Uniform asymptotic theory of diffraction by apertures

Simple and explicit formulas that represent uniform asymptotic approximations to double integrals are derived and applied to obtain a uniform asymptotic theory of diffraction by apertures. The formulas are used to evaluate the diffracted field or its angular spectrum in regions in which the results of ordinary asymptotic theories, such as the theory of the boundary diffracted wave and the geometrical theory of diffraction, break down. Thus our results remain valid in regions containing geometrical shadow boundaries or caustics of diffracted rays. To illustrate their general applicability, we apply them to two diffraction problems involving a circular aperture: For a diverging spherical incident wave we compute its diffracted field, and for a converging incident wave we compute both its angular spectrum and its diffracted field far from focus.