Elektromagnetik dalgaların iki parçalı ince dielektrik tabakalardan kırınımı

Abstract

Bu çalışmada, sonsuz geniş homojen bir uzay içerisinde bulunan iki parçalı bir dielektrik tabakadan elektromagnetik dalgaların saçılımı ayrıntılı olarak incelenmiştir. Göz önüne alman iki parçalı dielektrik tabaka yaklaşık sınır koşullan yardımıyla modellenmiştir. Bu model, iki parçalı hale ilişkin karma sınır değer problemini bir matrisel Wiener-Hopf denklemine indirgemektedir. Sözü edilen Wiener-Hopf sistemi, çekirdeği Daniele-Khrapkov yöntemi ile faktörize edilerek kesin olarak çözülmüştür. Bulunan bu çözüme dayanılarak alanın yüksek frekanslardaki asimptotik analizi yapılmıştır. Probleme ilişkin değişik parametrelerin saçılan alan üzerindeki etkisini açığa çıkarmak amacıyla bir takım sayısal uygulamalar da yapılmıştır.

Introduction The present work deals with the diffraction of a line source field by two-part thin transmissive slab. In this work the slab is simulated by a material plane with a set of approximate boundary conditions. These approximate boundary conditions are used by Rawlins and et al. [1] to treat the diffraction of a line source field by an acoustically transmissive half-plane. After simulating the dielectric slab by a material sheet with approximate boundary conditions, the problem is reduced to the solution of a 'matrix Wiener-Hopf equation' which can be treated by using the Daniele-Khrapkov method. 2. Formulation of The Problem The basic geometrical configuration considered in the present work is illustrated in figure- la. The problem consists in studying the line source field diffraction by the junction O of the two-part thin transmissive slab. In order to this end one considers an equivalent two-part material plane illuminated by a time harmonic line source with time dependence e~,wt and located at x - x0, y = yo>0, z e (-Qo, od) (see figure- lb). For the sake of analytical convenience, the total electrical field is expressed as follows: u(x,y) = u,{x,y),y>y0 u2(x,y),0<yy) >y<0 (o) (b) (1) Figure 1. a) Two-part dielectric slab b) Two-part material plane For uB(x,y) (n=l,2,3) which satisfy the Helrnholtz equatioa + jL+kAuH(x,y) = 0,« = U,3 Kâc2 + dy (2) where k is the free-space wave number, which is temporarily allowed to have a small imaginary part for convenience. It is appropriate to consider the following integral representation. >y>y<> u,(x,y)=\A{ayK[a)y~taxda -oo u2(x,y) = ][B{aYK[a)y + C{a)e~^yYia*da,0<yy0)--ru2(x>yo)= -mzs(x-x0) (5.a) (5.b) Im(a) ^ - L~ Re (a) t: Figure 2. Brunch-cuts and the intergatration line in the complex a ~ plane Here / and Z denote the current of the source and the free-space wave impedance, respectively. The following approximate boundary conditions simulating the two-part material plane aty=0 [I] are given by, M2(x,0) = o-i«3(jr,0) - «*(*,(>) =T,-U,(*,0) u2(x,0) = 0,x>0 (5.c) (5.d) (5.e) (5-f) vn In (5c-f) a j, Tj j=J,2 stand for Q (7a) In (6.a,b) X stands for with X = &L 1 1 = 2^'°g 1-g Kl + s. (7.b) (7.c) Ö - sign\ arg 1-g \ + s. (7.d) and vni %.+r»X*.+»*) From (7.b), (7.c) and (7.d) one concludes that 0<re(a)<="" im(£)="" g+(a),="" h+(a),="" h+(a)="" upper="" style="margin: 0px; padding: 0px; outline: 0px;"> Im(-£). By using the edge condition given by (7.a) one can easily show that when |a| -» oo in their respective regions of regularity. From (lOa-d) and (3.b,c) one can write G~ (a) + G+ (a) = B(a) + C(a) (12a) H(a) + H+(a) = D(a) (12.b) Ğ' (a) + Ğ+ (a) = iK(a)[B(a) - C(a)\ (12.c) H~ (a) + H+ (a) = -iK(a)D(a) (12.d) The elimination of B(a) and D{a) among (12.a-d) and the use of the relations obtained in (9a-d) yield the following " Matrix Wiener-Hopf Equation" (MWHE) which is valid in the strip Im(-£) < Im(a) < Im(£) : P+ (a) + M(a)P" (a) = F(a) (13.a) where **(«) = <**(«)' [**(«). (13.b) M(a) = ? - - (^2+^2) (a2 + r,) ^ t2 (a2 - a^i^a) 0~2 ~ ?"l) 1 xx iKypc) (13.c) and F(«) = -2 7* C(a) V (0-2+^2) V iK(a) (13) XI with C(a) being given in (8c). In order to obtain an explicit solution of (13. a), firstly one has to factorize the Kernel matrice M(a) given in (13.d) as the product of two invertible matrices, say M+(a) and M~(a), whose entries are regular functions of a with algebraic behavior for j«| -> oo in the upper and lower half-planes, respectively. To this end one writes M(a)-M+(a) M~(a) (14.a) C = rı (