Analysis of coupled inset dielectric guide structure

1. Analysis of Coupled Inset Dielectric Guide
Structure
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 The propagation and coupling characteristics of the inset dielectric guide
coupler are theoretically considered. Rigorous solutions for the dispersion relation
and the coupling coe cient are presented in rapidly-convergent series. Numerical
computations illustrate the behaviors of dispersion, coupling, and eld distribution
in terms of frequency and coupler geometry.
1 Introduction
The inset dielectric guide (IDG) is a dielectric- lled rectangular groove guide and
its guiding characteristics are well known 1, 2]. The coupled inset dielectric guide
consisting of a double IDG has been also extensively studied to assess its utility as
a coupler 3]. The IDG coupler may be used for a practical directional coupler and
bandpass lter due to power splitting and lter characteristics with low radiation loss
and simple fabrication. It is of theoretical and practical interest to consider the wave
1

2. propagation characteristics of an inset dielectric guide coupler which consists of a
multiple number of parallel IDG. We use the Fourier transform and mode matching
as used in a single IDG analysis 2], thus obtaining a rigorous solution for the IDG
coupler in rapidly-convergent series.
2 Field Analysis
Consider a IDG coupler with a conductor cover in Fig. 1 (N: the number of IDG).
The e ect of conductor cover at y = b is negligible on dispersion when b is greater than
one-half wavelength 2]. Assume the hybrid mode propagates along the z-direction
such as Hz(x;y;z) = Hz(x;y)ei z and Ez(x;y;z) = Ez(x;y)ei z with e i!t time-factor
omission. In regions (I) ( d < y < 0) and (II) (0 < y < b), the eld components are
EI
z(x;y) =
N 1
X
n=0
1X
m=0
pn
m sinam(x nT)sin m(y + d)
u(x nT) u(x nT a)]; (1)
HI
z(x;y) =
N 1
X
n=0
1X
m=0
qn
m cosam(x nT)cos m(y + d)
u(x nT) u(x nT a)]; (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)
where am = m =a, m =
p
k2
1 a2
m
2
, =
p
k2
2
2 2
, k1 = !p 1, k2 =
!p 2 and u( ) is a unit step function. To determine the modal coe cients pn
m and
qn
m, we enforce the boundary conditions on the Ex;Ez;Hx, and Hz eld continuities.
Applying a similar method as was done in 2], we obtain
N 1
X
n=0
1X
m=0
n
pn
mAmInp
ml qn
m BmInp
ml + a
2 cos( md) ml np m]
o
= 0;(l = 0;1; )(5)
N 1
X
n=0
1X
m=0
n
pn
m sin( md)Jnp
ml + a
2
Cm ml np] + qn
m
a
2
Dm ml np
o
= 0; (6)
2

3. where ml is the Kronecker delta, 0 = 2; m = 1 (m = 1;2; ),
Am = k2
2 k2
1
k2
1
2
am
! sin( md); (7)
Bm = k2
2
2
k2
1
2 m sin( md); (8)
Cm = k2
2
2
k2
1
2
1
2
m cos( md); (9)
Dm = k2
2 k2
1
k2
1
2
am
! 2
cos( md); (10)
Inp
ml = a
2
m ml np
m tan( mb)
i
b
1X
v=0
vf( v)
v( 2
v a2
m)( 2
v a2
l ); (11)
Jnp
ml = a
2
m ml np
tan( mb)
amal
b i
1X
v=1
(v
b )2
f( v)
v( 2
v a2
m)( 2
v a2
l ); (12)
f( ) = (( 1)m+l + 1)ei vjn pjT ( 1)mei vj(n p)T+aj ( 1)lei vj(n p)T aj; (13)
v =
p
k2
2 (v =b)2 2
, and m =
p
k2
2 (m =a)2 2
. A dispersion relationship
may be obtained by solving (5) and (6) for .
1 2
3 4
= 0; (14)
where the elements of 1, 2, 3, and 4 are
np
1;ml = AmInp
ml; np
2;ml = BmInp
ml
a
2 cos( md) ml np m;
np
3;ml = sin( md)Jnp
ml + a
2Cm ml np; np
4;ml = a
2Dm ml np: (15)
When N = 1 (a single IDG case), (14) reduces to (30) in 2]. When 1 = 2, (14)
results in the dispersion relationship for the rectangular groove guide in 4] and 5] as
j 2jj 3j = 0; (16)
where 2 and 3, respectively, represent TE and TM modes in the groove guide with
an electric wall placed at y = b. In a dominant-mode approximation (m = 0, l = 0),
(14) reduces to
j 2j = 0: (17)
3

