RUBIN:
PROPAGATION
CHARACTEBMTICS
OF
SIGNAL
LINES
531
[8]
[9]
[10]
[11]
C.
J.
Anderson,
“Electrical
properties
of
an
input-output
cable
for
Josephson
applications,”
Rev.
Sci.
Instrum.,
vol.
53,
pp.
1663-1666,
NOV.
1982.
A.
Feller,
H.
R.
Kaupp,
and
J.
J.
DiGracomo,
“
Crosstafk
and
reflections
in
high-speed
digitaf
systems,”
in
Prvc.
Fall
Joint
Com-
puter
Conf.,
Dec.
1965.
W.
T.
Weeks,
“Calculation
of
coefficients
of
capacitance
of
multi-
conductor
transmission
lines
in
the
presence
of
a
dielectric
inter-
face:’
IEEE
Trans.
Microwave
Theoq
Tech.,
vol.
MTT-18,
pp.
35-43,
Jan.
1970.
D.A.
Reitan,
“Accurate
determination
of
the
capacitance
of
rectan-
Barry
J.
Rubin
(S’72-M74-M82)
was
born
in
New
York,
NY,
in
1952.
He
received
the
B.
E.E.E.
degree
from
the
City
College
of
New
York,
New
York,
in
1974.
He
joined
the
International
Business
Machines
Corporation
in
1974
and
received
the
M.
S.E.E.
degree
from
Syracuse
University,
Syracuse,
New
York,
in
1978’
and
received
the
Ph.D.
degree
from’
the
Polytechnic
Institute
of
New
Y&kX
Brooklyn,
in
1982.
He
has
worked
on
power
transistor
design,
CCD
tec~ology,
circuit
design
g&r
parallel-plate
capacitors;
Journ.
Appl.
Phys.,
vol.
30,
pp.
172-176,
Feb.
1959.
and,
since
1976,
on
electncaf
packaging
analysis.
He
has
“been
awarded
five
US
patents.
Computer
Analysis
of
Dielectric
Waveguides:
A
Finite-Difference
M[ethod
EDGARD
SCHWEIG,
hlH?f13ER,
IEEE,
AND
WILLIAM
B.
BRIDGES,
FELLOW,
IEEE
.&tract
—A
method
for
computing
the
modes
of
diel~tric
guiding
structures
based
on
finite
differences
is
described.
The
numerrcaf
computa-
tion
program
is
efficient
and
can
be
applied
to
a
wide
range
of
problems.
We
report
here
solutions
for
circular
and
rectangular
dielectric
waveguides
and
compare
our
solutions
with
those
obtained
by
other
methods.
Limita-
tions
in
the
commonly
used
approximate
formulas
developed
by
Marcatili
are
discussed.
I.
INTRODUCTION
D
IELECTRIC
WAVEGUIDES
of
high
perrnittivity
(c;
>
10)
have
been
proposed
as
practical
waveguid-
ing
structures
for
use
in
millimeter-wave
integrated
circuits
(MMIC)
[1],
[2].
The
prospect
that
the
dielectric
material
could
be
a
hi@t-resistivity
semiconductor
raises
the
further
possibility
that
active
devices
could
be
fabricated
directly
into
the
transmission
line.
Various
practical
devices
for
millimeter-wave
applications
utilizing
dielectric
waveguides
also
have
been
suggested:
directional
couplers
[3],
balanced
mixers
[3],
phase
shifters
[4],
[5],
scanning
antennas
[6],
channel-dropping
filters
[7].
The
theoretical
analysis
of
these
devices
has
been
based,
in
the
case
of
rectangular
guides,
on
the
analytical
solutions
proposed
by
Marcatili
[8],
which
can
be
expressed
in
simple
closed
forms.
How-
ever,
Mamatili’s
quasi-plane-wave
analysis
is
based
on
Mamrscriut
received
Jnlv
7,
1983;
reyised
December
15,
1983.
This
work
was
s~pported
by
tie
Office
of
Naval
Rese&ch
under
Contract
NOO014-79-C-O-0839.
