This document analyzes the dispersion of multiple V-groove waveguides using Fourier transforms and mode matching. It presents a new rigorous dispersion relation in a fast-converging series for efficient numerical calculation. A closed-form dispersion relation based on a dominant mode approximation is shown to be accurate for practical applications involving double and triple V-groove guides. Field distributions are presented that confirm the validity of the dominant mode approximation.

1. Dispersion of multiple V-groove guide
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 multiple V-groove guide is rigorously analyzed with the Fourier-
transform and mode matching. A new rigorous dispersion relation is presented in
fast-convergent series which is e cient for numerical calculation. A closed-form dis-
persion relation based on a dominant-mode approximation is shown to be accurate
for most practical applications.
1 Introduction
A V-groove guide 1, 2] is an alternative to the rectangular groove guide 3] for
millimeter-wave applications. The overall characteristics of the V-groove guide are
similar to those of the rectangular groove guide except for the lower attenuation and
e ective rejection of higher order modes. A single V-groove guide has been extensively
studied with employing the conformal mapping technique 1] and step approximation
2], but the eld patterns were not presented. A double-V-groove guide has been also
1

2. studied in 4] using the transverse resonance method. It is, therefore, of fundamental
interest to understand the guiding characteristics and eld patterns of the multiple V-
groove guide. In this letter, we analyze the dispersion of the multiple V-groove guide
by introducing a new method based on the Fourier-transform and mode matching 3].
2 Field Analysis
Consider the multiple V-groove guides in Fig. 1. (N : the number of V-groove guides)
The TE-wave is assumed to propagate along the z-direction and the ei( z !t)
phase-
factor is omitted throughout. In region (I) (0 < x < d1), (II) (d1 < x < d1 +d2), and
(III) (d1 + d2 < x < b + d1 + d2), the Hz components are
HI
z(r; ) =
N 1X
n=0
1X
m=0
qn
mFm(r; ) (1)
HII
z (x;y) =
N 1X
n=0
1X
m=0
h
rn
m sin m(x d1) + sn
m cos m(x d1)
i
cosam(y nT + a1)
(2)
HIII
z (x;y) = 1
2
Z 1
1
~H+
z ei (x d1 d2)
+ ~Hz e i (x d1 d2)
]e i yd (3)
Fm(r; ) = J m( r)cos m( + 1)
8
<
:
r =
q
x2
+ (y nT)2
= tan 1
(y nT)=x
for the nth groove
(4)
where J m( ) is the Bessel function of a fractional order, m = m =( 1 + 2),
i = tan 1
(ai=d1), =
pk2 2
, am = m =(a1 + a2), m =
q
k2
a2
m
2
,
=
pk2 2 2
, and k = 2 = 0. It should be noted that an addition of region (II)
in the problem geometry simpli es the analytic formulation, thus substantially facil-
itating the numerical computation. To determine the modal coe cients qn
m, rn
m, and
sn
m, we utilize the same method as was done in 3] and obtain the simple dispersion
relation as
N 1X
n=0
1X
m=0
qn
m
"
a1 + a2
2
Fml(d1) np +
1X
k=0
F0
mk(d1)Inp
kl
k
#
= 0 (5)
where np is the Kronecker delta, 0 = 2, k = 1 (k = 1;2; ),
Fmk(x) =
Z a2
a1
Fm(r; )cosak(y + a1)dy (6)
2

