Uniform asymptotics of paraxial boundary diffraction waves

Abstract

Starting from the paraxial formulation of the boundary-diffracted-wave theory proposed by Hannay [J. Mod. Opt. 47, 121-124 (2000)] and exploiting its intrinsic geometrical character, we rediscover some classical results of Fresnel diffraction theory, valid for "large" hard-edge apertures, within a somewhat unorthodox perspective. In this way, a geometrical interpretation of the Schwarzchild uniform asymptotics of the paraxially diffracted wavefield by circular apertures [K. Schwarzschild, Sitzb. München Akad. Wiss. Math.-Phys. Kl. 28, 271-294 (1898)] is given and later generalized to deal with arbitrarily shaped apertures with smooth boundaries. A quantitative exploration is then carried out, with the language of catastrophe optics, about the diffraction patterns produced within the geometrical shadow by opaque elliptic disks under plane wave illumination. In particular, the role of the ellipse's evolute as a geometrical caustic of the diffraction pattern is emphasized through an intuitive interpretation of the underlying saddle coalescing mechanism, obtained by suitably visualizing the saddle topology changes induced by letting the observation point move along the ellipse's major axis.