ABSTRACT
The need for efficient methods to compute integrals of the form has long been recognised. Hopkins (1957) devised a method by which the integration domain is divided into rectangular subdomains, in each of which the phase function is expanded in a Taylor series about the midpoint. By discarding second- and higher-order terms in this Taylor series, and letting the amplitude function be equal to its value at the midpoint, we can express the integral over each subdomain in terms of a simple formula, and find the value of the integral by adding the contributions from all subdomains. A slightly different scheme has been proposed by Ludwig (1968). He linearises not only the phase but also the amplitude. In the case of a two-dimensional diffraction integral there is little to be gained by using second- or higher-order approximations to the phase function, since the resulting approximate integral cannot then be evaluated in terms of known functions.
