A perturbation theory based on a single-scattering approximation is developed from the rigorous differential theory of diffraction in cylindrical coordinates. It results in analytical formulas in the inverse space for the field amplitudes providing results that are in quantitative agreement with the results of the rigorous method, in both the near- and far-field regions, when a proper correction to the incident field inside the aperture is made by using the renormalized Born approximation. When working in reflection by a screen having permittivity high in modulus, the method proposes an equivalence with the simple model consisting of the emission by a single magnetic dipole excited inside the pierced layer, emission that is then transferred back into the cladding following the Fresnel's coefficients of transmission from the layer into the cladding. The theory predicts a directivity of the radiation pattern that increases for smaller values of modulus of permittivity, both for dielectrics and metals, thus independently of the possibility of plasmon surface wave excitation along the interface. The theory can take into account such surface wave resonances, as well as the waveguide supported by a dielectric slab, but cannot implicitly recognize the modes carried out by the cylindrical waveguide corresponding to the aperture. This fact limits its domain of validity when used in transmission, although the far- and near-field maps can be reconstructed sufficiently well within a multiplicative factor corresponding to the enhanced transmission due to the excitation of these modes.