CROSS-POLARIZATIONOF MICROWAVE
REMECTORANTENNAS
to the raculty
A thesis
submitted
Sciences
of the University
of Mathematical
of Surrey
for
and Physical
the Degree of Doctor
of Philosophy.
HURDIN A. ADATIA
1974
/
B",
est,,
*Co
py
Available
Variable Print Quality
\
ABSTRACT
This
thesis
characteristics
parabolic
involving
both with
prime focus
sources
identified.
mance are also
discussed.
Fina"y,
double off-set
suitable
and off-set
investigated,
and the
in each of these
Methods of improving
cassegrain
arrangement
their
adjustment
evaluated
It
numerically.
of the off-set
parameters,
perfor-
involving
of the antenna components is described,
properties
a
and its
is shown that
by
the cross-
from such an antenna can be reduced to an extremely
polarization
low order,
a modified
element.
feed and the conventional
of cross-polarization
configurations
radiation
symmetrical
feed systems are initially
cassegrain
principal
as the main focussing
surface
Antenna configurations
radiation
antenna systems which employ the
of reflector
reflecting.
reflectors,
the cross-polarized
examnes
making it
communications
highly
and tracking
attractive
applications.
for
both satellite
ACKNOWLEDGMENTS
I should
like
to thank Dr. S. Cormbleet
of the microwave group for
and the computer staff
technical
their
for
are also
discussions
of Defence for
rinally,
support
typing
and for
members
the
for
the text.
MY
of Birmingham
valuable
supplying-several
data.
I acknowledge with
Ministry
of Surrey
of the University
due to Dr. A. W. Rudge of the University
many helpful
experimental
help and encouragement;
and Mrs. S. N. G. Browne for
services,
thanks
their
all
and the fellow
I should
gratitude
the pursuit
like
the receipt
of a grant
from the
of the study.
to thank my wifes
Ruth,
for
her continued
and encouragement.
0
Page No.
CONTENTS
i-iv
INTRODUCTION
CHAPTER 1
THE CROSS-POLARIZED SCATTERED FIELD
Cross-polarization
1
Analysis
5
1.1
Definition
1.2
Reflector-antenna
1.3
Current
1.4
Edge Currents
1.5
The Physical-optics
Fields
of
Distribution
in
Reflectors
11
Diffracted
13a
REFERENCES
CHAPTER 2
7
13b
ANALYSIS OF AXIALLY SYMMETRIC PARABOLIC REFLECTOR
ANTENNA
CHAPTER3
Distribution
2.1
Current
16
2.2
Scattered
Radiation
18
2.3
Simplified
Model
19
2.4
Numerical
Results
21
2.5
Practical
Considerations
25
2.6
Feed Models
2.7
Typical
2.8
Circular
2.9
Conclusions
46
APPENDIX
48a
REFERENCES
48c
29
Antenna Performance
Aperture
Conical
Feedhorns
35
41
OFF-SET FED PARABOLICREFLECTORANTENNA
3.1
Introduction
3.2
Analysis;
3.3
The Induced Current
3.4
Aperture-field
49
Geometry
50
Method
Technique
52
55
3.5
CHAPTER 4
ADerture
Polarized
Distribution
Excitation
for
Linearly
57
3.6
Secondary
Radiation
3.7
Numerical
Examples
3.8
Experimental
3.9
The Effect
of
Cross-polarization
Radiation
the
3.10
The
Longitudinal
Currents
3.11
Radiation
Reflector
Excitation
Properties
of
Circularly
with
the Off-set
Polarized
3.12
Numerical
Examples
3.13
Experimental
3.14
Conclusions
88
APPENDIX
89
REFERENCES
90
Patterns
61
64
Confirmatioý
Effect
of
69
Primary-feed
on the Secondary
72
76
79
81
Confirmation
85
THE CROSS-POLARIZATION OF AXIALLY SYMMETRIC
CASSEGRAIN AERIAL
Introduction
PART I:
91
The Depolarization
Properties
of
Symmetric
the Axially
Hyperboloid
4.1
Analysis
4.2
Numerical
4.3
The Effect
of the
Cross-polarization
93
Results
97
Primary-feed
101
PART II:
4.4
The Diffraction
Main Reflector
4.5
Numerical
4.6
Concluding
4.7
The Off-set
Patterns
from, the
106
Results
REFERENCES
108
Remarks
Cassegrain
112
Aerial
113
115a
CHAPTER 5
CROSS-POLARIZATION CHARACTERISTICS OF DOUBLY
OFF-SET CASSEGRAIN AERIALS
5.1
Antenna Geometry
5.2
Calculations
of the Radiation
the Sub-reflector
116
from
118
Geometry of the Feed System in the
Elliotts-Antenna
124
5.4
Numerical
Computation
127
5.5
Numerical
Results
128
5.6
Radiation
Reflector
Patterns
5.3
from the Main
133
5.7
Discussion
143
5.8
Conclusions
145
REFERENCES
146
(i)
Introduction
The polarization
characteristics
by microwave aerials
radiated
considerations
in a given
depending
coverage
conditions)
into
free-space,
and as a consequence,
in the final
are present
the tolerance
implies
within
an angular
directive
acceptable.
distortions
cases,
condition
However, there
can be significant
the radiated
than the specified
The presence
aerial
has several
The radiation
patterns
levels
of diverse
or cross-polarized
of energy into
pattern.
over a wide angular
of the cross-polarized
effects
polarization
suffer
states
component
Most highly
of the main
is sufficiently
is usually
where polarization
applications
on the antenna axis
undesirable
normally
of antenna polarization
met, the performance
of cross-polarization
source
polarization
over a narrow region
are several
the antenna
of the aerial.
component on the axis
is normally
polarization
(boundary
the, fields
of orthogonal
function
generally
requirement
from the primary
of the antenna radiation
the cross-polarized
and this
small,
sector
antennas
beam and if
an allowance
A basic
constraints
fields
to operate
linear
throughout
elements,
pattern
radiation
In antenna engineering
essentially
focusing
via various
depolarization,
and ideally
in transmission
imposed on the fields
astronomy
from fixed
of the desirable
However, owing to several
region.
varies
application.
the purity
of space,
region
a specified
within
on the particular
is then to maintain
of the aerial
radio
the antenna is required
which typically
mode of polarization
to circular,
in radar,
application
In general,
and teleco=unications.
in the design
interest
are of major
of the antenna for
fields
of the electromagnetic
sector
and in these
components rather
are of importance.
in the radiation
patterns
on the performance
states
other
of an
of the aerial.
than the desired
(ii)
I
implies
polarization
energy is
that
is insignificant
illumination
losses
(in
Admittedly,
this
parabolic
This
radiation.
has received
why cross-polarization
input
compared to the losses
of the main aperture
due to spill-over
from the total
is impaired.
and thus the antenna efficiency
in efficiency
'lost'
energy
loss
due to non-uniform
reflector-aerials)
and
is one of the principal
little
comparatively
reasons
in
attention
literature.
In several
indirectly.
If.
circular
thus
of
for
the available
bandwidth.
of major
of a tracking
effect
aerial
states
locking"
axis,
errors
which differ
distortion.
coverage
for
signal
amounts
fields
communications
two comunication
in pointing
- usually
zone in order
to present
thus provides
a foundation
minimum interference.
Is to be found in
using monopulse facilities.
generally
contains
from the polarization
cross-polarization
of the aerial
to
of cross-polarization
polarization
aerial
fields
and the
in satellite
depolarization
from a complex target
substantial
the
making a more efficient
use of
I
it is of importance to
In such applications
echo returns
generates
and communication
of
to provide
on which the system may be optimised
the operation
a
band, thereby
of antenna
undesirable
radar
in
aerial
component
interest
over the entire
A study
A further
leads
This
the
operate
in
case
handed)
polarizations
each frequency
to
depolarization
weather,
a topic
years
the
is felt
of cross-polarization
desired
often
polarized.
mýLintain orthogonality
cross-talk.
is
(opposite
orthogonal
is the use of orthogonal
channels
it
mode as is
elliptically
In recent
instance,
inclement
during
to generation
are
for
polarization
application
the presence
applications,
accuracy
itself,
uncorreilated
in which the aerial
particularly
can result
The
near the
in "polarization
onto the peaks of the cross-polar
lobes.
(iii)
This has the effect
of swerving
hence the necessity
for
The depolarization
in
the
aerials
feed
for
using
monopulse
develops
the
a squint
which
cross-polarization
radiation
within
the
impairs
directly
with
in
the
is
the
asymmetry
[11
(boresight
the
main beam
performance
to
attributable
main beam and an axis
or the
receiver
used,
tracking
of
[2]9
a monopulse
'jitter'
either
the
to
previously
conjunction
polarization
properties
may be calculated,
jitter
in this
work described
of single
the
the
of
presence
of
antenna
cross-polar
was performed
theoretical
with
and nulti-element
as a function
systems
reflector
aerials,
of the geometrical
of a doubly
response
assess the possible
fed reflector
off-set
ways of minimising
and bore-
and to calculate
parameters,
Cassegrain
the
models on which the
2) to use these models to study the beam shift
phenomena in off-set
these effects
thesis
1) to develop
main objectives:
depolarization
sight
circular
are
owing
a tracking
are
reflectors
pattern.
The research
following
develop
seriously
These effects
aerial.
in
be compensated
If
comparator.
off-set
has been reported
reflectors
cannot
using
counterparts,
it
application,
which
aerials
symmetric
off-set
tracking
fluctuation)
their
In particular
geometry.
that
in
away from the main target,
in such applications.
purity
polarization
effects
in
than
more serious
the bore-sight
design
cross-polarization
3) to evaluate
and 4) to
in reflector
type aerials.
The research
to the classical
work started
problem of diffraction
surfaces.
Several
available
in literature.
the physical
chronologically
optics
techniques
current
for
with
by finite,
analysing
perfectly
the scattering
A commonly favoured,
distribution
a rigorous
method.
approximate
In this
approach
conducting
problems
are
method is
the scattered
(iv)
fields
by integrating
are obtained
the region
of the surface
directly
assumptions
in this
The principal
induced
and 3) to facilitate
field
of the primary
These assumptions
current
filaments
current
lines
order
practical
over
source.
1) the current
2) the effects
current
optics
assumed that
of the
are neglected
local
the interaction
between the adjacent
bending
and the peripheral
curvature)
in truncated
at estimating
the effects
currents
on the principal
and cross-polar
patterns
and inferences
of these
of
reflectors.
were thus directed
Results
is
the reflector
studies
antenna.
on an
radiator.
thus neglect
(finite
by the primary
of incidence,
is often
currents
are the same as that
surface
due to edge discontinuities
Initial
higher
it
analysis
optics
are:
on the physical
of the reflector
in the far
technique
plane at the point
target
truncation
illuminated
on the reflector
at a point
infinite
the physical
studies
of
of
are given
in Chapter 1.
Estimates
of practical
physical
obtained
reflectors
optics
distribution
from the above study
of moderately
is not only
but provides
the scatter
large
an excellent
an effective
indicated
dimensions
approximation
means for
that
for
numerical
the current
computation
fields.
by which the scattering
problems,
in efficient
may be resolved
programmes thus developed
The dearth
fed parabolic
analysis
Q 20 X) the
The second phase was thus devoted to study of numerical
surfaces
for
symmetric
and asymmetric
Computer
manner possible.
were then used to study the problem at hand.
of information
reflector
involving
techniques
an cross-polarization
prompted the study
in Chapter
in symmetrically
3.
The intrinsic
of
(v)
depolarization
is shown that
is that
secondary
as is often
ratio
feed itself
the primary
is the
in such a system and a method of
dominant source of cross-polarization
predicting
introduced
50 dB below the pattern
dependent on the f/d
The main conclusion
and it
studied
cross-polarization
can be typically
curvature
maxima and is not as critically
assumed.
belief,
to popular
contrary
by the reflector
are initially
of the reflector
property
field
from primary
peak levels
cross-polar
measurements is described,
A topic
on the radiation
errors
studied
importance
of practical
particular
on the crOss-polar
errors
distortions
are not affected
by such distortion.
however,
considerable
exhibit
of profile
that
reveals
of axially
symmetric
peak levels
cross-polar
The cross-polar
is
levels
side-lobe
degradation.
significant
cross-
isolation.
polar
In Chapter
3 the depolarization
properties
parabolic
reflector
are investigated
polarized
radiation
are numerically
characteristic
readily
model.
a simple
for
formula
as functions
linearly
of the
The method of stationary
by which the beamshift
may be
predicted.
Quantitative
different
calculated
fed
of the off-set
cross-polarization
of the reflector.
parameters
is
used to obtain
phase
will
emphasis on the effects
shape is proposed which achieves
A new reflector
profile
This problem
For the types
patterns.
computation
considered,
of systematic
of the aerial.
characteristics
with
rigorously
is the effect
estimates
modea of excitation
The analysis
be present
of the boresight
are initially
shows that
in off-set
polarization
fed reflectors,
jitter
phenomena for
obtained
on scalar
diffraction
sensitive
boresight
and depending
on the
errors
(vi)
polarization
characteristics
will
either
exhibit
boresight
of the target
calculations
of these errorsfor
shift
in both the planes.
occurs
Closed from expression
axis
for
defocussed
out to study
axially
defocused
The study
can he neutralized
function.
feed and the gain degradation
that
field
distribution
the depolarization
by the appropriate
The study
could eliminate
characteristics
suggests
that
the depolarization
an off-set
effect.
The
on the antenna
computation
is then
for
of the aerial
curves
for
are obtained.
partially
due to the reflector
choice
in the
or shift
are obtained.
distribution
Rigorous
the aerial
Numerical
parameters
the field
the radiation
of the aperture
suggested
typical
for
mull
simultaneously.
feed are obtained.
carried
dipole
on the difference
an increase
or both these effect
axis
echo returns,
of primary
section
tilted
off-set
illumination
of a hyperboloid
-1-
CHAPTER 1
THE CROSS-POLARIZEDSCATTEREDrIELD
1,1
Of Cross-Polarization
Definition
The far-field
field
Cartesian
co-ordinate
definitions
to nominally
of these terms as applied
leads to
polarized'
This has been brought
who discusses
study by Ludwig(')
in a recent
in terms
of the distant
'principally
of the terms
sources.
Is best described
'cross-polarizedlradiation.
and
(co-polarized)
fully
of the radiating
the description
system,
sources,
co-ordinate
system based on the same origin
in the definition
ambiguities
in terms of a spherical
distribution
or current
in terms of a spherical
fields
of electromagnetic
in the vicinity
located
the origin
Since the source
of a local
from a distributicn
radiation
is commonly described
or an antenna,
system with
I
three
out
possible
linearly
polarized
antennas.
