General integral expressions for converging spherical wave diffracted at a circular aperture

Starting from the Fresnel-Kirchhoff and Rayleigh-Sommerfeld diffraction integral formulas, respectively, and in consideration of the inclination factor, the general diffraction integral expressions for convergent spherical waves diffracted at a circular aperture are derived, which are not only used to compute focal shifts of the diffraction field, but also valid for both weak and strong focusing cases, and thus have the generally applicable advantage. The results are compared with those got using the Li-Wolf formula, and the Li-Wolf formula can be regarded as an approximation of the general expressions under certain condition. The results are useful for the study of the beam precise focusing.