Directed Study For DERIVATION OF DYADIC GREEN'S FUNCTION FOR MULTILAYER DIELECTRIC SUBSTRATES By Yook, Jonggwan Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan For the partial fulfillment of the EECS 599 Advisor: Professor Pisti B. Katehi Date of Submission; May 5, 1992

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Abstract To characterize a field in the multilayer dielectric structure, a generalized full-wave Green's function is derived using two-dimensional spectral-domain technique and it is transformed to space-domain. It is derived using a simple structure where the current source lies between two layers bounded by surface impedance boundaries called the "standard" structure. Reflection and transmission coefficients at the surface impedance boundaries are used to compute the coefficients of the Green's function and this can be achieved by simple iteration technique with the transmission line analogy. The multilayer dyadic Green's function derived in spectral-domain can be converted to spacedomain by use of Fourier-Bessel transformation. This space-domain Green's function is consistent with that of Sommerfeld approach for a grounded dielectric geometry excited by a horizontal Hertzian dipole. This method is versatile and can be used for the either closed or open boundary problem. While the Sommerfeld's approach is difficult to apply in multilayer structure, the method in this report can easily be adapted to multilayer geometry. The Green's function for multilayer dielectric structure as detailed here is used in the numerical modeling of monolithic microwave integrate circuit, dielectric waveguides, and multilayer microstrip antenna structures.

Contents 1 Introduction 1 2 Part 1: Decomposition of fields using electric and magnetic vector potentials 3 2.1 Geometry and general formulation of the problem...... 3 2.2 Dyadic Green's function in the spectral-domain....... 6 2.3 Space-domain solution of dyadic Green's function....... 9 2.4 Fresnel coefficients at the other boundaries........... 12 2.4.1 Grounded-substrate geometry.............. 13 2.4.2 Substrate-superstrate geometry............. 13 2.4.3 Two-layer-substrate geometry.............. 14 2.4.4 Substrate-air gap-superstrate geometry......... 15 3 Part 2: Decomposition of fields using magnetic vector potential 16 3.1 Magnetic vector potential................. 16 3.2 Space-domain solution of dyadic Green's function....... 19 4 Conclusions 21 5 Bibliography 22

List of Figures 1 Geometry of multilayer dielectric substrates with surface current source............................. 24 2 "standard" geometry for general formulation.......... 25 3 Transmission line analogy to calculate reflection coefficients.. 26 4 Grounded-substrate geometry... 27 5 Substrate-superate geometry............... 28 6 Two-layer-substrate geometry.................. 29 7 Substrate-air gap-superstrate geometry............. 30

1 Introduction The electromagnetic wave propagation in multilayered media, both isotropic and anisotropic, has been studied extensively by the use of full-wave analysis[l][5]. In [1], a generalized spectral-domain Green's function for multilayer dielectric is computed with iterative method to find the contribution of all other layers. In [2], a two-dimensional space-domain method of moments treatment of open microstrip discontinuities on multi-dielectric-layer substrates is presented. In [3], a dyadic Green's function in lossy media are investigated. While an operator approach in spectral-domain is presented in [4], in which a TE-TM decomposition and propagation matrices are used. In this report, a general formulation of the problem of a horizontal dipole in a multilayered environment is presented. The formulation is considerably simplified by seperating Green's function into a transverse-electric(TE) and transverse-magnetic(TM) terms. Moreover, this report contains derivation the Cartesian dyadic components as functions of cylindrical coordinates, which were found to be more handy in many cases related to planar structures which exibit a circular symmetry. This is done by means of a Fourier-Bessel transform, which provides a tractable form of the dyadic Green's function. The aim of this report, therefore, is to evaluate the dyadic Green's function for the multilayer planar structure, under a planar excitation, in the Fourier domain and, then, transform it to space domain using Fourier-Bessel transform. In fact, the dyadic Green's function by itself can provide useful information about the effects of the substrate, the characteristics of the radiated field, and, finally, the power losses in the layers.