In this
definition
thesis
we accept
is that
such a definition,
probe initially
antenna on boresight
This alignment
recorded.
rotating
in the co-ordinate
Is then kept fixed
a set of constant
planes
the polarization
by 90* and repeating
while
cross-polar
vector
from
Employing
of either
which
the polarization
patterns
are obtained
the probe,
of
of the test
antenna is rotated
radiation
patterns
the measurement procedure.
as that
system shown in rig.
the test
the associated
while
The corresponding
the polarization
with
in a
field
obtained
is defined
antenna range with
aligned
(0 a0
of the tadiated
field
electric
would be measured on a conventional
the distant
the most useful
measurement techniques.
standard
the co-polar
that
distribution
to the field
directly
using
conclusion
description
which provides
form which corresponds
actual"measurements
Ludwig's
or the test
1.1).
tbrough
are being
by axially
antenna
-
x
I
(P)
z
V
FIG 1-1
-
the field
If
field
components at an arbitrary
procedure, can be related
(1)
the following
:
E (0,4)
E (0
E(e,
m Eo(0,0)
(0
e)
xE9
1,
+E
sin
4) cos
-E
essentially
4) at any point
on the space-sphere
vector
respectively
by
is noted that
It
is,
element,
that
of equal
strength,
Therefore,,
according
radiator'
everywhere
it
a device
with
sin
1. b
9
the radiated
field
electric
in terms of two spatially
9P, aC are given
vectors
c
l. d
vector
unit
radiated
component aligned
its
P)
by a Huygen
with
radiation
pattern
dipole
the y-axis.
a Huygens source
to the above definition,
in the sense that
in fact
and magnetic
of an electric
the electric
(e
is an
is co-polariz*ed
in space.
source-currents
contribution
(0
reference
consisting
is necessary at this
will
lea
of the electric-field
the polarization
represents
$ideal
the co-polar
by
E
cos
sin
cos
ec(e,
(E.,
components
+ Cos
sin
e
C)
(09
specifies
components whose unit
(E
as measured by this
0, ý),
spherical
This definition
orthogonal
7bis
(r,
to the usual
along the
and the cross-polar
p)
point
is polarized
antenna
(E
then the co-polar
on boresight,
y-axis
by the test
transmitted
stage to relate
J to the polarization
help isolate
those sources
to the far-field
the polarization
of the field
of radiation
cross-polar
radiation.
it
E that
of the
radiates.
which provide
the major
-4-
Consider
an idealised
are the x, y and z directed
radiation
source taken as the reference
The polarization
is
current
in which the basic
situation
given by
f=
filaments,
current
polarization
sources
with
of
y-directed
current.
launched
of the electric-field
by the reference
(1)
(siný coso 0+ cosý ýb
Ey
p
co-polar
Ey. 2p = sin2o
=
Ey. e = siný
-C
,x.
pSimilarly,
its
into
which can be resolved
it
components as
Cose + COS20
cosý (cos 0
can be shown that
components of the fields
and cross-polarized
the co-polar
and the cross-polarized
by the x and the z-direated
launched
current
elements are
(x)
Ep=
,
sine cose (1-cos0)
COS2e (1-COSO)
=1-
E(z)
p
=-
siO
E(z)
c
=-
cosý sino
sinO
Comparision
of the expressions
shows that
the major contribution
from
arises
this
component
particular
their
the
label
contributions
x-directed
is
defined
is
affixed
to
the
for
to the far-field
current
as the
to
terms in each case
the cross-polar
Following
element.
cross-polarized
the
y-
cross-polarized
cross-polarization
and the
current,
z-directed
ratiation.
Ludwigg
whereas
currents
therefore,
no
despite
-
1.2
Analysis
Reflector-Antenna
The electromagnetic
perfectly
reflector-antenna,
I
distributibn
of the current
radiation
determined
at the surface
scattering
currents,
be made exact.
(i. e. incident
involves
plus
and hence,
of the reflector,,
problems
form by an application
is known, the prediction
can, in principle,
by the total
in a microwave
Once the surface
technique.
in the reflector
distribution
in a precise
can be formulated
which in a majority
current
of the scattered
However, the currents
fields
scattered)
a fbrmal
existing
for
equation
must be solved
of applications,
are
approach to the
to an integral
solution
from
scattering
such as the reflectors
surfaces,
conducting
problem involving
radiation
the
numerically.
Such techniques have been applied with somesuccess in solving certain
(2 )
involving
bodies of
dimensional
two
problems
problems
and
class of
(3)
A major
revolutions
it
involves
of
this
of the scatterer
however,
method,
work and is only
amount of numerical
considerable
when the dimensions
limitation
are small
is
that
practicable
in terms of operating
wavelength.
In dealing
problem is
currents,
with
is based on the assumption
that
on areas directly
source,
optically,
the currents
)
11
optics
are the same as they would
whereas in the
are zero.
That is the
are
2(n xH
=
the currents
were reflected
shadowed side of the reflector
j=
the induced
of the physical
fields
currents
approximation
for
the scattering
that
by the primary
the incident
antenna situations
being
the most common approximation
illuminated
postulated
reflector
by invoking
commonly resolved
This approximation
be if
practical
illuminated
the
on
on the shadow side
side
-6-
This approximation
of the incident
and that
reflector
the radii
that
presupposes
of the
of curvature
are large
wavefront
compared to a
wavelength.
Despite
this
provides
approximation
an accurate
radiation,
scattered
the dimensions
is now well
it
restriction,
in the forward
of the scattering
direction,
polarized
even when
geometry do not comply with the
(4)
In fact, this technique
of the physical-optics
assumptions
this
of the principally
prediction
particularly
that
recognized
answers that
yielded
such accurate
replace
a good deal of experimental
has
have begun to
the computed patterns
data in antenna development
and
analysis.
However, the validity
in computing
the cross-polarized
The main argument in this
in the specification
errors
is that,
of the surface
of the edges where the postulated
vicinity
from the true
current
is that
assumption
adjacent
current
the scattering
Therefore,
characteristics
filaments,
in some cases,
it
by the
particularly
in the
currents
can deviate
considerably
the mutual
it
is,
provided
currents,
source of error
ignores
that
as that
the predicted
ignores
in the physical-optics
interaction
the local
between the
curvature
of
body.
before
reflector
can be made on the performance
any observation
of reflector
of magnitude
paraboloid
Another
currents.
approximation
(5), (6)
been questioned
has often
radiation
connection
current
component is of the same order
cross-polarized
order
of the physical-optical
antenna,
of these errors
it
is of interest
particularly
which is of direct
practical
to examine the
in the case of a finite
interest.
-
1.3
In Reflectors
Distribution
Current
-currents
field
J proceeds
E (r)
=E
free
-Lk; L0fx
- 41r
s
(r)
(r)
=Hi
of the integral
+
H)ý + (n,
(n,
x H)
47r
for
equation
of the total
representation
(1)
S
surrounding
region
f
H(r)
upon a perfectly
from the Helmholtz
(Ej H) in the source
i
incident
S, the derivation
smopth, surface
conducting
(E
field
For an electromagnetic
iHi)
x H) . Vjjds
(1.2)
ds
V*
s
is the free-space
*(El, El)
where
function
Green's
by
defined
ikRI
R
and the quantities
for
(1.2)
the vector
integrand
with
product
n.
to lie
r
However, there
r
by a small
surface
AS and taking
This procedure
extracted
in the following
results
in the
is a singularity
is normally
element
and taking
on the surface
This singularity
for
it
surrounding
( 6)
process
point
from
directly
follows
H(r)
J=nx
the currents
the field
by allowing
(1.2).
in
Fig.
shown
are
as
and 2,
EI. K,
The equation
rl
by
the limiting
expression
for
the
surface currents.
J (r)
(r)
=J
1e
(r)
+
n.
21r
-ikR,
fj
xx
ds
S*
where
S*
implies
which excluded
in
(1.3)
in fact
10=
that
the integration
the singularity.
represents
2(n
is performed
The frequency
the physical
optics
over the surface
independent
current
tem
given by
J (r)
-8
p
S
qc-,
FIG 1-2
ometyq
Of' llne ýcaWo--rer.
-9-
writing
ve-lkR,
R,
(1.3)
ikRI
1+
R, -ikRI
R13
in a simplified
can be expressed
form as
(EI)(n
j
El)
- El(n.
1+ik
J(Z1»
-ikR ds
-e
R3
(1. L1)
It
is observed
an infinite
(1.4)
the induced
conducting
as it
vanishes,
case W.
in this
that
therefore,
optics
for
hand side of
the right
the fact
that
currents
plane
should,
the second tem
by the virtue
n) = (n .R)=0.
For this
on
of
geometry
by the physical-
are given precisely
currents.
For an arbitrary
the physical-optics
the interaction
tangent
may be interpreted
estimate
between the currents
Several
methods of improving
have been proposed in literature*
( 7)
and Tandit
on solutions
, relies
using the physical-optics
By, means of high
integrals,
these authors
the leading
tem
illuminated
surface
of revolution
current
frequency
of
(1.4)
estimate
planes
with
drawn
mathematically
for
by a spherical
the physical-optics
approximation
by Tartakovskii
as the initial
iterating
approximation.
of the resulting
explicit
the general
wave.
favoured
by successive
expansions
were able to derive
01)
k
infinite
is represented
One method,
asymptotic
(or order
in
the error
(1.4).
term of equation
possible
therefore,
as being associated
on adjacent
This error
to the curved surface.
by the integral
surface,
curved reflector
expressions
for
case of a conducting
For the special
case of bodies
with the primary source located on the axis of symmetry,
- 10
have shown that
the authors
The surfaces
zero.
f
the'ellipsoids,
identically
terms of 0(T')
are
k
the correction
include
considered
important
are
of which
all
hyperboloids
the paraboloids,
from the practical
and
of
point
view.
The authors have also extended their analysis
to evaluate the
]
for the special case of paraboloid
with
terms of 0.1
of revolution
k2
,
its
focus.
in
form
Huygen
located
the
at
source
of
a
element
primary
indicates
Their
estimate
term
AJ = (J - JO)
in this
that
to the true
of the correction
case the ratio
is
J
currents
a
of the order
of
V/2
sin2
(Kr)2
Where
V
is the polar
(see Fig.
focus
from
the
measured
Thus considering
length.
of 0,25
ratio
maximum deviation
is of the order
2.1)
with
their
aperture),
of the physical-optics
particularly
focal
to diameter
length
estimate
implies
that
f-rom the true
having
reflectors
Clearly
this
order
when one considers
that
only
as
focal-
is the reflector
F
estimate
of 0.2%, 0.05% and 0.02% for
of 10k 20X, and 3OX respectively.
negligible,
and
a paraboloid
(focal-plane
on the reflector
of a point
angle co-ordinate
the
currents
diameters
is
of r-orrection
the
f0%
factor
has been examined.
for
the case of paraboloid
that
the phase of these
Thus their
expected
net integrated
A detailed
reflector
currents
carried
illuminated
exhibit
contribution
to be even smaller.
study
highly
out by Watson"'
by a plane wave shows
oscillatory
to the scattered
behaviour.
radiation
can be
11
-
b
1&'4
Edge-currents
Another
finite-sized
currents
generated
sources
factor
which requires
caused by the presence
in addition
estimate
for
geometrically
problem provides
considering
were a part
where the incident
of a semi-infinite
to the linear
principle
distribution
fields
edge, for
of superposition,
to cross-polarized
currents,
into
fields
are either
Sommerfeld
which is customarily
used.
an obliquely
be used to include
the
of the reflector
by Sommerfeld
incident
edge
and integrating
plane,
polarized
the
The classical
around the periphery
obtained
current
currents.
of these
of the surface
an approximation
currents
every element of the curved reflector
Sommerfeld induced currents
The current
order
by the physical-optics
shadowed areas of the reflector.
This involves
it
The higher
can be made of the magnitude
the continuity
with
of the physical-optics
can give rise
to those generated
which are necessary
as if
of edges;
at such discontinuities
No direct
half-plane
is the perturbation
reflectors
when dealing
consideration
(10)
parallel
plane
for
(9)
the case
to or perpendicular
wave, can, by the
the general
case where the
12
-
incident
would
feed
can
fields
in
occur
are
the
illumination
Following
source.
for
be shown that.
at
angle
a
diffracting
analysis
given
edge by a primary
in
(9),
reference
cos
the
This
edge.
wave
a plane
to
the
a curved reflector
general
(1.5a)
cos
-ikr
(ýk xe
H
to
an angle
of
the
-ikr
incident
at
polarized
the
edge,
(1.5b)
exact
currents
in
the
half-plane
are
i
z
(2,1 kx
J=J0
7r
4(L. H )e
Ei
COS 2
2)
-?
V-2
rk
ikx(l
+ cosa)
+ iff/4
x
(1.6)
where
V2
ilT/4
tI
fe2
IT
T2
d-r
e0
Aa.
cos 7ta
a
sin
a
-1
.
it
13
-
that
current
distribution
at
Defining
orthogonal
0
in (1,7)
7r
)
(0 =
2
(1.3)
for
of incidence
a.
optics
with
either
below - 40 dB relative
currents.
practical,
large
amplitudes
tapering
reflector
towards
It
Also for
a given
when
1/4
occures
with
the case of
is
the distance
0 2-' w/4
seen that
antenna situation,
from the
the cross-polarized
currents
polarized
therefore,
in a
that
particularly
the edge, as in practical
from
and a selected
apart
to the principally
was concluded
(B = 0) or
parallel
edge.
currents
It
currents
no cross-polarized
of the edge itself,
in the vicinity
are suppressed
that
are inclined
of cross-polarized
of the angle
physical
implies
to the diffracting
the edge is shown in Fig.
region
vector
and
the maximum depolarization
of incidence,
The variation
value
field
sin2a)l
when the fields
perpendicularangle
of the magnetic
factor.
The expression
are generated
the
(1.7)
the inclination
is the normalizing
can be shown that
2
(COS20 + sin2a
tI=
J0.
value,
2
to the edge (i, e. z-axis),
respect
exact
kx)l
t(2v
denotes
the
have the forn,
cur: rents
sin 20 sin
1Jol
it
currents,
cross-polarized
edge (x -0--),
as the components of (1.6)
current
to the physical-optics
IJ, I
the
the physical-optics
approaches
the crossý'-polarized
edge-generated
where
away from
distances
large
We note
I
with
antenna,
field
the
-29
-30
96
OT
0
-M
0-40
CN
0( =.
A50
q=7,6*
*
15
K=
-5,0
36'
ýA
0
F1ci 1.3 E bc;EGE NJEP,
ff iEb LRo gs POLAP1Z
Lb
C0R
Pr-- A) TS
13a
-
degree of field
additional
1.5
in the first
to be ignored
currents
reduction
Diffracted
The Physical-02tics
would allow
instance.
Fields
Under the assumption that the current
can be adequately
n
if
reduced
it
is further
distance
at large
fields
the total
to obtain
from
In the far-field
and provided
estimate,
(1.2)
and
can be evaluated
is greatly
However, the problem
at any point.
E(r)
and H(r)
are to be evaluated
S.
the scattered
region
field
components,
s
(E
'0
H3)
the relation
satisfy
si",
(P)
H
s2A
where
(1.1)
of
assumed that
in the reflector
distribution
by the physical-optics
are known, the integrals
H1
and
represented
the edge generated
n
is given
lilil
-=
RxE
,
-ikR
E (P)
KR
defines
the vector
0
26
+
of the source currents
potential
J0,
and
by
4=-1,
fj
ds
(1.9)
p
13b
T, I'
rflflflnrVtYt,
"The Definition
LUDWIG A. C.,
Trans.,
2)
AP-21,1973,
Arbitrary
Vol.
AP-12, No. 6 pp. 746-754,
Antenna",
(1969),
.
7)
(1970),
and Acoustic
Ant.
Prop.
