Abstract
We performed theoretical analysis of sharp focusing of radially polarized beams when there are annular apertures at vortex phase elements in the focusing system. We obtained analytical expressions for the field in the focal region in case of one and two annular apertures. We also present asymptotic expressions for the focal field near the optical axis and at a distance from it. We showed that for one narrow annular aperture with a vortex phase the focal distribution is proportional to a superposition of Bessel functions of different orders. In this case, the focal field is an axially symmetric and does not depend on the longitudinal coordinate. When two narrow annular apertures are used an interference pattern of two vector Bessel-type beams is generated in the focal region. In the case of equal vortex phases in both rings, the field in the focal region remains an axisymmetric one, but a periodic dependence on the longitudinal coordinate appears. If in each of the rings there are vortex phases of different orders, we obtain an interference pattern of more intricate type leading to formation of spiral beams.
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