Spectral-domain dyadic Green‘s functions for surface current excitation in planar stratified bianisotropic

E.L.Tan and S.YTan

Abstract: A rigorous formulation of the spectral-domain dyadic Green’s functions for surface current excitation in planar stratdied bianisotropic media is presented. The media may consist of any number of layers bounded by optional perfect electric/magnetic conducting walls. The surface current can be impressed along any of the layer interfaces. Both electric and magnetic dyadic Green’s functions are derived simultaneously based on the principle of scattering superposition. For the primary dyadic Green’s functions associated with a two-layered semiinfinite medium, their source consequents are determined directly from the elementary boundary conditions across the interface. An alternative approach to obtain these source consequents is also devised by considering their relationship with those of unbounded dyadic Green’s functions. For the scattered dyadic Green’s functions, their scattering coefficient matrices are determined without cumbersome operations using the concepts of effective reflection and transmission of outward-bounded and inward-bounded waves. This approach provides good physical insights to the scattering mechanism and leads to compact and convenient representations. To illustrate the application of general dyadic Green’s function expressions, the configuration of a grounded bianisotropic slab embedded in isotropic half-space is considered explicitly.

1 Introduction

Over the past few decades, the dyadic Green’s function technique has been widely employed to investigate the interaction of electromagnetic waves with multilayered media [ 1 4 . For microwave and millimetre-wave integrated circuit applications, the analysis of microstrip antennas, planar transmission lines and waveguidmg structures can be readily accomplished once the dyadic Green’s functions in spectral domain have been obtained [5]. In the literature, there are large amounts of research work dealing with deri- vation and utllisation of spectral-domain dyadic Green’s functions. These Green’s functions relate the electric and magnetic fields to the surface currents in planar stratified media containing isotropic [6, 71, anisotropic [8-141 and bianisotropic [ 15, 161 materials. In recent years, advances in material processing and fabrication technology have mani- fested many new types of complex materials which could become potential candidates for the substrates used in microwave circuits and printed circuit antennas [17, 181. As the materials become more complex, the analytical determi- nation of their associated Green’s functions may become more tedious and intractable. This complication thus calls for the derivation and representation of Green’s functions in compact and convenient forms which at the same time also feature some physical interpretations such as those achieved for simple media.

OIEE, 1999 ZEE Proceedings online no. 19990448 DOL 10.1049hpmap:19!?3048 Paper received 22nd September 1998 The authors are. with the School of Elecbical and Electronic Engineering, Nan- yang Technological University, Singapore 639798, Republic of Singapore

394

In this paper, we present a rigorous formulation of the spectral-domain dyadic Green’s functions for surface cur- rent excitation in planar stratifed bianisotropic media. The media may consist of any number of layers bounded by optional perfect electridmagnetic conducting walls. Although general excitation source can be distributed any- where and may include normal components as well, we wdl restrict our analysis to sources that are located right at the layer interfaces and made up of transverse components only. Both electric and magnetic dyadic Green’s functions attributed to the surface currents are derived simultane- ously. Making use of the principle of scattering superposi- tion, each dyadic Green’s function is constructed in primary and scattered parts. The primary parts account for the direct waves excited by primary surface sources in a two-layered semunfiite medium. The scattered parts account for the scattered waves caused by other layers/ walls. Using the elementary boundary conditions across the interface, the primary dyadic Green’s functions are deter- mined directly in terms of dyads formed by eigenfunction antecedents and source consequents. An alternative approach to obtain the source consequents is also devised by considering their relationship with those of unbounded dyadic Green’s functions [19]. For the scattered dyadic Green’s functions, their scattering coeficient matrices are determined without cumbersome operations using the effec- tive plane wave reflection and transmission concepts [20, 211. To illustrate the application of general expressions obtained for dyadic Green’s functions, we consider explic- itly the configuration of a grounded bianisotropic slab embedded in isotropic half-space. Throughout the follow- ing analysis, e& time dependence is assumed and sup- pressed.