The
authors
are
with
the
California
Institute
of
Technology,
Depart-
ment
of
Electncaf
Engineering,
Pasadena,
CA
91125.
assumptions
that
are
not
met
when
the
permittivity
of
the
guide
is
high
compared
to
the
outer
medium.
Several
authors
have
proposed
methods
for
the
study
of
rectangular
guides:
Knox
et
al.
[1]
(modification
of
Marcatili’s
analysis),
Goell
[9]
(expansion
in
circular
harmonics),
Schlctsser
[10]
and
Solbach
[11]
(mode
match-
ing),
and
Yeh
[1;!],
[13]
(finite
elements),
With
the
excep-
tion
of
Solbach
[II],
they
limit
their
analyses
to
relatively
sm~l
values
of
ptm&tivity
(c;
=
2.5)
and
they
do
not
give
the
field
distributions
calculated
by
their
methods.
We
have
developed
a
numerical
technique
based
on
finite
differences
(FD)
for
computing
accurate
dispersion
characteristics
and
field
distribution
for
dielectric
wave-
guides.
This
method
is
computationally
more
efficient
than
finite
elements
(FE),
thus
allowing
the
use
of
finer
meshes,
a
desirable
feature
when
accurate
values
of
the
fields
are
required.
II.
VARIATIONAL
FOItMULATION
Both
the
finite-elements
method
(FE)
and
the
finite-dif-
ference
method
(lFD)
are
based
on
a
variational
principle
[14],
[19].
For
one-dimensional
problems,
the
two
methods
are
equivalent
[15
].
This
equivalence
is
dmintained
in
two-
dimensional
problems
that
have
simple
rectangular
boundaries.
The
advantages
of
a
variational
approach
are:
1)
the
method
does
not
restrict
the
shape
of
the
dielectric
interfaces
so
that
complicated
dielectric
cross-sectional
profiles
can
be
treated;
2)
the
procedure
is
numerically
stable;
and
3)
it
permits
the
use
of
a
graded
mesh
that
can
0018-9480/84/0500-0531
$01.00
@1984
IIEEE
532
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
M_rT32,
NO.
5,
MAY
1984
be
made
finer
in
regions
of
particular
interest
or
impor-
tance.
Consider
a
waveguide
uniform
in
the
z
direction
which
consists
of
isotropic,
lossless
dielectric
media.
Assume
that
the
cross
section
can
be
divided
into
several
subregions
over
which
the
relative
permittivity
is
constant.
Further,
assume
propagation
along
the
z-axis
of
the
form
exp
[
j(
tit
–
~z
)]
with
longitudinal
field
components
H=
and
E=.
In
the
subregion
SP,
Hz
and
E,
satisfy
the
wave
equations
(1)
and
v:
is
the
transverse
Laplacian,
c
is
the
speed
of
light
in
free
space,
Kp
is
the
relative
permittivity
of
the
pth
subregion,
and
CO
is
the
perrnittivity
of
space.
We
introduce
a
propagation
constant
normalized
to
free
space
and
a
parameter
7P
that
quantifies
the
discontinuity
be-
tween
regions
SP
and
the
outer
medium
(7)
The
outer
medium
is
assumed
hereto
be
free
space
(KA
=
1)
and
will
be
referred
to
by
the
subscript
“A”.
It
has
been
shown
[12],
[14],
[16]
that
the
following
variational
expression
for
k;
can
be
written:
where
J
is
a
quadratic
expression
of
the
field
values
r$,
~j
at
each
of
the
N
mesh
points,
The
expression
for
J
can
be
further
simplified
(by
use
of
the
divergence
theorem)
so
that
it
involves
only
first-order
derivatives
[16],
[17]
This
last
relation
is
the
basis
of
our
FD
procedure.
1“
,
--1
I
I
I
Fig.
1.
To
apply
the
FD
method,
the
dielectric
guide
is
enclosed
in
a
box
(metatic
watl
boundag
conditions),
and
the
area
of
interest
is
covered
by
a
rectangular
grid.
The
perrnittivity
inside
each
element
must
be
uniform.
h2
hz
3
hl
I
L.—.2
Fig.
2.
A
typicaJ
rectangular
element
~P
used
for
FD
approximations,
of
dimension
hl
bYh2,
relative
perrnittiwty
KP,
and
with
verticies
1,
2,
3,
and
4,
in
the
order
shown.
