Calculation of focusing properties of a radially polarized beam with annular intensity distribution

A tightly focused radially polarized beam is known to produce a smaller focal spot mainly composing of longitudinal electric field near the focal point. The bright spot near the focal point can be disappeared by the interference with an additional optical field with a π phase shift, and this region or `dark spot´ is expected to be used to applications such as optical trapping and particle guiding. Furthermore, a radially polarized beam with an annular intensity pattern is expected to form a much smaller spot. In this report, we demonstrate the calculation of the focusing properties of a radially polarized beam with a narrow intensity annulus, which is less than 1/40^th of its beam radius, as well as a spiral phase shift corresponding to the topological charge m = 1 and 2 based on the vector diffraction theory. The calculation shows that the radially polarized beam with spiral phase shift forms a hollow with a narrow width around the beam axis. In addition, the results is compared to that for an azimuthally polarized beam with an annular intensity pattern.