Abstract
A solution of the diffraction from a dielectric-metallic join due to an incident
Ez-polarized plane wave is presented. Direct diffraction, coupling, launching, and reflection
subproblems associated with a loaded parallel plate waveguide are treated via the dual integral
method. The results are subsequently combined in the context of the generalized scattering
matrix formulation to obtain the diffraction from the truncated parallel-plate waveguide with a
recessed stub and a dielectric loading extending to infinity. The stub is then restored to the
waveguide mouth to obtain the diffraction by the dielectric-metallic join. As expected, the
final expressions involve several Wiener-Hopf split functions and an efficient numerical
technique for their evaluation is given in an appendix. The convergence of the solution with
respect to the number of included modes is examined and a number of scattering patterns are
presented.

TABLE OF CONTENTS
Page #
List of Figures
I. Introduction 1
II. Scattering Matrix Formulation 3
III. Plane Wave Diffraction and Coupling 7
IV. Radiation and Reflection by a Waveguide Mode 20
V. Numerical Results 26
VI. Summary 39
References 41
Appendix A - Expressions for the Split Functions L1,2() and U1,2(X) 42
Appendix B - An Efficient Numerical Wiener-Hopf Factorization Method 44

LIST OF FIGURES
Figure # Page#
1. Geometry of the metallic-dielectric join. 2
2. An illustration of stub geometry (a) and associated individual problems.
(b) Direct diffraction. (c) Coupling. (d) Reflection from the stub.
(e) Reflection at the waveguide mouth. (f) Launching or radiation. 4
3. Illustration of the C and steepest descent path contours in the a-plane 10
along with the chosen branches for the roots -a and K+.
4. Mapping of the contours shown in Figure 3 in the a'-plane, where
cosa=iccosa'. 10
5. Illustration of the C contour in the X-plane, where X=cosa. 14
6. Convergence test of the solution given in equation (1).
(a) 2t = 0.95, er = 2, g = 1. (b) 2t = 0.95k, er = 5 - j0.5,
r = 1.5 - j0.1. (c) 2t = 0.95k, er = 7.4 - jl.l, r = 1.4 - j0.672. 27
7. Ez-polarization calculated echowidth family curves for 2t = 0.01, 0.1,
0.25, 0.5 and 0.75 wavelengths. The constitutive parameters of the
dielectric are er = 2 and gr = 1. (a) Backscatter case. (b) Bistatic with
0 = 45~. (c) Bistatic with o = 150~. 30
8. Ez-polarization calculated echowidth family curves for 2t = 0.01, 0.1,
0.25, 0.5 and 0.75 wavelengths. The constitutive parameters of the dielectric
are er = 5 - j0.5, gr = 1.4 - j0.1. (a) Backscatter case. (b) Bistatic, 0o = 45~.
(c) Bistatic, o0 = 150~. 33
9. Ez-polarization calculated echowidth family curves for 2t = 0.01, 0.1,
0.25, 0.5 and 0.75 wavelengths. The constitutive parameters of the dielectric
are er = 7.4 - jl.l and gr = 1.4 - j.672. (a) Backscatter case. (b) Bistatic, 0o = 45~.
(c) Bistatic, Oo = 150~. 36
B1. Illustration of C1 contour. 45
B2. Illustration of the C2 contour with the permitted values of 0. 45

I. Introduction
The problem of interest is the diffraction by a thick metallic-dielectric join
illuminated with an Ez-polarized plane wave, as illustrated in Figure 1. To the authors'
knowledge, this problem has not been attempted although various researchers have considered
related problems. For example, Bates and Mittra [1], and Uchida and Aoki [2] have studied
the grounded dielectric slab with a truncated upper plate with regard to surface wave excitation
by the dominant parallel-plate waveguide mode. Uchida and Aoki [2], as well as Fong [3],
have investigated the radiation cause by the incidence of the dominant waveguide mode upon
the waveguide mouth. Also, Aoki and Uchida [4] attempted a solution of the closely related
problem of a dielectric-dielectric junction by introducing a Fourier series representation of the
field and its derivatives at the interface. Using such an expansion, Wiener-Hopf equations
were generated in terms of unknown spectral functions related to the total field.
Unfortunately, the authors attempt to obtain an explicit expression for these was not realized.
Instead, their computation involves an iterative solution requiring knowledge of rather
complex integrals and functions whose evaluation is cumbersome and could only be done
approximately.
The dielectric-metallic join problem is of interest primarily as a canonical one in
microstrip structures, since a simple application of image theory yields the solution to a
dielectric slab recessed in a perfectly conducting ground plane. Further impetus is derived
when one considers the presence of composite materials on man-made structures where the
occurrence of material-metallic junctions is common place.
The solution to the diffraction by the metallic-dielectric join is obtained by first
considering the closely allied problem of a parallel-plate waveguide with a dielectric loading
extending to infinity and a perfectly conducting stub recessed a distance from the waveguide
mouth as shown in Figure 2a. Upon a solution of this, it is then a simple matter to extract the
solution for the metallic-dielectric join by setting the distance d to zero. The diffraction of the
1

Fig. 1. Geometry of the metallic-dielectric join.
2

recessed-stub geometry due to an incident plane wave is treated in the context of the
generalized scattering matrix formulation (GSMF) [5]. This method requires a solution to a
number of canonical problems, illustrated in Figures 2b-2f. Specifically, in addition to the
direct diffraction problem, one must also consider the coupling, reflection, and launching of
waveguide modes at the loaded waveguide mouth. Each of these will be analyzed via the dual
integral equation approach [6], which provides a reduction in complexity over the parallel
Wiener-Hopf technique. A crucial step in every one of these problems is the factorization of a
particular complex function into components regular in the upper and lower half complex
plane. Unfortunately, the factorization cannot be done analytically and to circumvent this
difficulty an efficient numerical method is introduced for the factorization of an arbitrary even
complex function.
In the first part of this report, the formal solution to the problem is presented. After
a short summary of the GSMF, the integral equations for Ez-polarization incidence are
formulated in a consistent manner by imposing the necessary boundary conditions. The
coupling and direct diffraction coefficients, as well as the launching and reflection coefficients
are then extracted from a solution of the appropriate integral equations last presented. In the
final part of the report, families of computed scattering patterns are presented for selected
material parameters to illustrate the scattering behavior of the dielectric-metallic join as a
function of slab thickness. The convergence behavior of the formal solution is also examined
with respect to the included number of modes.
II. Scattering Matrix Formulation
The problem to be considered is that of an Ez-polarized plane wave incident upon
the structure shown in Figure 1. In order to apply the GSMF procedure, the stub must be
recessed a distance d into the waveguide, as shown in Figure 2a, forming the genesis of the
3