III.
FINITE-DIFFERENCE
RELATIONS
To
treat
the
problem
of
a
dielectric
guide,
we
first
need
to
define
a
finite
cross
section
by
enclosing
the
guide
in
a
“box”
with
electrically
conducting
walls
sufficiently
large
so
that
it
will
not
perturb
the
modes.
Because
of
symmetry,
we
need
to
treat
only
one
quadrant;
the
longitudinal
electric
field
E=
must
be
either
symmetric
(magnetic
boundary
conditions)
or
antisymmetric
(electric
boundary
conditions)
with
respect
to
the
x-
and
y-axes,
while
the
longitudinal
magnetic
field
H,
has
the
opposite
symmetry.
We
then
define
a
mesh
that
covers
the
region
of
interest
with
rectangular
elements
(Fig.
1).
These
elements
are
chosen
so
that
the
permittivity
is
constant
inside
each
element
and
the
electric
and
magnetic
walls
each
divide
a
row
of
elements
in
two
equal
halves.
Consider
one
such
element
SP
(Fig.
2).
The
contribution
of
this
element
to
the
variational
expression
J
can
be
evaluated
by
using
FD
approximations
[17],
[18].
Using
@i
and
vi
to
denote
the
field
values
at
point
i,
we
have
(lo)
1
The
grid
lines
are
atigned
parrdlel
to
the
x-,
y-axes.
For
convenience,
we
assume
that
point
1
is
at
the
origin
of
the
coordinates.
SCHWEIG
AND
BRIDGES:
COMPUTER
ANALYSIS
OF
DIELECTRIC
WAVEGUIDES
533
For
the
first
term,
we
assume
that
13rp/dx
has
the
constant
value
(03
–
@l
)/hl
on
the
segment
1
–3
and
the
constant
value
(@d
–
I#Iz
)/hl
on
the
segment
4–2.
We
then
integrate
with
respect
toy
and
apply
the
trapezoidal
rule
+[(+)’+wd’lo
Likewise,
we
obtain
for
the
second
term
.(y)’+
pep)’].
(H,
A
similar
expression
is
obtained
for
the
int~gral
o~l
G,IJ12.
Finally,
each
term
of
the
integral
of
(v,+
X
V,+)Z
is
evaluated
as
follows:
(12)
and
(
J@2
*4–+3+@2–h
@4–@2+’$3
-@l
4
h2
h2
)(
hl
)
hl
“
The
variational
expression
J
is
then
obtained
as
the
sum
of
the
contributions
from
each
element
SP.
The
stationary
property
of
J
is
utilized
by
differentiating
with
respect
to
each
of
the
2N
variables
+,
and
I),.
In
this
way,
a
set
of
2N
linear
equations
is
derived,
which
can
be
written
in
the
form
AX=
k:BX
(13)
where
A
is
a
symmetric,
banded
matrix,
B
is
a
diagonal,
positive-definite
matrix,
and
X
is
an
ordered
vector
of
the
NW
N
NE
T
T
T
S4
s,
hN
w
P
E
hw
:
‘E
S2
hs
S3
Sw
s
SE
Fig.
3.
Rectangular
mesh.
By
adding
the
contributions
of
the
elements
SI,
S2,
S3,
and
S4,
we
find
that
the
FD
equations
at
the
node
P
are
equivalent
to
the
equations
resulting
from
the
application
of
the
five-
point
Laplace
operator
for
a
single
scafar
potential.
variables
@j,
tjj.
Detailed
expressions
for
A
and
B
are
given
in
the
Appendix.
By
a
transformation,
we
can
reduce
(13)
to
a
simple
eigenvalue
probleIn
A’X=k:X
(14)
and
A!=
B-1/2AB-l/2
(15)
It
is
important
to
note
that
the
matrix
A‘
is
also
symmetric
and
banded.
This
structure
allows
us
to
use
a
compact
storage
scheme
for
the
numerical
calculations
and
efficient
algorithms
for
computing
the
eigenvalues
and
eigenvectors.
