Elektromagnetik dalgaların bir yüzü mükemmel iletken bir düzlemle sonlandırılmış dielektrik tabaka üzerine yerleştirilmiş bir mükemmel iletken şeritten kırınımı

Abstract

Bu çalışmada mükemmel iletken bir düzlem üzerine konuşlandırılmış bir dielektrik tabaka ile yarı sonsuz başka bir dielektrik ortamın arakesit yüzeyine yerleştirilmiş mükemmel iletken bir şeritten düzlemsel elektromagnetik dalgaların kırınımı incelenmiştir. Problemin çözümündeki Fourier gösterilimi problemi önce modifıye Wiener- Hopf denklemine indirgemiş sonra bazı dönüşümlerle ikinci tip küple Fredholm integral denklem sistemi iteratif bir yöntemle çözülmüştür. Şeridin boyutlarının ve dielektrik tabaka kalınlığının kırınıma etkisi sayısal örneklerle grafik olarak verilmiştir.

1. Introduction: The problem of diffraction by strips and slits in composite media is an important subject in diffraction theory and is relevant to several engineering applications. An emerging interest arises in this class of problems since there exist dielectric/ferrite materials in many aerospace structures in the form of composites, although electromagnetic diffraction by strips and slits has been extensively studied by several authors following different analytical and numerical methods, almost all works are limited to the case where the strips and slits are located in a homogeneous medium. Taking into acount the realistic situation, there is a need to extend the problem of edge diffraction from strips and slits in a homogeneous medium to develop a new approach for the diffraction by strips and slits located in the proximity of medium discontinuties. In this paper the plane wave diffraction by a perfectly conducting strip located on a dielectric layer on a ground shield is investigated by the Wiener-Hopf technique (see Fig.l) Applying the boundary conditions to the integral represantations for the unknown scattered field, the problem is formulated in terms of the modified Wiener-Hopf equation (MWHE), which is reduced to a pair of simultaneous integral equations via the factorization and decomposition procedure. The method of successive approximations is then applied, and first and second order solutions correspond to the singly and doubly diffracted fields are presented. Uj (x.y) *1,H1 P(o.-d) Fig.l. Geometry of the problem. VI Finally, some numerical results concerning the variation of the diffracted field versus the observation angle are presented for different values of dielectric thickness and the slit width. 2. Formulation of the Problem: Let a strip (see Fig.l) be illimunated by a plane wave. y>0 and ye(-d,0) regions are characterized by the relative permittivity and permeability e" ^ and e2, H2 respectively, while y=0 is a perfectly conductor. The wave number are denoted by kx=oiJe^[ and fcj=o>^e2n2. In the following analysis we will consider the case The total electric field uT(x,y) be u£w)=uptey)+uj&y) (1) where up(x,y) is the primary field, for the case where the strip removed from geometry is defined by u(xy)4Ui(X'y)+RUi(X'~y) iy>0 (2) with ufay) =exp{-ik1(xcos$Q +ysin(|)û)} "t(*>y) =exp{-i£2(xcos(|)1 +ysin4>i)^ (3) R= ^x ° 1 (4) |A2ifelSm4)0(l-eö*^*1)+|i1^sİIı4>1(l+ei2*i*in,,,1) T_ 1*R T_-(UR)eik^^i In equation (3) 4>\ is the transmission angle defined by Iqcos^^ k2cos<^1, while b0 stands for the incidence angle, u,(x,y) is the field due the existence of the strip. Vll For the sake of analytical convenience, we will consider x^ as follows, \uXxfy) ; y>0 M^H«2(x,y); -d<y In the above expressions un(n=l,2) satisfy the Helmholtz equation A""^X=0 (7) For convenience of analysis, we assume the media, to be slightly lossy, as in k^Reik^+Hmik^ with (Klmik^ImlkJ. In view of the radiation condition un(x,y) are expressed as, oo+fc «1(x,y)= / A{a)e~K'{a)ye-iaxda (8a) -m+İC "+İC "2(*o0= f [BWe'^'^+Ctoe^^e-^da (8b) Kn(a)=f^tf, Kn{0>-ikn (9) 2.1. Boundary Condition and Wiener-Hopf Equation: In order to determine the unknown coefficients A(a), B(a) and C(a) in (8 a,b) one has to take into account the boundary and continuity relations which read u^xfiyu^xfi^O; xe(-°o,«>) (10a) UiOcOy-a+WufaO); xe(0,l) (10b) 1 ök, i 3u, 1-(pc,0)~-±(x,0)=0; xe(-oofl)U(l,°o) (10c) V-i ty V-2 ty Vlll uJx,-d)=0; *e(-oo,oo) (10d) In addition to the above equations to secure the uniqueness, following edge conditions must hold Using (10 a,d) one gets rather easily; with v ' +$ (a)+e,a<<&+(a)=- M(a) 2iti a -^cos^ M(a)=- +- ^ Hj n2 shxhK2d (11) (12) (13) $+(a)=_LjUl(x,0)c'o(x-() die (14a) $_(«)=- [u2(xy0)eiaxdx (14b) T-tJ i 3k, i k Hi 3y Hj dy e'***** (15) Equation (12) constitute a Modified Wiener-Hopf equation which permit us to solve P(a) and $,(a). 3. Solution of the Wiener-Hopf Equation: In order to have a Wiener-Hopf factorization of M(a) appearing in (13) in the form, M(a)=M+(a)M_(a) we can first write, (16) IX A/(a)=M1(a)M2(a) M^a) and M2(a) are given by (17) M1(a)=^^K1(a) MM= ^2 H2 ^ KAtii+e"*** Hi+H2 li1+n2lTI(a)i_e2l!tf«* (18a) (18b) Mi (a) is factorized easily as follows M1(«)=M1 (a)Mt (a) (19a) M1t(«) = Hi +H2 \ HiH2 Ja^-k^ (19b) In equation (18b) M2(a)-1 as |a|-«>. Thus we can factorize M2(a) by Cauchy Integrals as follows M2(a)=M2(a)M2(«) (20a) M2 (