d e
(a)
(b)
(c)
(d)
(e)
Fig. 2. Illustration of stub geometry (a)
and associated individual
problems: (b) Direct diffraction.
(c) Coupling. (d) Reflection
from the stub. (e) Reflection at
the waveguide mouth.
(f)^^^^/^^^^^^ ^ (f) Launching or radiation.
4

individual problems illustrated in Figures 2b-2f. At the end of the procedure, the distance d is
set to zero restoring the original geometry.
In accordance with the GSMF, the individual problems to be considered are as
follows:
1) Evaluation of the direct diffracted field by the substructure in Figure 2b due to
plane wave incidence. This field will be denoted as
e- jk
FtDD (41 Od S(DD ( 0) -/
where SDD is usually referred to as the diffraction coefficient and (p, 4) are the cylindrical
coordinates of the far zone observation point.
2) Evaluation of the field coupled into the loaded parallel-plate waveguide due to
plane wave incidence as shown in Figure 2c. Hereon we will denote the field associated with
the nth coupled mode as
Cn (Oo = Cn (O) e jkx
where Cn ( 0o) is usually referred to as the coupling coefficient and kn is the propagation
constant associated with the nth mode.
3) Evaluation of the modal field reflected at the stub. This will be written as
rn e w here r is the stub reflection coefficient of the nth mode to the mth mode.
mn Mn
4) Evaluation of the reflected field at the waveguide mouth due to the nth mode.
-jkx
This will be denoted as R e where R is the reflection coefficient of the nth mode to the mth
mn mn
mode.
5) Evaluation of the far-zone radiated field due to an incident nth mode at the
waveguide mouth. This field will be denoted as
5

e- jkp
Ftn ()~ Ln (0)
p
where Ln ()) is usually referred to as the launching coefficient associated with the
nth waveguide mode.
Accordingly, the total far-zone diffracted field by the recessed stub geometry in
Figure 2a is given by
- jkp
z ( ~;d) - SDD ( 0 + MOD' (; 0' r (1)
fp
where SMOD (4, 40; d) is associated with the presence of the stub and, therefore, includes the
contribution of the modal fields within the waveguide. It can be written in a matrix form as
[7]
OD (; d) [ Lm () I ][ Pmn mn] Pmn] [ Rmn]} [ Pmn] [mn] [ Pmn] [ Cn (
(2)
in which the brackets signify column or square matrices depending on whether one or two
subscripts appear, respectively. In addition, [I] denotes the identity matrix and [ Pmn] is the
modal propagation matrix whose elements are given by
4k d
e; m=n
=mn (3)
mn 0mn
t 0; men
Clearly to obtain the far-zone scattered field by the dielectric-metallic join it is only required
to set d = 0 in (1) and (2). In this case [ Pmn] reduces to the identity matrix and
SMOD (4), o) becomes
6

L T i FM4f \AI] [ I I MI
OD ( MOD (q); U) = [ Lm ] J mn mn mn n
(4)
III. Plane Wave Diffraction and Coupling
The plane wave (an eJ~t convention is assumed and suppressed throughout)
jk (x coso + y sino)
Ei=e
z
Hs =-Si jk (x coso+ + y sin~o)
=- Ysinmo e (5)
jk (x coso + y sino)
y= + Y sinO e
is assumed to be incident upon the structure in Figure 2b, where Y = 1/Z is the free space
admittance and 0o is the angle of incidence measured from the positive x-axis.
In the absence of the perfectly conducting half-planes, the plane wave (5) will
produce the following total field
z z
EZ+Ewz {y>0
Epw = 0 Em O> y > -2t (6)
Z Z
Er y <-2t
where
jk (x coso - y sin, o)
Erz = RE e (7a)
jkx cos (1 + RE)sin[k' (y + 2t) sino'] - TE sin(k' y sin4o') e2kt
Em=e -— (7b)
sin (2k' tsino') ]
7

tr jk (x cos+o + y sin o)
R EO(1- Pb Pa) (1- RC Pb 8
RE=- 2 -' E= 2 2 (8)
1 - RR PbP l-R2 PbP
EOFb a EO b a
j4kt cos cotto' -j2k't/sin'o j2kt cos(o, - {')/sin4
P =e ==e P e (9)
C
and REO is the usual plane wave reflection coefficient associated with a dielectric half-space
having relative constitutive parameters er and gr'. In addition
=k'i= =k k, (10)
where K is the refraction index and 0' is defined according to Snell's law as
kcos = k' coso'. (11)
The presence of the perfectly conducting half-plane causes the generation of the
additional field
E - y>< 0
Es = Eb2 -2t < y < O (12)
E3 y <-2t.
These may be represented in terms of an angular spectrum of plane waves [6]. Specifically, a
suitable spectral representation for them takes the form
8

E = J P1 (cosa) e- -o'( da, (13a)
c
F5 = rF B~L)j-jk'p cos(~ + a') -j2k't sinca -jk'p cos( - a)1 (13b)
= [Q (cosa)e + Q (cosa) e d (13b)
C
3 = P2 (cosa) 2kt sina -jkp cos( + a) (13c)
E^ = P^ (cosa) e e da. (13c)
c
where C is the contour on which cosa runs from +oo to -oo as shown in Figure 3 and P1,2(cosa)
with Q1,2(cosa) are the spectra which must be determined via the application of the necessary
boundary conditions. Note also that a and a' are different parameters whose relationship will
be established later. The introduction of the factors ei2kt sina and e-J2k't sina' is totally
arbitrary and could have been omitted. However, such factors are expected to appear in the
final expressions for the spectra and are therefore introduced from the start in order to reduce
the complexity of the resulting integral equations. The corresponding expressions for the
x-component of the scattered magnetic field are
2= Yj sina' [Q \(cosa) ejkP cos(A + a) Q2 (cosa) j2k't sina' ejk'p cos( - a')] da
if- = Y sina P1 (cos() e -jkp cos(e- a) da,(14
C
(14b)
T3 =Y sina P2 (cosa) J2kt sina -jkp cos(, + a) d (14c)
x3 -Y naP e da (14c)
c
where
9

B3a B4
B3b
B3a
Bla B2a2a
s /:/
-WE/2 0 T /2 71 3[1/2 -',r/t/2 n 37t/2
Steepest
Bb / Descent B2b Bla Bib
( / Path
B4a elb
B3b B4b
Fig. 3. Illustration of the C and steepest Fig. 4. Mapping of the contours shown in
decent path contours in the a-plane Fig. 3 in the a'-plane, where
along with the chosen branches for the cosa=Kccosa'.
roots sqrt(K-a) and sqrt(K+a).

