SCATTERING FROM THE JUNCTION OF A PERFECTLY CONDUCTING HALF-PLANE AND A RESISTIVE SHEET Giuseppe Pelosi, 1 Stefano Selleri, 1 and Giuliano Manara 2 1 Department of Electronic Engineering University of Florence I-50134 Florence, Italy 2 Department of Information Engineering University of Pisa I-56126 Pisa, Italy Recei ed 28 No ember 1997 ABSTRACT: A numerical analysis of plane-wa e scattering from the edge of a penetrable wedge illuminated at normal incidence is presented. The wedge is formed by a perfectly conducting half-plane and a resisti e sheet joining at an arbitrary angle. The diffracted field is determined through the application of the parabolic equation method in conjunction ( ) with a finite-difference FD scheme. The solution obtained is alid in the high-frequency region. 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 18: 8586, 1998. Key words: wa e scattering; numerical analysis; electromagnetics Ž . The finite-difference FD technique has recently been ap- plied in the framework of the parabolic equation method for determining a uniform high-frequency solution for plane-wave scattering from impedance wedges 1 3 . The procedure is extended for the first time in this letter to determine the scattering from an angled junction between a perfectly con- ducting half-plane and a resistive sheet. Reference solutions for this latter case are available only for planar junctions 4 . The geometry for the scattering problem is depicted in Figure 1. The edge of the resistive wedge is along the z-axis, and a harmonic plane wave impinges on the edge from a direction determined by the angle '. The observation point Ž . is at P , , and the exterior wedge angle is n . The wedge is obtained by joining along the z-axis a perfectly conducting half-plane and a resistive sheet; they will be denoted by S and S , respectively. Consequently, the space 0 n Ž . is subdivided into an exterior region region 1, 0 n Ž . and an interior region region 2, n 2 . The incident Ž . field is a TM -polarized H 0, e case plane wave imping- z z ing from region 1, 0 ' n . The case TE can be treated z Ž . in a similar way. The total field E , satisfies the z Helmholtz equation and the following boundary conditions at Figure 1 Geometry for the scattering problem the perfectly conducting half-plane and transition condition at the resistive sheet 4 : Ž . Ž. E , 0 1 z 0,2 Ž . Ž . 1 E , 1 E , z z n n Ž . Ž. jk E , 0 2 z n R where R is the resistivity of the sheet, and k and are the wavenumber and the impedance of free space, respectively. We note that the total field can be expressed as the sum of Ž . Ž the geometrical optics GO field incident, reflected, and . transmitted fields which can be determined a priori, and the diffracted field, which is actually the unknown of our prob- lem. The determination of the E go contribution may be z cumbersome in the most general configurations since multi- ple reflections may take place in region 2. However, it is important to point out that the number of internal reflections is always finite, and the locations of the corresponding shadow boundaries can be determined through simple geometrical considerations. In order to simplify the explanation, we will make explicit reference in the following to a penetrable wedge with a generic aperture, choosing the incident plane wave so that both faces bounding region 1 are directly illumi- Ž . nated Fig. 1 . This guarantees the absence of internal reflec- tions; two different reflected field contributions originate from the faces of the wedge in region 1, as well as a transmitted field term in region 2 through the penetrable Ž . sheet. In the above hypothesis, by defining f Ž . go Ž . Ž . Ž exp jk cos , we obtain: E , f ' f z . go Ž . Ž . ' for 0 ; E , f ' for 1 z 1 go Ž . Ž . Ž Ž . . Ž ; E , f ' R, ' n 1 f 2 z Ž . .. go Ž . Ž 2 n 1 ' for ; E , R, ' 2 3 z Ž . .Ž . go Ž . n 1 f ' for ; and E , 0 3 4 z Ž . for 2 . In particular, R, sin 1 4 Ž . Ž . Ž . Ž . 2 R sin and R, 2 R 1 2 R sin are the reflection and transmission coefficients for the resis- tive sheet at n . Moreover, ' denotes the 1 shadow boundary for the field reflected from the 0 face Ž . Ž . RSB ; 2 n 1 ' denotes the shadow boundary 1 2 Ž . for the field reflected from the n face RSB ; 2 3 n ; and finally, ' identifies the shadow boundary 4 Ž . for the transmitted field TSB . Ž . By introducing a scalar auxiliary function U , so that d Ž . Ž . Ž . E , U , exp jk and by assuming k 1, that z is, by resorting to a high-frequency approximation, the term involving the second derivative in of the Helmholtz equa- tion in cylindrical coordinates can be neglected. This process transforms the elliptic Helmholtz equation into the parabolic Ž . equation for the function U , . As pointed out previously 2 , the solution of this equation tends to the solution of the Helmholtz equation as the distance of the observation point Ž . from the edge of the wedge increases k 1. Ž . If the value of U , at 0 is known, the parabolic Ž equation can be solved in both open domains regions 1 and . 2 by a marching in space FD procedure. This value can be determined through the same procedure used in 1 3 . In Ž . u Ž . particular, U 0, B for 0 ; U 0, 1 u u Ž . u 2 u B for ; U 0, B Ý 1 1 2 i1 i Ž . u 3 u for ; U 0, B Ý for 2 3 i1 i 3 Ž . u 4 u ; and U 0, B Ý for 2 . 4 i1 i 4 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 18, No. 2, June 5 1998 85