Generation of multiple spherical spots with a radially polarized beam in a 4pi focusing system

We demonstrate the possibility of creating multiple spherical spots in a 4pi focusing system with a radially polarized beam. Using spherical waves to expand the plane wave factor in the Richards-Wolf integral, it is found that a proper spatial modulation in the amplitude of the input field with radial polarization can form multiple spherical spots with a focusing system satisfying the Herschel condition. These spots are distributed symmetrically about the focus on the optical axis with variable positions and intensities. Although we consider only the case of three spherical spots in this paper, generalization to the multiple-spots case will present no difficulty.