In part 1, the equivalent boundary value problem solved with the use of electric and magnetic vector potentials. In part 2, according to the nonuniqueness of resolution of Hertz vector potentials, dyadic Green's function is derived with magnetic vector potential only and shown that these two approaches produce consistent results.

2 Part 1: Decomposition of fields using electric and magnetic vector potentials 2.1 Geometry and general formulation of the problem The electromagnetic study of the structure, Figure 1, can be obtained from Maxwell's equations, with a time variation ejt, where only an electric current density J(f') is assumed to be present. With such a hypothesis the electric field can be obtained via the following integral equation: E(r) = j G (f') J(i') d' (1) where G (lI') stands for the dyadic Green's function. In the most general case G has 9 components, while for planar currents these 9 components are reduced to only 4 components. Moreover, the symmetry of the geometry makes it possibile to find some components of G in terms of the other components. The dyadic Green's function is the solution of the fields due to a point source and can be represented, in rectangular coordinades, by Gxx Gx + Gx + GZ, G = +G, G.yx + Gy,,y + G,,yz (2) +GsZox + Gzf'y + Gw kz where Gij is the i-th component of the field due to a unit j-directed current source 6(i - i')y. The well known relation between Hertz vector potentials

and the electromagnetic field is given by the following two equations E = -jkZV x fIm + k2fI + VV.e (3) H = jkYV x fle + k2fim + VV fJI (4) In addition to these equations, the relations between the Hertz potentials and electric and magnetic vector potentials make the above expressions as it follows: E = -V x F-jwpA+ -VVA (5) jWE H = Vx A-jweF+. VV-F (6) JW!l with i = A (7) jwe 1 m = F. (8) jWE These two vector potential functions satisfy the wave equations: V2A + k2A = - J (9) V2F + k2F = - i (10) where Ji and MA are electric and magnetic current sources. A field in any region can be completely defined by suitable components of {As, Ay, AZ, FT, Fy, Fz} and judicious choice of these two components make the field decomposed [1],[6]. In the present chapter, the field is decomposed using (A,,Fz), that is, A = A(z, y,z)2 (11) F = F,(x, y, z) (12)

For an arbitrary surface current distribution in the xy plane between 11 and 21 layers, with x and y components: J(x, y) = Jx(x, y) x + Jy(x, y) Y (13) the solution to E and Hft or A and F can be written in terms of the known Green's function as follows T(f) = Ji[ Gij. (ii )Jx(f') + Gpj (rlr')Jy(r') ]ds (14) Since the multilayer geometry in this case is infinite in x and y, the twodimensional function Psi(r) can be defined as I(k, k7y, z) = f (x, y, z)ei(kjz+ky) dxdy (15) (x,, z) = (27r)2_ (kx, k, z)e-j(kxx+kyy) dkdky (16) (2ir)2 J J-00 where'= (E/, H, J. A, AF). (17) Now, the individual components of the electric and magnetic fields can be written in terms of A, and F, as shown below: OF: 1 I2Az E, = -d + e OXO(18) OF: 1 O2Az Ey = + *v a2Z (19) 1 O2Az E = + 2A 2 (20) OA, 1 a2FZ Hx = +. (21) 591 1I ~a

HOA~ 1 O2FZ H, = - Ox + j Oyaz (22) 1 F2Fz - 1(O + k2F) (23) 2.2 Dyadic Green's function in the spectral-domain The field components in space-domain can be easily transformed to the spectral-domain using a two-dimensional Fourier transformation. Ex = -jkFz +- (24) WE Oz kv OAz Ey = jkFz Z+- - (25) WE Oz 1!92A, E,= ( + k2A") (26) JW k az2 H _= Az + - (27) wti Oz H1y = -jkAz +- a w z (28) 1 02F Y HZ + Zk2Fz) (29) -jw 0+z2 in which -represents Fourier-transformed component. In source free region, A and F satisfy the homogeneous wave equation V2[ A ]+k2 Az 0 (30) In the spectral domain this equation takes the form: (2-U) [u = 0 (31) jZ2~~

