Modeling the focusing of a radially polarized laser beam with an initially flat-top intensity profile

Radially polarized light beams present very interesting and useful behavior for creating small intensity spots when tightly focused, and manipulating nanostructures or charged particles via their longitudinal field. The modeling of the propagation of such vector beams, however, is almost always done using the lowest-order fundamental radially polarized beam or a single higher-order mode due to the complexity of vector diffraction theory. We show how a flat-top radially polarized beam, relevant for high-power lasers, can be modeled analytically using a sum of higher-order beams, describe a number of interesting qualities of such beams, and compare to numerically solved integral descriptions.