1. Analysis of Multiple Groove Guide
Hyo J. Eom and Yong H. Cho
Department of Electrical Engineering
Korea Advanced Institute of Science and Technology
373-1, Kusong Dong, Yusung Gu, Taejon, Korea
Phone 82-42-869-3436, Fax 82-42-869-8036
E-mail : hjeom@ee.kaist.ac.kr
Abstract Wave propagation along a rectangular multiple groove guide is rigorously
studied. The Fourier transform is used to obtain simultaneous equations for the modal
coe cients in rapidly-convergent form. The dispersion characteristics of a multiple
groove guide and its eld distribution plots are presented.
1 Introduction
A rectangular groove guide 1] is a low-loss and high-power guiding structure. A
double-groove guide has been extensively studied to assess its utility as a waveguide
or power coupler at 100GHz in 2]. It is of practical interest to understand guiding and
coupling characteristics of a multiple groove guide which consists of a nite number
of parallel rectangular groove guides. The purpose of the present short paper is to
present an exact and rigorous solution for a multiple groove guide by utilizing the
Fourier transform which was used to analyze a single rectangular groove guide 3].
The Fourier transform approach allows us to represent a solution in rapidly convergent
1

2. series. In next section, we present a rigorous dispersion relation for a multiple groove
guide.
2 Field Analysis
Consider a multiple rectangular groove guide in Fig. 1 (N : the number of groove
guides). Assume the TE-wave propagates along the z-direction such as H (x; y; z) =
H (x; y)ei z and the e i!t time-factor is suppressed throughout. In region (I) ( d <
y < 0), (II) (0 < y < b), and (III) (b < y < b + d), the Hz components are
HzI (x; y) =
N 1X
X 1
n=0 m=0
n
qm cos am (x nT ) cos m(y + d)
u(x nT ) u(x nT a)];
1
~
~
HzII (x; y) = 21
Hz ei y + Hz e i y ]e i xd ;
1
N 1
sn cos am (x nT ) cos m (y b d)
HzIII (x; y) =
m
Z
+
(1)
(2)
1
X X
n=0 m=0
u(x nT ) u(x nT a)];
q
p
(3)
, = k
, k = 0 and u( ) is
where am = m , m = k am
a
n
a unit step function. To determine the modal coe cients qm and sn , we enforce
m
the boundary conditions on the eld continuities. Applying the Fourier transform
R
( 1 (:)ei xdx) on the Ex continuity at y = 0, we obtain
1
N 1
n
~z Hz = X X 1 qm m sin( md)Gn ( );
~
H
(4)
m
n m i
where
m ia
(5)
Gn ( ) = i 1 ( 1) e ] ei nT :
m
am
Similarly from the Ex continuity at y = b,
N X 1
X 1
n
n
~
~
Hz ei b Hz e i b =
(6)
i sm m sin( md)Gm( ):
n m
Multiplying the Hz continuity at y = 0 by cos al (x pT ), (p = 0; ; N 1), and
integrating over pT < x < pT + a, we obtain
N X
X 1
n
fqm m sin( md)I + a cos( md) ml np m ] + sn m sin( md)I g = 0; (7)
m
2
n m
2
2
2
2
2
2
2
2
1
+
=0
=0
2
2
1
+
=0
=0
1
1
=0
=0
2

3. where
is the Kronecker delta,
Z 1
cot( b) Gn ( )Gp( )d ;
(8)
I = 21
m
l
1
Z 1
csc( b) Gn ( )Gp( )d ;
(9)
I = 21
m
l
1
8
< 2 (m = 0);
= :
(10)
m
1 ( m = 1; 2; 3; ) :
Applying the residue calculus, we transform (8) and (9) into rapidly-convergent series
as
I = a m ml npb)
2 m tan( m
1
i X v (( 1)m l + 1)ei v jn pjT ( 1)m ei v j n p T aj ( 1)l ei v j n p T aj] ;
bv
v ( v am )( v al )
(11)
I = a m ml np )
2 m sin( mb
1
i X( 1)v v (( 1)m l + 1)ei v jn pjT ( 1)m ei v j n p T aj ( 1)l ei v j n p T aj] ;
bv
v ( v am )( v al )
(12)
ml
1
2
1
+
(
2
=0
2
2
) +
(
)
2
2
+
(
2
=0
q
where v = k ( vb )
region (II) and (III),
N 1X
X 1
2
2
2
2
) +
(
2
2
. Similarly from the Hz continuity at y = b between
a cos( d)
m ml np m ]g = 0:
2
n m
A dispersion relationship may be obtained by solving (7) and (13) for .
n
fqm m sin( md)I + sn
m
m sin( m d)I1 +
2
=0
=0
1
np
;ml
np
2;ml
1
1
and
2
2
2
where the elements of
)
1
are
=
m sin( m d)I1 +
=
m sin( m d)I2 :
= 0;
(13)
(14)
a cos( d)
m
2
ml np m ;
(15)
(16)
When N = 1 (single-groove case), (14) reduces to (33) in 3]. When N = 2 (doublegroove case), (14) in a dominant-mode approximation (m=0), reduces to
00
1 00
;
00
2 00
;
=
(
3
10
1 00
;
10
2 00
;
);
(17)