where u= kx + k2 - k2, k = ko/4 (32) This homogeneous wave equation is not a partial differential equation but an ordinary differential equation and the general solution to the equation can be written as Az(k,, ky, z) = (e-UZ+ rAeu")a(kx, ky) (33) F (kx,, k,z) = (e-u+ rFeUZ)f(kx, k,) (34) As shown in Figure 2, for the "standard" structure with x directed current source between the interface of the two layer'11' and'21', the general solution to aij and fii is given by Al1 (kk, kyz) = (eullZ + rAll, e-UlZ)ajl(kz, ky) (35) AZ (k~, ky,z) = (e-U21z + rA2leU21Z)a2l(kx, ky) (36) Fzll (k, ky, z) = (eUlZ + rFl e,,ull)fll(kx, ky) (37) F,, (k, k, z) = (eu2lz + rF,, eU2)f2l(k, k ) (38) At the interface, tangential electric fields are continuous and tangential magnetic field are discontinuous due to the x directed electric current source as shown by the following equation: E,,(k, ky, z = O) E2,,(k, ky,,z = O) (39) Ey11 (k,0 ky, = =) E,21 (kx,ky, z = 0) (40) HxI1 (kky, Z=O) = HX21 (k, ky, z = 0) (41) HY21(k~, ky, z=O) - Hy1 (k, ky,, z=0) = (42)

With the above 4 boundary conditions the unknown constants rA and rF are determined and the functions all, a2l, fil and f2l may be written in the form: ky wpo(1 + rF21) k=2 +_ k2 [U11(1 - rF,,)(1 + rF2l, ) + U21(1 -rF2,, )(1 + rF, )] fk1- wso(1 + rF11) (44) -kx a j(k2 + k+2) ellU21(1 - A2(45 ) [ellU2l(1 + rA1,)(1 - rA2) + E21U1l(1 + rA21 )(1 - rA,)] a2 j(k2 + k2) e21U(1 - rA11) (46) [llU21(1 + rA1 )(1 - rA21) + E21Ull(l + rA21 )(1 - rAl)] As a result, the elements of dyadic Green's function for the electric field due to an infinitesimal electric current source in spectral domain are given as follows GEJ- =( - rA21 el )a2 + jky(eu2z - rF eu21)f21(47) W621 GB9JS- (e-12Z - rA2, eU )a2 - jk.(e-U2-z - rF21 eu2)f21 (48) WE22 21~ k; + k ( _21Z U2a GE1 = k (e-. + PrA2 eU2l)a2l (49) jWe21 for z > 0 and ~11 kxull U,, GJS = (eullz - rAl, e-u")all - jky(eul'z + rFlle-ul)fll (50) Wk11 G1 = kYull(eullz - rAle-u l)all + jk,(eulz + rFl e-Ull)fll (51) G =+(WE11 "k2,.a Gil k k(eulI + rA11 eU )all (52) — IJI

for z < 0. Finally, we note that the elements Gij of the spectral Green's function show very interesting properties, due to the geometrical symmetry of the structure around z-axis, that can be summarized in the followings[5]: GEJ,(- k, k., zlz') = -GEJ.(k, ky, zlz') (53) GEJ,(-ky, k,kzIz') = GE.j.(k.,kj, zlz') (54) GE, j,(-ky, k., zlz') = GEJr.(k, ky, zlz') (55) GE. J, (-ky, k, zz') = - GEyJ, (kx, ky, zIz') (56) GE,.(-ky, k, zlz') = GE.j.(k. ky, kZIZ') (57) Moreover, if there were z directed current source, using the reciprocity theorem and Parseval's theorem we can deduce the following result Gpq(kx, ky, zl z2) = Gqp(kz, ky, Z21ZI) V p q (p =x, y; q = z). (58) where zl and Z2 are the z coordinates of the chosen sources. We note that the above result express an index permutation property of the spectral dyadic Green's function obtained in the Fourier transform space. 2.3 Space-domain solution of dyadic Green's function To find space domain Green's function, a two-dimensional inverse Fourier transform is need. The fi, and aij are functions of kx and ky, so a transfor

