Asymptotic Evaluation of the Field at a Caustic (Classic Reprint)

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Excerpt from Asymptotic Evaluation of the Field at a Caustic The electromagaetic field which can he constructed by the methods of geometrical optics, following R. K, Luneburg, becomes infinite at caustics of the optical ray systems. In this paper we overcome this difficulty and obtain a finite value for the field at a caustic which is produced by the reflection of any incident wave from an arbitrary reflector (all in two dimensions for simplicity). We have exemplified these general results by considering the special cases of a plane wave incident on the concave sides of a segment of a parabolic cylinder and a segment of a circular cylinder. In the parabolic case, with incidence along the axis of the parabola, the entire caustic is a point, the focus of the parabola. The asymptotic behavior of the field at the focus is obtained explicitly and transition formulae which show how the field varies in the neighborhood of the focus are also obtained. These formulae involve integrals which we have evaluated numerically in some special cases. In the case of the circular cylinder, the caustic is a curve with a cusp. The asymptotic behavior at the cusp is different from that on the rest of the caustic. The variation of the field near the caustic is described in terms of Hardy's generalized Airy functions. Since our results yield exactly the leading term in the asymptotic expansion of the field, they constitute a check on the solutions proposed by Debye Picht and Luneburg. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.