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

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