Y= |2rY=Y Y
n
The total field due to a plane wave incidence in the presence of the configuration in
Figure 2b can now be expressed as
Eiot= EP+E, (15)
with EPW and Es as defined in (6) and (12) respectively. For its complete determination we must find
the scattered field E, implying an explicit knowledge of the angular spectra Pl,2(cosa) and Q1,2(cosa)
appearing in (13). This will be accomplished in the subsequent sections via the application of
the following boundary conditions:
1) The total tangential electric field is continuous over-o < x < oo, y = {2t}, implying
(B1) EZi =E over-oo < x <, y=,
(B2) EZ =Ez over-oo < x < oo, y =-2t
since EPW is already continuous.
z
2) The total tangential magnetic field is continuous over x <0, y = -2t implying
(B3) HSl =Hx2 over x<O, y=O
(B4) HI2=-Si over x<0, y=-2t
3) The tangential electric field vanishes on the perfectly conductive half-planes, implying
(B5) E'+ErZ+E =0O over x>0, y=0,
(B6) Etr+Es3=O over x> O, y=-2t
11

The application of boundary conditions (B 1) - (B2) gives
/2 2
kcosa = k' cosa' or k'sina' =kl -cos a, (16)
and that
P1 (cosa) = Q1 (cosa) + Q2 (cosa) e-j2kI (17)
2 2 1
Thus, (17) reduces the number of unknown spectra from four to two and a complete
knowledge of the scattered fields can be deduced from Q1,2(cosa) alone. Additionally, the
branch of K - cos a in (16) is choosen such that Im K - cos a < 0 for 0 < Re (a) <. This
defines a mapping from the a-plane to the a'-plane as shown in Figure 4.
Next we employ the boundary conditions (B3) - (B4) demanding that
f[sina Pl(cosa) + Ysina' Ql(cosa) - Y sina'Q2(cosa) ej2k't sina] e'jcosa = 0;
nneo[ = 0; (18a)
x<0,
sina P2(cosa) - Ynsina' Ql(cosa) ej sina + Ynsina' Q(cosa) ex cosa da = 0; (18b)
_00~~ 212
c _X F~)~-jd,=x< 0.
[QI(X)+ ~ Q2(X)] 2 3(X)e-jk d= x<0, (19)
g;0 4
where
12

Fosh - IC — - / " ---....
F()= = {^J77 cosi-i-tj )+q- -/77 - n J7 — )], (20a)
3 2 It 1)A>- 2
(r+ 1) j7-K e
and
2F4k)= [ 2 sinh (kt 2 K2)+ 2 K2 cosh (kt2 ]. (20b)
F4(X)= = --------- -----. (20b)
(tr + 1) -f;;K e 1 7
Note thatin deriving (16) wehave used the relations -1 = J 71- and K= jJ K2
In addition, the complex X-plane defined by the mapping X = cosa is shown in Figure 5.
The boundary conditions (B5) and (B6) imply the integral equations
00
1 \ jkx coso
1 P.l(X)ekx d( = - (1 + RE) e;x>0, (21a)
4-7
00
f 1 -j2kt sino jkx coseo
J _P 2 () ePkx dX = -TEe e; x>O. (21b)
1-k
1 F- (1 e;kx -j2kt sin9o jkx coso
Substituting (17) into (21) and adding and subtracting the resulting equations we find
J[Q (x).Q2 (X)]X d (1+RE~TE e 0;x>
]1.72 2
(22)
with
F1 () = 1 e-j2 /. (23)
2
13

S 1T:i
- K
C -1 -ko 0.-. K
4h
Fig. 5. Illustration of the C contour in the k-plane,
where k=cosa.

The dual integral equations (19) and (22) are now sufficient for a solution of
Q1 (k) ~ Q2 (X). However, before such a solution can be pursued, it is necessary that
F1,2 (x) and F3,4 (x) be factorized into functions regular (i.e. free of poles, zeros and branch
points) in the upper and lower half of the complex X-plane. Utilizing the factorization
procedure outlined in Noble [8], F1,2 () may be factorized as
F1 () = L1 () UI (2), (24)
2 2 2
where the functions denoted by L/U are regular in the upper/lower half of the 2-plane.
Expressions for L1,2 (), U1,2 (2) are given in Appendix A. On the other hand, the
factorization of F3,4 (k) into
F3 (.) = L3 (L) U3 (2), (25)
4 4 4
is much more involved. Notwithstanding, numerical and analytical techniques do exist for
accomplishing it [1,9]. The factorization of F34 (X) herein will be accomplished through a
recently developed numerical procedure with the final expressions of U3,4 (.) and L3 4 (.) in
terms of an integral over the convenient finite interval [0,1], as given in Appendix B. The
utility of this numerical technique also stems from the fact that it may be applied to a very
wide class of complex functions with no special preconditioning of these necessary (i.e.
involved treatment of poles, etc.). In passing, we note that for the special case of K = 1,
F1,2 (2) reduce to functions already encountered [10] and F3,4 (2) = 1.
15

