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Elektromagnetik dalgaların çok parçalı ince dielektrik tabakalardan saçılması

Multiple diffraction of a plane wave by a multi-part thin dielectric slab

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  1. Tez No: 39678
  2. Yazar: ALİ ALKUMRU
  3. Danışmanlar: PROF.DR. MİTHAT İDEMEN
  4. Tez Türü: Doktora
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Dalga yayılması, Dielektrikler, Elektromanyetik dalgalar, Wave propagation, Dielectrics, Electromagnetic waves
  7. Yıl: 1994
  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 ve üç parçalı bir dielektrik tabakadan elektromagnetik dalgaların saçılımı ayrıntılı olarak incelenmiştir. Göz önüne alman çok 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 faktorize edilerek kesin olarak çözülmüştür. Bulunan bu çözüme dayanılarak alanın yüksek frekanslardaki asimptotik analizi yapılmıştır, iki parçalı probleme ilişkin çözümün integral ifadesi esas alınarak, Spektral İterasyon Tekniği (SIT) ile, üç parçalı problemin asimptotik çözümü elde edilmiştir. Probleme ilişkin değişik parametrelerin saçılan alan üzerindeki etkisini açığa çıkarmak amacıyla bir takım sayısal uygulamalar da yapılmıştır.

Özet (Çeviri)

SUMMARY MULTIPLE DIFFRACTION OF A PLANE WAVE BY A MULTI-PART THIN DIELECTRIC SLAB 1. Introduction The present work deals with the multiple diffraction of a plane elec tromagnetic wave by a multi-part thin dielectric slab. The aim is to reveal the mechanism of the excitation of the second order terms between two edges. After simulating the dielectric slab by a material sheet with appropriate 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. By considering this result in the three-part dielectric plane problem one gets the terms up to the second order. 2. Formulation of The Two-Part Dielectric Plane Problem The basic geometrical configuration considered in the present paper is shown in figure- la. The problem consists in studying the plane wave diffraction by the junction between the thin dielectric slabs. In order to this end one considers an equivalent two-part material plane illuminated by a linearly polarized plane wave u*(x, y) (see figure-lb), namely: uifx y\ - e-ik(x cos <j>o+y sin (j>0) 0<<f>0<TT. (i) £o.Ho.0< £.11.0 o 'f^o* o (a) *»x (b) Figure- 1 a) Two-part dielectric slab b)Two-part material plane VIIn the case of TM illumination ul(x, y) signifies the Oz-component of the electric field while in the case of TE illumination it is the Oz-component of the magnetic field. Then the total field uT{x, y) satisfies the reduced wave equation 92 +-^ + k2]uT(x,y) = 0, (y^O), (2) dx2 dy2 where k = kr + ik{ is the free space wave-number which is temporarily allowed to have a small imaginary part for convenience of analysis. This equation is subject to the following boundary conditions which involve the constitutive parameters of the slabs shown in figure- la: uT(x, +0) - (JiuT{x, -0) = 0, x < 0 (3.o) - uT(x, +0) - n t- uT(x, -0) = 0, x < 0 (3.6) dy dy uT(x, +0) - a2uT(x, -0) = 0, x > 0 (3.c) - uT{x, +0) - r2 -z-uT(x, -0) = 0, x > 0 (3. d) dy dy <tj = 1- 2iKjtj/Nj, j = 1,2 (4.a) tj = 1- 2iKjtjNj, j = 1,2 (4.6) JCj = kyjejfij - cos2 <?i>o, j = 1,2 (4.c) {K j/(k/j.j sin <f> 0) for TMZ case (4.d) )Cj/(ksj sin <f>o) for Ti?2 case. E one writes uT(x,y) = uu(x,y) + u°{x,y), (5) where «u(x, y), signifies the“unperturbed field”which would be observed if the whole plane y = 0 in figure- la was occupied by a homogeneous slab with parameter (£2,^25*2) while u°(x,y) represents the effect of the material discontinuity resulting from the fact that the actual dielectric layer for {y = 0, x < 0} is characterized by (eı,/fı,tı). It follows from the definition of uu(x, y) that u^x^ + Rie-W*T5**-»*T^, y>0 uu(x,y)={ (6) me-ik(x cos (j>0+y sın M V<0 Vİİ where with andwhere İ22 and T2 are the reflection and transmission coefficients, respec tively: R2 = fl^H, 2İ--H-. (7.a,b) &2+T2 CT2+T2 As to the field u°(x,y), it satisfies also the Helmholtz equation (2) and has an integral representation as follows: fA0(a)e-^a^-iaxda ; y > 0 A*, *) = 4 j- B0(a)e^y-iaxda ; y < 0 c (8.a) with 7(a) = V“2 - k2. (8.6) The square-root function in (8.6) is defined in the complex a-plane cut as shown in figure-2 such that 7(0) = - ik. Im(a) k / kcos0o. - V Re ( a) 0 -? V Figure-2 Branch-cuts and the integration fine in the complex a-plane The integration fine £ in (8.a) is the infinitely long straight line shown in figure-2. The coefficients AQ(a) and B0(a) in (8.a) are to be deter mined with the aid of boundary conditions in (3. a, 6, c, d) which can also be written explicitely as follows: u°{x,+0)-(t1u0(x,-0) = ~2- - ?le-ifc*c°s^ ; x < 0 02 + T2 -u°(x,+0) - Ti-- u°(x,- 0) = 2iksm<f>o e oy oy cr2 + r2 2 1 - ifci COS ^0 « °(x,+0)-o-2u0(a;,-0) = 0 ; a: > 0 - u°(a;,+0)-r2~u0(x,-0) = 0 ; a; > 0. (9.a) x <0 (9.6) (9.c) (9.d) In order to obtain a unique solution to the problem (8.a, 6) and (9. a, 6, c, e?), it is also necessary to take into account the edge condition. By using Meixner's method, it is shown that *°(*.°> = -^+°(>) as x -* 0 (lO.a) Vlllwhere with and with \0=60\0 (10.6) Ao= 1 logf^- ^) (lO.c) So = sign I arg f ° J J- (10-d) (ffı -q-2)(ri -r2),. ao = l/(ffl+r2)(n+.2)- (1°-e) By substituting first (8. a) into (9. a, 6, c, eQ and then inverting the resulting integral equations one gets A0(a) - fflB0(a) = *+ (o) + -. (°”*“ ^ -r (ll.o) 7re (o-2 + t2)(ol - k cos 0O ) -7(a) [A0(a) + riBo(o)] = $&(a) - Efesin <£0 (n ”^ 7T (<r2 +t2)(q! - kcos<t>o) (11.6) ilo(a) - 02#o(a) = $oi(a) (1Lc) -7(a) [A0{a) + r2B0(a)} = $^(a), (ll.d) where <&*x(a) and $^2(0;) are certain unknown functions. What is known about these functions is that $01(0;) an(^ $02 (a) are regular functions of a in the half-plane Im(a) > Im(-k) while $^(0;) and $^2(a;) are regular in Im(a) < Jm(&cos^>o)- On the other hand from the edge condition in (10. a), one concludes *± (a) = O (a“*0) (12.a) $ 01 (a) = O (a-*0-1) (12.6) and *+(«) = fl^£İ_î_ + O fa-^”-1) (12.c) <T\- 02 1 o“2 + T2 ma when we let |a| - > 00 in their respective region of regularity. The elimination of Ao{a) and Bo(a) between (11. a, 6, c,d) yields a matrix Wiener- Hopf equation from which we shall derive the expressions of #^(a); namely: G.