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.

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The Huygens-Fresnel diffraction integral has been formulated for incident Gaussian laser beams by using the Kirchhoff obliquity factor with the wave front instead of the aperture plane as the surface of integration. Accurate numerical-integration calculations were used to investigate the Fresnel field diffraction region for the much-studied case of a circular aperture. It is shown that the classical aperture-plane formulation becomes inaccurate when the wave front, as truncated at the aperture, has any degree of curvature to it, whereas the newly developed wave-front formulation produces accurate results for as much as one aperture diameter behind the aperture plane. The wavefront diffraction integral was developed for both the classical paraxial and the recently developed exact solutions to the scalar wave equation for a Gaussian beam. Detailed comparisons of these two diffraction solutions show that they are essentially identical for the typical laboratory laser. The typical laboratory laser is defined as having a wavelength in the near-infrared-through-visible range, a beam diameter as large as several millimeters, and a beam divergence angle as large as several milliradians.