Using the factorizations (24) and (25) we may now proceed for a solution of the
spectra Q,2. Since (19) holds for x < 0, we may close the path of integration by a
semi-infinite contour in the upper half of the X-plane and employ Cauchy's theorem along with
(25) to deduce that
[Q1 (X) ~ Q2 ()]' L (h) U3 (X) = UA (X) (26)
/1-$2 4 B
where UA,B (X) are unknown functions regular in the upper half of the X-plane. Similarly,
(22) holds for x > 0 enabling us to close the path of integration by a semi-infinite contour in
the lower half of the X-plane and again invoke Cauchy's theorem along with (24) to obtain
-j2kt sinAo LA
__i ) 1 (1+R~Te S LA (X)
[Q, (O) Q2 (X)] L L ) (U (X) = 1 E
-l;,2 2 2icj X + LA (-cos0O)
2 2 2L^(-COS(o))
B
(27)
where LAB (x) are unknown functions regular in the lower half X-plane. Substituting (26) into
(27) it may be deduced that
-L1 (X)
LA (X) =,(28)
B L`3 ()K+
4
16

1 -j2kt sin, IC-_ _-_ U3 (,_) U3 40)4
UA(X)-(1 +RE~TEe ) 0-) e (29)
B 2icj + X+ U1 (o)U (X)
2 2
since LA,B and UAB are associated with different regions of regularity. Finally from (29) and
(26) it follows that
Qi(~)=1 41+X 1-h 4~-~ I+RETE+T
Q1 ((X) =-1 1T F { (i + Sife) U3 (X)
2 4tj: + ) + Xo L3 (X) U1 (X) U1 (X0)
(1 +RE-Te-j2kt sin) U4 (X) (30)
L4 (x) U2 (X) U2 (Xo)
and from (17)
P1 (x) 1 E i +XV {^ (1 +R + T e-j2kt sin (1 (X) U3 (X0)
P1 (X) - /'7Y j'3J j 1{+RE+TEe I ) —:U
2 47ij +: + E E L (X) U, (xo)
/ lREE -j2kt sino) L, (X) U4 (,o) 1
~(1 +RE-TEe L) 4 e (31)
L4 (X) U2 (Xo)
where XO = cosO.These may now be substituted into (13) to obtain the field scattered by the
loaded parallel-plate waveguide. This requires an evaluation of the resulting integrals as
described next.
To compute the field diffracted by the geometry in Figure 2b, (31) is substituted in
(13) and a steepest-descent-path approximation is performed for large kp. Noting that the
pertinent saddle point is at a = ) when ) < X and at a = 2n - ) when 4 > n we find that
-jkp
ix-SD[ (*, ~') Cp1
17

where SDD (), 0,) is the direct diffraction coefficient given by
ei/4 /1 + cosi4.n....
DD 22k /lc + cos cos) + cos +e 2kt
k( L1 (-32 cos)) (oU -2 (cos) ) U4 (cosp)d
{(1 +RE+TEe — ) _+(1 + R -Te 0),
L3 (cos)) U1 (cos(o) L4 (cos) U2 (cos,)}
(32)
in which the upper sign holds for 0 < ) < i and the lower for i < ) < 2x.
For the computation of the field coupled into the waveguide (x>O), kp cannot be
assumed large. Therefore, one must employ a technique other than the steepest descent
method for its evaluation. A standard procedure is to transform (13b) to the X-plane giving
ES = Es 1 = j { ( +RE +T e-j2kt sin)o U3 (x0)'-"z z2 2cj F + Te
^2 L^j " cK + (k + x0) L3 (X) UI (x) U1 (xo)
cos [k(y+t) Y ] +j (1 +R e-TEj2t s) U4( sn [k (y+t) 7 ] }
e-jkt;-_ ejkx dX. (33)
Since x>O, the above integrals can be evaluated via the residue theorem after closing the path
of integration by a semi-infinite contour in the lower half of the X-plane. In doing this, it
should be remarked, that the above integrand does not have a branch at x = K. Noting now
that U1,2 (X) have zeros at
18

k= = K2- (IC2, (34)
we find
sCos nr ( -jkx Xn
^ES=E2= Y cnsi[ y - +t)]e (35)
z z2 n 2t
=l1,2...
where
n ":jc n2k~t - (1 + RE 2kt si
rC ( M0 je 2 ( ) CS+oxR +~TE
e flln +
U3 (cos o)
---— 4 (36)
U1'(Xn)U (cos )o)L3 (an)
2 2 4
are the coupline coefficients. In the above, the subscripts e denote odd and even n, respectively, and
dU1 (X)
Ul (,n)= 2
2 dX X=Xn
2 aCl ^ \n
As a check, we note that when er = gr = 1, (32) and (36) reduce to the known
expressions given in [10]. In passing, we also note that if we were concerned with the modal
fields in the region-x < 0, we would also have to consider the residues of the poles
corresponding to the zeros of L3,4 (cosa). These are precisely the surface wave modes of a
dielectric waveguide. Furthermore, any branch-cut contribution would also have to be
included.
19

IV. Radiation and Reflection by a Waveguide Mode
This problem is illustrated by Figures 2d, 2e, and 2f. The modal field (see (34))
i cos nm %n
COSin - (y + t) eJ n
ZOV~n sinl 21
Hi..Y mn sin Ajkx X
Xo — T ^co ~( +t) e
is now assumed to be incident toward the waveguide mouth and present throughout the region
-oo < x < oo, O > y > -2t.
Our solution for the radiated field and that reflected back into the waveguide will
follow the same general steps employed in the plane wave incidence analysis. The sum of the
radiated and reflected fields are now the scattered fields and since they are solely caused by
the currents on the perfectly conducting half-plane they can again be represented by (13) -
(14). In addition, all of the boundary conditions (B 1) - (B6) stated earlier are still valid.
Their mathematical forms are now given by
(Bl) E =E2 over -oo <x<, y = 0,
(B2) E2 =Esz3 over -< x<oo, y=-2t,
(B3) Il1=Hx2+ H over x<0, y=0,
(B4) Hfx= IH2+ H over x<O, y=-2t,
_X3 x2 x
(B5) Ez =0 over x > 0, y=O,
(B6) E>3 =0 over x>0, y=-2t,
since E' is already zero over the perfectly conducting half-planes.
20