IEE Proc.-Microw. Antennas Propug.. Vol. 146, No. 6, December 1999

2 Primary dyadic Green's functions

Consider the planar multilayered bianisotropic media con- sisting of N layers stratified in $ direction with infiite transverse dimensions (Fig. 1). The outermost (labelled N) and innermost (labelled 1) layers can be semiinfinite or bounded by perfect electric/magnetic conducting (PEC/ PMC) walls. These optional boundary walls as well as the layer interfaces are denoted by p = Pf cf = 0, 1, ..., N).

t a PEC/PMC wall (optional) - p=P,

layer f = N

P=pN., f>st { layerf=N-1

P=pN-2

s t p=P

P=P,

P = ps-,

P=P*

P=P,

//771//////// P=Po

layer f=st

layer f=s

layer f=2

layer f = 1

PEC/PMC wall (optional)

Fig. 1 Geometry o f p h stratfzd bianisotropic media

Within each layer f cf= 1,2, ..., N), the medium is homoge- neous and characterised by constitutive relations of the

- form [2]

- D f = Z f . E f + $ f . H f (1)

B f = < f . E f + z f . Z f (2) - -

where Z5 Er, g f and Ef represent respectively the medium permittivity, permeability and magneto-electric dyadid pseudodyadics. In general, the electric and magnetic fields in each layer consist of superposition of four sets of eigen- functions. Each eigenfunction can be expanded in Fourier domain in terms of a complete set of R, R, L vector wave functions (of T and k) [4] as (j = 1, 2, 3, 4) - - - E,f = ae,fM($; k j f ) + b e j f N ( F ; x j f ) + C e j f z ( F ; I c , f )

(3)

H , f = %,f M ($; k, f ) +bh3 f X(F; E j f ) +Chj f IC, f ) (4)

- - -

The ubc expansion coeficients can be determined from source-free Maxwell equations as detailed in [19] although their explicit expressions are not required for general treat- ment in ths paper. Denoting th_e unit vectors transverse to $ as f, and f2, the wave vector kfcan be decomposed into

- - - k,f = kt + k,,fl j kt = ktlt"1 + kt2& (5 )

with k f obtained from the quartic dispersion equation [22, 23f

(7 being the idemfactor). Henceforth, we assume that kplf and kp2f with positive imaginary parts correspond to out- ward-bounded waves E , HV and EZr, which remain bounded as p - +W. Smilarly, kp3f and kHf with negative imaginary parts correspond to inward-bounded waves R3f and whch are still bounded as p - 4. Note that it is not very appropriate to term the waves as 'out- going' and 'incoming' since for general bianisotropic media, incoming waves may exist at infinity. Still, the radiation condition requires all waves to be suffcienty bounded at infinity [24]. With the electric and magnetic eigenfunctions available, we are able to construct the expansions of dyadic Green's functions in terms of these eigenfunctions.

Assuming a surface electric current source Ips is impressed along the interface p = Ps (s = 1, 2, ..., N - 1) of layers s and st = s + 1. Owing to the linearity of Maxwell equations, the electric and magnetic fields in layer f can be related directly to the surface source via

- Ef (V) = // ds'EdfPq$,F') . J p , (v') (7)

S'

- H f ( $ ) = / / d s ' E E p s ) ( ~ , ~ ' ) .Jp,($') (8)

where the integration with respect to source (primed) coor- dinates extends throughout the surface occupied by Ips(?'). G e m s ) and G m $ s ) are, respectively, the electric and mag- netic dyadic Green's functions due to surface current exci- tation. According to the principle of scattering superposition, each dyadic Green's function can be consid- ered as superposition of primary and scattered parts as

S'

where the symbol 6 with two indices in the sukscript denotes Kronecker delta. The primary parts G$tps), G $bps) and $'J), 3) correspond, respectively, to the outward- bounded waves (in layer st) and inward-bounded waves (in layer s) that are excited by the surface source at p = P, in a two-layered semiinfinite medium, i.e. assumingother lyersl walls are absent in Fig. 1. The scattered parts G ep), G 9) represent the additional contribution from the scattered waves caused by other layers/walls. In the following, we wdl first obtain the expressions for the primary parts and defer the derivation of the scattered parts until next Sec- tion.

