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.

Sharp-edge diffraction under Gaussian illumination: a paraxial revisitation of Miyamoto-Wolf’s theory

“…The application of the plane wave paraxial BDW theory has already produced some interesting results. An important connection with catastrophe optics has already been pointed out in [12][13][14], where it is also shown how to build up analytical estimates of the diffracted field at observation points which are located in the neighborhood of the field singularities. The latter are the geometrical boundary shadow and the caustics produced at the geometrical evolute of the projection of the diffracting aperture on the transverse observation plane.…”

“…First of all it is worth recalling Hannay's formulation of paraxial BDW theory under plane wave illumination following the formulation and notations used in [12,13].…”

“…On further invoking paraxial propagation, the derivation of the BDW decomposition can be achieved starting from the two-dimensional Fresnel in a way that almost resembles an academic exercise, as first shown by Hannay [11]. Quite recently [12][13][14], Hannay's formulation has been used as an effective starting point for revisiting Fresnel's diffraction theory within the fairly new theoretical framework of the so-called catastrophe optics [15,16].…”

“…A possible route to develop a more manageable theory is to invoke paraxial approximation from the beginning. This has already been done in [11][12][13][14] as far as the Young-Maggi-Rubinowicz theory is concerned, for which the mathematical formulation considerably simplifies within paraxial approximation.…”

“…This is exactly the aim of the present work, which is structured as follows: in Sec. II a brief résumé of the paraxial version of the Young-Maggi-Rubinowicz BDW theory developed according to the prescriptions given in [11][12][13][14] is given. This will help readers to familiarize with the geometry of the problem and the main notations used throughout the paper.…”