GENERAL
PHYSICS
I.
MICROWAVE
SPECTROSCOPY
Academic and Research Staff
Prof. M. W. P. Strandberg
Prof. R. L. Kyhl
J. D. Kierstead
Graduate Students
J. U. Free, Jr.
J. K. Raines
A. Tachagumpuch
A.
PROPAGATING WAVES IN HIGH DIELECTRIC WAVEGUIDE
OF SQUARE CROSS SECTION
Joint Services Electronics Program (Contract DAAB07-71-C-0300)
J.
K.
Raines
Work reported previously
1
has continued, and we have obtained field solutions for
propagating waves in a slab of high dielectric material.
This waveguide is of square
cross section, there are no metal walls, and it is surrounded entirely by free space.
The permittivity is sufficiently great that displacement current outside the waveguide
may be neglected.
The transverse electric field inside the waveguide is described by a homogeneous
differential integral equation.
2
kE
z
T
T
)
d r
0E
T T
2T
k E2_
T VT(V
zT
Toz
/)i
K (k R)
2
+
0i
d 2 r' VT[Ko(kzR)V
E
1 j,
where
ET : ET( r T)
E (r'
R
K
T
o
= modified Bessel function.
Solutions to this equation must also satisfy the boundary condition
QPR No. 111
(1)
l
FOR SQUARE HIGH
DIELECTRIC WAVEGUIDE
"TM
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1-2o
S"TM"o1 0
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DISPERSION RELATION
STE"
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ASPECTRATIO
12
b=
12
S"TM "3 0
S"TE" 32
0
Fig. I-1.
10
Dispersion relation for square waveguide and for E z of even symmetry
in both transverse dimensions.
oa=kza
2
0
8
10
12
14
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QPR No. 111
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(a) Magnetic field inside and outside the guide for "TM"10 mode
and Xz/2a = 1.
(b)Magnetic
ZTM"30
field for
mode and z/a
= 1.
(b) Magnetic field for "TM" 3 0 mode and Xz/2a = 1.
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Fig. 1-3.
(a) Electric field for "TE"12 mode and Xz/2a = 1.
(b) Magnetic field for "TE" 1
2
kz/2a = 1.
mode and
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Fig. 1-4.
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(a) Electric field for "TM" 1 mode and
()
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z /2a = 1.
(b) Magnetic field for "TM"12 mode and X /2a = 1.
QPR No.
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(a) Electric field for "TE"32 mode and kz /2a = 1.
(b) Magnetic field for "TE"3 2 mode and z /2a = 1.
Fig. 1-5.
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Fig. I-6.
QPR No. 111
Snow
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(a) Electric field for "TE"12 mode and X /2a = 0. 3.
(b) Magnetic field for "TE"12
12z mode and kz/2a = 0. 3.
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(a) Electric field for "TM"12 mode and kz/2a = 0. 3.
(b) Magnetic field for "TM"12
12 mode and zz/2a = 0. 3.
Fig. 1-7.
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(b)
Fig. I-8.
(a) Electric field for "TE"3 2 mode and Xz/2a = 0. 3.
(b) Magnetic field for "TE"3 2 mode and Xz/2a = 0. 3.
QPR No. 111
I
_
_
_
_
I,
(I.
MICROWAVE SPECTROSCOPY)
(2)
^
(2)
ET=
n.
Using a collocation technique,
we transform the equation
into a
straightforward
matrix eigenvalue problem of the form
i1
=
(3)
.
m2
The eigenvalues are normalized frequencies
2
oEab
2
(4)
2r
S
2Tr
The eigenvectors A4 and B are coefficients in the expansions for the transverse field
components E x and E y , respectively. The expansions used in this study were in terms
2
2
of Tchebyshev polynomials, weighted by factors such as (1-x ) or (1-y ), so that Eq. 2
is satisfied identically.
Equation 1 and the procedure for solving it are valid not only for a square waveguide
but also for a rectangular waveguide of arbitrary aspect ratio. They are also valid for
a dielectric waveguide of arbitrary cross section, provided that the problem is done in
Cartesian coordinates.
Equations 1 and 2 determine the transverse electric field, and the remaining field
components follow directly from Maxwell's equations
V
EE = 0
H=-
V XE.
Figure 1-i is a dispersion relation obtained for the square waveguide and for Ez of
The first five modes were obtained for
even symmetry in both transverse dimensions.
field expansions of 4 or 5 terms in each dimension.
1-2 through I-8 are vector plots of selected electric and magnetic fields.
What is shown is the projection on the transverse plane of an array of unit vectors for
Figures
displaying the direction only of the field indicated.
the field vector is mostly in the z direction.
Where the marks are very short,
No indication of field magnitude is shown.
The fields on some of the figures seem to spiral.
This is
only an optical illusion
because the array points are at one end of each of the marks.
This work may be extended to dielectric resonators.
Eq.
1 has been obtained.
QPR No. 111
2
A formulation analogous to
(I.
MICROWAVE
SPECTROSCOPY)
References
1.
J. K. Raines, "Propagating Waves on a High Dielectric Slab," Quarterly Progress
Report No. 108, Research Laboratory of Electronics, M. I. T., January 15, 1973,
pp. 23-27.
2.
J. K. Raines, "Application of Integral Equations to Dielectric Waveguides of Rectangular Cross Section," Ph.D. Thesis, submitted to the Department of Electrical
Engineering, M.I.T., September 1973.
QPR
No. 111