(«)*t-(a) = *f(a) + j-L (13.a) ixwith Ğo(a) *5”(«) = and cr2 +T2 _$o~2(°0. F - X in o“l +t2 0”! - (72 7(a).7(a)(rı-r2) n + <r2 r*3i(«)_, #+(o) = cr2 +r2 -ifcsin^o(Tı - T2) 0“2 +T2 (13.6) (13.c,d) (13.e) The solution of (13. a) requires the factorization of the kernel matrix Go(a) as the product of two non-singular matrices, say Gq”(q;) and Gq~(o:), whose entries are regular functions of a with algebraic behaviour for \a\ - > 00 in the upper and lower half-planes, respectively. Although the Wiener- Hopf factorization of an arbitrary matrix still remains at present an open problem, the kernel matrix Go (a) given in (13. b) belongs to class for which the Wiener-Hopf factorization can be accomplished through the Daniele- Khrapkov method. Indeed, if one writes G0(a) = C0W(a) with Cn = and W(a) = er2 + r2 1 0 0 1 + <ri + r2 0 0 ti + cr2 0 60 72(«) (14.o) (14.6) 7(a) 60 0 (14.c) where ao is given by (10. e) while 60 stands for 60 = '(<7i -cr2)(ri +cr2) (o“ı +7-2)(ri -r2) then the matrix W(a) can be factorized immediately to give 60sinhxo(a) ”(U.d) W+(a) = (l-ag)1/4 coshxo(a) 7(a)sinhxo(«) 7(a) coshxo(o;) (15.a)and In (15. a, 6) we put and W“(a) = W+(-a). a Xo(a) = -i^Q arccos - a 7T - arccos - k. Xo(-a) = -iXo Furthermore, it can be easily shown that one has f (±a)x° bQSQ(±a)~x°-1 W±(a) ~ _f^(±ûO*°+1 (±a)~Xo as |a| - > oo in the upper and lower half-planes, respectively. (15.6) (16.o) (16.6) (17) Since the Wiener- Hopf factorization of the kernel matrix is accom plished, the equation in (13. a) can be rearranged as 1 i. W-(a)^-(a) - I0->) = [Co W+(a)] *+(a) + I+(a) (18.a) with and î0 (a) = [C0W+(*cos6))] Fp a - k cos <j>Q ~ [CoW+Ctt)]”1 - [CoW+(fccos^0)]-1Tn 10 (a) = ; ; 10. (18.6) (18.c) a - k cos <po The standard asymptotics show that I* (a) are of O (a-1) as |a| - > 00. The left-hand side of (18. a) is regular in the lower half-plane Im(a) < Im(kcos<^o) while the right-hand side is regular in the upper half-plane Im{a) > Im(-k). Hence, by analytical continuation principle they define an entire function Po(a)- By taking into account the order relations one concludes from LiouvihVs theorem that Po(o;) is a constant matrix of the form Po(a) = Po (19) Thus, the solution of (13.a) reads: *o~(«) = *o2(«). [W(a)] -1 Joi(°0 JÖ2(a)+P0 (20) In order to determine the constant po one has to consider (12. a, 6, c), (17) and (20) for \a\ -> 00, which yields Po = -7-(a-fccos^o)Ioi(°0- oo (21) XIProm (11. c, d) and (20) we get (i - 4)-v* A0(a) { 0-2 r2coshxo(- «) + T_sinl1Xo(- «) Vi(«) {<r% + r2) - [r2b0 sinhxo(-a) -f 02 coshxo(-a)] [l0l(a) +p0] > (22.a) j(c and So (a) (i-^)“1/4 { coshxo(-a:) + t~ sinhxo(-a) öo ^OlO) (CT2 + T2) + -rr [bo sinhxo(-a) - coshxo(-a)] |/m(ûs) +P°] (. (22-&) By using these expressions of Ao(a) and Bo (a), the singly diffracted field by the junction O can easily be determined by evaluating (8. a) through the saddle-point technique as follows: u0(p,<j>) = V2rce where -i./J1-^) V4e^ j>iQMo), <^e(0,7T) (^2+r2) y/Tp \ V2{4>Ao), ^e(7T,27r) (23.a) Vi(<f>,<j>o) =ksm^).(?2 r2 cos Ao^ - i-r~ sin Ao<^ ”0 IQ1(-k cos <^>) [t2&o sin \0<j> + icr2 cos X0</>] [l02(-k cos <£) + p0] (23.6) and ^2(^,^0) =&sm<£ cos Ao^i + - sinAo^ 6o I01(- &cos<^) [60 sin \0(j> - i cos Aq<^] |/02(- & cos <j>) + p0]. (23. c) 3. Formulation of The Three- Part Dielectric Slab Problem The geometrical configuration pertaining to the three-part problem is illustrated in figure-3. xn* y Figure-3 a) Three-part dielectric slab b)Three-part material plane Since the second junction (edge) is in the transition region of the first one, the field illuminating the second junction after being diffracted from the first exibits the non ray-optical behaviour. Hence, the double diffraction can not be obtained by the traditional Geometrical Theory of Diffraction approach which consists of