Numerical evaluation of one-dimensional diffraction integrals

An efficient technique is described for performing one-dimensional (1-D) integrals of oscillatory complex functions, such as might arise in the numerical solution of diffraction problems. The upper integration limit may be infinite, provided that the phase of the integrand varies rapidly beyond some point. The integration is done in segments chosen so that both the phase and amplitude may be approximated as cubic polynomials on each segment, with a small cubic term for phase, and continued to infinity from the last segment. Three methods are given for integrating at each step: 1) an asymptotic-series method; 2) an “almost-quadratic-phase” method; and 3) an “almost-linear-phase” method. Criteria are given for choosing among the methods. An example calculation verifies the method in the special case of wave diffraction over two knife edges