Application of the boundary conditions (B1) - (B2) again result in the relations given
by (17). Thus the determination of Q1,2 (cosa) is our only remaining task. To find these, we
proceed with the application of boundary conditions (B3) and (B4), then add and subtract the
resulting equations to obtain
X M-j im jkx n
co 2 2 ~lul-J ) sin(T) e
[QL (X) + Q2 ( iJ ({) Ut () eix);x<O
e e
0
(37)
J- * 0 r+1 kt 2x <
(38)
By enforcing boundary conditions (B5) - (B6) and again adding and subtracting the resulting
equations, we also have that
00
fj[Q () +Q2 ] 1 L] (L)U ())e'xdk=0; x>O, (39)
e e \/;r
00
J[Ql ( -Q20)] 1 L U2 (x) e'jkx d = O;x>O > (40).~ e e
Equations (37) with (39) and (38) with (40) form again a coupled set sufficient for
the solution of Q1,2 (X). In proceeding with this solution, we note that since (39) - (40) are
21

valid for x > 0, the path of integration may be closed by a semi-infinite contour in the lower
half of the X-plane, giving
[Q1 (X) + Q2 (X)] 1 L (X) U1 () LA () (41)
e e e
[Q1 ()- Q2 (X)] - L2(X)U2 () =LB () (42)
~~~~e e 5 / ~e
where LA (X) and LB (x) are again unknown functions regular in the lower half X-plane.
Similarly, because (37) - (38) apply to x < 0, we may close the path of integration with a
semi-infinite contour in the upper half of the X-plane resulting to!t~~~It
-n in(T) UAO()
______ (UX) +t Q2~h (-) X
[Q () + Q20 ()] 2K L3 (X) U3 (k) = ( + 1) 2kt + X U () (43)
e e f 7
UAe () EA ()
UB (X) E (X)
[QL1 )- Q2 (x))] U() = cos( ) U (X) (44)'e e 4 X|"(r + 1)2kt X+ X UB (-n)
where UA, UB are unknown functions regular in the upper half of the X-plane and EAB (k) are
unknown entire functions whose justification for the appearance of will soon become apparent.
Substituting (41), (42) into (43), (44) respectively, and equating regions of regularity we find
22

U3,() JK-X
UUA () = U ()= (45a)
e U (X)
U4 (k) -
UB (X)= UB (X) =, (45b)
-sin ( - ) ___- _ L1 L1 (_n)
I, nre i-^-*
(r + 1); JK(:+X ) L (X) L3(n) (4)
Lo(k)= (45c)
L1 (X) E (
IC (X)K+
L,(| )
L2 (k.)
L4 ( +) K: +
LiB () = nx (45d)
cos (2 n L2 () L2 (45n)
lr + 1) (: + + n) L4 (X) L4 (n)
We may now use (17), (43), (44), and (45) to determine the spectra P12 (X) as
pl_(X) =E (I) m- ( 1- C r- L (I)L1 (Xn)
I(C)S 2______ ______ _________ _n 2 2_
e 27 L4(X) A (i r+1)27t K+ +n +3( L Xn)
3 4 4
(46)
23

1 cs 1+ - Xn 2 2
(2 (-) EB (" --
e F + m' Xt -'"
e 2/Kc+L 4) A ({r+1) + 2:h 4+ + n L(X)L3n)
3 4 4
(47)
with the evaluation of EA (k) and EB (X) remaining. From a straight forward examination of
the field behavior at the plate edges, it can be shown that [9] P (X) - X-1/2 as Re (k) - oo and
since L1,2 (X), L3,4 (X) - 1 as Re (X) -> oo, one concludes that
EA()=EB (X)=. (48)
Consequently, we find that
pos () I+r pJ1-j I7-7 L1 (X) L2('n)
P, (X)=+P2 () =;+ Y 2 (49)
0 (k ) + 2) 0 -.. 2
e e (r + 1) 27 / + X n Ul () L3 (X) 3 (n)
4 4
(50)
descent path evaluation of the resulting integrals for large kp we obtain
24

e-jkp
zE (7) (51)
e e p e
where the launching coefficients Ln ()) are given by
n0
e
Asin( ) 4I
je-J i4 1 TS /2 sin ( 2) KX L (cos)L (n)
L je 2 + coS 2 2 2
no r+ IS L
e BcOS(n) | (r + 1);C_ COS<_ + n L3 (COS)) L3 (:)
4 4
Bcos( ) 4 4
inwhichA= -1,B= 1 whenO< ) < andA=B= -e -2kt si when <)<2n.
The field reflected back into the waveguide is E2 in (13) and is evaluated by the same
procedure employed for the coupled field. Specifically, we transform the integration path to
the k-plane and invoke Cauchy's theorem after closing the path of integration with a
semi-infinite circle in the lower half of the X-plane to obtain
c c \r COs mni -jkx km
Ez = g= Rn 2( + t) s e (53)
e e m=1,2... e e
where
4kt ^cs"0') I~ - I Tm c- 1 () \ - - T
cos 2 n 2
m ono nXn+Xm
me me (~r + 1) + m U' ('m) L3 (m) L3 (Ln)
2 4 4
(54a)
25

R =R =0 (54b)
m1e^ meno
are the reflection coefficients.
Finally, the matrix elements rmn due to reflection from a perfectly conducting slab
are given for the Ez-polarization by
-1, m=n,
F[r~~~~~~~~~~~ mn~= m~~n,(55)
mn 0, m~n,
implying that - [ rmn] is the identity matrix.
This completes the analysis required for the evaluation of the diffracted field Es by the thick
metallic-dielectric join shown in Figure 1. Below we present some numerical data which
describe the scattering behavior of the metallic-dielectric join as function of thickness and the
dielectric's constitutive parameters.
V. Numerical Results
Before proceeding with the computation of the diffracted field by the join as given
in (1), it is essential to first determine the minimum number of modes required to achieve
convergence of the infinite sum implied in (4). Such a test was performed in [10] for the thick
perfectly conducting half-plane, equivalent to the dielectric-metallic join with er = gr = 1.
However, the conclusions in [10] are incorrect due to a programming error in the computation
of the higher order modal fields and thus one cannot draw expectation from these results.
The convergence behavior of the backscattered field is presented in Figure 6 for a
dielectric-metallic join.95-wavelengths-thick and for the three sets of material parameters
(Er = 2-j.0, Lr = 1-j.O), (r = 5-j.5, gr = 1.5-j.1), and (er = 7.4-jl.1, [r = 1.4-j.672). In each case,
26

co I I' I I
0 modes
Backscatter 0 modes
E,= 2, = 1 o —- 1 modes
o L, 2t=0.951 — A 2 modes
e~ 2t = 0.955L