multiplying the single diffraction coefficient of successive junctions. In the analysis that follows, a spectral iteration technique (SIT) will be employed to obtain the doubly diffracted fileds. According to the SIT, the doubly diffracted field by the junction Q say uoq, will be obtained by assuming that the junction of the half- planes {x < l,y = 0} and {x > l,y = 0} with parameters (£2,^2,^2) and (£3,^3,^3) is now illuminated by the singly diffracted field from the junction O, i.e. u°(x,y). For the doubly diffracted field uoQ{x,y) we assume the following integral representation : r / AOQ{oc)e-^y-iaxda ; y > 0 uoQ(x, y) = J J BoQ(a)el{a)y->axda. y<Q (24) v c The spectral amplitudes Aoq(oc) and Boq{ol) are to be determined so that the appropriate boundary conditions are satisfied. Following the same procedure as in section 2, one gets Aoq(oi) = -ial (<r-2 +r2) r2$u((x) -<T2 *&(«) Boq{oc) = *£(<*) + 7(a). *&(«)' 7(a). (25.a) (25.6) (ö“2 + T2) The functions $^(0;) and $i”2(a) aj^e *° De determined through the Wiener- Hopf equation (26.a) - iotlt Gi(a)*+(a) = $l(a) - c-,a,Gi(o)#J-(a) XUlwhere *+(«) = *&(«) and Ğx(«), *r(°) = 03 + r2 0“2 +^2 *ii(«) *r2(°;) &3 - &2 (26.6, c) 7(«). (26.d).7(a)(^ -r2) T3+0”2. Notice that $i“(or) and #j~(a) are regular functions of a in the half-planes Im{a) > Im{-k) and Im{a) < Im(k), respectively. The application of the standard Wiener-Hopf technique enables one to write the solution of (26. a) as *?(*) *fi(«) ^[V+Ca)]-1 î+(a) + P!(a) (27.o) where V+(a) - (1 - a\) 2\l/4 coshxi(a) 7(a)sinhxi(a) 6isinhxi(«) 7(a) coshxi(oO (27.5) with _ /(03 -qr2)(T3 -r2) V (°3 + T2)(r3 +cr2) 61 '(0-3 -o^Xts +0-2) (o-3 +r2)(r3 -r2) and while a Xı(a) = -i^i arccos - k 1 /1-ai 2717 V 1 + a\ (27. c, d) (27.e) (27./) The term Pi (a) appearing in (27.a) is a yet unknown constant matrix of the form Pı(«)=Pı \ (28) resulting from the application of Liouville's theorem during the Wiener- Hopf procedure. And also, in (26. a) the expression I* (a) is given by I+(a) = e-in/4 [ClV-(-Jb)]_1 (^ ~ ff*)®0l(~k) e«”*f 27r(a + fe) (<r2+r2) [ii(r8 - ^^(-fc) J V^' (29.a) XIVwhere with V-(a)=V+(-a) Xi(-or) = -*Ai a 7T - axccos - k (29.6) (29.c) The constant pi is to be adjusted in such a way that the doubly diffracted filed uoq, satisfies the edge condition uOq(x,±0) = -u°(1,±0) + o((x-1)'x^ x -> /. (30) By proceeding as in the previous section one finds that Pi = £(a + k)I+(a). (31) Aoq{oc) =- (To r2 coshxi(a;) + - sinhxi(a) 0\ #(«) From (25.a, 5) and (27. a) we get eftt*(l - a\ )~^4 (ö2 + T2) 7-r [r26i sinhxi(a) + er2 coshxi(a)] [i«(a) + pi] \ (32.a) and #oq(o0 eiQ'(l - a?)“1/* - cosh xi (a) + - sinh xi (a) °i #(«) (0*2 +7”2) + -rr [61 sinhxi(a) - coshxi(a)] [j£(a) + pi] 1 (32.5) Since Aoq(oc) and Boq(ch) are now completely determined, the doubly diffracted field by the junction Q can easily be determined by evaluating (24) through the saddle-point technique to give uOQ{r,il>)~yj2^e~T1* where Si(tp,<f>o) =ksmij) h-afy1/4 eikr (Si(?Mo) ; ^?(0,tt) (<T2+T2) y/k^\S2{i,M. ^?(ff|2ff) (33.a) r2 cosAi(7r - i>) - i- sinAi(7r - V>) "1 ?Tn(- kcostp) - [r2fei sinAi(7r - r/>) + ecr2 cos Ai(7r - ^>)] [.^i2(- kcostp) +pi] (33.6) and ^(Vs^o) =A;sin'0 cos Aı (ît - ^>) - - sin Aı (ît - ^) Iii(-kcostp) - [61 sinAi(7r - if)) + icos Ai(7r - t/>)] [l^2(- kcostj)) +pi] (33.c) XV

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