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Elektromagnetik dalgaların iki parçalı ince dielektrik tabakalardan kırınımı

Başlık çevirisi mevcut değil.

  1. Tez No: 55881
  2. Yazar: M.GÖKSEL AYDEMİR
  3. Danışmanlar: DOÇ.DR. İBRAHİM AKDUMAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Elektromanyetik dalgalar, Kırınım, Electromagnetic waves, Diffraction
  7. Yıl: 1996
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET 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.

Özet (Çeviri)

SUMMARY DIFFRACTION OF A LINE SOURCE FIELD BY A TWO-PART THIN DIELECTRIC SLAB 1. 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<y<y0 Mx>y) >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<y<y0 -oo 00 *h(x,y) = J l^aY^^da,y<0 (3.a) (3.b) (3.c) with VIK(a) = Jk2-a' (4) The square root function in (4) is defined in the complex cc-plane cut as depicted in figure-2, such that K(0) = k. The unknown spectral coefficients A(a), B(a), C(a) and D(a) appearing in (3.a-c) are determined by using the definition of the source at y = y0 which reads: -j-^(x>y0)--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) = <rzui{xSi) - u,(*,0)=r2-^(x,0),x<0,*<0,x>0,x>0 (5.c) (5.d) (5.e) (5-f) vnIn (5c-f) a j, Tj j=J,2 stand for <Tj = l-2iK/j/Nj (6.a) Tj = \-2iKjtjNj (6.b) with and Kj = kJsjjUj - cos2 ^0 Nj =Kj/(kSjSia4o) (6.c) (6-d) In order to obtain unique solution it is also necessary to take into account the edge condition at *=0. By following a method similar to that described in [1], it is easy to show that u(x,0) = ö(x1},x->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)<I (7.f) By substituting first (3a-b) into (5a-b) and than inverting the resulting integral equations one gets A(a) - B(a) = C(ay2K{a)y° (8.a) A(ay B(a) = - «* e**-*M* - C(a)e“2K^ (8.b) v ' v ' K[a) v ' From (8.a) and (8.b), C(a) can be solved to give C(a) = _J^e<H+*(»W (8c) 2Kl(x) Consider now the Fourier transform of the boundary conditions in (5.c-f) namely G-(a) = alH(a) (9.a) G-{a) = TxH-(a) (9.b) G+(a) = a2H+(a) (9.c) G+(a)=r2#+(«) (9.d) IXIn the above expressions G±(a), #*(#), G±(a) and #*(#) stand for G-(a)' H-(a) 2nK u2(x,6) u3(x,6) ilaxdx (lO.a) 1 eo tt2(jr,0) m3(jc,0) eto<& (lO.b) /r(«)_ 1 ° 2xt - u2(x,0) e,axdx (lO.c) G+(«) 2* i -^u2(x,0) -u2(x,6) e,cadx (lO.d) In accordance with the analytical properties of Fourier integrals, the functions G~(a), H~(a), G~(a) and H~{a) are regular in the lower half-plane Im(ör) < Im(£) and the functions G+(a), H+(a), G+(a) and H+(a) are regular in the upper half-plane Im(ör) > Im(-£). By using the edge condition given by (7.a) one can easily show thatwhen |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) XIwith 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 (<r2 + r2 ) 0 0 a2+r2 (14.b) and W(a) = i Q-2-Q 1 tx (er, +T2)iK(a) 1 (14.c) The matrix W(or) given in (14c) belongs to the class for which the Wiener-Hopf factorization can be accomplished through the Daniele-Khrapkov method which can be factorized immediately to give Xll1 cosh^«) -iTlbK(a)swh.%(a) iswhz(a) ( v - / / cosh4«) TybKya) x ' (15.8) and W-(a)=W+(-<z) (15.b) where e is given by (7.e), while b stands for (15.c) In (15.a,b) %(a) and x(-<*) are given by Aa) - -Uarccos- A: (16.a) z{-a) = -^ ^ - arccos- V kJ (16.b) with X defined by (7.c). Furthermore, by considering the known asymptotics it can be easily shown that one has W+(a) = ax Tipda 1+1 rfi -ax~l ax (17) for \a\ -> oo in the upper and lower half-planes, respectively. Now by using the well-known factorization technique, MMWHE obtained in (13.a) can be reduced to the following pair of equations xm[cw»]vm=s7 n^wr'S^ +p (18.a) F(r) w(«)*-(«)-5^[cw-(r)]-il2*., (18b) Here /? appearing in (18a) and (18b), is an unknown constant resulting from the application of the Liouville' s theorem in the Wiener-Hopf procedure. In (18.a,b) P+(a) and P~(#) are regular in the upper Im(«) > Im(-£) and lower Im(<z) < Im(£) half-planes, respectively and the positions of the integration lines Z* are indicated in figure 2. From (18a) P~(#) can be solved as p-(a) = [w-(a)]”1(a)- p (19.a) where 1(a) is the integral which is evaluated asymptotically through the saddle- point method to give l(a) = h(a) I2(a\ -~n*AMz_ T2 = e [CW+(^cos^0)]_1 *$2n (a2 + r2) a - £cos^0 -/£sin^0 1 Jkp* -jkpo klZt, (<?2 + *l) K(4l 1 xK{a\ e^a^***]*^H[ln(a - k cos^o)] (19.b) In order to determine the constant p one has to consider (ll.a-h), (17) in (18.a-b) for \a\ -> oo, which yields p = -&vxb. lim a. I2 (a) (20) XIVThe substitution of (9a-d) in (12a-d) gives the expressions of the spectral coefficients B(a), D(a) in terms of P(a) as follows: Bd(a) PA"). (<T2 + T2) irxK(a) (ri-Q j(o-!-o-2) cr2 -<x, P;(a) (21) By using these expressions of B(a) and Dfieç), the diffracted field by the junction O can easily be determined by evaluated (3.b,c) through the saddle-point technique as follows: (22.a) Dx(M0) = jîn -in/ {°2 +72) (T2{rx-T2)P^(-kcos IT, 4- A: sin 0t2 (al - a 2 )Pdi (-£ cos ^) (22.b) D2(<f>J0) = yl2x in/ /4 (o-2 + *%) t.-^KHcos *7, + A: sin ^(cr, - er2 )Prf~ (~k cos ^) (22.c) XV

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