Im}(\mathbf{E}^{*}\times\mathbf{E}), \tag{6}\]  
where \(\omega\) is the angular frequency of the light (the constant \(1/(8\pi\omega)\) will be then omitted), sign \"*\" means complex conjugation. It should be noted that the spin angular momentum  
Figure 3: Intensity \(I\) (**a**) and its individual components \(I_{x}\) (**b**), \(I_{y}\) (**c**), and \(I_{z}\) (**d**) at the focus of a beam with azimuthal polarization (4). (6) is opposite, and differs in absolute value only by a factor from the magnetization vector:  
\[\mathbf{M}=i\gamma\{\mathbf{E}\times\mathbf{E}^{*}\}, \tag{7}\]  
where \(\gamma\) is the magneto-optical susceptibility. Previously [29], it was shown that the components of the electric field for azimuthally polarized light have the form:  
\[ \begin{array}{l}E_{x}=-\sin\varphi[I_{1}-I_{2}],\\ E_{y}=\cos\varphi[I_{1}-I_{2}],\\ E_{z}=0,\end{array}  \tag{8}\]  
where  
\[ \begin{array}{l}I_{1}=\frac{\pi f}{\lambda}\int\limits_{0}^{\alpha}T(\theta )\sin\theta(1+\cos\theta)B(\theta)\mathrm{e}^{ikz\cos\theta}J_{1}(x)d\theta, \\ I_{2}=\frac{\pi f}{\lambda}\int\limits_{0}^{\alpha}T(\theta)\sin\theta(1-\cos \theta)B(\theta)\mathrm{e}^{ikz\cos\theta}J_{-1}(x)d\theta.\end{array}  \tag{9}\]  
In (9), \(x=k\)rsin\(\theta\), \(J_{m}(x)\) is the \(m\)th order Bessel function, and \(B(\theta)\) is a real function describing the input field amplitude in the plane of the entrance pupil of the aplanatic system, depending only on the angle \(\theta\).