Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: spherical waves

The Huygens–Fresnel diffraction integral has been formulated for incident spherical waves with use of the Kirchhoff obliquity factor and the wave front as the surface of integration instead of the aperture plane. Accurate numerical integration calculations were used to investigate very-near-field (a few aperture diameters or less) diffraction for the well-established case of a circular aperture. It is shown that the classical aperture-plane formulation degenerates when the wave front, as truncated at the aperture, has any degree of curvature to it, whereas the wave-front formulation produces accurate results from ∞ up to one aperture diameter behind the aperture plane. It is also shown that the Huygens–Fresnel–Kirchhoff incident-plane-wave-aperture-plane-integration and incident-spherical-wave-wave-front-integration formulations produce equally accurate results for apertures with exit f-numbers as small as 1.