# Vector Beams with Only Transverse Intensity at Focus
## 3 Results
### Beam with a V-Line of Polarization Singularity--Sector Azimuthal Polarization  
It is possible to obtain other beams that do not have the \(z\)-component of the electric field. It is possible if, in addition to the V-point singularity, the beam contains V-line singularities [26]. The polarization functions of the initial field (1), in this case, are [30]:  
\[E(\theta,\varphi)=\left(\begin{array}{c}a(\varphi)\\ b(\varphi)\end{array}\right)=\left(\begin{array}{c}\cos(2\varphi)-1\\ \sin(2\varphi)\end{array}\right). \tag{11}\]  
Substituting (11) into (2), we can show that:  
\[\begin{array}{l}P_{z}(\theta,\varphi)=\\ =-\sin\theta\cos\varphi(\cos(2\varphi)-1)-\sin\theta\sin\varphi\sin(2\varphi) =\\ -\sin\theta\left\{-\cos\varphi+\underbrace{\cos\varphi\cos(2\varphi)+\sin \varphi\sin(2\varphi)}_{\cos\varphi}\right\}=0.\end{array} \tag{12}\]  
Figure 5 shows the distribution of the intensity and its individual components in the focal plane for the beam (11). Figure 6 shows the distribution of the longitudinal component of the SAM at a distance \(\lambda\) from the focus: directly at the focus for this beam, all SAM components are equal to zero [30]. If a beam contains regions of nonzero longitudinal SAM, then the polarization in these regions is elliptical (or circular). Figure 6 shows that the plane contains four local regions of approximately 500 nm width, in which the light is circularly polarized along different diagonals. The initial beam has no points with circular polarization in its section (Figure 1). This conversion is a particular case of the Hall effect at a tight focus [30; 31; 32; 33].  
Figure 5: Intensity \(I\) (**a**) and its components \(I_{x}\) (**b**), \(I_{y}\) (**c**), and \(I_{z}\) (**d**) at the focus of the beam with the V-line of polarization singularity (11).  
Figure 6: Distribution of the longitudinal component of the spin angular momentum SAM\({}_{z}\) at a distance \(z=\lambda\) from the focus of the beam with the V-line of polarization singularity (11).  
A similar beam with a rotated V-line of singularity has the form (Figure 1c):  
\[\begin{array}{l}a(\varphi)=\sin(2\varphi),\\ b(\varphi)=-\cos(2\varphi)-1.\end{array} \tag{13}\]  
This beam has the same distributions of intensity and longitudinal SAM, but rotated by 90 degrees.  
Beam (11) (Figure 1b) can be experimentally obtained using the experimental setup shown on Figure 2. To do this, it is possible to pass a beam linearly polarized along the \(x\) axis through element (10); however, the polarization of the lower segment should differ in sign from the upper segment. Then, the polarization for the upper beam segment can be described by the Jones vector:  
\[E=\begin{pmatrix}-\sin\varphi&\cos\varphi\\ \cos\varphi&\sin\varphi\end{pmatrix}\begin{pmatrix}1\\ 0\end{pmatrix}=\begin{pmatrix}-\sin\varphi\\ \cos\varphi\end{pmatrix} \tag{14}\]  
and for the bottom sector by the Jones vector:  
\[E=\begin{pmatrix}-\sin\varphi&\cos\varphi\\ \cos\varphi&\sin\varphi\end{pmatrix}\begin{pmatrix}-1\\ 0\end{pmatrix}=\begin{pmatrix}\sin\varphi\\ -\cos\varphi\end{pmatrix} \tag{15}\]  
To obtain polarization described by Equation (11) or Equations (14) and (15), a vertically divided screen was displayed on the SLM in such a way that a phase shift of \(\pi\) occurred in the lower part of the beam. The result of focusing is shown in Figure 7.  
Figure 7 shows that despite the small numerical aperture of the focusing lens, the intensity distribution pattern at the focus is consistent with the simulation (Figure 5). The slight asymmetry could be explained by the imperfection of the experiment. The beam center should be precisely positioned on both the SLM center (Figure 2) and the center of vector wave plate.