Converging and Diverging Beam Diffraction by a Wedge: A Complex-Source Formulation and Alternative Solutions

In a previous publication, the problem of 2-D beam diffraction by a wedge has been solved via the complex-source (CS) approach. However, the straightforward CS formulation may be applied only when the incident beam is diverging as it hits the edge, but not when it is converging as it hits the wedge. In this paper, we generalize the CS setup, so that it can address both problems. The surprising result is that the CS approach can be applied for the converging beam case, but only if the CS coordinates are defined in a specific fashion. We then formulate the angular harmonics and the spectral integral representations for both cases, and also derive uniform asymptotic expressions for beam diffraction by a wedge. The validity of the results is verified by calculating the diffracted field via each one of these formulations, and comparing them with yet another approach, wherein the field of the incident diverging or converging beam is synthesized using a plane-wave integral, and the diffracted field is then calculated via multipole expansion. The overall goal of this paper is the derivation of techniques for the analysis of 3-D beam diffraction by a cone.

I. Introduction

The CS method provides a canonical setting for the rigorous study of beam diffraction phenomena in the frequency or in the time domains [1]–[11]. In [12], we have used this approach for the analysis of 2-D beam diffraction by a wedge as a function of the beam parameters: collimation, direction, and displacement from the edge. The emphasis in [12] has been placed on exploring various analytical techniques, with a specific goal of extending them to a 3-D cone diffraction problem (see initial results in [13]–[16]). However, as has been alluded to in [12], the straightforward CS formulation applies only for the case where the incident beam is diverging as it hits the edge but not in the case where the incident beam is converging as it hits the edge [see Fig. 1(a) and (b), respectively]. This paper explores the application of the CS method to these problems. Although we consider a 2-D configuration, the results derived can be generalized to the 3-D problems of beam diffraction by a wedge and by a cone [14]–[16]. Fig. 1.

(a) A diverging incident beam (DIB) versus (b) a converging incident beam (CIB).