mation to polar coordinates is made both in coordinate space and k space. x = pcosq, y = psinG (59) k - Acos(, ky = A sin, (60) with kTx + kyy = Apcos(( - 0). (61) Then, the Fourier transform defined in an earlier paragraph becomes (P, q Z) = (21)2 j Ii p(A, (, z)eiAPco8(C-O) d(AdA. (62) The function ejAPcoS(C~-)-jwt represents a plane wave whose propagation constant is A, traveling in a direction which is normal to the z axis and which makes an angle C with the x axis. Each plane wave is multiplied by an amplitude factor I(A, C, z) and then is summed with respect to the propagation constant, or space frequency A. If we take a close look at the components of dyadic Green's function, we see that in (A, ()-domain the components can be separated into A and 5 functions, respectively. That is, for any componens the following is true: i(Ax, ~, z) = A(A, z) B(~) (63) and as a result, for a surface current source, the kernal of the Green's function has this form: -Alcos24 + A2sin2C -(Al + A2)cosCsin] -(Al + A2)cos(sin~ -Alsin2C + A2cos2C (64) A3cosC A3sin(

In equation (64) A1 U21Ul (e-UZ -2 _rA2l e"21z) 1 (65) iw TA A2 jwpo(eu21z - rF2 euZz)1 + PF1 (66) TF A3 -Ull (eU21 + A2 eu21) A, 1 (67) TA for z > 0 and A1 - U21 (euz-rA e-u1z) rAT1 (68) jw TA A2 = -jwpo(eu1'z + rF e- Z) + (69) TF A3 c= Ue, -) - rAe ) T (70) W TA for z < 0, with TA = E11U21(1 + rA1,)(1 - rA21) + 21Ul11(l + rA2 )(1 - rA,) (71) TF - 11(1 - rF11)(1 + rF21) + 21(1 - rF2,)( ++ rF,,). (72) Now, one may use the following integral 2ir cos mO e-j"zco dO = 27r(-j)mJm(z) (73) 27 sin mo e-jz co dO = 0 (74) to evaluate the (-integration. The final results are in the form: GEJ.(p,, zlz' = 0) = I [(-A1 + A2)Jo(Ap) +(A1 + A2) cos2q J2(Ap) ]AdA (75) 1 o GEJ. (P, 0 ZIZ' = O) = I (-Al + A2) sin2o Jo(Ap)dA (76)

GE.j.(P, X, ZIz' = o) =2 A3 cosq J1(Ap)A2dA (77) GE.j,(P,,zlz'= 0) = GE,,j (P,q, zz' = 0) (78) 1 r0~ GEJ,J,(P, q, IZ = 0) = 4r [ (-Al + A2)Jo(Ap) -(Al + A2) cos2o J2(Ap) ]AdA (79) GEZJY(p, X, ZIz' = 0) = tanO GEJ: (P, q0 zjz' = 0) (80) 2.4 Fresnel coefficients at the other boundaries The F's of the field expression are found to be reflection coefficients for a transmission line which is terminated by the load of different characteristic impedance. For the magnetic vector potential(A), the the equivalent transmission line characteristic impedance is equal to /jlcijq which is identical to the TM wave impedance. For the electric vector potential(F), the equivalent transmission line characteristic admittance is equal to #/ilj/ij which is identical to the TE wave admittance. In addition, the reflection coefficient for A /F is equivalent to that of a current/voltage wave of a transmission line. With these analysis and Figure3, the reflection coefficients are determined: rA, = ri. e-2uiidii ZAij - ZT = I ZAA+-ZJ e-2uijdj (81) rFij = r, e-2ijidi YFij Fi e_2uijdii (82) yF __ U. e- ~T~(82) 12i