In
the
case
of
the
FE
method
[12],
[13],
an
equation
similar
to
(13)
is
obtained,
but
the
matrix
B
is
banded
rather
than
diagonal,
with
the
same
bandwidth
as
A,
and
therefore
the
eigenvalue
problem
is
more
complicated,
An
estimate
of
the
errors
in
this
method
has
been
made,
with
the
result
that
errors
in
the
eigenvalues
will
be
of
0(h2),
where
h
is
the
largest
dimension
of
a
rectangular
element
in
the
case
of
a
graded
mesh.
This
property
was
verified
in
the
course
of
our
numerical
calculations
on
the
rectangular
guide
by
comparing
the
eigenvalues
corre-
sponding
to
successively
refined
grids,
all
other
parameters
remaining
unchanged.
The
accuracy
of
the
eigenfunctions
computed
numerically
is
much
more
difficult
to
assess,
but
it
is
usually
assumed
to
be
one
order
of
magnitude
smaller
than
the
accuracy
of
the
corresponding
eigenvalue,
i.e.,
O(h).
A
detailed
discussion
is
given
in
[19].
IV.
SOLUTION
OF
THE
MATRIX
EIGENVALUE
EQUATIONS
The
solution
of
the
matrix
eigenvalue
equation
is
sought
in
the
standard
form
A
‘X
=
AX,
where
~’
is
a
symmetric,
banded
matrix.
If
we
use
a
mesh
of
~
x
3/
rectangular
elements
(Fig.
3),
then
the
order
of
A
is
2(
N
–
1)2
and
the
bandwidth
(the
number
of
subdiagonals
including
the
main
diagonal)
is
2~.
For
a
typical
calculation
N
=
15,
and
thus
the
order
equals
3’92
and
the
bandwidth
equals
30.
The
numerical
algorithms
chosen
were
specifically
de-
signed
for
a
symmetric,
banded
matrix;
the
memory
re-
quirements
are
minimized
by
storing
only
the
nontrivial
534
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
MTT-32,
NO,
5,
MAY
1984
elements.
The
routines
employed
are
part
of
the
well-known
EISPACK
package,
[20].
The
solution
proceeds
as
follows.
a)
A’
is
reduced
to
tridiagonal
form
by
a
series
of
Givens
rotations
that
eliminate
successively
each
subdiagonal
while
maintaining
the
band
form
(routine
BANDR)
[20],
[21].
b)
The
eigenvalues
in
an
interval
of
interest
(that
is,
the
negative
eigen~alues
that
are
closest
to
zero)
are
de-
termined
by
the
routine
BISECT
[20],
[22].
The
number
of
eigenvalues
in
the
given
interval
is
computed
from
Sturm
sequences.
Next,
the
eigenvalues
are
evaluated
by
refining
the
input
interval
by
a
bisection
process.
c)
The
eigenvectors
are
computed
by
inverse
iteration
(routine
BANDV)
[20],
[23].
The
system
(A’
–
1)X
=
b
is
solved
by
Gaussian
elimination:
The
right-hand-side
vector
b
is
chosen
to
obtain
a
proper
eigenvector.
It
is
important
to
note
that
by
these
methods
it
is
impossible
to
omit
any
eigenvalue,
which
is
a
very
desir-
able
feature
in
comparison
with
iterative
methods.
If
the
finite-elements
formulation
had
been
used,
the
resulting
eigenproblem
A
‘X
=
A
BX
would
involve
two
banded
matrices.
The
minimum
storage
requirements
are
doubled
and
the
numerical
methods
available
are
all
iter-
ative
[12],
[13],
[24]
V.
MODE
DESIGNATION
All
the
possible
modes
of
the
rectangular
guide
may
be
divided
into
four
classes
depending
on
the
symmetry
of
the
longitudinal
fields,
since
a
longitudinal
field
H,
of
even
symmetry
with
respect
to
one
particular
axis
is
always
associated
with
a
longitudinal
field
E=
of
odd
symmetry
with
respect
to
the
same
coordinate.
These
four
classes
are
designated
in
this
work:
HE:’,
HE:”,
HE:O,
HE:e.