4. When N = 2, (17) yields a simple dispersion relation as
00
2;00 = 10
2;00; (18)
where each sign corresponds to the odd and even modes in 3]. To calculate
the coupling coe cients between the guides, we introduce the eigenvector Xs as-
sociated with the eigenvalue = s (s = 1; ;N) , where the elements of Xs
are x0;x1; ;xN 1]T and xn = HI
z(nT; d) =
P1
m=0
qn
m: Note that qn
m is ob-
tained by solving (5) and (6) with = s determined by (14). The eld at z,
hn
z(z) , HI
z(nT; d;z) in the nth IDG, is related to the eld at z = 0 through a
transformation with the eigenvector 6]
2
6
6
4
h0
z(z)
...
hN 1
z (z)
3
7
7
5
= ~X1 ~XN]
2
6
6
4
ei 1z 0
... ... ...
0 ei Nz
3
7
7
5
~X1 ~XN]T
2
6
6
4
h0
z(0)
...
hN 1
z (0)
3
7
7
5
(19)
where ~Xs = Xs=jjXsjj is the normalized eigenvector and ( )T denotes the transpose
of ( ). We de ne the coupling coe cient at z = L between the ith and jth guides as
Cij = 20log10 jhi
z(L)j; (20)
where hp
z(0) = pj. For instance, N = 2, h0
z(0) = 1, and h1
z(0) = 0, (20) reduces to the
coupling coe cient (39) in 3]. Applying a dominant mode approximation for N = 2
and 3, we obtain eigenvectors as
X1
= 1 1]T
= 1
; X2
= 1 1]T
= 2
; (N = 2) (21)
X1;3
= 00
2;00 2 10
2;00
00
2;00]T
= 1; 3
;X2
= 1 0 1]T
= 2
: (N = 3) (22)
Fig. 2 illustrates the dispersion characteristics for IDG couplers (N = 2;3;4), con-
rming that our solution for N = 2 agrees with those in 3] within 1% error. Note
that an increase in the number of IDG causes an increase in possible propagating
modes. Fig. 3 shows the magnitude plots of Hz and Ez components for 3 fundamen-
tal HEp1 modes (p = 1;2;3), where the subscripts p, 1 denote the number of half-wave
variations of Hz component in the x and y directions, respectively. The eld plots
illustrate that Hz remains almost uniform in the x-direction within the groove, thus
4

5. con rming the validity of a dominant-mode approximation in all cases considered in
this paper. In Fig. 4, we compare the behavior of the coupling coe cients for N = 2
and 3 versus frequency. When N = 2, we calculate the coupling coe cient (20)
using the measured ( ) and calculated ( ) propagation constants in 3]. Note that
our theoretical calculation agrees well with the results based on 3]. When N = 3,
the couplings between adjacent grooves are almost the same (C12 C21), while the
coupling to the far groove is far less (C31 C21). Fig. 5 illustrates the coupling
coe cients versus z = L for N = 4. As the excitation wave propagates in the rst
groove (n = 0), it is coupled to the adjacent grooves (n = 1;2;3) in an ordered man-
ner. Note that a maximum power transfer occurs from the rst guide to the adjacent
ones at L = 44:0cm;67:3cm; and 108cm, successively.
3 Conclusion
A simple, exact and rigorous solution for the inset dielectric guide coupler is presented
and its dispersion and coupling coe cients are evaluated. Numerical computations
illustrate the eld distributions and coupling mechanism amongst the guides. A
dominant-mode approximation is shown to be accurate and useful for inset dielectric
guide coupler analysis.
References
1] T. Rozzi and S.J. Hedges, Rigorous analysis and network modeling of the inset
dielectric guide," IEEE Trans. Microwave Theory Tech., Vol. MTT-35, pp. 823-
833, Sept. 1987.
2] J.K. Park and H.J. Eom, Fourier-transform analysis of inset dielectric guide
with a conductor cover," Microwave and Optical Tech. Letters, Vol. 14, No. 6,
pp. 324-327, April 20, 1997.
5

