# Vector Beams with Only Transverse Intensity at Focus
## 1 Introduction  
Currently, there is a growing interest of researchers in the use of azimuthally polarized light [1]. Initially, azimuthal polarization was proposed to obtain compact spots [2, 3]. However, later other applications were demonstrated. For example, in [4, 5], an azimuthally polarized beam was used to generate the third harmonic in ensembles of silicon nanoparticles (quadrumers and trimers). It is azimuthally polarized light that turned out to be the most suitable for the excitation of Mie resonances in the oligomers under study: generation was improved by two orders of magnitude compared to using linear polarization. In [6], a fiber-optic sensor was proposed in which the azimuthally polarized mode propagated as the fundamental mode.  
Quite intriguing is the use of azimuthally polarized light for manipulation of magnetization, which is calculated as the vector product of the electric field and its complex conjugation: \(\mathbf{M}\)-\(\mathbf{E}\times\mathbf{E}^{*}\). The increased interest in the study of magnetization is caused by their possible application in optomagnetic memory systems [7, 8]. Due to the absence of a longitudinal component of the electric field, the magnetization in an azimuthally polarized vortex beam is always directed along the propagation axis. In this way, it has been possible to obtain spots [9], needles [10, 11, 12], chains [13, 14, 15], and arrays of magnetization [16]. In [17], an increase in the magnetic field was observed at the center of quadrupoles and hexapoles made of silicon microspheres when they were illuminated by light with azimuthal polarization.  
It should be noted that in a classical azimuthally polarized beam, the polarization direction changes continuously [1]. In this paper, we will show that similar properties to azimuthally polarized beams can also be obtained for sectoral azimuthally polarized beams containing V-lines of singularity. Previously, similar piecewise continuous polarization states were investigated in [18, 19, 20]. In [18], optical vortices with cylindrical polarization of half-integer orders were studied. In [19], the focusing of cylindrical vector beams of fractional orders was numerically simulated, and in [20], a vector beam that was formed fromtwo other vector beams was considered--the upper part of the beam had one order, and the lower half had another order. Initially, there was no longitudinal spin angular momentum in these beams [19,20]: the polarization at each point of the initial beam was linear; however, with tight focusing, a nonzero longitudinal component of the spin angular momentum arose, which indicated the appearance of regions in which the polarization is elliptical. This effect is a special case of spin-orbit interaction in a tight focus. Beams with discontinuities in polarization can be created using metasurfaces based on superoscillations [21] or subwavelength gratings [22].  
In this work, tight focusing of vector beams with azimuthal polarization and beams with V-lines of polarization singularity was simulated numerically using the Richards-Wolf formula. It was demonstrated that in a tight focus for these beams, there is no longitudinal component of the electric field. Previously, a similar effect was demonstrated for azimuthally polarized beams only. The absence of a longitudinal component of the electric field leads to the absence of transverse components of the spin angular momentum and transverse components of magnetization. This characteristic of the investigated beams can be used to obtain pure longitudinal magnetization distributions, for example in optomagnetic memory systems [7,8]. The possibility of creating sectorial azimuthally polarized beams using vector wave plates and spatial light modulators was experimentally demonstrated. The intensity distribution patterns measured in the experiment were consistent with the simulation results.