General integral expressions for on-axis spherical waves diffracted at a circular aperture

Starting from the Fresnel-Kirchhoff diffraction integral formula and taking into consideration the inclination factor and wave-front integration, the general expressions for on-axis convergent and divergent spherical waves diffracted at a circular aperture are derived. Our results are valid for the case |f| > a, where f and a are the focal length and aperture radius, respectively. The numerical results are consistent with the experimental data of divergent spherical waves diffracted at a circular aperture. For convergent spherical waves the results are compared with those by using the Li-Wolf formula, showing the more general applicable advantage of our formulae.