2t -=t-+ 3 modes
3_
A
QQo
r'0 f
U
LU
-c
CU
~U (
uJ
cU7t)~ ~~27
U
U,
Cu
Ce
co
0.00 30.00 60.00 90.00 120.00 150.00 180.00
Rngle in Degrees
(a)
Fig. 6. Convergence test of the solution given in equation (1).
(a) 2t = 0.95., e = 2, pr = 1.
(b) 2t = 0.95X, es = 5- j0.5, $xr = 1.5 j0.1.
(c) 2t = 0.95X, er = 7.4 -jl.1, gr = 1.4 -j0.672.
27

Backscatter 0 modes
r= 5.0 - j0.5, r= 1.5 - j0.1 -- - 3 modes
c= 2t = 0.95k - A- 6 modes
~ --
-N:3t
A
c —
r_ Il
L-.
o._ L L \ I,
CU
Ic
0'00 30.00 60.00 90.00 120.00 150.00 180.00
Li
(%J'0.00 30.00 60.00 90.00 120.00 150.00 180.00
Rngle in Degrees
(b)
Fig. 6. Cont.
28

Backscatter 0 modes
r =7.4 - j1j0.672 gr 14 modes
- I -
A
co 00 30.00k 60.00 90.00120.007 modes,
C:I
-c'
— I
3"
-C\
U / \
LLJ
I
(-3V)~~~~~~~~~()
Fig. 6. Cont

Backscatter 2 t
E=2-, = 1 0.01
o -— 9 —- 0.1
m A!| —---- 0.25
_ —'^ -- 0.5
A - -X —- Q 0.75
CC
LLi
(a) Backscatter case. (b) Bistatic with O = 45~ (c) Bistatic with O = 150~.'30 30.00 60.00 901
(a)
Fg7E-oaitoclltdcwdha le30= 010
-i'
c, 02
-oo~~~~~~~~~~~~~a
C:3. 7. — oaiaincluae coit aiycre o t=00,01.5. n
r'-~~.7' " eegh.Tecntttieprmtr ftedeecrcaeE,t,=
-— Ia akcatrcs.()Bittcwt,= 5.()Bsaicwt 50
li'='. ~~~~~3

3 I
cm
Bistatic, 0 =45~ 2tl/
0.01
Er 2, pT 1
c m — 0 — 0.1
NI ~t || -- & — 0.25
|- -- + 0.5' O0.75
- -
A' c
4-J
331
LCd
0.00 60.00 120.00 180.00 240.00 300.00 360.00
Rngle in Degrees
(b)
Fig. 7. Cont.
31

Bistatic, = 1500 _ 2t'0.0.0.01
ngle in D2, egrees
o) 0 -- - 0.125
0.5
- x —- 0.75
(c)
3Ci
4c4'0.00 60.00 120.00 180.00 240.00 300.00 360.00
Rngle in Degrees
(c)
Fig. 7. Cont.
32

cD
Backscatter 2t/x
0.5
Lr = 5- j0.5, r = 1.4- j 0.1
+ ----- 0.75
0.00 30.00 60.00 90.00 120.00 150.00 180.00
CD C=C'~~~33
rc"
0-'Ui
4-Jc
Co
cm'0.00 30.00 60.00 90.00 120.00 150.00 180.00
Rngle in Degrees
(a)
Fig. 8. Ez-polarization calculated echowidth family curves for 2t = 0.01,0.1, 0.25, 0.5 and
0.75 wavelengths. The constitutive parameters of the dielectric are er = 5 - j0.5,
Pr = 1.4 - j0.1. (a) Backscatter case. (b) Bistatic, %, = 450. (c) Bistatic, ~o = 1500.
33

Bistatic, 4 = 45~ 2t
0.01
erI = =5-j0.5, r 1.4 - j0.1 I
c=0.00 60.0 120.00 180.00 240.00 300.00 360.00
ngie in Degrees0.2
()0.5
--— SC~ — x 0.75
A
C=
— 34
LL —
CI
0.00 60.00 120.00 180.00 240.00 300.00 360.00
Rngle in Degrees
(b)
Fig. 8. Cont.

Bistatic, % = 150~ 2t0/
0.01
E =5- j0.5, r= 1.4- j0.1
a ---- 0.25
0.5
------ 0.75
A
c(
=6
(c)
Fig. 8. Cont.
35

C r l
2t/X
Backscatter — 0.01
= er = 7.4- jl.l, r = 1.4 - j0.672 l 0.1
SC~ ----- 0.25
j| — ==- 0.5
- I,-. I -. —,X —-- 0.75:3
A
QJ 2I
L4-JU
Li
Lm
(1)
0.00 30.00 60.00 90.00 120.00 150.00 180.00
Rngle in Degrees
(a)
Fig. 9. Ez-polarization calculated echowidth family curves for 2t = 0.01,0.1,0.25, 0.5 and
0.75 wavelengths. The constitutive parameters of the dielectric are r = 7.4 - j. 1
and Ir = 1.4 - j.672. (a) Backscatter case. (b) Bistatic, 4o = 45~. (c) Bistatic,, O= 1500.
36

2t/X
Bistatic, PO = 450~ 01
Er= 7.4 -j l. l| | _- 0.1
| —A — 0.0125
=1.4-jO.672 | 0.5
nge in Der —-— s75'C:3
4-J
=3
L-Hc —
0.00 60.00 120.00 180.00 240.00 300.00 360.00
Rngle in Degrees
(b)
Fig. 9. Cont.
37

Bistatic, % = 150~ 2t/
0.01
r = 7.4-jl.l, = 1.4 -j0.672 0.
" | - -- 0.25
0.75
m
(c)
co
4-JO
o3
cm
LIJ
0.00 60.00 120.00 180.00 240.00 300.00 360.00
Rngle in Degrees
(c)
Fig. 9. Cont.
38

it is seen that the scattered field is well-converged with the inclusion of modes up to the first
evanescent one. It is also apparent that an increase in material loss decreases the significance
of the modes, allowing the converged field to be more closely approximated by the direct
diffracted field.
Having established the convergence of the solution (1), we may now proceed with
the computation of the field scattered by the dielectric-metallic join given in Figure 1.
Backscattering and bistatic echo width curves are given in Figures 7 - 9 for dielectric-metallic
joins of thickness varying from 2tfX =.01 to 2t/A =.75. Each figure includes backscatter and
bistatic curves corresponding to one of the material parameter sets chosen above.
Unfortunately, comparison with alternate solutions is not possible since to the author's
knowledge neither experimental nor alternative analytical results are presently available.
VI. Summary
The dual integration approach has been used along with the generalized scattering
matrix formulation to obtain the field scattered from a dielectric-metallic join for an
Ez-polarized incident plane wave. This was accomplished by first considering the diffraction
from the loaded parallel-plate waveguide with a recessed stub (Figure 2a) and subsequently
restoring the distance between the stub and the waveguide mouth to zero. The solution to the
recessed-stub geometry was formulated via the GSMF, requiring in turn solution of the five
subproblems illustrated in Figures 2b-2f. We initially considered the subproblems of the direct
diffraction and mode coupling of an incident plane wave on the loaded parallel plate
waveguide. The problems of radiation and reflection due to waveguide mode incident upon
the waveguide mouth was considered next. In both cases the scattered field was expressed by
a suitable angular spectrum representation involving unknown spectral functions. These were
then determined via application of the necessary boundary conditions and explicit expressions
