Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams

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 wave-front 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.