The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture

Based on the vectorial Rayleigh-Sommerfeld integrals, the general propagation integral expressions for on-axis nonparaxial vectorial spherical wave diffracted at a circular aperture are derived. The results are strict integral formulae for the light field on the axis, valid for either strong or weak focusing, both far and near zones, and for the systems in which the size of the aperture is comparable to or smaller than the wavelength. Thus, it has the advantage for general application. For convergent spherical waves, the numerical calculation results are compared with those obtained by using the integral formulae of scalar paraxial approximation and scalar nonparaxial approximation, confirming the consistence in the situation of scalar approximation.