7014 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 67, NO. 11, NOVEMBER 2019 Astigmatic Gaussian Beam Scattering by a PEC Wedge Ram Tuvi and Timor Melamed , Senior Member, IEEE Abstract—We are concerned with the scattering of a 3-D time harmonic astigmatic Gaussian beam from a perfectly electric conducting wedge. The incident wave object serves as the wave propagator of the phase-space beam summation method, which is a general framework for analyzing propagation of scalar and electromagnetic fields from extended sources. We perform asymptotic analysis for the total field including fields in the transition regions identifying the corresponding wave phenomena such as reflected beams and diffraction beams. Index Terms— Asymptotic analysis, Gaussian beams (GBs), wedge diffraction. I. I NTRODUCTION B EAM methods has been intensively investigated in the past few decades due to the advantages arising from the mutual spatial-directional locality of the beam wave objects. These wave objects serve as a basis for decomposing scalar and EM fields into a set of Gaussian beams propagators (GBPs) that are emanating over a discrete (phase-space) grid of locations and directions [1]–[3]. By applying locality con- siderations for some generic scattering problem, the scattered field due to each GBP can be obtained for a wide class of canonical problems [4]–[17]. By using such a library of canonical solutions, the scattering of a generic EM field from a complex scatterer can be obtained asymptotically by summing over each local interaction. Such a canonical problem is the scattering from a PEC wedge that is investigated here and presented in Fig. 1. The diffraction of plane waves (PWs) by wedges has been explored in the literature [18], [19]. Uniform asymptotic solutions were derived in [20] and [21], where the dyadic diffraction coefficients were obtained in the form of Fresnel integrals (for a review of several asymptotic techniques in connection with electromagnetic wave scattering from wedges please refer to [22]). Beam summation representation of the edge field of a half- plane due to a PW was obtained in [23]. In [24] and [25] a 2-D Gaussian beam (GB) scattering and a 3-D GB scattering were investigated, respectively. In this method, the diffraction Manuscript received April 13, 2018; revised June 8, 2019; accepted June 15, 2019. Date of publication July 9, 2019; date of current version October 29, 2019. (Corresponding author: Timor Melamed.) The authors are with the School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 8410501, Israel (e-mail: timormel@bgu.ac.il). Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2019.2925928 Fig. 1. Physical configuration. A GBP is impinging on a PEC wedge. The GBP is emanating from the point ( y, z ) = ( ¯ y, ¯ z) over the plane x =¯ x . The scattering object is a PEC wedge with a head angle of α. field is represented by a sum of GBs, which propagate in all directions. Other publications involving beam methods and wedge scattering can be found in [26]–[28]. II. PROBLEM DEFINITION We briefly review here the main results of the EM frame- based beam decomposition, which was introduced in [2]. The frame-based beam summation for propagating an EM field from the x =¯ x aperture is constructed on a discrete frame spatial–spectral lattice ( ¯ y , ¯ z , ¯ κ y , ¯ κ z ). The lattice unit-cell dimensions satisfy 1 ¯ y 1 ¯ κ y = 2πν/ k , 1¯ z 1 ¯ κ z = 2πν/ k (1) where 0 ≤ ν ≤ 1 is the overcompleteness (or oversampling) parameter. The lattice is overcomplete for ν< 1, critically complete in the Gabor limit ν ↑ 1, and for ν ↓ 0, the discrete parametrization attains the continuity limit. The EM aperture field E 0 ( y , z ) over the plane x =¯ x is propagated into x > ¯ x via a summation over GBPs. These GBPs are emanating from the frame lattice set of points over the aperture plane ( ¯ y , ¯ z ) and in the discrete set of directions that is determined by the spectral tilts ( ¯ κ y , ¯ κ z ). The expansion coefficients a y and a z for each point over the frame lattice are obtained by the inner product of the aperture field with the so-called dual frame, via a y ˆ y + a z ˆ z =  dydz E 0 ( y , z )ϕ ∗ ( y -¯ y , z -¯ z ) × exp[- jk ( ¯ κ y ( y -¯ y ) +¯ κ z (z -¯ z ))] (2) where under the framework of Gaussian window frames ϕ( y , z ) = (-ν 2 k Im0/π) exp [- jk 0( y 2 + z 2 )/2]. (3) 0018-926X © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.