The
first
superscript,
o
ore,
indicates
the
symmetry
of
Hz
with
respect
to
the
x-axis,
while
the
second
superscript
denotes
the
symmetry
of
Hz
with
respect
to
the
y-axis.
The
sub-
script
n
indicates
the
order
of
the
given
mode
in
its
class.
The
correspondence
between
these
exact
modes
and
the
designations
adopted
by
Marcatili
for
his
quasi-plane-wave
modes
is
discussed
later.
VI.
CALCULATED
DISPERSION
CURVES
AND
COMPARISON
WITH
OTHER
WORR
Using
the
FD
method
described
above,
we
wrote
a
computer
program
that
builds
the
matrices
A
and
B
for
a
guiding
structure
with
a
given
distribution
of
relative
per-
mittivity
and
a
given
mesh.
The
details
are
given
in
[19].
The
dielectric
waveguide
has
a
core
of
relative
permittivity
KI
and
is
surrounded
by
an
infinite
medium
of
relative
permittivity
K2.
ln
addition
to
these
parameters,
we
specify
a
value
of
/3/k0
(in
the
range
~
–
~
for
a
guided
mode),
which
in
turn
determines
~
=
k~/k~.
Then
the
program
computes
the
dominant
eigenvalues
k;,
that
is,
the
negative
eigenvalues
closest
to
zero.
For
each
of
these
eigenvalues,
the
corresponding
value
of
the
free-space
wave-number
kO
is
determined.
This
computation
is
re-
peated
for
a
set
of
values
of
/?/kO
to
obtain
a
complete
dispersion
curve.
To
obtain
the
fields,
the
program
com-
putes
the
corresponding
eigenvectors.
J=fj
J=M
I
I=M
I:N
Fig.
4.
Approximation
of
a
round
guide
by
a
square
mesh.
The
hatched
elements
are
assigned
a
dielectric
constant
KI,
while
the
remaining
ones
correspond
to
K2
(
<
K1
).
1
0,
ROu
No
GUIDE:
FD
–Af=p
ROxIMATIONS
,
d’
0
0
2
4
6
8
10
NORMALIZE
FREQUENCY:V
Fig.
5.
Dispersion
characteristic
for
the
lowest
order
mode
of
a
round
waveguide:
the
continuous
line
corresponds
to
the
exact
theoretical
solution
for
the
H,EII-mode,
while
the
dashed
line
is
obtained
from
the
FD
method.
The
theoretical
solution
has
no
cutoff,
and
it
blends
smoothly
with
the
V-axis
to
V
=
O.
The
dispersion
curves
are
presented
in
the
B
–
V
descrip-
tion
commonly
used
for
optical
waveguides
(see
[25],
for
example).
V
is
the
normalized
frequency,
defined
by
(16)
for
a
waveguide,
of
dimensions
a
X
b,
and
B
is
the
normal-
ized
phase
parameter
~=
(W’ko)2–
L
KI–
K2
“
(17)
Our
results
using
this
finite-difference
method
are
com-
pared
below
for
three
cases:
a)
the
exact
solution
for
a
dielectric
waveguide
of
circular
cross
section,
b)
the
mode-matching
solution
of
Solbach
[11]
for
the
rectangular
guide,
and
c)
the
quasi-plane-wave
mode
solution
of
Marcatili
[8]
for
a
square
guide.
A.
Round
Guide
A
comparison
with
the
round
dielectric
guide
is
particu-
larly
indicated
because
it
is
the
only
case
for
which
the
exact
solution
is
known.
We
defined
the
permittivity
of
the
elements
of
a
square
mesh
in
such
a
way
that
they
ap-
proximate
one
quadrant
of
a
circular
fib~r.
In
Fig.
4,
the
hatched
squares
are
assigned
a
permittivity
KI
while
the
SCHWEIG
AND
BRIDGES:
COMPUTER
ANALYSIS
OF
DIELECTRIC
WAVEGUIDES
1.0,
0
:
/
O24681OI2
14161820
NORMALIZED
FREQUENCY.
V
Fig.
6.
Comparison
between
the
FD
and
the
mode-matching
solutions.
The
mode
nomenclature
for
the
mode-matching
solution
is
in
accor-
dance
with
Marcatili’s.
