Accurate non-paraxial approximation to the vectorial Rayleigh-Sommerfeld diffraction integrals for elliptical, radial, azimuthal and other polarized illuminating fields

Some formulas are developed to simplify the calculations of the vectorial Rayleigh-Sommerfeld (R-S) diffraction integrals when several types of polarized fields are used as illuminating light. It is proved that when rotationally symmetric functions are considered altogether, the two-dimensional fields of the R-S expressions can be obtained with single integrals, speeding its calculations, maintaining its accuracy, and giving new insight related with symmetries of the diffracting patterns. It is also proved that the new formulas can be used confidently to study diffraction properties of converging waves even of high numerical aperture.