"Analysis
of
Reflector
of Scattered
Fields",
1971.
pp.
"Current
and TANDIT V. L.,
of a Reflector
7, No. 8,
Antenna",
Distribution
on
Radio Engng. and Electronic
pp. 53,1960.
WATSONW. H.,
"The Field
No. 6, pp. 746-754,
Ant.
Prop.
Vol.
AP-12,
Nov. 1964.
by the
POGORZELSKI S. s "Diffraction
Elect.,
in the Focal Plane of a
Distribution
I. E. E. E. Trans.
Reflector",
gommcfmo,
10)
Vol.
183-184.
TARTAKOVSKII L. B.,
Paraboloid
Characteristics
"Polarisation
Letters,
Electronics
by Simple
Scattering
Company. pp. 30-31.
Publishing
North-Holland
and SENIOR T. B. A.,
Rozprawy
Cylinders
Nov. 1964.
and POTTER P. D.,
"Electromagnetic
Phys. 5,6,
9)
Metallic
SENIOR T. B. A. and USLENGHIP. L. E. (Ed. ),
the Reflector
8)
53-61,
pp.
I. EX. E. Trans.
KNOTTE. F.,
April
of
Academic Press.
Shapes".
6)
Cross-sections",
BOWMAN
J. J.,
5)
from Parallel
with
RUSCH W. V. T.,
4)
Solution
1967.
M. G., "Scattering
ANDREASEN
3)
"Numerical
I. E. E. E. Spectrum,
Problems",
Electromagnetic
I. E. E. E.
Polarization",
116-119.
and ANDREASEN M. G.,
TANITER R. L.,
Sept.
pp.
Cross
of
V: )l.
A. -J.w
13,1967,
edge of
an Antenna
Reflector",
p. 451.
Opilcsl: Acaciemic Pyrsv,inc.,
New.voy)c,
- -1
- 14 -
CHAPTER 2
ANALYSIS or AXIALLY SYMMETRICPARABOLICRErIXCTOR ANTENNA
The symnetrical
parabolic
source feed at its
point
to the high-gain.
solutions
of construction,
microwave link
provides
for
is well
radiations
known for
polarized
electric
field
parallel
and perpendicular
A study
element.
been carried
for
simplicity
satisfactory
type of antenna
from small
dishes.
symmetric
and the magnetic
parabolic
located
at its
has also
who
.
focus,,
the'reflected
of the dipole
of polarization
of these
investigated
of the antenna due to illumination
by a
of components both
consists
patterns
dipoles.
is illuminated
when the reflector
to the direction
of the radiation
a
the case of such analytically
in the aperture
out by Jones(')
characteristics
that
dipole
distribution
this
of the axially
as the electric
electric
and generally
radio-astronomy
studied
example,
linearly
Owing its
of frequencies,
characteristics
having
one of the simplest
a number of applications
antennas to large
have been frequently
primary
2.1)
high efficiency
choice
The polarization
It
(Fig.
antenna requirements.
moderately
has been a popular
reflector
or the paraboloid,
over a wide spectrum
performance
simple
focus
reflector,
components has
the aperture
by magnetic
field
dipoles
and Huygen elements.
However, while
providing
paraboloid
an understanding
have proved useful
reflecting
obtained
in
of the
properties
by the use of these sources
to assess the performance
of an antenna in a
situation.
A realistic
assessment of the cross-polar
under such conditions
may arise
of the basic
the predictions
surface,
cannot be generalized
practical
devices
these theoretical
from all
requires
consideration
performance
of an antenna
of the contributions
the components which comprise
which
the antenna system.
-15 -
/I
)o-
FIG 2-1 Gt:0',,
OF
THE
PARABOLOID
IETVýY
JLI
16 -
These include
such as the feed support
feed and peripherals
with
associated
2.1
Consider
a perfectly
conducting
In this
source can be written,
focus
(Fig.
fields
source so that
from the source
radiated
by the
-iKp
+E2.
Cp
8P
e,
Ot
-iKp
P-
4)
unit
orthogonal
e
are two arbitrary
-C
sphere and E
and E
are components of
pC
vectors
and
directions,
E
=f
(2.2)
tangent
to the
along these
respectively.
e
p
and
are selected
e
p
(rig.
2.1).
the cross-polar
to be the directions
sirg a+
cosC Z-C
a
cos;
sin;.
2-
defined
by
2
then, for a primary feed having its
y-axis
The reflector
as
or equally
If
a linearly
in the form
+ EC (ý 9oa)a cp
radiation
2.1).
of the distance
case, the electric
either
with
reflector
of the primary
region
as the inverse
varying
are of interest.
paraboloid
at its
source located
primary
the fields
where
and the hardware
structure
it.
is assumed to be in the far-field
only
the primary
distribution
Current
polarized
by the reflector,
those components created
principal
electric
EP and EC correspond directly
components of the feed radiation:
vector aligned with
to the co-polar
and
- 17 -
Ep(*,
(ý,
E*
sinc
0=
C) + COSCE4(*,
0
(2.2)
E (ý,
C)
the dimensions
Assuming that
the operating
the incident
wavelength,
C) - sinC Ec (ý,
of the reflector
the current
wave can be approximated
IR
j2x
-0
ror
co sC E (*,
IP
=
(a
z0
a paraboloid
xE
ff
in terms of
are large
in the reflector
distribution
to the physical
optics
due to
currents,
)l
(2.3)
which is defined
surface
C)
bY the equation
2f
+ co#
it
is readily
reduces
an incident
for
shown that
wave in the form (2.1),
(2.3)
to
2(l
+ cosO
K(*
0z0tp
-ikp
ýý
(2.4a)
where
Kx(ýs 0
Y, 41
y
(E
0E
sinc
COS*
Ep
The relationship
cosi
in (2.4b)
currents
the crqss-polarized
demonstrates
(K. ) in the reflector
fields
effects
quito
clearly
that
are excited
(Ec(ý, C)) of the incident
important
but
obvious,
the cross-polarization
o4b)
E,
*12
tan
-
t2
to one rather
+Ec
C(ý
C)
(2
Kz (Vo C)
cross-polarized
EE
sinC
cosC
*
the
directly
wave.
by
This leads
conclusion that in order to minimize
in symmetrical parabolic
reflector
antennas,
- 18 -
feed should be designed
the primary
0
E
COP.
for
does not radiate
to be zero
any
constraints
components E. 9 EC of the feed radiation,
on the spherical
that
it
imposes certain
This condition
energy.
cross-polarized
so that
which requires
I
tanc
One example of a source which satisfies
is,
that
element,
crossed electric
impedance.
free-space
In this
(1 t co#)
(i
satisfy
2.2
Scattered
If
rig.
'ideal'
then,
Es (010
pIy
of practical
feed condition
which approximately
nature,
are provided
in the following
point
is
defined
sections.
A2) that
'0
0,
shown in
y,
by the co-ordinates
of the vector
the normalised
field
diffraction
theory
it
can
s)
(E
and the crossp
co-polar
radiated
by the reflector
can
as
COS2012 IIY
(09o)(l+COS20
=
(090)
I
tan2 6/2)*" Ix (ego) tan2 e/2 sin2ý
21 (0, o) tanO/2
z
s
ECOSO
by the
case
(Es) components of the electric
c
be expressed
related
sinC
by an application
be shown (Appendix
polar'
dipoles
is a Puygen
radiation
a far-field
2.1,
and magnetic
condition
+ CosO cosc
Examples of sources,
the
this
[1
e/2
cOs2
(0, o)(1-COS2o tan26/2)
=Iy (010)
x
sinýl
(2.5a)
-I
y
(ego) tan2 6/2 sin2o
+Mz(O, ý) tanO/2 COO]
(2.5b)
19
-
where
ffE(*,
(010=
- sinO siný
co#
C)e-iKp[l+cosO
the vector
is seen from (2.5)
It
ror
from all
arises
radiation
p1y
2.3
for
(2.6))
angles
of diffraction,
as arising
the co-polar
directly
and
from the
y
respectively.
of the diffraction
integrals
by assuming that
the spherical
of the primary
E(*)
cOsO/2
Model
circumstances
EC(*, C)
patterns
small
can be regarded
fields
The computations
certain
however,
case
components of the currents,
_Simplified
where
of the boresight,
ix(oso)
y
is seen that
x
in the reflector.
(em
Ec
si(evo
and
to the cross-polar
(0,ý)
y
(0,0)
Esom
the cross-polar
the contribution
the components of the currents
and in this
tanO/2 -1' 0,
It
that
in the vicinity
points
K.
of the source currents
Potential
(2.6)
d* dý
psin*
defines
cos(C+ý)]
(*gC)
=
(V. 0
= cosC H(V)
HM
,
sint
can be performed
Ev (*909
components
in separable
under
form as
E(*)
correspond
of the feed.
can be expressed
radiation
can be made simple
directly
In these
analyticallly
to the E- and the H-plane
cases the integration
over
4
by the use of the formula
radiation
(in
- 20 27r
e
i0cos(c+ý)
The types
sin
cos(nýTý)dC
such as the electric
elements
and, in general,
physical
circular
radiated
by these feeds is given by
planes
primary
for
the expressions
to the following
os
20
(balanced
in (2,7)
patterns
feed radiation).
type are being
and the cross-polar
considered,
in
to the integrals
A
cos24)II(8)+(tan2.
2
-itan
2 [tan2
c2y2z
which possess
and the phase of the radiation
y
sin24C0B2
(2.6)
an
reduces
components in
(2.5)
j
jt(O)+jxt(9)+i
ý
+cos2e)I
i
sin2eI
2
tan
11
Z
2-1l(0)1
,
(2.8a)
(2.8b)
y,
where the reduced
il(0)
I'M
x0
integrals,
jýMi
(ß)
=
f Mi
22
II, if,
xy
(E+li)
(E+2H'
I
Z'
are
psineeict
dý
(2.9a)
psin*e
dip
(2.9b)
fýMJj(O)E(*)02sin2*eic'd*
11(0)
z
in which
= -1
4f
(2.9c)
0
the
Huygen
form
[(1+tan2
lit(0)1
dipoles,
include
the cross-polarization
noting
feeds of this
the co-polar
simple
"i
procedure
and magnetic
is worth
are equalized
of the relation
application
(2.7)
cosC
when the amplitude
Assuming that
P
It
symmetry.
in the principal
by this
waveguide or horn radiators
Ec (*, C) = sinC
which vanishes
(O)S"(n+l)ý
n
Cos
of feeds which can be treated
radiators
elenentary
T21r
(jýnj
=
21
-
1+
47r (fix)
4ff (fl,,
(f
for
which
the
square
'm Ot 7r/2)-
in
it
the
(i. e.
terms of (2.8)
Numerical
were carried
effects
diagonal
the
the
of
in
fields
cross-polarized
is identically
essentially
space.
any crossThe remaining
zero.
the cross-polarization
of
geometrical
feed does not radiate
of the paraboloid
terms
illumination,
feed
planes
produced as a result
surface.
Results
of the integrals
computations
out to an accuracy
for
radiationpand
Initial
this
purpose,
and (2.9)
places
effects
the primary
using
103,
Algorithm
were directed
calculations
of the diffraction
the contributions
decimal
Simpsons procedure,
(2).
in reference
in (2.8)
appearing
than four
of better
of the adaptive
which is described
cross-polar
functions
primary
the primary
then describe
an improved version
at estimating
are
H) then Ix (0)
E=
of the diffraction
Numerical
the
of
is assumed that
polarization
2.4
and the
in
The ý-independent
which
reflector
in the principal
vanishes
occurs
n=1,3,5,7.
(2.8),
distribution
the
If
where
brackets
of
cross-polarization
The maximum radiation
nv/4
parameters
control
) sin* sinO
1+ eosý
is seen from (28) that
it
planes
coso cosip
I+
COS*
to the overall
source was assumed
to be a Huygen element.
Typical
plots
of rig.
radiation
to diameter
results
obtained
(2.2).
The'diagrams
in the diagonal
ratio
33A, 84X and 20OX.
from these computations
(f/D)
It
planes
illustrate
of reflectors
of 0.357 (*
is seen that
M=
are shown in the
the cross-polarized
having
the focal
720) and aperture
the diffractive
length
diameters
contributions
of
to the
16
-22--
r<
cr) -'T
-00
.-on.
C:)
C)
CN
z
CL
it
0
CL
V)
V)
0
w
3
0
0--l
z
<
C
\
i ci
CL
*%
04
Ln
.00
Cý
04
UL
-
. 1-1
LO
ILO
OIP) JO-/AOdOAOIIDI08
00
Ir"m
i
- 23 -
cross-polar
radiation
to vanish
in the limit
that
these levels
It
into
is extremely
as VD approaches
currents
(as demonstrated
patterns
shown.
earlier),
However, in considering
that
contributions
a perfectly
the higher
order
source
edge currents
deviation
the slightest
'ideal'
of the first
'ideal'
the E- and the H-plane
by a small
amount such that
of the feed radiation
co-polar
this
are localised
waveguide with
this
aperture
of this
which shows the cross-polar
in section
diffraction
which may be
To examine
surface.
in (2.9)
of the
diffraction
were allowed
solution,
to differ
in the diagonal
plane
by some 40 dB below the peak of the
of 0.7
computation
Whereas
currents
functions
pattern
A discussion
2.8.
for
t
mode circular
on the primary
-
are shown plotted
patterns
selected
from a TE
to those of radiation
diameter
is unlikely
from that
the peak cross-polarization
type is provided
The results
it
near the rim of the reflector,
on the physical-optics
was suppressed
correspond
may, in fact,
feed illumination
fucntions
conditions
in practice.
can be realized
must
priority
For example,
crder.
(The E- and the H-plane
beam.
computation
feeds-of
pattern
the radiation
and which,
can produce cross-polarized
of such perturbation
to cross-polarized
rise
effects,
order
over the major area of the reflector
the effect
do not take
of the assumed idealized
of the primary
feed condition
distributed
can give
situation,
reveal
due to. the truncation
which can modify
to the perturbations
tend
increased.
the above calculations
these higher
in a realistic
which may arise
provide
that
also
is
as the f/D ratio
Edge discontinuities
be given
Calculations
of the edge discontinuities
the effect
of the reflectors.
clearly
and as might be expected,
zero.
even further
diminish
must be remarked however,
account
small,
in rig.
(2.3),
of the same reflectors
-241<
e<
-4
00
e<
CY)
CV)
z
Lx
Ld
C)
C)
G5
r-
(0
If)
--T
m
Cý4
CY)
C*-4
IT-
CD
(D
LO
C3
(D
(OP) JDMOd
(D
t-OA41)Vsd
CD
00
0
LL
- 25 -
when illuminated
of the cross-polar
diameter
'non-ideal'
by this
of the reflector,
feed.
primary
in the variation
is only
of t4e diffraction
in any case,, are some 70-80 dB down.
2.5
Practical
blocking
The construction
direct
for
structure
In this
analysis
situation
into
the eleciromagnetic
the higher
radiation.
order
by and large,
and the secondary
and the effect
scattering
the
reflector
of
immersed in
the
a formidable
in dealing
with
this
defined
cast well
radiation
of the shadowed areas
surface.
