Dispersion equation for groove nonradiative dielectric waveguide

1. A dispersion equation for groove nonradiative
dielectric waveguide
Yong H. Cho and Hyo J. Eom
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 A rigorous expression of the dispersion equation for a groove nonradia-
tive dielectric (GNRD) waveguide is obtained with the Fourier transform and mode-
matching. The image theorem is utilized to transform a GNRD waveguide into an
in nite number of GNRD waveguides, thereby representing the dispersion equation
in fast-convergent, numerically e cient series.
1 Introduction
The groove nonradiative dielectric (GNRD) waveguide is a one of major transmission
line components in millimeter-wave applications. The waveguiding characteristics for
the groove nonradiative dielectric waveguide 1, 2] have been extensively studied, using
the numerically-intensive techniques. Although the dispersion characteristics for the
GNRD waveguide are well understood, it is also of fundamental interest to obtain a
dispersion equation in a rigorous and analytic closed-form. Our dispersion equation,
based on the technique of the Fourier transform, mode matching, and image theorem,
1

2. is rigorous, analytic, and rapidly-convergent series, so that it is e cient for numerical
computation. The use of simple, analytic dispersion equation may facilitatethe design
of millimeter-wave antenna and directional couplers using the GNRD waveguides.
2 Field Analysis
Consider a GNRD waveguide as shown in Fig. 1(a). Applying the image theorem
with the electric walls placed at x = (a T)=2, we transform the original problem
of Fig. 1(a) into an equivalent problem of Fig. 1(b). The propagating longitudinal
eld components are represented as Ez(x;y;z) = Ez(x;y)ei( z !t)
and Hz(x;y;z) =
Hz(x;y)ei( z !t)
. Then in each region, the guided waves are
EI
z(x;y) =
1X
n= 1
1X
k=1
pn
k sinak(x nT)e i kyun(x) (1)
HI
z(x;y) =
1X
n= 1
1X
m=0
qn
m cosam(x nT)e i myun(x) (2)
EII
z (x;y) = 1
2
Z 1
1
~E+
z ei y + ~Ez e i y]e i xd (3)
HII
z (x;y) = 1
2
Z 1
1
~H+
z ei y + ~Hz e i y]e i xd (4)
EIII
z (x;y) =
1X
n= 1
1X
k=1
rn
k sinak(x nT)ei k(y b)
un(x) (5)
HIII
z (x;y) =
1X
n= 1
1X
m=0
sn
m cosam(x nT)ei m(y b)
un(x) (6)
where am = m =a, m =
q
k2
0 a2
m
2
, =
q
k2
2
2 2
, k0;2 = !p 0;2, un(x) =
u(x nT) u(x nT a), and u( ) is a unit step function. To determine the unknown
modal coe cients pn
k; qn
m; rn
k; and sn
m, we enforce the boundary conditions on the eld
continuities. We apply the Fourier transform (
R1
1( )ei xdx) to the tangential E eld
continuities at y = 0 and b to obtain
~E+
z + ~Ez =
1X
n= 1
1X
k=1
pn
kFn
k ( ) (7)
~H+
z ~Hz =
1X
n= 1
1X
k=1
1
i pn
kAkGn
k( )
1X
n= 1
1X
m=0
1qn
mBmGn
m( ) (8)
2