Since the primary dyadic Green's functions satisfy the source-free Maxwell dyadic equations within layers st and s, they can be represented with eigenfunctions in eqns. 3 and 4 being the antecedents as

J k t

IEE Proc.-Microw. Antennas Propag., Vol. 146, No. 6, December 1999 395

whereSkt implies ST: dktl $_’,” dktz and Slips are the source consequents to be determined as functions of source (primed) coordinates. Across the interface containing sur- face current at p = P,, we have the elementary boundary conditions to be satisfied by the primary dyadic Green’s functions as

(15)

where 7 , is the transverse (to j3) part of idemfactor and St is the two-dimensional transverse Dirac delta function. These boundary conditions constitute the fundamental equations from which Slips and hence the dyadic Green’s functions can be determined explicitly. In particular, substi- tuting eqns. 11-14 into eqns. 15 and 16 and applying some orthogonality relationships via two linearly independent transverse vector functions V t l , V t 2 , we obtain

where

and ss implies 1:: dtl $T2 dt2 Solving eqn. 17, we arrive at

and hence all dyadics in eqns. 11-14 have been deterrningaJ. Later, we will devise an alternative approach to obtain S g in terms of the source consequents for unbounded dyadc Green’s functions and the reflectiodtransmission matrices to be defined below.

3 Scattered dyadic Green’s functions

In this Section, we make use of the effective plane wave reflection and transmission concepts to construct the scat- tered dyadic Green’s functions. The expressions of the local (superscripted I ) and global (superscripted g ) reflection and transmission matrices can be derived for planar stratified bianisotropic media following the approach described in [20,21]. Specifically, the local matrices which correspond to the reflection and transmission at an interface separating two semiinfiite bianisotropic layers can be found as

-9 -1

L (24)

where E$ and E: are given by eqns. 18 and 19, respec- tively. The global matrices whch also incorporate the effects of multiple-reflections can be shown to be

N - q (26) q - 1

I = 1 , 2 , ...,

Applying eqns. 25 and 26 recursively, one can find the global reflection and transmigion matricg for all layers provided the initial matsices R&,,+l and Rt0 are specified as

( G layer N open

( 5 layer 1 open

Having determined the reflection and transmission matn- ces, we will use them to relate the scattered fields in layer f to the surface source at p = P,. Since the location of field point can be arbitrary, all cases off= st,f= s, f > st andf < s are to be considered. Then, the results of each case are combined to form compact expressions for the scattered dyadic Green’s functions. A. Case f = st, s

Let 2 x 1 column vectors iif and ii? denote respectively the unknown amplitudes to be determined for the scattered outward-bounded and inward-bounded waves in layer f. Referring to Fig. 1 (f = st, s), at the interface p = Pq,the scattered inward-bounded wave in layer st is related to the total (primary + scattered) outward-bounded waves via the global reflection matrix as

- (30) -< = p

a s + st ,st+l . [g> +a;] where ? stand for the amplitudes of primary waves deriv- able from Szy. At the interfacep = P,, the constraint condi- tion requires that the scattered outward-bounded wave in layer st is a consequence of the local transmission of the scattered outward-bounded wave in layer s plus the local

IEE Proc.-Microw. Antennas Propug., Vol. 146, No. 6, December 1999 396

reflection of the scattered inward-bounded wave in layer st, i.e.

Similarly, at the interface p = Ps_l, the scattered outward- bounded wave in layer s is related to the total inward- bounded waves as

- -> a, = p s,s-1 . [s< + a,<] (32)

while at the interface p = P,, the scattered inward-bounded wave in layer s satisfies the constraint condition of

Manipulating eqns. 30-33, one can readily solve a,; and ass in terms of the known primary amplitudes 3". Their explicit expressions will be included in the unified represen- tations to be given later. B. Case f

Consider now the case when the field point is located in layerfbeyond layer st, Fig. 1 (f> st). Within this layer, the scattered outward-bounded wave can be related to the total (primary + scattered) outward-bounded waves in layer st via the global transmission matrix:

st, f < s

At the interface p = Pr, the scattered inward-bounded wave is related to the scattered outward-bounded wave via the global reflection matrix:

- - = Rgf,f+l -a;>,, (35)

Sdar ly , for field point located in layer f beneath layer s, Fig. 1 (f< s), one can follow the same steps to reach

- - 8if\, = T5,f . [S< + a:]

a f <s = Rgf, f -1 . q < s

(36)

(37) -

-> -

Eqns. 3437 can be readily solved for ZiS, and a& in terms of s".