4. where sign corresponds to the TE and TE modes, respectively. It is trivial
to nd by using a root-searching scheme with an initial guess based on a singlegroove case. Fig. 2 illustrates the dispersion characteristics for a multiple groove
guide, con rming that our solution agrees with 2] when N = 2. Our computational
experience indicates that a dominant-mode approximation with m = 0 is almost
identical with a more accurate solution including 3 higher-modes. Fig. 3 shows the
magnitude plots of Hz component for the TE p modes (p = 1; ; 4) where p signi es
the number of half-wave variation of Hz component along the x-axis. The eld plots
illustrate that Hz remains almost uniform in the x-direction within the groove, thus
con rming the validity of a dominant-mode approximation.
12
11
1
3 Conclusion
A simple, exact and rigorous solution for the multiple groove guide is presented and
its dispersions are numerically evaluated. Our numerical computation for the doublegroove dispersion characteristics agrees with other existing solution. The eld plots
for a quadruple groove guide illustrate the eld distributions of various modes existing
within the groove guide.
References
1] A.A. Oliner and P. Lampariello, "The dominant mode properties of open groove
guide: An improved solution," IEEE Trans. Microwave Theory Tech., Vol. MTT33, pp. 755-764, Sept. 1985.
2] D.J. Harris and K.W. Lee, Theoretical and experimental characteristics of
double-groove guide for 100GHz operation," IEE Proc., Vol. 128, Pt. H. No.
1, pp. 6-10, Feb. 1981.
3] B.T. Lee, J.W. Lee, H.J. Eom and S.Y. Shin, Fourier-Transform Analysis for
Rectangular Groove Guide" IEEE Trans. Microwave Theory Tech., Vol. MTT43, pp. 2162-2165, Sept. 1995.
4

5. y
b+d
III
b
PEC
III
n=0
Region III
.....
n=1
n=N-1
Region II
z
0
-d
a
I
T
T+a
I
. . . . . (N-1)T
PEC
(N-1)T+a
Region I
Figure 1: Geometry of a multiple groove guide.
5
x

6. β, phase constant [ rad/m ]
2028
2026
2026
2024
2024
2022
2022
2020
2020
Triple groove (N=3)
TE12
TE13
TE12
TE13
2018
2018
5
5
Double groove (N=2)
TE11
Quadruple groove (N=4)
TE14
o o Harris experiment in [2]
for double groove (N=2)
10
15
20
10
15
20
T, groove separation [ mm ]
25
25
Figure 2: Behavior of phase constant of TE , TE , TE and TE modes versus
groove separation for o = 3:08mm; a = 5mm; b = 10mm, d = 2:5mm and N =
2; ; 4.
11
6
12
13
14

7. 1
11
1
0.50.5
0.5
0.5
00
−0.5
-0.5
00
b/2
x
0
-1 −1
b/2
3T
-d
x
3T
0
0
y
-d
(a)
1
0
y
(b)
1
1
1
0.50.5
0.50.5
00
0
−0.5
-0.5
0
-0.5−0.5
-1 −1
b/2
-1
3T
0
-d
−1
x
b/2
x
3T
0
0
-d
y
(c)
0
y
(d)
Figure 3: Hz eld distributions for (a) TE mode, (b) TE mode, (c) TE mode
and (d) TE mode when N = 4.
11
14
7
12
13