Under this
by the feed and the supports
which may be significant
are generally
radiators,
upon the far-field
the contributions
of the full
effect
they are,
the feed and the supports
by detracting
from the contribution
of the supports
A common practice
resolved.
is to assume that
account
which can modify
due to these components presents
of the scattering
shadows on the reflector,
taken
dimensions
of both the primary
problem and not easily
feed and the
surface.
and since
of a wavelength
require
feed at or near the focus
of the fields
scattering
generally
both the primary
situation,,
Since the cross-sectional
antennas
the primary
from the reflector
radiation
the near fields
0
reflector
mounting
cause additional
the order
which,
effects
of practical
of th e paraboloid.
supports
levels*
at wide
considerations
Aperture
supporting
from the
apparent
'side-lobe'
of the cross-polar
in the
contributions
are now swamped by the contributions
The effect
the peaks
to the variations
the diffractive
that
suggesting
to the cross-poiarization
angles
are now insensitive
radiation
is seen that
It
source.
only
assumption,
is taken
at wider
as
anle of
- 26 -
is clear
It
of their
the
the
of
additional
shadowing
the
bore-sight,
Fig.
provides
fields
within
(2.4)
of diameter
d. with
In the computation
is taken into
,
to
the focus.
Similarly,
the optical
tan-I
about
rectangular
by changing
found
that
characteristic
the
antenna
here.
of the
of a selection
which may arise
a central
cross-sections
the blocking
fields
disc
of width
W,
of the central
effect
the lower
in
limit
in the *
subtended by the disc
at
is given by
(2f )
'd .
due to the radial
(2.4)
it
struts
is taken
into
those areas of the surface
For the rectangular
shadows of the struts.
is r.eadily
shown that,
account
affected
by
shadows of the
the shadowed regions
by
tn 4(*)
'c
'aCB(*)
where * varies
strut,
main
be implemented
is the semi-angle
from the integrations
type shown in Fig.
are defined
simply
*B
ýB' where
the blocking
by excluding
having
account
angles
to the conditions
of the model,
This angle
*=2
B
cone of
has been
the
of
since
the
of
is assumed to include
structure
struts
indication
will
concerned,
it
plane projections
which approximate
The support
integration
method
shows the focal
practice.
is
Nevertheless,
a small
as the predictions
depolarization
possible
a fair
and as such this
shadowed surfaces
disc
the
as far
radiation
scattering.
method
radiated
inadequate
cross-polarized
ignores
essentially
this
model is
this
upon the
effect
model
due to
that
from *B to ý
'c Cn + ACIP)
Cn denotes the angular position
m"
aud At B is the half-angular
of the nth
width of the shadow which is given by
2-ý
/
T--
CL
4----
Fig
4
EAoC-kt.
Aýe--ftuce
Aj
C-oVIjLqLLVCCtLCIY'LS
28
-
sin-1
B
Thus considering,
strut
as an example,
*M
d*
diffraction
a typical
21r
cl -Ac BM
f
F(ý, C, G, ý)dC
+f
r(*, c,m)d;
"B
(2.10)
I+Al: B('P)
10r.
where F4, CO90)
(2.4),
single
is compuied in the form
in (2.6)
f
located
the case of an arbitrary
in
rig.
type
the
shown
of
configuration
integral
ItW
cOtV/2]
4f
incorporating
is the complex quantity
the terms of the
in (2.6).
integrand
Primary-feeds
A variety
paraboloid
radiators.
cross-polar
It
response
in the literature
mode conical
interest
of considerable
can be expected
with
the
and several
types
of feeds
are the TE10 mode
application
horns and the TEll
is therefore
use in conjunction
Two of the simple
polarization-diversity
aperture
for
suitable
have been described
have been reported.
for
rectangular
types
of primary
reflector
applications
suitable
sources
or circular
waveguide
to know what typical
from an antenna which employs these
of feeds.
The overall
of the reflector
patterns
from the general
relationships
at the reflector
are known.
be defined
the purpose
as the far-field
Approximate
rectangular
In the majority
simple
mathematical
the radiated
represent
expressions
and circular
and (2-6)
once the fields
describing
aperture
homs
fields
expressions
can be found
incident
and for
are required
of the feed-types
the radiation
computed
these may
of applications,
components of the feed radiation,
of computation,
which adequately
in (2.5)
antenna can be readily
involved.
from both the
in
literature
(3)
- 29 -
In each case, however,
inconsistencies
considerable
between the cross-polar
exist
in literature..
data reported
experimental
from the reflector
radiation
feed illumination,
dependent upon the primary
feed model must be used if
accurate
In view of this,
model for
a suitable
it
is evident
of horn aperture
the primary
that
an
are to be valid.
at this
considering
cross-polar
to be critically
the predictions
is worth
made in the treatment
assumptions
establish
it
and the
Since the overall
has been demonstrated
but
components
by each model and also between these predictions
predicted
2.6
model is available,
more than one theoretical
stage some fundamental
and thereby
radiators,
feed.
Feed-models
which is frequently
One method of approximation
from waveguides
of radiation
field
principle
equivalence
that
states,
distribution
if
certain
electric
is known over a closed
continuity
Applying
detemined
principle
equivalent
radiated
fields
electric
(E
and/or
surface
provided
tangential
magnetic
then the field
the surface
field
at any exterior
fields
satisfy
conditions.
this
mathematically
to the problem
of the
and horns is based on an application
(3)
The principle
to the radiating
structure
the tangential
can be uniquely
point
applied
to the horn radiator,
expressions
to be calculated
) or the tangential
tan
approximate,
can be obtained
which enable
from a knowledge of either
magnetic
field
(H
but
tan)
the
the tangential
or a combination
- 30 -
distributions
of these
It
across
is noted that
radiation
known Chu's treatment
I
C't'),
described
by
Silver(locas
problem,
the well
example of an application
In demonstrating
we shall
the different
whereby the radiated
flare
as one
fields
and the magnetic
can be obtained
the case of radiation
consider
angle so that
the aperture
of TE10 mode in an infinite
waveguide.
In this
that
results
in the dominant TE
mode.
10
horn excited
small
can be regarded
in the aperture.
distributions
procedures,
of the principle,
of the waveguide
from an assumed knowledge of the electric
are obtained
field
the mouth of the horn.
case, two equivalent
by these
from a rectangular
The horn is assumed to have a
fields
may be approximated
to that
models are commonly employed:
Chu-model
This model is based on the use of Chu's equation
form a rectangular
The aperture
waveguide.
to consist
of the TE10 mode tangential
tangential
magnetic
the transverse
field
component,
E=
ýM-t
Z.;It
y
ct
is assumed
distribution
to the electric
impedance of the waveguide mode; that
in the z-direction,
propagating
the radiation
distribution
electric-field
related
for
the assumed aperture
is,
for
fields
and a
component by
the mode
are
Yx Et)
=
--x -
010
impedance
is
the
freemode
to
with
the
normalized
respect
a=
wnere
K
space impedance.
- 31 -
For a horn with
by this
model can be expressed
[sin2C
Ep 0"0
=
Ec (*, c)
=
siný
inthe
(H-plane).
xb
fields
components of the radiated
and the cross-polar
co-polar
of a (E-plane)
dimensions
aperture
the
predicted
form as
normalized
1)
(atcos*)
F(*gt)
t cos2ý(l+acosf
(2.11a)
F(*, c)
cosc(l-a)(1-cos*)
where
sin(ir
a sin* sinO
wa sin* sin;
(ii)
E-field
is that
of this
cosC)2
the radiated
principle,
field
Assuming that
the aperture
electric-field
TEIO waveguide modes the co-polar
fields
by the use
obtained
model can be shown to be given by
(sin2c
Ec4*0
=
is seen that
Chu equations
dipole
COSO
knowledge of the electric
components of the radiation
E
p
It
siný
of the unperturbed
and the cross-polar
E-field
from a
in the aperture.
distribution
1-(2b
consequence of the equivalence
may also be predicted
distributions
b sin*
model
As a direct
fields
cos(w
vanishes
model predicts
having
(i
t
- cos*)
cos*)
cos2c
sinc
that
a directional
component predicted
as a -o-1 (plane
horn behaves essentially
pattern
(2.12b)
cosc F(*, 4)
whereas the cross-polar
in the limit
(2.12a)
F0,0
by the
wave source),
the
as a short
electric-
FQ), C).
1.11
- 32 -
In Figs. (2.5a& 2.5b)
are shown superimposed
dimension
has an aperture
It
than 10".
the E-field
somewhat in its
but differs
that
circular
level
reflector,
for
of magnitude
in the diagonal
hand,
the peak
It
patterns
(± 64")
reasonable
component
distribution.
spatial
angle of
On the other
low.
the cone of angles
the cross-polar
(with
the cross-polar
order
This horn
both models
while
radiation
the measured co-polar
although
symmetry within
that
horn
= 64*).
X and a flare
is significantly
the right
model predicts
worth noting
the co-polar
results),
by the Chu equations
predicted
level,
for
the experimental
agreement with
X, b=1.11
and (2.12)
a rectangular
of 0.4(*m
is seen fxom these results
predictions
similar
provide
of a=0.92
for
data
F/D ratio
with
from (2.11)
obtained
on the experimental
to feed a paraboloid
designed
less
the predictions
is also
exhibit
good
subtended by the
plane
is relatively
high.
It
must be remarked that
Chu-equations
inaccurate,
does not necessarily
but merely
from the actual
and this
fields
If,
with
model relies
in the apertureq
field
example,
only
order
only
accurate
as indicative
in the aperture
is
differ
modes exists,
of the electromagnetic
explain
the low cross-polar
hand, since
On the other
on the knowledge of electric-field
the comparatively
distribution
from the use of
over the aperture
higher
then the omission
by the Chu-equations.
obtained
the Chu-analysis
that
these modes may well
model may be interpreted
electric
for
appears very likely,
associated
imply
the assumed fields
that
fields.
component predicted
E-field
the poor results
result
distribution
obtained
of the fact
the
that
is substantially
from this
the tangential
linearly
C
E-Plang
-11
1
14
w
M
1
0/.
u
w
1111
---
13
H-planes
-1
-20'
-so
-60
0
-30
30
so
so
DECREES
radiation
Fig-2-50LCo-polar
flare-angle
small
0.92X x 1.11X
fields
in
rectangular
F_.-FiCLD
CVAU_ fyIODSL_
the principal-planes
horn with aperture
from a
dimensions
S4
C')
-J
w
0
a
-so
r
-60
30
0
-30
so
so
p
DECREES
in the
fields
Fig. 2-5tRadiation
0.92X x 1.11X rectangular
predicted
diagonal
horn.
planes
measured
from
(9,
the
UrjC%E)
cý%umocleL
i
- 35 -
polarized,
and approximates
reasonable
to suggest
transverse
horns propagating
H-field
the
about
made
solution
for
defined
the rectangular
In satisfying
continuity
at the aperture,
the solution
type which are in general
evanescent
to the radiated
contribute
This solution
further
fields,
indicates
the y components of the magnetic
2.7
Typical
It
antenna
hom of the type
feed for
To provide
Rudge.
section
described
the TE
mode
contribution.
10
the aperture
discussions
in the previous
to the overall
both the x and
contains
cross-polar
a practical
feeds,
that
if
the rectangular
were used as primary
section
the feed itself
an example of the typical
for
the guide and which may
field.
from an antenna which employs these
in this
to waves of mixed
rise
2erformance
the parabolic-reflector,
significantly
gives
type 11,
of the field
the requirements
but include
from the foregoing
is evident
of the magnetic
within
that
a more general
the dominant mode can
propagating
potential
can be
This view is
who has shown that
jwu Vx11,
by E=-
assumptions
in the horn aperture.
waveguides
vector
from rectangular
the radiation
modes no simple
( 4)
of Lewin
from the Hertz
be obtained
with
electric
distribution
by the analysis
supported
in dealing
that
is therefore
It
of TE10 mode.
to that
would contribute
radiation
from the antenna.
performance
that
numerical
results
can be expected
are presented
antenna used in the experimental
program of
- 36 -
The antenna under consideration
of 0.4 and a diameter
f/D ratio
support
structure
struts,
each of width
of three
consists
in Fig.
(2.6),
account
by the shadow diffraction
The blocking
The predicted
diagonal
(2.9)
plane
fields
The slight
model to predict
cross-polarized
in the secondary
primary
described
The
in Section
and the cross-polar
(2.5).
of the horn and
in the
radiation
fields.
of the cross-polarized
fields
cross-polarized
disagreement
into
(2.7)-(2.9).
of Figs.
are the measured
between the predicted
in part
the precise
to the inability
distribution
spatial
the peak level
to note that
pattern
is approximately
feed cross-polarization
in peak gain can be regarded
and
of t he
of the incident
3 dB below the peak level
intercepted
as being
of cross-polarization
by the reflector.
of
This loss
caused by the poor focusing
of the paraboloid.
efficiency
Theoretical
are reduced
predictions
to provide
the peak level
radiation
also
show that
if
the feed aperture
a -10 dB edge illumination
of cross-polar
by approximately
secondary
of 2 1.
energy.
is interesting
It
rectangular
diameter
from the combination
measured data can be attributed
E-field
with
The feed
of these components is taken
the peak level
which contains
is the
(2.6).
placed
with
case is of the type illustrated
technique
shows the co-polar
of Rudge.
disc
in this
effect
Superimposed upon the predicted
results
symmetrically
are shown in the plots
the reflector
Fig.
radiation
reflector
The primary-feed
1 X, and a central
shadowing configuration
aperture
the
of 40 X.
a parabolic
in our example of Section
horn described
rectangular
comprises
radiation
2 dB, and the level
patterns
incident
in the principal
at the reflector
of cross-polarization
show similar
dimensions
increase.
planes
increases
in the
This suggests
that
-11
-
0
-10
-20
-41
-
UL1
F 19 1
:we
-38-
0
'-I
pJG
2-8:
fields
Radiation
40AparaboloId
the
by a
supported
I4
in
tripod
the
with
diagonal
rectangular
structure.
from
plane
feed
--20
13
-40
05
or=C3REES
FICj7_-'(j
Cross-polarised
fields
in the diagonal-plane
radiation
from the 40% paraboloid
feed
with the rectangular
supported by a tripod
structure.
0
predicted
measured
-40-
co-polar
-1
U)
J
w
0ý-
Lq
Q
cros!
ý-polar
F.
-GO
-0-0
00i:
SO
60
0
G R, ZEEs
r
Fig.
ZlORadiation
in the
fields
1-57k x 2,1', 'i\ rectangular
predicted
diagonal-plane
horn
from
the
measured
( Qu D
I
- 41 -
the overall
of the paraboloid
performance
cross-polar
feed
by
the
radiation
of
measurement
assessed
general
pattern
rule,
the peak level
will
be approximately
the antenna design
will
for
suitable
introduced
large
F/D ratio,
employs a reflector
in the diagocal
horn with
rectangular
2-3 dB below the peak level
be proportionately
patterns
radiation
feeding
the peak cross-polar
aperture
a paraboloid
level
with
required
to illuminate
large.
