Some reflections on ray methods

Summary form only given. This article begins with an account of how Professor Robert E. Collin's work has impacted on the author's own teaching and research. Particular note is made of his contributions to dyadic Green's functions and to his book Field Theory of Guided Waves. It then moves on to a description of how an important discovery impacted on the author's research. In a paper by Keller (1953) edge and vertex diffracted rays were introduced using a generalization of Fermat's principle. But no expressions were given for the fields which propagate along the ray paths, so it was impossible to apply this intriguing method at the time. However, a few years later Keller published expressions for the fields of rays diffracted from a wedge in the scalar acoustic case and in the electromagnetic case when the wedge is perfectly conducting. He then referred to the method as the geometrical theory of diffraction (GTD). His expressions for the diffraction coefficients were accurate away from the shadow and reflection boundaries, but they failed at and near these boundaries. Clearly, new expressions were needed and they could be found from the solution of the canonical problem only if uniform asymptotic methods were used. The author and his graduate students derived uniform diffraction coefficients for plane, cylindrical, conical and spherical wave illumination of the wedge.