for the scattered fields were subsequently obtained by employing an asymptotic or residue
39

series evaluation of the pertinent integrals. Implicit in this analysis is the Wiener-Hopf
factorization of several functions and in some cases this was accomplished via numerical
means using a new technique described in Appendix B.
At the end of the report, the convergence behavior of the scattered field with
respect to the number of included modes was examined for three different material
compositions and it was found that inclusion of modes up to the first evanescent one provided
a well-converged result. It was also seen that an increase in material loss de-emphasized the
modal contribution to the diffracted field. Finally, families of backscattering and bistatic
echowidth curves for join thicknesses ranging from 0.01X<2t<.752 were presented for various
sets of er, gLr
40

REFERENCES
1. C.P. Bates and R. Mittra, "Waveguide Excitation of Dielectric and Plasma Slabs,"
Radio Science, Vol. 3, No. 3, pp. 251-266, March 1968.
2. K. Uchida and K. Aoki, "Radiation From and Surface Wave Excitation by an
Open-Ended Dielectric-Loaded Parallel-Plate Waveguide," The Transactions of the
IECE of Japan, Vol. E-67, No. 4, pp. 218-224, April 1984.
3. T.T. Fong, "Radiation From an Open-Ended Waveguide With Extended Dielectric
Loading," Radio Science, Vol. 7, No. 10, pp. 965-972, October 1972.
4. K. Aoki and K. Uchida, "Scattering of a Plane Electromagnetic Wave by Two
Semi-Infinite Dielectric Slabs," The Transactions of the IECE of Japan, Vol. 62-B,
No. 12, pp. 1132-1139, 1979.
5. J.R. Pace and R. Mittra, "Generalized Scattering Matrix Analysis of Waveguide
Discontinuity Problems," Proc. Symp. Quasi-Optics, Vol. 14, Brooklyn, N.Y.,
Polytechnic Inst. of Brooklyn Press, pp. 177-197, 1964.
6. P.C. Clemmow, "A Method for the Exact Solution of a Class of Two-Dimensional
Diffraction Problems," Proc. Roy Soc. A., Vol. 205, pp. 286-308, 1951.
7. S.W. Lee and R. Mittra, "Diffraction by Thick Conducting Half-Plane and a
Dielectric-Loaded Waveguide," IEEE Transactions on Antennas and Propagation,
Vol. AP-16, No. 4, pp. 454-461, July 1968.
8. B. Noble, Methods Based on the Wiener-Hopf Technique, Chapter 1, Pergamon, 1958.
9. R. Mittra and S.W. Lee, Analytical Techniques in the Theory of Guided Waves,
pp. 4-11, MacMillan, 1971.
10. J.L. Volakis and M.A. Ricoy, "Diffraction by a Thick Conducting Half-Plane," IEEE
Transactions on Antennas and Propagation, Vol. AP-35, pp. 62-72, January 1987.
11. R.D. Coblin, "Scattering of an Electromagnetic Plane Wave From a Perfectly
Electrically Conducting Half-Plane in the Proximity of Planar Media Discontinuities,"
Ph.D. dissertation, Univ. of Mississippi, 1983, pp. 122-129.
41

Appendix A
Expressions for the Split Functions L1,2(X) and U1,2(X)
The split functions U1,2 and L1,2 arise in the factorization of the functions
_j2kti'2_t
(1 + e ) as follows:
1 + e= U1() L(). (Al)
1I - (A1)
2 2
The U functions are free of branch cuts, poles and zeros (i.e., regular) in the upper half of the
X-plane shown in Fig. 5. Similarly the L functions are regular in the lower half of the X-plane.
These functions may be derived using the procedure given by Noble [8]. The appropriate
expressions for the eJ~t convention employed in this paper are
-.l-i i ~~~,"~ ~~(f-j xp2kt (A2)
U(X) = Ll(-X) = exp[-T(X)-XI(X)](.l 1 (- n) exp(j2k/n) (A2)
U2(=) L(= L(-l) = ei exp[-T(X)-X2(X)l ( X-) ) exp(j2kt2ni),
L-2(-X)~~~ e n=2,4,6 k nic
(A3)
where
T(W) -= 1- os K (A4)
X2()=- 0.4228 +In ( ln2 +- (A)
42

and
X = K-. (A6)
The branch of the above logarithm is defined such that -KT < Im(ln) < n, while all other branches
in the above expression are explicitly defined in Figures 3, 4, and 5.
43

Appendix B
An Efficient Numerical Wiener-Hopf Factorization Method
A crucial and major step in obtaining a solution to a Weiner-Hopf equation is the
factorization/splitting of an even function F(a) into a product of two functions such that
F(a) = L(a) U(a) (B1)
where a = a+jr. In the above, U(a) is free of zeros, poles and branch cuts (i.e., regular) in the
upper half of the a-plane (T>T.) shown in Figure B 1, while L(a) is regular in the lower half of
the a-plane (t<t+), where T. < T+. To accomplish the factorization (B1) we must generally
assume that F(a) is regular within the strip T. < T < T+, where ti are allowed to approach
vanishing values. If we further demand that F(a) -+ 1 uniformly as Iol -- oo within the strip,
then U(a) and L(a) are formally given by [9]
U(a) = L(-a) = exp[H(a)], Im(a) > 0 (B2)
where
H(a)= 1 h[F( )]d (B3)
2ij C -a
- 1
with C1 as shown in Figure B1. Note, however, that this last condition on F(a) does not
necessarily restrict its form since any F(a) can be modified as such in a recoverable manner.
Additionally, due to the even property of F(a), we may set t. = -t+ implying that the contour
44

Jk~
* a
C1
Figure B1. Illustration of C1 contour.
O,,
ZA
* 82
Figure B2. Illustration of the C2 contour with the
permitted values of.
45

C1 (T = 0) remains within the strip 1I1 < T+ as T+ become arbitrarily small.
Despite its general applicability, (B2) contains several barriers to its direct numerical
implementation. In addition to displaying infinite limits of integration, the contour integral