Fig.
is now reduced
flare
angle
X which is
SCCVA
It is/, that
of 0.95.
the overall
significantly;
feed will
this
the reflector
of 1.57 Xx2.14
F/D ratio
with
then
(2.10) shows the
of a small
plane
dimensions
from the antenna using
radiation
in the secondary
of cross-polarization
of the feed aperture
the dimensions
efficiently
As a
characteristics.
feed.
by the primary
If
can be
antennas
thus show similar
improvement*
2.8
Circular
The theoretical
dominant
conical
aperture
feedhorns
for
predictions
mode circular
aperture
the far-field
from the
radiation
horn or waveguide radiation
conical
are
from the
based upon the use of Chu-equations for the radiation
(3)
As mentioned earlier,
this model is
open end of a circular-pipe
frequently
based on the assumption
in the horn aperture
distribution
related
to the electric
fundamental
exists
of a TEll
field
waveguide mode.
between the predictions
mode tangential
and the tangential
by the transverse
Once again
knowledge
assumed
of the tangential
an
on
alone*
(Theoretical
expressions
from these two models are provided
for
magnetic
component,
impedance of the
howevers considerable
from this
obtained
field
electric
disparity
model and those based
electric-field
the far-field
in Appendix A. 2),
distribution
radiation
obtained
- 43 -
Fig. (2.11 a/b)
diameter
conical
F/D ratio
of 0.4.
predicted
fields
horn which is suitable
for
planes
data for
whereas the E-field
be almost
(2.11b).
symmetric
has in fact
is noteworthy,
however,
been achieved
around the. outside
recentlý%ý
of a small
walls
from an assumed
walls
of the guide,
in both the theoretical
Similar
the radiation
patterns
radiation
reasonable
interesting
It
for
is
agreement with
feature
be
should
dB
below
the peak of the co-polar
-40
that
the performance
of this
the use of peripheral
aperture
circular
the contributions
between the predicted
comparisons
of 2.3 X.
should
order
chokes
The
waveguide.
of current
flow
of which are neglected
models.
(2.12)
in
Fig.
are shown
diameter
from
considerably
main purpose of the chokes appears to be the suppression
in the outer
it
alone.
tends to predict
and the cross-polar
by at least
the radiation
than the measured fields,
of cross-polarization
that
are the
in the diagonal
radiation
The Chu-equation
model suggests
suppressed
It
fields.
the cross-polar
level
circularly
consequently
from both models deviate
the predictions
higher
for
distribution
aperture
with
patterns
guide and (b) those predicted
as shown in Fig.
significantly
a reflector
from (a) the Chu-equation
obtained
knowledge of TE mode electric-field
11
the experimental
feeding
from a 1.15 A
Superimposed upon the experimental
from a TE
mode
circular
11
is seen that
fields
shows the measured radiation
and the measured- fields
the case of a conical
seen that
in this
the experimentally
of the radiation,
horn with
aperture
case both the models are in
derived
however,
patterns.
is that,
An
in comparison
to
-'-rl"-
0
L
E-Plane
J
tn
I
.9-160
-7r
LU
0
H-plane
\I
-60
a
-30
60
30
0
DEGREES
predicted
mea
su re d
I
fields
Co-polar
radiation
for a linearly-polarised
in the principal-planes
conical-horn
of diameter
1-15X
c
co-polar
cn
-i
LU
U.
C:2i
cross-polar
-Ar
60
IN
00
-j
60
30
DEGREES
--0-
Fig. ()L)
preclicted
MOasurecl
Cý-- ý'
----
F-
M
(R U Dc, C)
fields
Radiation
linearly-polarised
in
the diagonalrplane
for
of diameter
conical-horn
the
1-15X-
-
0
co-polar
LU
m-
20
z;
w
cross-polar
-/l
a
-30
30
60
DEGREES
-v. .
pre
dicted
measured
---
C6-'ý"
"l
(RUDCAE)
in the dia-onai
fields
Fig., 2. (l Radiation
linearly-polarised
2-3% diameter
plane from
conical-horn
the
- 46 -
to the small
high.
consequently
in fact,
of the cross-polar
feeds cannot be made unless
however,
characteristics,
examples that
assessment
a realistic
fed by small
conical
aperture
fields
can be
measurement of the feed radiation
A direct
will
of
properties
trend.
feed radiation
the primary
confidence.
with
to the radiation
a
is
of cross-polarization
from paraboloid
radiation
feed produces
circular
shows the opposite
from the foregoing
is evident
predicted
aperture
is in contrast
This
horn which,
the rectangular
It
the large
main beam apd the level
elliptical
markedly
horn,
aperture
provide
indication
a fair
of the expected
performance.
2.9
Conclusions
The principal
source
reflector
antenna
reflector
is
the
introduce
higher
particularly
to
antennas
The dominant
mode conical
polarization
which
tend
to
Therefore,
peak
smaller
employing
itself.
importance
only
with
levels
smaller
of
horn
an increasing
feeds,
assuming
in
the
which
function
reflectors
that
than
the
feeds
other
of
parabolic
of
that
sense
apertures,
on the
levels
cross-polar
The F/D ratio
primary
feed,
the
a symmetrical
the
with
the
apertures.
radiates
cross-
horn-aperture
with
large
its
smaller
smaller
applies
rectangular
hand,
the
consequently
This
cross-polarization.
employing
feeds,
these
with
higher
exhibit
is
feed
feeds
primary
in
cross-polarization
primary
of
a parameter
require
ratios
is
of
diameter.
F/D ratios
ratio
edge illumination
will
counterpart
remains
the
same.
The cross-polar
reaches
planes.
its
peak
value
radiation
from
in
planes
The peak value
corresponding
the
inclined
of the dross-polar
to approximaýely
is
a paraboloid
DB
points
-12
at
zero
450 to
radiation
on boresight
I
the principal
occurs
of the co-polar
and
at a point
main beam.
- 47 -
The aperture
shadowing by the feed and its
effect
upon the co-polar
lobes
can be significantly
side-lobe
designed
model selected
to represent
its
effect
selected.
far-field
to meet low side-lobe
the blocking
caused by the currents
upon the cross polar
radiation
However, the contribution
cross-polarization
become important
to a low order.
once the primary
effect
principal
These
especially
when
The crude
specification.
is unable to account
induced
in the struts,
for
and thus
cannot be assessed by the model
from the strut
is expected
has its
of the antenna.,
by the shadowing,
modified
the antenna is
the depolarization
structures
supports
currents
to be small,
feed cross-polar
to the
but which can
radiation
is suppressed
- 48a -
APPENDIX A2
Derivation
the paraboloid
the far-zone
for
Expressions
ir
reflector
with
associated
vector
forthe
can be derived
ikp
Jo
.
'A
to the co-ordinate,
reflector
fields
by
radiated
from the Hertzian
vector
potential
in the reflector:
ds
A2.1
geometry shown in Fig
and the differential
normal,
paraboloid
electromagnetic
the currents
4,
Ur.4, ; xf J
la% -0
s
Referring
integrals
of radiation
surface
element
(2.1),
the unit
ds
on the paraboloid
are
t cos*)a
sin*
p
t _-
da
=
p2 siný
d* dC
=t
P'2 sin* dý dC
(i t COS*)
A2.2
(n^
p)
Also,
A2.3
sin* sinO cos(4+0) - cosO cos*
Using equation
(2.4a),
together
with
f
e-ikp[l
sp
A2.2 and A2.3,
(A2.1)
+ cose cos* - sin*
sin*
reduces
to
sinO cos(ý+; )]
d* dC
A2.4
- 48b -
The spherical
are related
components
to the Cartesian
E0=
E.,
E
of the radiated
components of
Kllrx cose cosý -wy
electric-field
w by the relation
cose siný
twz
A2.5a
sine]
I[17jr
E=-K,
where
x
KI is a constant.
and the cross-polarized
Ep=
siný
twy
siný
Resolving
expressions
reduce to those
E0+
the normalized
given
(A2.5)
components according
cosý E0-
for
A2.5b
cosýl
into
its
principally
polarized
to the definition
cosý E
siný
E
radiation
in equations
patterns
of these
components
(2.5a) and (2.5b) respectively.
48c
Drr,
1)
2)
3)
kTnrc
Reflector
Antennas"s
I. R*E* Trans. 9 AP-2,3,
SILVER S.,
(1949),
McGraw-Hill,
pp.
COLLIN R. E.,
LEWIN L.,
"Microwave
Lens
pp. 119,1954.
Antenna
Theory
and Design",
334-347.
and ZUCKERF. J.
(1969).
"Antenna
Theory"
"Advanced Theory of Waveguides",
(1951),
COWANJ. H.,
Part
596-S97.
Iliffe
and
Company.
"Dual-band
Re-use Applications",
pp.
and Hyperboloid
N. Y..
Sons Publishing
5)
"Paraboloid
JONES E. M. T.,
McGraw-Hill
4)
rDr,
Reflector-feed
Element for
I. E. E. Electronics
Letters,
Frequency
1973,9t
I
- 49 -
rWAPTPR
--l
OFF-SET FED PARABOLIC REFLECTOR AERIAL
3.1
Introduction
The circular
paraboloid
such that
of the reflector
point
with
collinear
the feed is
the paraboloid
located
from the intensely
that
reflectors
use a feed at the focal
the peak of the primary
from the disadvantage
suffer
axis
in the path of the reflected
illuminated
central
portion
rays that
are loss
in gain,
increase
in sidelobe
The aperture
blockage
problem may be overcome by using
technique
(1].
at the paraboloid
focus
reflector
with
surface
surface
is
feed or its
radiated
without
an arrangement
that
reception
in satellite
The clear
The problem
only
located
of the
a portion
by the
collimated
from any part
and increased
has led to the development
feeds and the open Cassegrain
attractive
the off-set
the feed system,
the enerey
obstruction
such as the hcrn-reflector
application
mismatch.
power and significant
to illuminate
the result
and results
of the primary
accessories.
The low noise
designs
of the aperture
In such a configuration
is tilted
This
of the reflector.
of the feed blocks
out a portion
that
originate
position
feeding
is
pattern
exit
feature
for
of achieving
mode of excitation
[2],
[4]
using
communication
design
and without
impairinZ
single
feed,
element
for
and radar.
by the geometry
split
using
aerial
dual element
of monopulse feeds
desired
cff-set
(3],
of such
capabilities
of several
aerial
aerial
path afforded
bandwidth
sensitivity
is a particularly
for
tracking
with
the officiency
application
the difference
of the sum-mode
- 50 -
in the centre
fed configuration
four
to the conventional
horn cluster
[5].
[6]
arrangement
of a multimode-multihorn
by the centre-fed
offered
can be overcome by adding extra
While the off-set
geometry offers
in distortion
results
several
aperture
are thus of direct
of the aerial
in satellite
application
interest
communication
for
area.
advantages
of sy=etry'in
one
in the
polarization
The resultant
of the main reflector.
blockage
practical
the loss
of the primary
is the use
degree of freedom not
-a
owing to increased
reflector
in terms of compactness and flexibility,
plane
An alternative
horns
radiation
patterns
dual polarization
for
and particularly
tracking
application.
In this
the radiation
chapter
properties
set fed paraboloid
reflector
feed configuration
and th e characteristic
geometry.
The analysis
integration
technique
estimate
3.2
Analysis
Fig.
(3.1)
are examined as a function
carried
radiation
of the primary
of the reflector
parameters
field
method in order
from the longitudinally
directed
patterns.
: Geometry
illustrates
off-
out using both the current
and the aperture
the contributions
on the secondary
is
of the vertically
the geometry of the aerial.
to
currents
If
Olt
+
plý5.1 -rvL.
e oll-SCL
- 51 -
feed polar
The primary
to the paraboloid
respect
axis
The rectangular
ip,
axis
of the off-set
at the cettre
is tilted
,
zf
(X
yp,
p,
with
directed
S1.
section
reflector
00
the main beam is
so that
co-ordinates
by an angle
zp)
(Xfs Yfs Zf)
and
by
are related
x
Cos
y
Yf
p
The equation
0zf
sin 00 xf
ozf-
system used to describe
The co-ordinate
is a spherical
sin
(3.1)
Cos
z
p
xf+
(r.
system
co-ordinate
of the paraboloid
09 ý)
with
in this
section
feed pattern
the off-set
axis
polar
zf
system of co-ordinates
is given by
2f
2f
r
f
where
is
the
focal
length
The intersections
surface
The projection
circles
with
the
contours
of these curves
onto
parent
0=
paraboloid.
constant
with
which are elliptical
N,
PP
Cos
y)
plane,,
the paraboloid
in shape [2].
however,
are
at
centres
2f sin
x03.3a)
c
of
of the curves
reflector
generate
sin 0 s-
1+ cos a dose
+ (z.p
(3.2)
Cos
0
+ Cos
and radii
2f sin
e
CoseT Cos
(3.3b)
52
-
between
The relationship
projections
(x
onto
xp=
r-x
yp=
r- yp
The co-ordinate
is
a spherical
focus
p,
Fig.
(3.1).
3.3
The Induced Current
the
expressed
as
far-field
chosen
to
(R,
system
in
axis
cos ý+sine
0
(3.4a)
cos a'
sin e sin
=r
system
polar
and their
reflector
are
60 sine
r(cos
p=
on the
points
plane
co-ordinate
and the
Let
yp)
the
the
(3.4b)
describe
the
01ý)
aa
negative
secondary
with
the
at
origin
the
as shown in
direction
zp
patterns
Method
fields
radiation
of
the
primary
source
be
ikr
Ee (e
induced
The currents
due to the incident
2
z
n
where
is
density
for
fields
Ef
illuminated
(3.5)
side
can be obtained
of
the
reflector
from
Ef
xQx
(3.6)
0
an outward
(3.5)
the
transverse
on the
(e,
[ý
Substituting
expression
+E
e
r-
unit
direct
unit
into
(3-6)
normal
vector,
and normal
to the
vector
norm.al to the surface.
and using
the
(x
P,
the
appropriate
the
com.ponents
of
yp)
are given by
plane
current
- 53 -
1ý
2e-:
(2t)'
ikr
t+Kz
z0
F
PI
where
Ec sin
ý E' + E, (b-a
ýp
x
cos
(3.7a)
[(b-a
cos ý) EO -c
t
[Ee
KZP
E
Here, the following
(sin
e0
sin
0 sin
6 sin
sin
cos 0 cos
abbreviations
cos 0 cos
a+
0E
sin
t sin e dos 00)
(3.7b)
0z
01have been used
0
60
cos e+ cose0
c
(a-b
cos 0)
The differential
surface
ds
element
expressed
,
in the tilted
is given by
co-ordinate
ds
=
(201
(a-b cos -ý-T r2 sin
The Hertzian
vector
potential
00
do
associated
with
the
source
currents
is thus
7r
=
127r f em -
ýt+z e
ikr(I-r^.
00p
where
reflector
Om is the half-angle
periphery,
and
subtended
) r sin 0 dO d$
P^,.
(a-b cos
at týe focus by a point
(3.8)
on the
- 54 -
(r^. R^) =
sin
0a Cos ýa (Cos 0
-
sin
0a sin
t
cos ea (sin 0 cos ý sin eo - cos e cos eo)
The far-region
ýa sin
field
scattered
0 Cos ý
sin
6 sin
sin eo Cos 0)
(3.9)
ý
s
"S
E0
components
i
can be obtained
from
Tir
'OL
T XP
e-ik
Es1
and
Cos 0 Cos
aaaa
ity cos 0 sin
(3.10a)
t
7r
sin
zp
0
a]
-ikR[
Es
0-2,1
Ra
7r 9 Ir . 7r
yp
XP
ZP
the
7r
7
xp
the
are
sin
0+
components
of
Computer programmes were developed
double integrals
listing
Es0,
Es
employed for
this
with
typical
The results
primary
in equation
arising
components
technique
yp
Cos
the
(3.10b)
a]
potential
corresponding
K
K
and
XP ,K yp
ZP
currents
field
7r
output
obtained
feed configuration
an alternative
approximate
given
to evaluate
the three
complex
(3.8)
and to yield
the scattered
in equation
(3.10).
