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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 : [email protected]
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
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
�
H
z
(x; y; z) = H
z
(x; y)e
i�z
and
�
E
z
(x; y; z) = E
z
(x; y)e
i�z
with e
�i!t
time-factor
omission. In regions (I) (�d < y < 0) and (II) (0 < y < b), the �eld components are
E
I
z
(x; y) =
N�1
X
n=0
1
X
m=0
p
n
m
sin a
m
(x� nT ) sin �
m
(y + d)
�[u(x� nT )� u(x� nT � a)]; (1)
H
I
z
(x; y) =
N�1
X
n=0
1
X
m=0
q
n
m
cos a
m
(x� nT ) cos �
m
(y + d)
�[u(x� nT )� u(x� nT � a)]; (2)
E
II
z
(x; y) =
1
2�
Z
1
�1
[
~
E
+
z
e
i�y
+
~
E
�
z
e
�i�y
]e
�i�x
d�; (3)
H
II
z
(x; y) =
1
2�
Z
1
�1
[
~
H
+
z
e
i�y
+
~
H
�
z
e
�i�y
]e
�i�x
d�; (4)
where a
m
= m�=a, �
m
=
p
k
2
1
� a
2
m
� �
2
, � =
p
k
2
2
� �
2
� �
2
, k
1
= !
p
��
1
, k
2
=
!
p
��
2
and u(�) is a unit step function. To determine the modal coe�cients p
n
m
and
q
n
m
, we enforce the boundary conditions on the E
x
; E
z
; H
x
, and H
z
�eld continuities.
Applying a similar method as was done in [2], we obtain
N�1
X
n=0
1
X
m=0
n
p
n
m
A
m
I
np
ml
� q
n
m
[B
m
I
np
ml
+
a
2
cos(�
m
d)�
ml
�
np
�
m
]
o
= 0; (l = 0; 1; � � � ) (5)
N�1
X
n=0
1
X
m=0
n
p
n
m
[sin(�
m
d)J
np
ml
+
a
2
C
m
�
ml
�
np
] + q
n
m
a
2
D
m
�
ml
�
np
o
= 0; (6)
2
where �
ml
is the Kronecker delta, �
0
= 2; �
m
= 1 (m = 1; 2; � � � ),
A
m
=
k
2
2
� k
2
1
k
2
1
� �
2
�a
m
!�
sin(�
m
d); (7)
B
m
=
k
2
2
� �
2
k
2
1
� �
2
�
m
sin(�
m
d); (8)
C
m
=
k
2
2
� �
2
k
2
1
� �
2
�
1
�
2
�
m
cos(�
m
d); (9)
D
m
=
k
2
2
� k
2
1
k
2
1
� �
2
�a
m
!�
2
cos(�
m
d); (10)
I
np
ml
=
a
2
�
m
�
ml
�
np
�
m
tan(�
m
b)
�
i
b
1
X
v=0
�
v
f(�
v
)
�
v
(�
2
v
� a
2
m
)(�
2
v
� a
2
l
)
; (11)
J
np
ml
=
a
2
�
m
�
ml
�
np
tan(�
m
b)
�
a
m
a
l
b
i
1
X
v=1
(
v�
b
)
2
f(�
v
)
�
v
(�
2
v
� a
2
m
)(�
2
v
� a
2
l
)
; (12)
f(�) = ((�1)
m+l
+ 1)e
i�
v
jn�pjT
� (�1)
m
e
i�
v
j(n�p)T+aj
� (�1)
l
e
i�
v
j(n�p)T�aj
; (13)
�
v
=
p
k
2
2
� (v�=b)
2
� �
2
, and �
m
=
p
k
2
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
= A
m
I
np
ml
;
np
2;ml
= �B
m
I
np
ml
�
a
2
cos(�
m
d)�
ml
�
np
�
m
;
np
3;ml
= sin(�
m
d)J
np
ml
+
a
2
C
m
�
ml
�
np
;
np
4;ml
=
a
2
D
m
�
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
2
jj
3
j = 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
2
j = 0: (17)
3
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 X
s
as-
sociated with the eigenvalue � = �
s
(s = 1; � � � ; N) , where the elements of X
s
are [x
0
; x
1
; � � � ; x
N�1
]
T
and x
n
= H
I
z
(nT;�d) =
P
1
m=0
q
n
m
: Note that q
n
m
is ob-
tained by solving (5) and (6) with � = �
s
determined by (14). The �eld at z,
h
n
z
(z) ,
�
H
I
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
h
0
z
(z)
.
.
.
h
N�1
z
(z)
3
7
7
5
= [
~
X
1
� � �
~
X
N
]
2
6
6
4
e
i�
1
z
� � � 0
.
.
.
.
.
.
.
.
.
0 � � � e
i�
N
z
3
7
7
5
[
~
X
1
� � �
~
X
N
]
T
2
6
6
4
h
0
z
(0)
.
.
.
h
N�1
z
(0)
3
7
7
5
(19)
where
~
X
s
= X
s
=jjX
s
jj 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
C
ij
= 20 log
10
jh
i
z
(L)j; (20)
where h
p
z
(0) = �
pj
. For instance, N = 2, h
0
z
(0) = 1, and h
1
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
X
1
= [1 � 1]
T
�=�
1
; X
2
= [1 1]
T
�=�
2
; (N = 2) (21)
X
1;3
= [
00
2;00
� 2
10
2;00
00
2;00
]
T
�=�
1
;�
3
; X
2
= [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 H
z
and E
z
components for 3 fundamen-
talHE
p1
modes (p = 1; 2; 3), where the subscripts p, 1 denote the number of half-wave
variations of H
z
component in the x and y directions, respectively. The �eld plots
illustrate that H
z
remains almost uniform in the x-direction within the groove, thus
4
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 (C
12
� C
21
), while the
coupling to the far groove is far less (C
31
� C
21
). 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
[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
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) HE
31
, (b) HE
21
, (c) HE
11
(H
z
�eld) and (d)
HE
31
, (e) HE
21
, (f) HE
11
(E
z
�eld), when N = 3.
Figure 4 : Behavior of the coupling coe�cients C
ij
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 C
ij
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
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
8 8.5 9 9.5 10
200
210
220
230
240
250
260
270
200
220
240
260
β,p
ha
se
co
nsta
nt
[ra
d/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 10frequency [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
−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
-d0
b
1
0.5
0
-0.5
-1
0
a+2T-d
0
b
1
0.5
0
-0.5
-1
0
a+2T
-d0
b
1
0.5
0
-0.5
-1
0
a+2T
-d0
b
1
0.5
0
-0.5
-1
0
a+2T
-d0
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
xx
x x
y
y y
y y
Figure 3: Field distributions for (a) HE
31
, (b) HE
21
, (c) HE
11
(H
z
�eld) and (d)
HE
31
, (e) HE
21
, (f) HE
11
(E
z
�eld), when N = 3.
10
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
-16co
up
lin
g c
oe
ffic
ien
ts [
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 C
ij
where PTFE(�
r
= 2:08); a =
10:16mm; d = 15:24mm; T = 11:86mm; b = 15mm;L = 250mm, and N = 2; 3.
11
0 50 100 150−30
−25
−20
−15
−10
−5
0
0 50 100
0
-5
-10
-15
-20
-25
-30
co
up
lin
g c
oe
ffic
ien
ts [
dB
]
150
L [ cm ]
C 31
C41
C 11
C 21
Figure 5: Behavior of the coupling coe�cients C
ij
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