3. Inp
ml = a1 + a2
2
tan( mb)
m
m ml np
+ i
b
1X
v=0
vf ( 1)m+l + 1]ei vjn pjT ( 1)mei vj(n p)T+a1+a2j ( 1)lei vj(n p)T a1 a2j]g
( 2
v a2
m)( 2
v a2
l )
(7)
v = (v + 0:5) =b; v =
q
k2 2
v
2
, and F0
mk(x) = @Fmk(x)=@x. In a dominant-
mode approximation (m = 0), (5) reduces to the closed forms as
00 = 10 (TE11;12 modes for N = 2) (8)
00 = 20 (TE12 mode for N = 3) (9)
00 ( 00 + 20) = 2 2
10 (TE11;13 modes for N = 3) (10)
where
np = (a1 + a2)F00(d1) np + F0
00(d1)Inp
00 (11)
F00(x) = 2 1X
k=0
( 1)kJ2k+1( r)sin (2k + 1) ]
y=a2
y= a1
(12)
where f(y)jy=b
y=a = f(b) f(a). When a groove separation (T) of the multiple V-
groove guide (N = 2;3; ) is larger than 3 0, it is di cult to nd since the
propagation constants for di erent modes are close each other. In this case, the
dominant-mode solutions (8) through (10) are very e ective to get an initial guess
for to use a root-searching algorithm. Our computational experience indicates that
the dominant-mode approximations (m = 0), (8) through (10), are almost identical
with a more accurate solution (m = 0;1;2), (5), in most practical cases. Using the
dominant-mode approximations (8) through (10), we illustrate the behavior of phase
constants for N = 2;3 in Fig. 2. Note that the dispersion relation converges to the
single V-groove solution when a V-groove separation (T) is larger than 6 0. The
dispersion characteristics of the multiple V-groove guide are seen to be very similar
to those of the rectangular groove guide. Fig. 3 illustrates the magnitude plots of
Hz elds corresponding to the N = 3 case in Fig. 2 with T = 15mm. Note that the
subscripts m and n in the TEmn mode denote to the number of half-wave variations
along the x- and y- directions, respectively. The Hz elds within the V-groove are
3

4. seen to be almost unchanged along the y-direction, thus con rming the validity of
our dominant-mode approximation.
3 Conclusion
A simple and rigorous dispersion relation for the multiple V-groove guide is obtained.
Our numerical computation for the double V-groove guides agrees well with other
existing solution. The eld distributions for the triple V-groove guide illustrate the
TE mode wave guiding characteristics. A closed-form dispersion relation, based on a
dominant-mode approximation, is found to be accurate and useful for most practical
millimeter wave applications.
References
1] Y.M. Choi, D.J. Harris, and K.F. Tsang, Theoretical and experimental char-
acteristics of single V-groove guide for X-band and 100 GHz operation," IEEE
Trans. Microwave Theory Tech., vol. MTT-36, pp. 715-723, Apr. 1988.
2] Z. Ma and E. Yamashita, Modal analysis of open groove guide with arbitrary
groove pro le," IEEE Trans. Microwave Guided Wave Lett., vol. 2, no. 9, pp.
364-366, Sept. 1992.
3] H.J.Eom and Y.H. Cho, Analysis of multiple groove guide," Electron. Lett., vol.
35, no. 20, pp. 1749-1751, Sept. 1999.
4] S.F. Li, Z.X. Shen, and X.M. Lou, A simple analysis of single- and double-V-
groove guides," IEEE Trans. Microwave Theory Tech., vol. MTT-39, pp. 1413-
1416, Aug. 1991.
4

5. . . . . .
x
-a
b
Region I
y
PEC
n=0 n=1
Region(III)
I
n=N-1
. . . . .
PMC
z
Region(II) (II)
Region(I)
(I)
1 a2
d2
d1
T+a2 (N-1)T+a 2
(II)
(I)
0
Figure 1: Geometry of a multiple V-groove guide.
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β,phaseconstant[rad/m]
double V-groove
triple V-groove
rectangular groove guide
theory [4] for double V-groove
T, groove separation [ mm ]
x x
TE11
TE13
18
TE12
TE11
TE12
TE13
N=2
N=2
N=3
N=3
N=3
Figure 2: Behavior of phase constant of TE11, TE12, and TE13 modes versus groove
separation for o = 3:08mm; a1 = 5mm; a2 = 5mm; b = 5mm, d = 2:5mm, and
N = 2;3.
6

7. 0
0.5
1
0.5
0
1
y
x
0
2T
0
d
b+d
1
1
(a)
−1
−0.5
0
0.5
1
y
0
2T
0
1
x
b+d1
0
d1
-1
(b)
−1
−0.5
0
0.5
y
0
2T
0
1
x
b+d1
0
d1
-1
(c)
Figure 3: Hz eld distributions for (a) TE11 mode, (b) TE12 mode, and (c) TE13
mode when N = 3. 7