The numerical
calculation
and the general
programme
elata are to be found in APPERDIX A(Pl).
from the
induced
are compared with
technique
current
method for
the results
which is described
s,)ecific
obtained
below.
from
to
- 55 -
ý2erture-field
3.4
Technique
Considerable
fields
knowledge of the tangential
to consist
of the projected
conducting
screen.
the field
Several
alone.
By applying
integrals
(7 ],
on a finite
data for
small
the boundary
in the forward
infinity
knowledge of the electric
This model effectly
field
extends
being
that
is
it
fields
and the magnetic
reasonable
typical
aperture
agreement with
the fields
and the radiation
but are based on only
region,
scheme are
of diffraction.
on the screen
conditions
Kirchoff's
objection
model adopted here,
In the particular
over the aperture
However, for
angles
on the screen,
conditions
the electric
is known to give
the theory
experimental
both
of a surface.
portion
problems,
the boundary
the principle
to prescribe
not permissible
bounded by a perfectly
aperture
to the original
objections
field
which is assumed
are reduced to the integrals
by Jones
discussed
and the magnetic
reflector
In
from an approximate
of the reflector,
plane
can be
formulation.
are derived
electric
in the focal
distributions
problem
diffraction
aperture
the radiation
treatment
this
Kirchoff's
by using
achieved
of the scattering
simplification
of validity
satisfy
at
condition
an approximate
in the projected
distribution
the region
exactly
aperture.
of the Kirchoff's
[71.
scheme
The reflected
is
electric
to the induced
related
[j,
j2
where
j
Er
=
j0
(n-E
is the direction
field,
current
Er
at the surface
density
j
of the reflector
by
A
'**
E-r(n *S
rd
of propagation
(3.11)
of the reflected
field.
56
-
For
a paraboloid
reflector,
(3.12a)
sz
-P
and
6. z^
(n^. r^)
Employing
the above relationship,
field
electric
(3.12h)
in the projected
Ea
F
E21b-a
2f
-a
to facilitate
In order
desirable
direction.
is the angular
(x
co-ordinates
in the aperture
plane
co-ordinate
p,
x
)l
cos
fields,
of the aperture
co-ordinates
measured from the
can be made by noting
yp) of the projected
are related
to the aperture
that
x
.P
the
(section
circles
co-ordinate
(r
c
3.1b) by
Cos
xp=x+r
r
where
E0 (0, ý
sin
in terms of auxilliary
(3.13)
This transformation
rectangular
study
qualitative
to express
ý1
where
ý EO(O, ý) - (b -a
sin
-
by (Fig.
aperture,
(6,
Ee
ý)
ý) +c
cos
[c
is
in terms of the (0, ý)
system is found to be given by
co-ordinate
it
compon'ents of the
the transverse
xc,
rc
sin
ýj =r
are as defined
The relationshps
between
systeMs
as derived
co-ordinate
(3.14a)
ýl
0 sin
sin
in equations
the
sinusoidal
frcm
(3.14b)
ý
equation
(3.3al
(3.3b).
and
functions
(3.14).
in
the
two
3.1)
- 57 -
c sin ý,
7-+ b cos
sin
b+a
Cos
(3.15a)
cos
a+b
(3.15b)
cos
form
simpler
fields
the aperture
Using the above transformations,
assume much
by
given
E (0,
Sin
ar
[Cos
- lp
EO(Oq
cos
E (09
+ sin
reflector
expressed
E6
(el
where
2f(a
+b
cos
C2
is the equation
of the off-set
and
co-oranates
11
1
E012 + JE 2]
ýl
Q=I
r
The fom
'distorted'
3.5
(3.16)
of eqn.
in a symmetrical
fields
paraboloidg
ror
Distribution
electric
reference
vector
that
fields
unit
A
e^
p9ec,
vector
directions
for
the
components are
Excitation
polarized,
along the
Xf
with
axis
its
or along
of the horn may then be resolved
which
co-polar
linearly
either
field
geometry.
Polarized
is
orientated
The radiation
the y-axis.
the
except
Linearly
the feedhorn
Let us assume that
along
to the form of the aperture
is identical
due to asymmetry in the reflector
Uorture
principal
in the transforved
respectively
fields
define
and the
the
cross-polar
polarization
fields
- 58 -
of the primer-I
Ef
where
=
El
c
E
of the fields,
A (09ý)
Ept ýp
Ecl
+
are the principal
and the cross-polarized
components
and
[
2p
?
Thus,
radiation.
[ sin
Cos
0
s in
c
Cos
in
Cos
Cos
sin
correSDond to the reference
directions
polarization
y
along
or
x.
for
the principal
linear
axis.
I
In terms of the spherical
conponents
components can be written
as
Cos ýj
p
sin
0
Cos
(3.4.8)
it
can be readily
shown that
of the horn the main field
of polarization
and the cross-polarized
(3.17b)
sin
Using the transformations
plane
(3.17a)
Cos
Cos
either
the orthogonal
sin
s in
Et
E, ,E
component
C
in the aperture
for
component
P
of the reflector
assume the form,
m El[(b
p
+a
cos ý, ) coo ýj +C
ElUb
c
+a
cos 01) sin
sin2ý1]
C sin
cos ý, ]
(3.18a)
- 59 -
+a cos ý, ) sin
m Et[(b
p
c sin
cos
(3.18b)
a cos ý, ) cos ý,
'c'[(b
in the reflector
The cross-polarization
as a consequence of the depolarization
Icomponent and the depolarization
It
of the feedhorn.
of the horn,
will
is
generated
the overall
through
response
of the aerial
of the horn owing to the total
geometry.
The "cancellation"
can therefore
be regarded
by considering
The field
feature
the role
Consequently
vary with
respect
to this,
of the
symmetry of the aerial
configuration
as a consequence of the compensation
is examined in subsequent
of the aerial
fields.
sections
examples.
some specific
components
of the
asymmetry of the incident
asymfnetry by the polarization
This particular
fields.
is independent
in the off-set
effect
the cross-
axis,
in contrast
aerial
orientation
cross-polarization
polar
will
Note that
of a centre-fed
response
one orientation
of the principal
be reversed.
of the feedhorn.
to the orientation
polarized
cancel
90* about its
will
cross-polarization
the cross-polar
and partially
by the depolarization
cross-polar
for
to note that
arises
components
components of the primary
the same horn is rotated
of the primary
therefore
aperture
of the cross-polarized
main fields
augment the aperture
reflector
11
of the principally
interesting
the depolarized
polarization
If
sin2ý
)
f2f
FC2r2j
m
where
+c
El 3, El
pc
of a linearly
polarized
horn can
be expressed as
H(e,
EI(O,
p
0=
sin2ý
+
E(e.
sin
E(es
ý cos
(3.19a
COS2ý
H(eq
(3.19b)
i
- 60 -
For the special
case of physically
are independent
ý)
E(O, ý)s H(e,
to the usual
form
the
ý,
function
and
correspond
,
of the horn.
pattern
and (3.19b)
(3.19a)
the amplitude
of the co-ordinate
distribution
form of the aperture
The explicit
of
H-plane
E-plane
homs
circular
with
(3.15)
through
transformation
after
excitation
is,
and some manipulation,
E(G$
P=m
(b +a
Cos
+c
cos
(3.20a)
sin
H(eq
E(Os
E(Gl
C=m
)
ý,
cos
b+a
sin
H(e,
C
H(Ov 01)
Cos
E(Ol,
(3.20b)
The intrinsic
depolarizatimpproperty
by assuming ideal
be studied
of the off-set
geometry will
of the form
feed excitations
F(69
EI
F(6,
where
=
ý, )
is the scalar
in the transformed
P=
C=
It
MF(es y
:tm sin
is seen that
component vanishes
regarded
fields
co-ordinate
as arising
[(b
system.
For this
+a
00=0
everywhere
direcly
due to the off-set
radiation
cos ý1)
ý1 F(Ot ý1)
when
of the horn expressed
pattern
cos
[(b ta
(b = 0. a=
in the aperture,
ý,
+c
c)
]
(3.21a)
]
ý,
Cos
(3.21b)
sin2ý,
)
ý,
cos
-c
the cross-polarized
and consequently
from the depolarization
geometry.
case,
can be
of the primary
- 61 -
that
diverge
lines
the field
which consequently
generates
the cross-polarized
aperture
field
with
handed cartesian
to a right
respect
As the plane of polarization
field
the electric
vertical,
in Fig
(3.2b) when the plane
for
its
sense of distribution,
is
distribution
cross-polarized
In each case the vector
in the vertical
cross-polarization
secondary
patterný
is the plane
3.6
shown
becomes horizontal.
it
component retains
commuted through
rotation.
sum of the cross-polarized
this
plane will
fields
not exhibit
The paak cross-polarization
can be expected
to occur
in the horizon tal
vanishes
any
in the
plane which
of maximum antenna asymmetry.
The far-zone
aperture
in the aperture
on radiation.
SecondaýZ Radiation
derived
away from the
the sense of the skew-symmetric
and therefore
plane
system
the distribution
the main field
polarization
either
co-ordinate
point
generates
of polarization
is seen that
positive
at a given
and finally
in the same sense,
is taken
in the positive
of the hom is rotated
vector
rotates
distribution
as indicated.
at the Centre of the aperture,
situated
is aligned
vector
the
symmetry
The sense of the distribution
when the electric
as positive
direction
of reflector
for
Note
plane.
away from the line
diagram.
shown in the adjacent
(3.2a),
in Fig.
in the vertical
is
horn
the
plane polarized
case where
of the
and the orientation
is sketched
in the aperture
vector
polarization
lines
field
The form of the electric
electric
on the basis
fields
Ea
Patterns
field
Er
of
of the aperture-field
by the following
the
reflector
method,
expression
antenna,
is related
as
to the
612-
F-
'VERTICA.
+L
L.-; Y POLAPýIZED
COMPONEEINT
I
=---j
±i1_
CRCSB-_Po/zD(HorizoNTAL)
C.OM RCN
VERTICALLY
POLLARIZEC)
FEED
FIG
NT
3-2 a
I+
HCRIZONT, al-l--Y
COMPONENT
I
CROSs-
-1
_;_
roi-ARq,
H C-)P.,
IZCNT/%LLY
PCL-AR, j!
D
-'l'7-.
FIG 3,2b
s
rp
7(V. -FITICAL)
- 60 ik R [t,
r
E
is
where
and
is
p
Kirchoffs
a
treatment
is
valid
Lx-pressed
in
fiý21d
apýýrture
radiation
transformis
contraf,
radiation
icor all
tha
01
+N
y
X,
the
above
the
of
r
"r
-0Eof
th,..
form
the.
asswrae
sin
to
N
y
N,
X-
componcia-cs
(3.22)
t
point,
hýýT-isphere.
trins-fo-r-mr-,
spher-r-I
from
observation
pro"!.,,lemý the
forward
the
Fourieýr
the
of
(3.23a)
cos
-ikR
1
Cos 0
the
It
(3,1).
in
points
-ikR
-7zlý
givým
In
ap,-ýrture
o.-)tained
are
the
the
are
T:
'ij7.
of
of
(3.22)
cýjý,
direction
nlamý.
COMDOnents,
(0
the
aperture
Y,
.
ET
shown in
in
the
r1e
E7
0
wberý!
x
in
torms
fields
ik(p.
.-a
vector
unit
a v--ýctor
expression
" xfz,
T _k,
h
ayax
abservation
-Far-field
is
shown in
(3.23h
14 sin
COS
ýmgles
measurcd
(3.1),
that
Oaryc cos
Lc
+r
Appendix
cl
the
as
Fourior
lijy
sin
-ik
00x
-sin
ccs
8dOdlýj
(3.24)
M
xyy
where
ai)erture
Th,-
rofers.
fieldq
fi,
thýý., polar,
is
z2tion
for
patterns
radiEtion
instancu,
thý,
either
x
thý?
or
compommt
of
tho
resp-ctively.
thý. ' cross-polar
for
to
ld
_,
component
obtainiýd
reference
by
tho
-P
s in 6ý
+ Cos
F -a
11
?c
Cos
aaa
for
Ec,
of
12
_ID
a
'a
the
the
of
fields
c.efined
-C
Ep,
component
v-polarizetion
resolution
dircýctions
sin
fiold
tl-, e copolar
(3.23)
bv
and
horn
64
-
TIhe resolution
yields
ý'! Fsin2
Ep
2a
COS
ýa+
COS a
+Nx
sln
cos
sin
cos
(1--cos
a
cl
T (cos2
e;
+ sin
2
cos e+-(1-
0
2-
0
cc,'3
.
(3.25)
We note
Cos 0 ',ý 1) , the
a2
'.,)e --Iirz2ctl,
for
that
small
co-polar
ss(-)c-iatýA
.1-.
anglas
with
. c.
in
thýý ,apD-rcxim, )tJ on
and tl,,.e crccss-polar
compon-mt
tia
(i
of f --axis
field
aperture
corresponding
comPOnent car,
cc! 'Monerits
i. f-:.
:: 14
EN
CX
Tho
computer
facilitate
modified
to
iw_ýn in
-,
,
ellUatiOn
(3.24)
petterns
gi-ven
radiation
programmes
are
not
edl i-TI the
drýscri',
Drogrammc.
nu:-ierical
and
in
evaluations
to
of
th2
yielO
since
was
intezrals
cro3s-polar
For
tho
(APl)
th, 3 fl*eld
copolar,
(3.25)
equation
liere,
do-scrib2d
lT-)pciT-idix
sake
JDrevity
of
is
:-,olification
-the
Straight--
forward.
An o'-)v-i.,:)us advanta; -.- of
computer
evaluated
to
3.7
the
efficiency
for
thrEýc,
Numerical
of
a cr)rrTI-otý.
int(2, Trals
In
view,
cjýjscription
t'i, -ý induced
is
field
th, ý-t c;nly
tochnique,
two intograls
th-:
-radi;:: tion
curren
c metho'll.
of
from
the
need bi2
fi. ýlds,
in
contrast
Examples
The diffrýýction
the
print
th.:! aperture
induce, d--current
integrals
lwý
mo(lol
arisin
Z, from
wereý, numericaliv
3perture-ficid
evn. luated
over
mcde.1 anc!
a wide
ra. nge
- 65 -
of
antenna
Prý--liminprv
in -t,----r-,s,t
I
stu, 'ios
de,,-)o1ariz, --;tic)-i
this
initial
purpose
Or ct sin
7T a sin e
w1jere
a
taDcr
for
ILs a parýlmetar
feed
the
and
for
the
case when
diam2tar
aDerture
for
tapýýr
to
Since
of
th-ý
+1
the
Howevar
it
th(-. path
will
exist
given
to
M, iq
'T/2
planes
in
(3.3)ý
Fi7-
has a projected
was
a
is
r, ýfiector
that
frcm
in
the
tile
this
ý,. consequenc,
as
focus
asy-inmetry
in
thOo'r(! ticFzlIv
patterns
noting
maximum
lOdB
-
7rovide
will
')(ý henceforth
z,:,ro
in
the
to
in
vertical
the
refl,:
plana
ý- of
ýctor
are
planýý,
shown.
not
there
periphery
ril,
cxpressed
pý, ran,--,
d-,.'fe-rence
the
apcrtijzný-dist
ind
the
ution.
in
this
decibels
by
(dB)
or
worth
lengths
is
and
radiation
an amplitudo
asymmetrv
is
20X
for
45c' --rp-f lector)
the
is
in
ýD
0=6
the
shn,,,7n superimpos-d
The reflector
= 450.
