A MAT LAB-Based Vi rt u a l Too l fo r the
E l ectro m ag n et i c Wave S catte ri n g from a P e rfectly Refl ect i n g Wed g e Feray Haclvelioglu1,2, M. Alper Uslu1, and Levent Sevgj1
1 Dogw;; U n iversity, Electronics and Communications Engineeri ng Department Zeamet Sokak 2 1 , ACl badem - Kad l köy, 34722 I sta nbu l , Turkey E-mail:
[email protected] .tr 2Department of Mathematics , Faculty of Science Gebze I n stitute of Technology, Kocael i , Tu rkey
Abstract
A MATLAB-based virtual d iffraction tool, using analytical exact as wei l as wel l-known high-frequency asymptotic ( H FA) tech niques, is i ntrod uced . Electromagnetic wave scattering from a wedge-shaped object with perfect electrical conductor (PEC) bou ndaries, under both line-sou rce and plane-wave illuminations, can be a utomatically analyzed . Effects of various parameters on the reflection , refractio n , and d iffraction can be i nvestigated . Comparisons among Geometrical Optics (GO), U n iform Theory of Diffraction (UTD), Physical Optics (PO), Physical Theory of Diffraction (PTD), and Parabolic Equation (PE) models are possible through many scenarios and illustrations. Keywords: Electromag netic scattering; d iffractio n ; Geometrical Optics; g raphical user interface (G U I ); exact series ; exact i ntegrals; MATLAB si mu lations; Parabolic Equation Method ; Physical Optics ; Physical Theory of Diffraction; h igh-frequency asym ptotics; nu merical i nteg ration ; reflectio n ; refractio n ; shadow d iffractio n ; U n iform Theory of D iffractio n ; wedge d iffraction
1 . l ntrod u ction
tially investigated for many decades. It was recently E reviewed with a tutorial on the canonical, two-dimensional lectromagnetic (EM) wave scattering has been substan
(2D) wedge problem in [ 1 ] . The numerical difficulties of vari ous analytical models based on complex integration as weIl as series summation were also discussed in detail in [2] . Com parisons with numerical techniques, such as the Finite-Differ ence Time-Domain (FDTD) Method, were given in [3] . In aIl of these studies, wave aspects such as reflection, refraction, and diffraction, which are the components of scattering, were revisited through analytical exact as weIl as high-frequency asymptotic (HFA) methods, such as Geometrical Optics (GO); the Geometrical Theory of Diffraction (GTD); its uniform extension, the Uniform Theory of Diffraction (UTD); Physical Optics (PO); the Physical Theory of Diffraction (PTD); ele mentary edge waves (EEW); and the Parabolic Equation (PE) Method [4-20] . In this paper, a .MATLAB-based virtual tool, WedgeGUI, for EM wave scattering from a perfectly electrical conducting 234
(PEC) wedge is introduced in two dimensions. First, the problem is posed briefty, and critical wave regions are outlined. Mathematical equations for both plane-wave and line-source illuminations are then presented, for the sake of completeness. There are many different forms of these models, but the forms presented here are the most numerically efficient forms. FinaIly, the virtual tool WedgeGUI is discussed, together with some examples. The non-penetrable wedge-diffraction problem is canoni cal. It plays a fundamental role in the construction of high frequency-asymptotic techniques, as weIl as for numerical tests. The exact solution to this scattering problem was first obtained by Sommerfeld [4] in the particular case of a half-plane. For a wedge with an arbitrary angle between its faces, the solution was obtained by Macdonald [5], and later on by Sommerfeld, who developed the method of branched wave functions [6] . The two-dimensional wedge-scattering scenario is pic tured in Figure 1 . The semi-infinite wedge with PEC boundaries is located in vacuum. The polar coordinates r, rp, z are used throughout the paper. The z axis is aligned along the edge ofthe
IEEE Antennas and Propagation Magazine, Vol. 53, No. 6, December 20 1 1
wedge. The angle rp is measured from the top face of the wedge. The exterior angle of the wedge equals a . The wedge is illuminated by a line source (LS) at a distance ro from a direction rp rpo . In other words, the source and observation points are given by ( ro , rpo ) and ( r , rp ), respectively. =
The scenario for the first line-source illumination, LS- 1 ( o < rpo < lL ), belongs to single-sided illumination (SSI), where the top face is always illuminated. In this case, the two dimensional scattering plane around the wedge may be divided into three regions, in terms of critical wave phenomena occurring there. In Region I ( 0 ::; rp < lL - rpo ), all the field components - incident field, reftected field, and diffracted field - exist. The angle rp lL - rpo is the limiting boundary of the reftected fields and Region I (reflection shadow boundary: RSB). In Region 11 ( lL - rpo ::; rp < lL + rpo ), only incident and diffracted . fields exist. The angle rp lL + rpo is the limiting boundary of the incident field and Region 11 (incident shadow boundary: ISB). In Region III (i.e., in the shadow region, lL + rpo ::; rp ::; a ), only diffracted fields exist. =
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