Analytic Theory of the Evolution of Circle Pearcey Beam

With the research of abrupt autofocusing beams deeper than ever before, the application of that turns wider, such as the particle manipulation, biomedicine, optical communication, vortex beams and so on, arousing scientists’ interest further. However, the analytical solution of circle Pearcey beams (CPBs) in paraxial propagation has not yet been proposed so far. In this work, we first propose the analytical solution of circle Pearcey beams (CPBs) transmitting on axi, and obtain semi-analytical solution of which transmitting off axis under paraxial condition. Particularly, The complex amplitude and analytical expressions of the circle Pear-cey beam (CPB) at any point in the free space under the paraxial approximation are obtained by means of the stationed phase method, the asymptotic theory and decouple theory–separation variable method, which contains separation variable method. It’s fantastic that the analytical solution we acquire is well consistent with the numerical solution gained by three-step Fourier algorithm. Furthermore, compared with the three-step method, we theoretically get the self-focusing distance of circle Pearcey beams (CPBs), expounding the phenomena of autofocusing and the law of spatial evolution under paraxial approximation