1-1
cross-pclarization
off-set.
desired
th2
tc, achieve
fi-, Icýs in
ire
illumination.
as the
and f-or
tlrikz! r-, (Iiat. i. on natterns
calculýqTions,
t7,io models,
of
the
roferrurll
Was -idjus--DO
th(-- cross-polar
r the
= 7T/4 , i:,.,,
D,
ý
geometry,
0)
whicri
these
fil, ýlds
c,,---polar
the'
Dattern.
As an exa.T.,ple. of
the
cxalmini-nz
41
-he form
of
excitation
practical
ý,ut by -assuminw ciircularl-ý,
was carried
s in
F(e)
at
-.f th, ý off-set
proT; er. ty
of
configur-ýtlcns
wýýr,ý dir"-cted.
computatýon
bal, 7nced feed
symmetric
feed
and primary
parameters
WE in
this
plane
thu
antenna
the
=
20 loclo
case of
can therefore
axis.
+ ccý3(em 5-0)
the
y+ Cis
40-reflector.
le
m
The radiation
Qx-Pectod to he sliLhtly
patterns
asymm"trical
in
ahout
-
6-(,
0
(CLI (
-10
-20
-30
m
0
-40
-50
-6c
3
ANC%LC
OFF-AXIS , DEG
0
- 67 -
7he worst
in this
cross-polarization
of the off-set
to the plane
orthogonal
in the far-field
for
plane
in
the
whereas
7r4-plane the peak occurs
cross-polarized
fields
fields
at a level
of -20dB
in phase quadrature
are nearly
The
of -23dB.
with
polarized
the principal
the
outside
of symetry.
plane
A characteristic
is
aerial
feature
the amplitude
of the cross-polar
-6dB power points
and reached
of the principally
imposes stringent
consequently
off-set
increases
fields
peak at approximately
This gives
pattern.
near the axis
requirement
the
in centre-feed
its
polarized
isolation
to poor cross-polar
in
cross-polarization
to cross-polarization
away from the boresight
rapidly
the
of
in contrast
that,
configuration,
rise
at a level
the bean. is elliptically
and therefore
The peak level
as expected.
the above geometry occurs
in the Dlane
occurs
of the aerial
on the pointing
and
accuracy
of the antenna.
The depolarization,
parameters
and the
aperture
polarization
also
for
(3.4 a).
shows that
Om #
values
of
the parameter
00
maximum cross-
0M
of
is
shown in
a' was adjusted
the edge illumination.
the taper
angle
Computation
to 20dB reduces
the maximum
by about IdB.
level
For a fixed
rapidly
with
valua
increase
of
sources
flowing
e01
it
is seen that
in the pattern
increased
the
of
consequence
current
The variation
some typical
for
increasing
the feed tilt
viz:
For each plot
a lOdB taper
by the two angle
characterized
geometry,
angle
00
with
curves of Fig.
to provide
property
of the off-set
half
is
over
angle
contribution
the
now larger
the peak level
This is
0m-
rises
a direct
from the cross-polarized
area
of
the
paraboloid
the
-6s--
-10
()C=
50
11C
=3
m
c
0
`Gý=:
20
-
20
0
CL
0
FIG
3-4a:
15
30
0
45
/'5
Dependence
of the peak cross-polar
e, and the illumination
angle
angle
radiation
go
on the
dog
off-set
I,
-- 0
FIG
-
3-4b:
Variation
radiation
of the peak
reflector
with
cross-polar
F/D ratio.
so
I
0.
-
0. ý
o
FI -D
- 69
in contrast
surface.
Note that
the centre
fed paraboloid
independent
of the pattern
is kept
illumination
It
full
for
values
with
reflector
of
00=0M,
the plot
the reflector
F/D ratio
results
geometry
suppressed
below -30dB for
this
values
an off-set
section
-23dB.
is
the secondary
3.8
which results
lost
through
for
D is the diameter
).
2(e
For
em
0+
versus
For this
can be
cross-polarization
In practice
than 0.6.
of greater
angle is
and its
of
usually
required
For instance,
hardware.
Although
the Cassegrain
(See
length,
from the sub-
to avoid blockage
of the clear
the presence
of high
about
path achieved
through
lobes
cross-polar
in
Confirmation
Towards the conclusion
notice
were conducting
+6m)
radiation.
Experimental
the author's
(0
in peak cross-polarization'of
In such a case the advantage
the off-set
to
a long focal
angle required
the off-set
47.5*,
for which
of a paraboloid.
Chapter
is
e
little
employs horn and a sub-reflector
provides
reflector
introduce
of off-set
feed system effectively
4),
will
in curve of Fig. (3.4b).
from the feed elements
the open-Cassegrain
to illuminate
of
the edge
provided
of maximum cross-polarization
F/D ratios
large
a relatively
to avoid blockage
is
where
equal
is seen that
particular
however,
feed excitation
and
cone angle
example when
it
6
F/D ratios,
large
paraboloid
response
(or F/D ratio)
angle
the off-set
for
i. e.
is small;
unbalanced
the cross-polar
constant.
is seen that
cross-polarization
the
with
to this,
that
similar
use as an objective
of the present
Rudge et.
studies
for
research,
al of the University
of the off-set
multibeam
sattelite
it
was brought
of Birmingham
paraboloids
particularly
communication
aerial.
to
- 70 -
In order
results
to obtain
were
of the author's
verification
the
with
compared
results
of
theoretical
predictions,
30GHZ experimental
their
programme. .
has the
The reflector
One type
a TEj,
in
used
of horn
(E-plane)
= 2.14
bl
and
approximated
are closely
conjunction
horn with
mode rectangular
(sin
f=
parameters
the
with
aperture
X (H-plane).
30.5X
60=
0m
30%
Birmingham
dimensions
is
aerial
a'
= 1.5 X
fields
The radiation
350.
of the horn
by
+ cos ý cos ee) F(e,
I-
where
0 sin
sin(7ral
sin
0 sin
wiil sin
F(O,
With these aperture
dimensions
symmetric
main beam with
Theoretical
calculations
The diffraction
field
r2b'
C -.
7r
the feed produces
a l2dB taper
indicate
sin
an almost
ý)
__
2
0 cos
circularly
at the edges of the reflector.
an approximately
patterns
of the aerial
for
the CO-polar
-33dB peak cross-
as predicted
fields
(3.5).
the experimental
the performance
antenna radiation
experimental
The agreement between the theoretical
result
is
seen to be very good.
of a practical
(as predicted)
off-set
and the cross-
frr,, m the "induced-current"
programme are shown super imposed on the provided
in Fig.
8 cos
sin
in the horn radiation.
polarization
polar
cos(vbl
aerial,
prediction
and
As a way of illustrating
the main features
are given helow:
data
of the
0
ei i
R
decjFiý
'k
xpov ovi
IIc,-r,iý:c -rI
- 72 -
Polarization
: Horizontal
co-polar
ist
co-polar
2nd
Cross-polar
isolation
isolation
isolation
3.9
Peak Level
_(
_dýT
kýni, ý of f-axis
-27
2.9"
-30.8
4.50
-26
1.3"
side-lohe
it
main-lobe
24
0.951,
-6dB
19.7
1.3 6
-lOdB
16.7
1.6"
-3dB points
The, Effect.
of
the
Primary
Feed Cross-polarization
on the
Secondary
RadiAtign
It
in
the previous
shown
was
that
section
from the feed system can, to some extent,
depolarization
this
effects
of the reflector
the cross-polar
we consider
when fed by a
to the rectangular
horns,
markedly
for
elliptical
aperture
principal
plane
approximation
TE
11
patterns
(the
In contrast
feedhorn.
conical
feeds generates
circular
high
cross-polarization
than about 0.9X (theoretically).
of the horn are to a sufficient
The
degree of
given by
10,11
[Cos
0+M-
J, (ka,
sin
0)
(ka,
sin
0)
J, (al
H(O)
The quantities
of the 30*-reflector
the dominant-mode
greater
to illustrate
In order
off-set.
main beam and correspondingly
diameters
or enhance the
neutralize
response
Birmingham aerial)
the cross-polarization
Cos e+
arising
sin
rpil
rk
a-, sin
above have been previously
2
0
defined
in Chapter
2.,
- 73 -
(2.1.1).
are shown in Fig.
2a, = 2.3X
in the feed patterns
fields
of the horn for
fields
The radiation
of -19dB and the principal
in
E-plane
24dB
and
to lOdB in the H-plane
are tapered
of
The peak cross-polarizations
in the level
occur
diameter
an aperture
at an
of 30%
angle
diffraction
The predicted
(3.6).
are shown in Fig.
data obtained
with
(curve
x-axis
about its
The two patterns
the feed orientated
a) and that
polar
axis
of the aerial
patterns
for
obtained
horizontal
with
this
the numerical
represent
to be polarized
along the
the horn through
by rotating
(curve
polarization
feed
90*
b) of the feed.
i
It
is seen that
the aerial
performances
cross-polar
particularly
is
feed
the
polarized
when
however,
and higher
to interpret
the main fields
points
the horizontal
this
Assuming that
in the azimuth
plane
the two orientations
vertical
of radiation
In this
by narrower
are characterized
result
let
configuration
beam width
us examine the distribution
fields
and the cross-polarized
of the reflector.
field
plane.
in the
side-lobes.
In order
for
at wide angles
in the vertical
fields
the co-polar
marked improvement
exhibits
the dominant
plane arise
of reflector,
in the aperture
from the source
the source
fields
of the horn are.:
polarization
P=-
mc
a2
C=
mb
a2
[(b;
E + C2H) - b2(E -H=
mcH(O)
[(1)21;
+ C2H) _ C2(H - E)]
to the far-
contribution
= mbE(O)
distribution
in this
plane
in
for
4L
-t
-7
-10
l's.
-20
-30
ao
a
--40
-50
0
ANG,
LG C>FF-A)(15,
Ft9 3.
Cleg
746-
ATioN
r-r-GFD PoLJ44Ri-e.
-10-
PEGI)
Pc>LAR(zA-fjot4
AL
C) W Ct
OqLoNrA
-T':
-TWC-
4)a
=%
-20-
CO-POLAR
I
-3
Co
(3
-40
-501
a
0
2
-j
OcF-PII
Pig
ctq9
C- )OP- AA'
-J
If
- 75 -
horizontal
]2olarization
[(b2H
ME
P
+ C2E) - b2(E
mcE(e)
-H
a2
(3.2b)
II
(b2H + C2E) + C2(H
-E
21
C
mbH(G)
az
Note that
in
has
the quadrantal
a
maximum
which
These cancel
(or enhance)
the depolarized
amplitude
amplitudes
horn belongs
to that
the reflector
of the reflector
major axis
generate
field
periphery)
the beam with
distribution
its
tapered
the geometry of the aerial,
is,
tapered
asymmetry will
in the previous
i. e.
field
in
in the gain
as a consequence of its
with
reflector,
the shape
the
the horn is orientated
in the horizontal
plane),
plane
(that
to
is,
the "orthogonality"
the overall
asymmetry in
When the shapes are aligned
to be "parallel"
as restoring
distribution
be augmented, resulting
case.
If
in the vertical
of the two shapes can be regarded
(that
The circular
at the focus of the off-set
plane.
major axis
by the
from the difference
are also. elliptical,
in the vertical
lying
the
plane.
shapes as seen by the feed (i. e.
profile
of the horm.
polarization,
is determined
planes,
When placed
beam-shape ellipticity.
aperture.
which the cross-polarization
as arising
in the principal
main fields
either
interpretation.
of feeds for
can be regarded
radiation
of its
class
for
in the vertical
to an interesting
Itself
lends
This
its
fields
of the primary
fields
principal
distribution
cross-polarized
after
plane of the reflector
Also note that
ofIthe
((E - H) term)
of the feedhorn,
plane
appears in the horizontal
depolarization,
fields
in primary
the cross-polarization
in the horizontal
in higher
plane)
the
cross-polarization
than
76
-
Note,
that
(E(O)
since
will
main lobe,
but will
The above conclusion,
the primary
even when the reflector
in the horizontal
plane
that
symmetrical
Cassegrain
analysed
in Chapter
by modifying
ellipticity
is also
response
of the aerial.
3.10 The Effect
The aperture
essentially
response
of a given
An alternative
total
almost
feed geometry.
or the
cannot be
4).
by the two angle
characterised
seen to have a critical
Cassegrain
cancellation
In this
effect
arrangement
can be achieved
case the beam-shape
on the cross-polar
'
Of Longitudinal
field
determines
radiation.
(Chapter
dominantly
of
the distribution
which essentially
feed configurations
the Cassegrain
illumination
of the front-feed
particularly
5 shows that
remain valid
plane,
not affect
clearly
the cross-polar
of the geometry.
parameters
side-lobes
can be reduced
uniform
in the secondary
and is
reduced significantly,,
will
of the aperture;
aerial;
reflector
cross-polar
the co-polar
in vertical
is shaped to achieve
As noted aboves however,
cross-
The energy
the cross-polarization
illuminations
the peak cross-polarization
off-set
into
near
peak levels.
This expedient
the aperture.
only
peak levels.
distributed
show higher
which consequently
by tapering
the radiated
reduce the wide-angle
is
value
by the primary
affect
not significantly
from these lobes
extracted
has significant
the cancellation
the edges of the reflector,
polarization
H(e))
-
Currents
method differs
in two respects:
a) it
from the induced
neglects
current
the contribution
method
from the
- 77 -
longitudinal
components of the radiation
current
source fields
assumes the transverse
(i. e.
reflector
to be distributed
of integrations
the surface
and b) it
over a planar
than on the curved surface
rather
equi-phWo-surface,
sources
of the paraboloid
are different
in the two
models).
of (b)
The effect
of integration
it
regioni
for
is apparent
a given
andimprove
identical
the path-lengths
observation
from the axis
the differences
plane,
of the
be larger
will
based on the two models can
only
at wide angles
(On the axis
reflectors.
from the longitudinal
radiation
the feedhorn.
For example,
fields
Ep(e
IYZ )=
7r
y
7V2)
(0
cax ,
(or
(0
for
in the plane
a9
vertical
vary with
will
cross-polarized
p
in the two cases.
differ
angle
in the far-field
of obliquity,
the two techniques
fields).
