Analysis of interference of cylindrical laser beams generated by ring optical elements with a vortex phase at sharp focusing

We theoretically analyze sharp focusing of cylindrically polarized beams by a focusing system containing ring apertures and a vortex-phase optical element in the pupil. Analytic expressions for a field in the focal area for radially and the azimuthally polarized beams for one- and two-ring apertures are obtained. Asymptotic expressions for the focal field at a distance from the optical axis are also derived.
It is shown that for a single vortex-phase narrow ring aperture, the field distribution in the focal area is proportional to the superposition of Bessel functions of different orders. The field is shown to possess the axial symmetry and be independent of the longitudinal coordinate. The structure of the interference field for radial and azimuthal polarization is investigated.
With two narrow ring apertures in the pupil of the sharp-focus system, the interference pattern of two vector Bessel-type beams is formed in the focal area. If both ring apertures have the same vortex phase, the field in the focal area remains axis-symmetric, although becoming periodically dependent on the longitudinal coordinate. If each ring has a different-order vortex phase, the interference is more complex and leads to the formation of rotating beams. The three-dimensional structure in the focal area can be controlled by varying the polarization and phase distribution in each of the rings.