3. ~E+
z ei b + ~Ez e i b =
1X
n= 1
1X
k=1
rn
kFn
k ( ) (9)
~H+
z ei b ~Hz e i b =
1X
n= 1
1X
k=1
1
i rn
kAkGn
k( ) +
1X
n= 1
1X
m=0
1sn
mBmGn
m( ) (10)
where Ak = k2
2 k2
0
k2
0
2
ak
! ; Bm = k2
2
2
k2
0
2 m; Fn
k ( ) = ak ( 1)kei a 1]
2
a2
k
ei nT, and
Gn
m( ) = i 1 ( 1)mei a]
2
a2
m
ei nT: We multiply the tangential H eld continuities at
y = 0 and b by cosal(x pT), substitute ~Ez and ~Hz from (7) through (10), and
integrate from pT < x < pT + a, then
1X
k=1
pkAkI1 + rkAkI2]
1X
m=0
qm(iBmI1 + a
2 m ml) + smiBmI2] = 0 (11)
1X
k=1
pk(I3 ia
2Ck ks) + rkI4] +
1X
m=0
qm
a
2Dm ms = 0 (12)
1X
k=1
pkAkI2 + rkAkI1]
1X
m=0
qmiBmI2 + sm(iBmI1 + a
2 m ml)] = 0 (13)
1X
k=1
pkI4 + rk(I3 ia
2
Ck ks)] +
1X
m=0
sm
a
2
Dm ms = 0 (14)
where ml is the Kronecker delta, 0 = 2; m = 1 (m = 1;2; ); Ck = k2
2
2
k2
0
2
0
2
k,
and Dm = k2
2 k2
0
k2
0
2
am
! 2
. Note that the modal coe cients pn
k; rn
k; qn
m, and sn
m become
independent of the GNRD location n, as the number of GNRD waveguides approaches
in nity (thus, pn
k ! ( 1)nkpk; ; and sn
m ! ( 1)nmsm). A dispersion equation is
thus obtained from (11) through (14) as
j 1 + 2j j 1 2j = 0 (15)
where
1 =
2
4 AkI1 iBmI1
a
2 m ml
I3 ia
2
Ck ks
a
2
Dm ms
3
5 (16)
2 =
2
4 AkI2 iBmI2
I4 0
3
5 (17)
I1 I2 = a
2
m ml
m
h 1
tan( mb)
1
sin( mb)
i i
b
1X
v=0
1 ( 1)v] vfml( v)
v
(18)
3

4. I3 I4 = a
2 k ks
h 1
tan( kb)
1
sin( kb)
i
akas
i
b
1X
v=1
1 ( 1)v](v
b )2
fks( v)
v
(19)
fml( ) = ( 1)m+l + 1] 1 + ( 1)mei T] ( 1)m + ( 1)l] ei a + ( 1)mei (T a)
]
( 2
a2
m)( 2
a2
l )(1 ( 1)mei T)
(20)
m =
q
k2
2 a2
m
2
, and v =
q
k2
2 (v =b)2 2
. In a dominant-mode approxi-
mation (m = l = 1; a << 0), the dispersion relation is simpli ed to
iB1(I1 + I2) + a
2 = 0 (21)
Fig. 2 illustrates the normalized phase constant versus b=(2 0) and con rms the
validity of our series solution. In our computation of (15), we use k = 1; 3 and
m = 1; 3, implying that our series solution is numerically e cient and useful for
practical purposes. In addition, our dominant-mode approximation (21) is seen to
agree well with (15) in a low-frequency limit.
3 Conclusion
A rigorous dispersion equation for the groove nonradiative dielectric waveguide is
presented, using the mode matching, Fourier transform, and image theorem. Our
solution is useful for the design of millimeter wave antenna and passive devices using
the GNRD waveguides.
References
1] C.E. Tong and R. Blundell, Study of groove nonradiative dielectric waveguide,"
Electron. Lett., vol. 25, no. 14, pp. 934-936, July 1989
2] Z. Ma and E. Yamashita, Wave leakage from groove NRD structures," IEEE
Microwave Guided Wave Lett., vol. 3, no. 6, pp. 170-172, June 1993
4

5. y
x
z
T
bRegion (II)
Region (III)
Region (I)
a
ε2
ε0
ε0
(a) GNRD waveguide
T
bRegion (II)
Region (III)
Region (I)
a
y
x
z
(I)
(III) . . . . .
. . . . .
. . . . .
. . . . .
(b) In nite number of GNRD waveguides
Figure 1: Geometry of a GNRD waveguide.
5

6. 0.15 0.17 0.19 0.21 0.23 0.25
0
0.3
0.6
0.9
1.2
1.5
0.15 0.17 0.19 0.21
0
0.4
0.8
1.2
1.4
Normalizedphaseconstant,
Ma’s result in [2]o o
1.6
β/0
b/(2 )0λ
ε = 4ε02
ε = 2.56ε02
0.23 0.25
k
equation (15)
equation (21)
Figure 2: Normalized propagation constant versus b=(2 0) for a GNRD waveguide
with a = 9:6mm;b = 12mm, and T = 12mm.
6