The
mode
nomenclature
for
the
FD
calculations
is
derived
from
the
symmetry
properties
of
the
longitudinal
magnetic
field
with
respect
to
the
x-
and
y-axes.
R
is
the
aspect
ratio
a\b.
Fig.
7.
other
curve
along
~
!r
K,
=120
KZ=I,O
m
R=1O
~08
w
+
w
EH,,
>06
a
/
—
FD[N:12
M.8)
m
g
/
---
FD(N=15
M=5)
04
--
MODE
MATCH(NG
%
a
y02
/xHE~
,!’
0
024681012
141618
NORMALIZED
FREQUENCY
V
Comparison
among
the
FD
and
the
mode-matching
solutions
for
a
higher
relative
permittivity,
K1
=12.
elements
correspond
to
Kz.
The
exact
dispersion
for
the
dominant
HE1l-mode
is
presented
in
.Fig.
5
with
the
FD
calculation.
The
total
number
of
ele-
ments
is
N2
while
the
parameter
M
defines
the
number
of
elements
in
the
circular
section
(see
Fig.
4).
If
M
is
chosen
too
small,
the
round
guide
is
poorly
approximated.
For
N
=15
and
M
=10,
the
FD
calculations
are
in
very
close
agreement
with
the
theoretical
curve,
for
larger
values
of
B
where
the
mode
is
well
confined.
For
smaller
values
of
B,
the
fields
extend,
further
outside
the
dielectric,
and
the
mode
is
influenced
by
the
outer
box
(metallic
walls).
To
obtain
the
dispersion
curve
in
this
frequency
region,
we
need
to
increase
the
ratio
N/M.
As
a
benchmark,
we
note
that
for
N=
15,
the
CPU
time
required
on
an
IBM
3032
to
compute
the
first
five
eigenvalues
corresponding
to
one
value
of
B
is
about
1.4
min.
For
N
=
20,
this
time
increases
to
approximately
4
min.
B.
Mode
Matching
Mode
matching
has
been
used
to
compute
the
modes
of
high-permittivity
image
lines
[11].
We
compared
the
disper-
sion
characteristics
obtained
by
this
method
to
our
FD
calculations
in
Figs.
6
and
7.
The
mode
designations
used
by
Solbach
[11]
are
in
accordance
with
those
ghwn
by
Marcatili
[8].
The
modes
are
designated
EHPq
when
the
longitudinal
field
is
mainly
electric
and
HEPq
when
the
longitudinal
field
is
mainly
magnetic.
The
indices
indicate
the
number
of
maxima
of
the
dominant
electric-field
com-
ponents.
We
computed
by
FD
the
modes
belonging
to
the
535
DOIWNANT
MODE
OF
A
SQUARE
GuIDE
(.O
—
(
581
“’’;’’’’’’O/
--’-”
+
~(,,,,
/’
‘+
0.6
/
a
‘m
—
FD(N=IO
M=7)
HE;”
//
2
~4
---
FD(N=IO
M=4)
HEY
g
/’
h4ARcATl
LlS
APPROXIMATION.
Efi-mode
a
/’
ROUND
FIBER
[same
ore.
m
,Z
0.2
/
qua,,
gu,
del:
HE,
,-
mode
/
,~”
I
‘1
0024681012
14161820
NORMALIZED
FREQUENCY!
V
Fig.
8.
Comparison
among
various
solutions
for
the
lowest
order
mode
of
a
square
dielectric
guide.
For
obtaining
an
accurate
FD
solution
at
low
frequencies,
it
is
necessary
to
use
a
higher
ratio
N/M,
The
solution
for
a
round
fiber
of
equivalent
cross-sectionaf
area
is
shown
for
comparison.
[)
OMINANT
MODE
OF
A
SQUARE
GUIDE
K,=I,O
------
(
,J
,
/’
/
/
/
/
—
FD(N=IO
M=7)
HEY
P
t’
--
FD(N=10M~4)
HEY
4’
h’
--
MARCATIL1’S
APPROXIMATION.
/
ROUND
FIBER
(some
0’,0
os
qua,,
g“,del
HE,
,
mode
-.
E;
.OL
!’