with Z = 1 - ZAPj+Z (83) ZJ+ 1 1 + rAi,+1 y T _1i+= 1 + ij++ (84) 2.4.1 Grounded-substrate geometry Since there are no reflection from the upper layer in Figure 4, the reflection coefficients rll, rFl, rA1, rF1 are identical to zero. On a perfect electric conductor(pec) the normal component of electric field is doubled by its image, however, normal component of magnetic field is cancelled by its image. Therefore, the following relations are true: rA,= 1, PrF, =-1 (85) and rA21 = e-2uodl (86) rF21 = -e-2uod (87) 2.4.2 Substrate-superstrate geometry For a substrate-superstrate geometry, Figure 5, the r's are calculated step by step from the farthest layer as shown below: rA,, = e-2ulldl (88) rF1, = -e-2ulldl (89) 13

For the upper layer, the reflection coefficients are given by: rA21 = rA2 e2U21 d2 (90) rF2, = Ile-221 d2 (91) where ZA1 ZT rA21 T ZA21 + ZA22 (21/l21) - (P22/622) (92) (/321/e21) + (P22/e22) F YF21 F21 YF21 + YF21 (#21//'21) - (P22/L22) 93 (021//21) + (P22//122) (94) and uij = A2 - k kj = ko0i/j. (95) 2.4.3 Two-layer-substrate geometry As shown in Figure 6, for an upward looking case, rA21 and rF21 are equal to zero. For a downward looking case, however, rA12 =e-2u2 d2 (96) rFl2 = -e-2ul2d2 (97) and ZA12 = j tanh(ul2d2)ZA,2 (98) YF2 = -j coth(u12d2)YF12 (99) 14

Finally, the rA,, and rFF1 are given as follows: rA11 = e-2u1d1 ZA11 -j ZA12 tanh(ul2d2) (100) ZA11 + j ZA12 tanh(u12d2) rF1 = e-2ulldl YF11 + j YF2 coth(u2d2) (101) YF1I -j Y,12 coth(u12d2) 2.4.4 Substrate-air gap-superstrate geometry For a downward looking case, Figure 7, the reflection coefficients are given by rA1 = e-2ulldl (102) rFl1 = -e-2ul di, (103) while, for an upward looking case are identical to the form: PA22 = e-2U22d3 ZA22 - ZA23 (104) ZA22 + ZA23 rF2 e-2u22d3 YF22 - YF23 (105) 2 YF22 + YF23(105) with ZA2T 1 - rA22 (106) A22 1 rA22zA2 YT 1- rF ZF22 (107) F22 1 + rF22 Finally, the r's at the air-gap layer are given by the following expressions rA2 = e-221d2 ZA - A22 (108) ZA21 + ZA22 rF2, = e2U21d2YF21 F22 (109) 15

3 Part 2: Decomposition offields using magnetic vector potential 3.1 Magnetic vector potential To derive the generalized Green's function for multilayer structure, a combination of Sommerfeld's resolution[7] of a magnetic vector potential (or Hertz potential He) A = (A,,O,A,) for the x-directed dipole source on the imperfect ground plane, and the spectral domain technique are investigated. Moreover, the equivalence between the former approach and this method is also investigated. In contrast to the previous chapter, the potential Green's function is defined as E(f)= j(ki + VV) )G () j(') J d'. (110) The field equations with the vector potential A have the following form; 1 O2AX 1 O2A, Ex = -jwA., + 2+. (111) SjW a 2 JWE aX6Z 1 92A, 92AZ E% = j-( + ay A) (112) JWE ayax 9yaz 1 a2A, 1 a2Az Ez = -jwpAz + A X + O2Z (113) jWE8dadz jWE C2z aA, HA = (114) ay aA:, 2A,,y = a A aAZ (115) Hz = A(116) ay Using the Fourier transform, the field expressions and wave equation for

