# Vector Beams with Only Transverse Intensity at Focus
## 4 Discussion  
In this paper, we investigated vector beams with azimuthal (Figure 1a) and sector azimuthal polarization (Figure 1b-d). Sector azimuthal polarization contains V-lines of polarization singularity. The absence of the longitudinal component of the electric field for these beams follows directly from the Maxwell Equation (3).  
Tight focusing characteristics of the studied beams were investigated using the Richards-Wolf integral (1). The absence of the longitudinal component of the electric field was numerically proven (Figures 3d, 5d, and 8d). The intensity pattern of the transverse component differs for the investigated beams. Azimuthally polarized light (Figure 1a) has a doughnut-shaped structure (Figure 3a); however, the beams with V-line singularity (Figure 1b,d) have a peak-shaped focal spot (Figures 5a and 8a, consequently). In other words, all the studied beams do not have a longitudinal component of the electrical field; however, at the same time, they have different intensity shapes--peak or doughnut--and as a result, they could be used in different applications, where different shapes of transverse intensity without longitudinal intensity are needed.  
Experimentally, the investigated beams with V-line polarization singularity (Figure 1b-d) could be generated using a vector wave plate and a spatial light modulator. The vector wave plate rotates the direction of polarization and converts the linearly polarized light to the beam with azimuthal polarization (Figure 1a). The example of an azimuthally polarized beam obtained after passing the linearly polarized beam through a vector wave plate is shown in Figure 4. Moreover, if the incident beam has sector linearly polarized light (for example, the Gauss-Hermite beam (0.1)), then after passing the vector wave plate, the output beam has sector azimuthal polarization. A beam with sector linear polarization could be generated using a spatial light modulator. Therefore, a spatial light modulator combined with a vector wave plate could be used to obtain the investigated sector azimuthal beams (Figure 2). The example of the focused beam with V-line singularity is shown in Figure 7. The intensity distribution pattern at the focus was consistent with the simulation.  
Figure 8: Intensity \(I\) (**a**) and its components \(I_{x}\) (**b**), \(I_{y}\) (**c**), and \(I_{z}\) (**d**) at the focus of the beam with two V-lines of polarization singularity.  
Figure 9: Distribution of the longitudinal component of the spin angular momentum SAM\({}_{\mathrm{z}}\) at the focus of the beam with two V-lines of polarization singularity.  
It should be noted that due to the absence of the longitudinal component of the electric field \(E_{z}\), the investigated beams do not contain transverse components of spin angular momentum (\(S_{\rm x}=S_{\rm y}=0\)). It is equivalent to the absence of transverse components of magnetization because the magnetization contains the same non-zero components as SAM. Moreover, the longitudinal component of SAM of the pure azimuthally polarized beam (Equation (4), Figure 1a) is always equal to zero. It could be proven analytically by Equation (8). The longitudinal component of SAM of the beam with V-line singularity (Equation (11), Figure 1b) is equal to zero at the focal plane; however, it is non-zero in the vicinity of the focal plane (\(z\
eq 0\), Figure 6). The longitudinal component of SAM of the beam (Equation (16), Figure 1c) is non-zero both at the focal plane and in the vicinity of the focal plane (Figure 9). This effect of the investigated beams can be used to obtain pure longitudinal magnetization distributions in optomagnetic memory systems [7,8].  
The investigated beams (Figure 1) have local linear polarization in the initial plane; however, in tight focus, there are regions with elliptical polarization (Figures 6 and 9). This effect is a particular case of the Hall effect (or spin-orbit conversion) at a tight focus [30, 31, 32, 33].