2468101214
1618
NORMALIZED
FREQUENCY
V
Fig.
9.
Comparison
among
various
solutions
for
the
lowest
order
mode
of
a
high-permittivity
square
dielectric
guide.
The
solution
for
a
round
fiber
of
equivalent
cross-sectional
area
is
shown
for
comparison.
same
class
of
symmetry.
For
a
perrnittivity
of
2.22
(Fig.
6),
the
two
techniques
give
slightly
different
values
for
the
first
mode,z
but
these
are
in
closer
agreement
at
higher
values
of
V
for
the
SeCOnld
mode.
For
KI
=
12.0,
the
agreement
between
the
two
methods
is
excellent
for
the
lowest
mode,
as
shown
in
Fig.
“7.
C.
Marcatili’s
Approximation
In
Figs.
8
and
9,
we
compare
the
dispersion
curves
for
the
lowest
order
mode
of
a
square
guide
of
permittivity
KI
=
2.1
using
three
different
methods:
1)
Marcatili’s
quasi-plane-wave
solutions,
2)
the
exact
solution
of
the
HE1l-mode
of
a
round
fiber
that
has
the
same
area
as
the
square
guide,
and
3)
the
FD
method
(N2
is
the
total
number
of
elements
and
&fz
is
the
number
of
elements
defining
the
dielectric
guide).
The
quasi-plane-wave
modes
are
designated
E~q
and
E&.
The
superscript
indicates
the
direction
of
polarization
of
the
dominant
electric-field
component
while
the
sub-
script
denotes
the
number
of
maxima,
respectively,
in
the
x-
and
y-directions.
Table
I
lists
the
symmetry
properties
of
the
longitudinal
Inagnetic
field
11,
for
the
four
lowest
order
EJq-
and
E:~-modes.
These
properties
determine
the
corre-
2
The
relative
difference
in
@/kO
M
of
the
order
of
4
percent,
which
is
difficult
to
measure
experimentally.
536
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
MTT-32,
NO.
5,
MAY
1984
TABLE
I
SYMMSTRY
PROPERTIES
OF
THE
FOUR
LOWEST
ORDER
E;q-
AND
E;q-MODES
node
E
Ey
0
x
.
x--i.
y-axis
x-axle
J-axie
x-axis
y-axis
&
11
e
e
e
o
%
0
e
-
-
0
0
#
12
e
o
-
-
e
e
#
22
0
0
0
e
~;l
.
.
e
e
.
e
#1
o
e
,
e
J2
e
o
0
.
#2
o
.
e
.
NORMALIZED
FREQUENCY:
V
Fig.
10.
Dispersion
curves
for
a
square
guide.
The
continuous
lines
correspond
to
the
FD
solutions,
while
the
dashed
lines
are
Marcatili’s.
spondence
between
the
quasi-plane-wave
mode
designation
and
the
more
general
mode
designation
used
in
this
work.
The
B-V
curves
are
compared
in
Fig.
9
for
K1
=
13.1.
At
high
frequencies,
~>>
1,
most
of
the
field
energy
lies
inside
the
guide,
and
the
dispersion
curves
computed
by
1)
and
2)
cannot
be
distinguished.
This
gives
us
a
very
good
check
on
the
accuracy
of
our
FD
calculations:
they
agree
very
well.
Because
the
fields
are
well
confined,
the
metalic
walls
can
be
relatively
close
to
the
guide.
For
small
values
of
~,
we
know
that
Marcatili’s
solution
is
not
correct,
since
it
erro-
neously
predicts
a
cutoff
frequency
greater
than
zero.
In
this
region,
the
fields
extend
far
outside
the
guide,
and
we
expect
that
the
dispersion
curve
of
the
square
guide
will
be
very
similar
to
the
curve
of
the
round
guide.
This
is
confirmed
by
our
FD
calculations.
In
this
frequency
range,
we
needed
to
remove
the
metalic
walls
farther
from
the
guide,
i.e.,
increase
the
ratio
N/M.
However,
for
the
domi-
nant
mode,
the
fields
are
expected
to
vary
slowly
inside
the
guide,
so
we
can
achieve
the
increase
in
N/&f
by
decreas-
ing
M.
