Autofocusing circular symbolic umbilic beams

In this paper, we report a new class of autofocusing beams, circular symbolic umbilic beams (CSUBs), based on the catastrophe theory. The analytical propagation expression and the evolution of the 2+1-D SUBs are firstly introduced and then the autofocusing properties of CSUBs are theoretically and experimentally studied in detail, including the amplitude distribution of the light field, the autofocusing capability and the focusing length. Compared with the previous circular Airy, Pearcey or swallowtail beams, CSUBs can focus twice without extra phase term or optical component. The autofocusing capabilities in two focal planes are remarkable as well as comparable and the two focusing lengths are symmetrically distributed around a fixed position. The influences of related parameters on CSUBs are also analyzed. Furthermore, by applying a chirp phase on CSUB, the beam exhibits a thrice focusing behavior during propagation. Our experimental results match well with theoretical predictions.