Diffraction calculations in the near field and the validity of the Fresnel approximation

The general theory of the diffraction and propagation of light is considered. Particular attention is paid to the case of uniformly illuminated rectangular or circular apertures, for which the derivative of the field given by the Fresnel approximation is evaluated exactly. The limitations of the Fresnel approximation are considered, and thus a useful, simple way of understanding its validity is produced. This treatment yields an estimate of the accuracy of the Fresnel approximation in the general case and a method by which a modified Fresnel treatment can be used over a substantially extended region.