With the aid of Kronecker delta and Heaviside unit step functions, one can unify the above results to obtain the final solutions for scattered dyadic Green's functions as - G e s = I { plf,Z2f] 3'. [s;p,:SI,pa]T

k t T - + p%f,E2f] 37. &,,Skps]

+ P3f,E4f] .z F1p3,SbJT -

+ F3f, Eq] .E . F 3 p 8 , q p , ] '} (38)

Note that the utilisation of effective reflection and transmis- sion concepts has avoided complicated formulation of the

391 IEE Proc-Microw. Antennas Propag., Vol. 146, No. 6, December 1999

scattering coeffcients and at the same time has provided good physical insights to the scattering mechanism. Moreo- ver, this approach has also led to compact and convenient representations of eqns. 38 and 39 in the forms involving matrices of size 2 x 2 only. Together with eqns. 11-14, we have thus obtained the complete dyadic Green’s functions in eqns. 9 and 10 for surface current excitation in planar stratified bianisotropic media.

4 Alternative derivation of source consequents

So far, the source consequents of a two-layered semiinfiite medium 3s have been determined cllrectly from the ele- mentary boundary conditions and are given by eqns. 21 and 22. In this Section, we wdl devise an alternative approach to obtain 3s by considering their relationship with the source consequents of unbounded dyadic Green’s functions. For medium characterised by constitutive parameters of layerf= st, s, the unbounded dyadic Green’s functions have been expanded in the form [ 191

(49)

kt

k t

The source consequents 3kof eqns. 50 and 51 (containing three components) can be derived from the discontinuity relations associated with the expansions of dyadic Green’s functions across p = p’ [19]

1 aw

- - - p x (G&, f - f ) = rtst - -$I: f Vtbt x pp’

- - - -It.(g:fFf + g R f t f ) * @ ’ s t

(53) This approach is analogous to deriving Sjps of eqns. 11-14 (containing two components) from the boundary con&- tions across p = Ps, eqns. 15 and 16. Indeed, using the same P, of eqn. 20 and restricting to sources with transverse

398

components, we find from eqns. 5&53,

-1 - - s;f = [E;. @;)-I ,E; - “;I P=P‘ . V t (55)

where

Notice that eqns. 54 and 55 for S$ bear some resemblance to those of eqns. 21 and 22 for S$. In fact, by examining the expressions of local reflection and transmis- sion matrices in eqns. 23 and 24, one can show that for P‘ = ps7

- - - sgs = Ti,,, . 3gs = sgst + RL,,, f S&

s, - st,s ’ STst = SA + STS (58)

(57) - -

< - p -< - -> -

Eqns. 57 and 58 thus reveal the intimate relationship between the source consequents of a two-layered semiinfi- nite medium and those of an unbounded medium. Moreo- ver, these equations also provide an alternative and more direct approach for determining SSs from the knowledge of S s which can usually be obtained in a simpler manner via Lorentdmodified reciprocity theorem [2, 31

t; layer 2 - isotropic

layer 1 - bianisotropic

PEC wall

z=Z,

TTTTTTTTTTT77 ==z, Fig. 2 A graolded blimimopic slab

5 Application to a grounded bianisotropic slab

To demonstrate how to simplify the general expressions presented above, let us consider a practical configuration (for microstrips) which consists of a grounded bianisotropic slab embedded in isotropic half-space, Fig. 2. Assuming j3 = I , f, = 2, f2 = 9 and M , N, L are constructed with I as the pilot vector [4]:

- - M(F; k ) = [p&, - ~ a J C , ] e i k z ~ + i k , Y + i k c , ~ (59)

1 5

- N( f ; x ) = - [ -Pk,k, - rjjk,k, + Z k ; ] e i k ~ z + ~ b Y + i k * z

k2 = k: + k i , k: = lc; + k i (60) - L ( F ; ~ ) [pis, + j j ik, + 2 i k z l e i k ~ z + i k , y + i k z ~ (61)

In bianisotropic slab (layer l), the electric E.1 and magnetic Rjl eigenfunctions are expressed in terms oflabc coeffcients as in eqns. 3 and 4 for j = 1, 2, 3, 4. IQ the isotropic half- space (layer 2) characterized by F2 = ~ ~ 1 , F2 = h I and E 2 = g2 = 0, the abcs are usually selected such that - - E;2 = X T ( F ; & f k,2&) Ej2 = N(F; Et & k , z i )