6. 3] S.R. Pennock, D.M. Boskovic, and T. Rozzi, Analysis of coupled inset dielectric
guides under LSE and LSM polarization,"IEEE Trans. Microwave Theory Tech.,
Vol. MTT-40, pp. 916-924, May 1992.
4] 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. MTT-43,
pp. 2162-2165, Sept. 1995.
5] H.J.Eom and Y.H. Cho, Analysis of multiple groove guide," Electron. Lett., Vol.
35, No. 20, pp. 1749-1751, Sept. 1999.
6] A. Yariv, Optical Electronics in Modern Communications. New York : Oxford,
1997, pp. 526-531.
6

7. 4 Figure Captions
Figure 1 : Geometry of the inset dielectric guide coupler.
Figure 2 : Dispersion characteristics of the IDG coupler lled with PTFE ( r = 2:08)
for a = 10:16mm;d = 15:24mm;T = 11:86mm;b = 15mm, and N = 2;3;4.
Figure 3 : Field distributions for (a) HE31, (b) HE21, (c) HE11 (Hz eld) and (d)
HE31, (e) HE21, (f) HE11 (Ez eld), when N = 3.
Figure 4 : Behavior of the coupling coe cients Cij where PTFE( r = 2:08);a =
10:16mm;d = 15:24mm;T = 11:86mm;b = 15mm;L = 250mm, and N = 2;3.
Figure 5 : Behavior of the coupling coe cients Cij versus z = L where PTFE( r =
2:08); frequency = 8 GHz, a = 10:16mm;d = 15:24mm;T = 11:86mm;b = 15mm,
and N = 4.
7

8. x
0
-d
a T T+a
Region (I)(I) PEC
z
(I)
(N-1)T (N-1)T+a. . . . .
b
Region (II)
y
PEC
. . . . .n=0 n=1 n=N-1
µ, ε1
µ, ε2
Figure 1: Geometry of the inset dielectric guide coupler.
8

9. 8 8.5 9 9.5 10
200
210
220
230
240
250
260
270
200
220
240
260
β,phaseconstant[rad/m]
Double IDG (N=2)
Triple IDG (N=3)
Quadruple IDG (N=4)
experiment in [3]o o
for a coupled IDG (N=2)
Pennock solution andx x
8 9 10
frequency [GHz]
Figure 2: Dispersion characteristics of the IDG coupler lled with PTFE ( r = 2:08)
for a = 10:16mm;d = 15:24mm;T = 11:86mm;b = 15mm, and N = 2;3;4.
9

10. −1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
(a) (d)
(b) (e)
(c) (f)
y
x x
x
x
x x
y
y y
y y
Figure 3: Field distributions for (a) HE31, (b) HE21, (c) HE11 (Hz eld) and (d)
HE31, (e) HE21, (f) HE11 (Ez eld), when N = 3.
10

11. 8 8.5 9 9.5 10 10.5 11
−16
−14
−12
−10
−8
−6
−4
8 9 10
-4
frequency [GHz]
11
-6
-8
-10
-12
-14
-16
couplingcoefficients[dB]
Pennock’s results in [3] (N=2)o x
C (N=2)12
C = C (N=3)12 32
C (N=3)21
C (N=3)31
Figure 4: Behavior of the coupling coe cients Cij where PTFE( r = 2:08);a =
10:16mm;d = 15:24mm;T = 11:86mm;b = 15mm;L = 250mm, and N = 2;3.
11

12. 0 50 100 150
−30
−25
−20
−15
−10
−5
0
0 50 100
0
-5
-10
-15
-20
-25
-30
couplingcoefficients[dB]
150
L [ cm ]
C31
C41
C11
C21
Figure 5: Behavior of the coupling coe cients Cij versus z = L where PTFE( r =
2:08); frequency = 8 GHz, a = 10:16mm;d = 15:24mm;T = 11:86mm;b = 15mm,
and N = 4.
12