The method of stationary phase applied to the boundary wave formulation of Kirchhoff's diffraction theory can be used to approximate the diffracted field from a curved edge that is illuminated by a spherical wave. The resulting equations for the asymptotic field are very simple (only basic trigono-metric functions and square roots are involved), but diverge when the observation point approaches the edge, the geometrical shadow boundary or the focus of the incident beam. These singularities can be removed if additional terms are introduced that force the asymptotic field expression to match the known values for the actual field in these regions. The resulting approximate expression for the total field is continuous and finite throughout space even though it is conveniently expressed as the sum of two discontinuous fields, a geometrical optics component and an edge diffraction component. Numerical calculations indicate that this formulation agrees remarkably well with exact "closed" form solutions that require the evaluation of complex Fresnel integral functions. In addition, the method can be directly applied with little modification to problems involving Gaussian beam illumination or multiple edges (e.g. slits and circular apertures).
Alan W. Greynolds,
"Simple Formulas For Calculating Near-Field Diffraction Profiles", Proc. SPIE 0818, Current Developments in Optical Engineering II, (1 January 1987); doi: 10.1117/12.967437; https://doi.org/10.1117/12.967437