(62)

(63)

- Hi2 = q2N(F; Ict f k,22) = q2M(F; Ict f k , z i )

IEE Proc.-Microw. Antennas Propag., Vol. 146. No. 6, December 1999

where k22 = k? - k? (Im kz2 0), q2 = k21iwp2 and k z = 6 Conclusion CL?&.F~. With reference to eqns. 5941, we choose

- 1 --ikLx--ik&y ut2 = - [ -kk ; + $ k i ] e 47r2 lC:2

for eqns. 18 and 19 and find

This paper has presented a rigorous formulation of the spectral-domain dyadic Green’s functions for surface cur- rent excitation in planar stratified bianisotropic media. Both electric and magnetic dyadic Green’s functions have been derived simultaneously by malung use of the principle of scattering superposition. For the primary dyadic Green’s functions associated with a two-layered semiinfinite medium, their source consequents have been determined directly from the elementary boundary conditions across the interface. Alternatively, these source consequents can also be expressed in terms of the source consequents for unbounded dyadic Green’s functions whch can usually be obtained in a simpler manner via Lorentdmodified reci- procity theorem. For the scattered dyadic Green’s func-

(67) tions, their scattering coeficient matrices have been determined using the concepts of effective reflection and transmission of outward-bounded and inward-bounded waves. This approach has avoided cumbersome operations and has also provided good physical insights to the scatter- ing mechanism. Moreover, the resulting expressions have been represented in compact and convenient forms involv- ing matrices of size 2 x 2 only. As an illustration for the application of general dyadic Green’s function expressions, the confirmration of a mounded bianisotronic slab embed-

(64)

(65)

(66)

(68)

(69)

Where kejl = qkzjIlkjI)$ejI + icejI> khjl = <kzjI/kjlZhjl + &hjl and k j Substituting these E?, - pfs into eqns. 23, 24 and 45 and noting that R f 3 = 0, the scattering coefficient matrices for the scattered dyadic Green’s functions reduce to

= k: + kzjI.

f= 1:

-1 - - - - - - D = [I - . E:,0] . . - - _ - A=C=O (70)

B = S ; , , . R y , , A = C = D = O (71)

f = 2: - _ _ _ _ _ _ _ - _ - -

where Elo is given by eqn. 29 for EEC at zz 2,. Notice from above that once EFand RFhave been

worked out from the eigenfunction expansions, the local and global reflection and transmigion matfices can be cal- culated readily. Moreover, these and p? also specify the source consequents S$ directly as stated by eqns. 21 and 22. Alternatively, the source consequents can be expressed in terms of those of unbounded dyadic Green’s functions by means of eqns. 57 and 58. For instance, let us choose the unbounded medium to be characterised by con- stitutive parameters of isotropic layer since its unbounded dyadic Green’s functions can be obtained easily based on Lorentz reciprocity theorem [3]. Then, corresponding to eigenfunctions in eqns. 62 and 63, the source consequents S’ of eqns. 50 and 51 for f = 2 are given by

- - S’ = C 2 B (F’; -k t k z 2 f )

- 3 - Sa2 = C2N (F’; -kt k Z 2 f ) (72)

where C2 = 4w&8n?k?kz2). Extracting their transverse parts according to eqn. 56, one can readily determine SIPS using the local reflection and transmission matrices as rn eqns. 57 and 58. With the source consequents known, we have thus obtained all primary and scattered dyadic Green’s functions for our microstrip configuration.

IEE Proc.-Microw. Antennas Propag., Vol. 146, No 6, December 1999

v v

ded in isotropic half-space has been considered explicitly. Although the main emphasis of the present paper is on seelung the general analytical treatment whch also feature some physical interpretations, there is still much work to be carried out on the numerical aspects as well as on the inves- tigation of specific materials, e.g. those recently proposed novel materials. Besides, since general excitation sources are tridimensional containing normal in addition to transverse components, their associated dyadic Green’s functions for arbitrary source location should be determined as well for completeness. All these works are currently in progress and will be reported in the near future.

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