possesses an integrand which may become singular depending on the contour's location. Further,
care must be exercised to insure that a proper branch of the logarithm is taken so that ln[F(f3)] is
continuous on C1. Under certain circumstances, however, these problems can be largely
alleviated via appropriate modification of (B2).
To this end, suppose that in addition to being regular in the strip Itl < T+, F(3) is also
regular in the angular sector ([3(t) = teiJ0; 0< <0, 0 < t < o00 for some Q0 and F(PF(t)) - 1
uniformly as t - oo throughout the above sector. Since F(3) is even, this also implies it will
have the same properties in the additional angular sector ([3(t) = tei; Kt < 0 < C+O0, 0 < t < 00.
Further, it should be noted that in general most functions requiring factorization in diffraction
theory are of this type and, therefore, this is not a significant restriction on F(a). With the above
provisions on the regularity of F(3), it follows that C1 can be rotated counter-clockwise about
the origin by an angle 0(0 < 0 < Q0) to contour C2, as shown in Figure B2. H(a) is thus
modified a
H(a) = h(O - a) ln[F(a)] +1 J [F() d, (B3)
2cj 3-a
2
where
Ofor 0<a,
h(0 - Oa) - (B4)
1 for 0>0 a,
and
45

Im(a)
Ha = arctan Re(a (B5)
Addressing the singularities associated with the integrand of (B3), it is clear that the
numerator becomes infinite when 3 = On, where 3nn = 1,2,...N are the N zeros of F(3).
By virtue of the stipulated regularity of F(P3) in the angular sectors defined above, Pn are,
however, precluded from lying upon the contour C2. Nevertheless, as 0 approaches 0 or 00, it
is possible for any of the Pn to become arbitrarily close to C2 (for 0 - 0, this is true if te tend
to zero, which is often the case). Fortunately, the resulting singularity is logarithmic, implying that
it need be removed only a small distance from the contour C2 to substantially reduce the
singularity of the integrand. The obvious solution, therefore, is to restrict the permissible angular
variation to 61 < 0 < 0 - 62, where 812 are small angles which may be determined empirically.
This scheme will work provided On are not in the vicinity of the origin, which prevents the
contour in being distanced from the pertinent On via a simple rotation.
In contrast to the numerator which may display a multitude of singularities, the denominator
contains a single zero at 3 = a. Recall that the only condition on a is that Im(a) > 0, admitting
the possibility of a lying close to or upon C2. The simplest method to prevent this is to again
Im(a) 1
impose restrictions on, so that 0 e [ a+ e, 0 a- e],where 0 = arctan IRe) and e is a
a [Re(a)
small angle to be determined empirically. However, the involved singularity is of higher-order
than the one previously encountered, implying that e >>. This greatly restricts the permitted
range of 0 if 0 < Oa < Oo, and is undesirable since the quantity ln[F(f3)] may exhibit different
rates of convergence as P(t) - oo among the admissible 0. One would therefore prefer to
47

choose an optimum path C2 from the standpoint of numerical accuracy. Hence, it is of interest to
reduce the restricted range [0o + e, 0a - e] so that the likelihood of this range superimposing
itself upon a region of optimum convergence is minimized. An appropriate modification to (B3)
for accomplishing this is considered next.
The integral in (B3) is obviously not convenient for numerical implementation and it is
therefore necessary to rewrite it for that purpose. By introducing the substitution P = teJ along
with the even property of F(3) we obtain
H(a) = h(O -0 )n[F(a)] + ln[F(t)] dt, (B6)
jri~A'20 2
J o t2e20 -_ a
which presents a numerical difficulty when the pole at t = ae-j is near the real axis. This can be
treated via an addition-subtraction process, provided a 0 C2. Specifically, we add and subtract
to the numerator its value at t = ae-J. By evaluating the additive term analytically, (B6)
becomes
H(a) = ln[ F ( t~) lnle(a)] dt (B7)
2 Xj i tJ 0 -a
where the integrand is now regular at t = aei-O. This effectively reduces e to the order of 5,
increasing the range of allowed 0 and thus eliminating the concerns noted in the previous
paragraph.
A final obstacle to the numerical evaluation of H(a) is the infinite upper integration limit of
2
(B7). This may be remedied via the change of variables [11] v = 2 axctan t to obtain
48

1J 1iln F{e" tan J ] - ln[F(a)]
H(a)= ln[F(ac)] + 2j- dv. (B8)
0 2. 2 _; 29 2 2 v
sin - a cos 2
The integral can now be easily evaluated especially if 0 is chosen such that F(v) exhibits a rapid
decay as it increases from 0 to 1.
Expressions (B2) along with (B8) provide a complete prescription for factorizing an even
function regular on the strip 11t < T+ and the angular sectors 0< < 0, < < 0 < < + Q0. The
integral in (B8) is over a convenient finite interval, and will be numerically tractable for 51 < 0 <
00 < 82, provided 0 e [Oa + e, Oa - e] if 0 < Oa < 00 and the zeros of F(3) are not too close
the origin. This allows a selection of 0 such that the numerical accuracy of (B2) and (B6) is
optimized. Additionally, care must be taken in defining the branch of the logarithm in (B8) so that
FIr F\ ej tan (- 2- remains continuous on the path of integration, eliminating a branch cut
contribution.
49