vector potential can be transformed to the k-domain. The boundary conditions can also be transformed to the k-domain. Ex = -(jw +` ))AX + k (117) k~k- ky OAz Ey = Ax + (118) jW- W, aE kx aAz 1 a2A, Ez = -jtAz - - + (119) we 9z jwC az2.H = jkAz (120) H = a Azr (121) iz = -jkyA, (122) In addition to the above, the following equation is true. (-2 -u )A = 0 (123) where u2= k2 + k2 - k2, k - koV/ (124) From the geometry of "standard" problem, the solution of the wave equation in the spectral domain in the both side of the interface can be assumed to be A1,,,(k,,k,,z) = (eullz + r,,,e-u11z)Pll(kx, k) (125) A,,1(k., k, z) = (eu11z + rz,,e-U1)Qll(kx, ky) (126) A21 (kx, k, z) = (e-U21Z + r,21 eU21)P21(kZ, kI) (127) Az21(k,k~,Z) = (e-U21- + rz2l eU21Z)Q2,(k:, ky) (128)

Moreover, the boundary conditions in the spectral domain at the interface z = 0 are the same as those of the previous chapter. From the electric and magnetic fields boundary conditions, the four potential boundary conditions shown below are obtained. A2(k, k, z = ) = (k, ky, z = 0) (129) Az (kk, ky, z = 0) =Al (k, ky,z= ) (130) aA, aAkz (131) - _ 1 (131) Oz az 1 A2 1 OA" A2x A" (- Z ) - jkx( - )= O (132) E21 aZ el1 aZ E21 e11 As a result, the four unknown function P's and Q's are determined in terms of r's in each region. P, (k, kv) = + r2, (133) + (133) P21(k, ky) = + r(134) 1 1 Ql,(kx, ky) = (1 1 ) 11 C21 jkT(1 + r.,11)(1 + rX21 )(1 + r.2,) (135) Sl,(Ull, 21, r.l, r,21 )S2(U, U2, r,,,1 r, (13 Q21(kI ky) 1 + rz1' Ql(k., k.) (136) where S Si (Ull,U21,rll,r=lT) = -[u21(1 - r21)(1 + rP,1) + u(1 - r,,1)(1 + r=,21)] (137) S2 = S2(Ull,u21,rll, r,2l) U21,(1 - rz21)(1 + r,,,) + u-(1 - rz1)(1 + rZ21) (138) /21 l11 18

These results are similar in the form with TA and TF of the previous chapter, however, the r is defined somewhat differently. 3.2 Space-domain solution of dyadic Green's function The space-domain Green's function of a vector potential d with x and t directed current source is derived using 2-dimensional inverse Fourier transform defined in the previous chapter. Moreover, the symmetry of the structure considered here allows us to deduce some components of Green's function from the other components. The final results are given as following: 1 fo0 G (21 (p q z) = 1 j( e(U2lZ + r221 ee21") P2 (A)Jo(Ap) AdA (139) i roo AGz;J~(p,7,z) = j2cos] (e-"'2z+ rz2, eu2)Q21(A)Ji(Ap)A2dA (140) for z > 0, and G~AJ(p,q, z) = 92 (e"llz + r,,,e-u"lz) Pl(A)Jo(Ap) AdA (141) AGx i(p,, z) = 2Cos0] (eullZ + rz,,e-u11z)Qll(A)Jl(Ap)A2dA (142) for z < 0. The P's and Q's are determined in the section 3.1, and rP's and rF's will be determined by the boundary conditions at the other interfaces, that is, tangential electric field must be zero on a pec and the normal component should be enforced. For example, on a perfect electric conductor, r' = -1 (143) =r' =1 (144) 19

in which the rs are reflection coefficients at the interfaces. For a multilayer problem, the rs are calculated by iterative method as it has been demonstrated at the end of the previous chapter