When
the
ratio
N/M
is
too
small
for
the
frequency
range
studied,
the
dispersion
curve
appears
to
drop
more
rapidly
than
it
should.
For
the
same
two
values
of
permittivity,
we
have
also
computed
B-V
curves
for
the
higher
order
modes
(Figs.
10
Fig.
11.
10
K,
=131
K,=10
----------------
m
R=(O
LK-
..-
&
0.8
-----
,.-’
.
+--
,-----
K
,,
y
,,-
----
,/,
.,”
.-’
~
0.6
,.,
.-
/.’
.Ep
,/
E;,
,/
~
/>
\
~(’;,y
/
,’
,,
a
.
‘
,,
.[:,,
/,
“Cy,
z’
.,
Y
~04
,,’
m
‘
/.cy
,/”
u
‘
E;,
,,’
,/
I
,’
n
,/
,,%:2
,,
0,2
,/
,,
,
‘y’
,’
$%
,’
0
,1,
/
02468,0
,214
(c.
1820
NORMALIZED
FREQu
ENc
Y:v
Dispersion
curve
for
a
square
guide.
The
continuous
lines
are
the
FD
solutions
and
the
dashed
lines
are
Marcatili’s.
Dom!nont
Mode
of
o
Rectongulor
Guide
I
1
K,
=131
__
-.-—
—-
----
~
m
!3
K2=10
P
/-
,.-
&
,.
Ii
,.-
+
!+6
~,,~
/’
,’
/’
<
R=30
/’
E
,’
.
a
R=20
CL
.4
,’
/
w
,’
In
,’
a
‘/
~
,2
‘,
,’
,,
.,
OJ
;
‘;’
;
I;
!
10
12
14
16
18
20
NORMALIZE
FREQUENCY
V
Fig.
12.
Dispersion
curves
for
rectangular
high-permittivity
wavegwdes.
R
is
the
aspect
ratio
a/b
for
the
rectangular
cross
section.
As
the
index
ratio
is
increased,
Marcatili’s
solution
(dashed
hnes)
becomes
a
better
approximation,
as
verified
by
our
FD
calculations
(continuous
lines).
and
11)
and
compare
them
with
Marcatili’s
modes
that
have
the
same
axis
symmetries.
Because
the
particular
waveguide
that
we
considered
is
square,
the
E~q-
and
E~P-modes
are
degenerate.
ln
the
case
of
FD
modes,
the
HE;
O-modes
are
degenerate
with
the
HE~e-modes.
In
Figs.
10
and
11,
we
observe
that
only
the
dominant
~~l-mode
agrees
well
with
our
FD
computation.
The
dis-
agreement
becomes
progressively
worse
for
higher
order
modes.
We
recall
that
the
propagation
constants
are
the
stationary
values
of
a
functional
(15).
Therefore,
for
the
dominant
mode,
deriving
the
eigenvalues
from
an
assumed
form
for
the
eigenfunction,
inherent
in
Marcatili’s
methods
still
leads
to
very
good
values
for
the
propagation
parame-
ter.
We
note
that
the
modes
closest
to
Marcatili’s
E~-
and
13:2-modes
are
not
degenerate
in
our
FD
calculations,
Corresponding
to
each
of
these
modes,
we
obtain
two
modes,
HE~O
and
HEje
(with
n
equal
to
1
and
2,
respec-
tively),
that
become
degenerate
only
for
very
large
values
of
P,
for
a
very
tightly
confined
mode,
the
fields
corre-
sponding
to
these
two
modes
are
not
much
influenced
by
the
outer
dielectric
interface,
and
the
fields
can
be
superim-
posed
by
a
450
rotation.
We
expect
that
Marcatili’s
solutions
will
become
more
accurate
as
the
guide
aspect
ratio
R
=
a/b
increases.
This
is
confirmed
by
Fig.
12
that
indicates
that
for
R
=
5
we
cannot
distinguish
the
FD
solution
from
Marcatili’s.
VII.
CALCULATED
FIELDS
From
the
preceding
section,
it
appears
that
Marcatili’s
solutions
are
relatively
good
for
the
propagation
constants