4 Conclusions In this report, two types of Green's functions, one is the electric field Green's function and the other is the potential Green's function, are derived for multilayer substrates geometry and shown that these two are equivalent. According to the non-uniqueness of the resolution of the Hertz vector potential, the former approach use TM - TE decomposition with electric(F) and magnetic(A) vector potential. While the later use only magnetic vector potential with two components, in which one component is parallel to the current source and the other is perpendicular to the current source and its surface. A two dimensional Fourier transform pair and Fourier-Bessel transformation are used to convert the spectral domain Green's function to space domain form. The versatility of the spectral solution for multilayer structure and the possibility of inverse transform to space domain allow us to develop a powerful method of analyzing multi-dielectric layer structure regardless of open or closed geometry. Moreover, the second kind of Green's function can be used to extend ready-made CAD program for a multilayer structure.

5 Bibliography [1] Nirod K. Das and David M. Pozar,"A generalized spectral-domain Green's function for multilayer dielectric substrates with application to multilayer transmission lines," IEEE Transactions on Microwave Theory and Techniques, vol.MTT-35, pp.326-335, March 1987. [2] William P. Harocopus Jr. and Pisti B. Katehi, "Characterization of multilayer dielectric substrates including radiation losses," IEEE Transactions on Microwave Theory and Techniques, vol.MTT-37, pp.20582066, December 1989.Correction: vol. MTT-38, p.825, June 1990. [3] Luc Beyne and Daniel De Zutter, "Green's function for layered lossy media with special application to microstrip antennas," IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, pp.875-881, May 1988. [4] T. Sphicopoulos, V. Teodoridis, and F.E.Gardiol,"Dyadic Green's function for the electromagnetic field in multilayered isotropic media: an operator approach," IEE proceedings, vol.132, Pt.H, No.5, pp.329-334, August 1985. [5] Lucio Vegni, Renato Cicchetti, and Pasquale Capece,"Spectral dyadic Green's function formulation for planar integrated structures," IEEE Transactions on Antenna and Propagation, vol.AP-36, pp.1057-1065, August 1988. [6] Ahmed Esteza,"Non-uniquiness of resolution of Hertz vector in pres

ence of a boundary, and the horizontal dipole problem," IEEE Transactions on Antenna and Propagation, vol.AP-17, pp.376-378, May 1969 [7] A. Sommerfeld, Partial Differential Equation in Physics, New York, N.Y. Academic Press, 1949.

0 ~2j d 2j ~22 I d 22 J (x,y) E21 d 21 11 dll z E12 d12 0 k I d lk Geometry of Multilayer Dielectric Substrates with Surface Current Source Figure 1: Geometry of multilayer dielectric substrates with surface current source 24

e-u1Z eU2lz r E21 J(x,y) E11 eUllz ~uliz ri " Standard " Geometry for General Formulation Figure 2: "standard" geometry for general formulation 25

I I I I I I I I I I ~~I ~ZA.. I I T i YFij I r' +1 I l I Transmission-Line Analogy to Calculate Reflection Coefficients Figure 3: Transmission line analogy to calculate reflection coefficients 26

-21 u J(X,y) e11 d pec Grounded Substrate Geometry Figure 4: Grounded-substrate geometry 27

E22 =0 21 J(x,y) 12 pec Substrate - Superstrate Geometry Figure 5: Substrate-superate geometry 28

~22 =O J(x,y) 811 812 pec Two - Layer - Substrate Geometry Figure 6: Two-layer-substrate geometry 29

~23 =0 ~22 d3 821 J(x,y) d2 e1 d1 pec Substrate - Air Gap - Superstrate Geometry Figure 7: Substrate-air gap-superstrate geometry 30

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