The contribution
and for
plane
will
increasing
with
to disagree
Oflatter'
for
cross-polar
rays are drawn from the surfaces
The predictions
F/D ratios.
be expected
therefore
yield
that
increase
will
and for
small
in terms of rays of
in the two models to an observation
This difference
aerial,
understood
if
For example,
optics.
geometrical
can be readily
respect
-xf)
ý
to the polarization
of asymmetry assume the form,
6a-
it
z
sin
0a
polarization,
7rxl
as
component to the
the principal
y-polarization,
ir'
7r/. )ý
72)
Cos
current
Or' Cos eyaza
irl sin
0)
of
and the
- 78-
It
is seen that
do not contribute
It
currents,
in certain
actually
and this
currents,
the integrals(assuming
for
difference
for
that
to conclude
minor role
in the determination
of the primary
between the numerical
it
peak level;
data,
lobes
secondary
is therefore
current
of cross-polarized
source.
components play
a
of the
radiation
reflector.
off-set
The secondary
current
components is best
the induced-current
technique
all
of the reflector
effect
between the co-polar
predicts
angles
fields
illustrated
that
diameter
fields
as predicted
techniques.
The latter
In the case of the 450-reflector
and the F/D ratios
phase difference
the peak of the main cross-polar
lobe is
are small),
(6.9*
by
off-axis).
(note
it
is
between the two components at
90.1 (electrical)
degrees
and 89.4 degrees at the peak of the first'cross-polar
off-axis),
'side-lobe'
the phase relationship
the two components remain in phase quadrature
of radiation.
the absolute
by comparing
field
and the longitudinal
curvature
and the cross-polar
and the aperture
both the reflector
found that
(2.3
balanced
symmetric
the longitudinal
reasonable
was chocked by
which the cross-polar
were some -50dB below the co-polar
from
found in the
The effect
orientations
the
of radiation
the contribution
was certainly
circularly
the two polarization
regions
cancel
integrals.
was found to occur at angles
that
to the
that
However, the only perceptible
for
however the contribution
is also possible
of the radiation
excitation)
sources.
current
from the y- and the z-directed
the cross-polarized
computation
case the
both
from the z-currents
contributions
plane,
arise
radiation
cross-polarized
evaluating
of the horn,
polarization
In this
radiation.
from the x-directed
purely
components
in the azimuth
radiation
of the co-polar
arise
cross-polarizations
vertical
the longitudinal
polarization,
to the cross-polarized
the fields
but modify
ror
horizontal
for
- 79 -
It
is therefore
and the cross-polar
the co-polar
The technique
Radiation
Properties
Polarized
Excitation
In order
to study
it
excitation,
the properties
be assumed that
will
by the reflector
for
time
a complete
and horizontally
vector
Circularly
for
this
mode of
the feedhorn
the cone
within
of angles
subtend
from the horn
The radiation
from a balanced
as arising
The resultant
time quadrature.
For
on transmission
at the focus.
periphery
polarized
of vertically
Reflector
of the aerial
polarization
can then be regarded
aperture
aerials.
of computation
need be evaluated
Of The Off-Set
pure circular
generates
prediction
of the off-set
saving
the
of the fields.
description
3.11
an accurate
radiation
integrals
interest,
of practical
some significant
also provides
only two diffraction
since
regions
method may be used to provide
aperture-field
for
for
that
concluded
distribution
Huygen elements
polarized
of the horn
fields
radiation
in
then assume the form,
ýf
U
where
polarized
defined
=0 r(e. o)
V2
unit
are the right-
-P
-c
vector,
[Right
respectively.
here as clockwise
rotation
or the left-circularly
hand of polarization
of the electric
vector
is
of a receding
wave].
The resultant
this
distribution
mode of excitation
aperture distributions
with
due regard
in the aperture
can be O)tained
generated by its
to the phase quadrature
of the reflector
by vectorially
orthogonal
relationship.
for
adding the
linear
components
If
this
- 80 -
is followed
procedure
(P
where
Cy)
x,
then the aperture
A-);
(P
x+cyypy
j (P I-c^)
xxp
-P
and
(P
ysC)
denote the amplitudes
fields
and the cross-polarized
feed
of the main fields
linear
principal
respectively.
x- and y-axis
assumed balanced
For the
for
of the aperture
of the horn along
polarization
are given by
fields
condition,
=P
C
:C
yx
from which
jy
(P + ic)
ýA
p
(3.27)
w'2
P
where
and
C are as given
With respect
(x
-Y
p,
p.
z)
denotes the direction
z
and the left
wave, the right-
reflected
be defined
(3.26)
handed Cartesian
to a right
, where
in eqn.
circular
co-ordinate
system
of propagation
unit
vectors
of the
will
now
by
+ jy
j ýD
-P
It
is seen from ()
aerial,
the fields
but will
excitation
that
will
for
circularly
remain circularly
have opposite sense of rotation;
there is no cross-polarized
of the
polarized
excitation
polarized
in the aperture
also for balanced feed
circular
component in the aperture.
- 81 -
(JE
The amplitude
can be
fields
) and the phase (ip) of the aperture
Al
as
written
12
EA
=1
(pi
r
t C2)1
and
t (b+a cos ý, )
c cos
tan"
+ cos ýI(b+a
c sln2
sin
cos ý, )
(3,28)
is
The phase-shift
circular
polarization.
across
the aperture,
one or the other
in
the
opposite
of the geometrical
of the incident
in this
the phase variation
Maximum displacement
the phase deviation
The displaced
two hands
the
the main beam in the far-field
of the radiation
displacement
for
As a consequence of the variation
side
hand of polarization
sense
patterns
is
plane
is expected
fields;
circular
incident
the
of the phase
be displaced
will
of aerial
axis
of
depending
there
on
on the
is no
in the plane of the off-set
since
zero.
in the horizontal
to occur
plane where
is maximum.
radiation
pattern
for
points
near the antenna axis
Ik sin
0a [r
is
given by
fo2jr f0Om
I(o
a'
ýa)
=
F(O, ý, )e
x
c
eWos
cos(ý,
0rc
-ýa)+xc
cosýal
dýj
do
sin-0
(3.29)
3.12
Numerical
Examples
The predicted
by two opposite
The aerial
radiation
from the off-set
handed circularly
has the same geometrical
polarized
reflector
fields
parameters
due to excitation
are shown in Fig'* (3.7).
as considered
in the case
D
u
Cr
u
-J
U
U
LA
C)
0
-0
Lr)
Q)
CD
LL
C
Lli
_0
I
r
LO
00
-It
z
ui
z
0
a-
cl-I
0
L)
CD
0
LU
N
b--4
c'J
cr-
U
C-0
-82-
- 83 -
of
450-reflector.
the
component has -3dB beamwidth at 3.8*;
Each circular
are displaced
the two patterns
Ae = 0.35*
the beamwidth of the circularly
Note that
.
is equal to that
pattern
of the principally
polarization
is used;
displacement
of the principal
direction
i. e.
the
of
can be obtained
aerial
0m,
the
approximation
of
the
pattern
degree
of beam-shift
by noticing
term
errors
phase
when linear
is obtained
by physical
by an amount
60
in the
to the plane of the off-set.
perpendicular
An estimate
linear
pattern
polarized
pattern
polarized
the circular
by an amount
of the axis
side
on either
the peaks of
for
for
that
in (3.2 9) will
any parameters
typical
values
of
This
allows
be small.
the
of
e0
and
the
form
ýd
tan-lg(().
where
g(e.
+ (b +a
c Cos
01)
Cos
g(o)
sin
+ COS
c sin2
In the plane of asymmetry,
for
+a
Cos
example, where maximum phase deviation
is expected
Simple
the
calculation
maximum phase
An upper limit
approximating
b/C=
7ý2
900
sin 0 sin 0
Cos
shows that
error
is
for
less
+ Cos
00<
than
0
4509 0m<
0.35
,P(O, 01) =±k
c
sin
across
.
for
instance,
radians.
to the beam displacement
the phase variation
450
can therefore
the full
aperture
be obtained
by
(3.30)
by
- 84 -
The validity
of this
compares the exact
ý,
for
= n/40M
,
is justified
approximation
expression
some typical
Under
this
plane
can be obtained
the
approximation,
from
(3.30) in the planes
with
range
of values
radiation
integral
the
(3.8).
by rig.
0
of
ý, = TV2 and
0.
and
patterns
for
(3.29),
which
which
ýa=2 IV
the
now assumes
the form
2Tr
Om
0aff
I(sin
Irc,
F(6,
-i
ý, ) eI
sin
0
0±b
a
sin
r
ck
6d ed ý,
sin
(3.31)
However, sinceg
2f
Cos 6t
the integral
2f b
0
sin
Cos ec
00
can be expressed
sin 0
as,
sin
27r em
I(sin
By replacing
sin
[sin
sin 00
0a-2fk
integral
will
NO,
0aIf
in
a
,
ocýur
it
is
at
points
I
sin-
or expressed
in
terms
the
of
sin-I
ý)
argýment
of
for
sin
sin
eca2
the
shown that
readily
left-hand
the
peak
side
of
the
by
radiation
which
sin 0
I
the
diameter
aperture
sin 0
Ga
r
00
sin
0m
at
0
OM+ Coseo)
W(Cos
iI
85
-
For
small
off-axisq
angles
The displacement
(3.9)
of
computation
computation
the primary
integrals
illumination
given
larger
in
plotted
of the geometry.
from the rigorous
(ror
in (3.29).
was assumed to circularly
the displacement
as predicted
10% of the value
but within
than,
is
this
symmetric,
at the edges of the reflector).
a lOdB taper
is found that
(3-32)
parameters
The agreement between the two predictions
It
from
data obtained
are numerical
of the diffraction
(3.32)
(radians)
of the two angle
as a function
Superimposed on the curves
with
normalised
beam as predicted
the
is,
beam-shift
sin 00 sin 6m
ff(cos 0mt Cos 00)
AG
"/d
Fig*
the
is seen to be very good,
is always
by the formula
by computation
predicted
of
the integral.
3.13
Confirmation
Experimental
The beam-shift
observed
originally
The experimental
of theoretical
to provide
procedure
confirmation
and Compton [8]
was
in 1954.
presented
by these authors
are illustratives
in nature,
and consequently
direct
patterns
data with
radiation
polarized
by Clare
and reported
than quantitative
rather
circularly
with
effect
their
cannot be made.
results
of the theoretical
used by the authors,
predictions,
and the results
of this
comparison
However,
the experimental
work are briefly
described.
The reflector
of
57A
0m
.=
the
and
,
30.40.
used in the test
angle
The feed
parameters
(the
details
had a projected
of
the
of
had the
aerial
which
are
diameter
aperture
values
not. given)
00=
was fitted
42.2%
- 86 -
with
left-eircularlY
polarized
to be separated
into
The radiation
it
two channels
radiation
from a distant
in each of the two channels.
as having
components,
by reciprocity,
should,
of this
are shown in the plots
radiation
Since
two equal and opposite
the patterns
in the
received
the radiation
represent
is transmitting
experiment,
patterns
in the respective
of
circularly
for
a) in the vertical
The diagrams
the cases when the incident
b) in the horizontal
plane,
the displacement
is
of the axis
polarization
c) inclined
plane and
of the circular
of the incident
the peaks of the patterns,
as quoted
displacement
as 7.5 minutes,
displacement
of about 8 min.
the expression
However, the authors
the turn-table
mechanism available
the theoretical
figures
between
is about 6 minutes.
feed excitation
of the peaks of the patterns
positions
with
while
of the displacement
by the authors
(assuming balanced
for
fieldsý
The magnitude
computation
components on either
in the same sense and has the same magnitude
any plane of polarization
comparison
the
represent
of 450 to the horizontal.
Note that
Numerical
from reference
as reproduced
(3.10-
of rig.
obtained
patterns
at an angle
with
by the horn
were checked in the usual
plane polarized
may be regarded
polarized
The results
precise
and the
modes.
polarized
side
with
when the aerial
the aerial
is
of the aerial
the signals
fields
handed circularly
the right
the energy received
components'of
properties
and plotting
the incident
which enabled
two receivers.
manner by illuminating
source,
duplexer
and a polarizer
a polarizer
predicts
(3,22) predicts
remark that
were difficult
the
the
the
to measure
to them and consequently
quoted here is not strictly
valid,
OF
A.
CHANNEL
P Z, AR I"'ý'%TI-'
vEr., -riCALL)'
A
HD
PL
tNcNT
r
LL_r
.
40.
0
0E
FFE0
Cý77ýý=7`0
r,
5
C5RE-E-Sp
T; -ý'H
CF 7P, 77 F'E! -Lf--C-rCR
L5--
-510
TALhAAV
A PERI (A CA-\
A, 17)
A,
%
, k..
kc) IA
!7
7-
-D
LLAR F- A ra b C-0 MP 1-0
Cl
- 88 -
[Recently
Chu and Turrin.
(9)
experimental
data which confirm
For example,
their
practical
show displacement
section.
(60 = em = 450),
a 19X dish
between the peaks.
in (3.29) predicts
disadvantage
The principal
its
its
of the reflector
performance
sufficient
radiate
For the same
the displacements
as 0.69",
results].
exhibit
of generating
Cassegrain
off-set
with
linearly
field
example,
feed system.
feeds.
shows, however,
polarization
illumination
This technique
polarized
primary
illuminates
squint
and the focal-length
This
order
the
can be achieved
wave-guide
properties.
modes
Another
method
is by the use of an off-set
is
fully
feeds,
the reflector.
which depends chiefly
of the parent
which
to compensate for
reflector.
the antenna does not have any cross-polarized
a small
polarized
distribution
by the use of higher
the required
With circularly
develops
configuration
can be improved by the use of primary-feeds
the desired
cross-polarization
reflector
amount of cross-polarization
for
in practice,
the
aperture
caused by the off-set
depolarization
which
of
performance
poor cross-polar
Examination
angle
in this
and
Conclusions
3.14.
that
some theoretical
presented
measurements with
by about 8% of their
which is smaller
is
the results
2A6 = 0.75*
of
the formula
geometry,
have published
described
the overall
in Chapter
The radiated
no
beam however,
upon the feed off-set
paraboloid,
from
radiation
components provided
S.
- 89 -
APPE14DIX 3.2
integral
A typical
field
radiation
in the calculation
which arises
components from the projected
of the
aperture-field
method
is
N(O
aaf
e
ý1)
dA(el
A. 3.2.1
A
The differential
Jacobian
the
ý1)
iik(p.
is the area of integration.
A
where
Wel
fi)
of
surface
the
dA(Gj ý1)
(f2f
LCI
vector
aperture
co-ordinates
R,
r sin
in
expressed
(x
0
sin
(a +b
p,
yP,
cos ýx+
aa
from
can be obtained
and (3.4.8),
and is
given
by
1
av
ax
2
301
Del
(2f 12,
P.
The unit
(3.4.7)
transformation
ý, )
dA(01
element
dOdol
0 d0#1
sIin
cos
e de dýj
terms
A. 3.2.2
the
of
limit
vector
the
of
is,
zp)
sin
-p
0
a
sin
cos 0z
a -p
y-P
and
A
(x
The phase factor
COs 01)
tr
ccpc
in
(A. 3.2.1)
in
f
(A. 3.2.2)
and (A. 3.3.3)
3.24 ]., are readýiy.
sin
0 sin
.p
is, thus
6 [x cos ý+x
acaca
sin
Substituting
x^ tr
obtained.
in
cos(ý,
(A. 3.3.1),
-ý
)]
the integrals
A. 3.3.3
given
90
REFERENCES
1)
(1949)
SILVER S.,
A. B., 'HOGG D. C.,
CRAWFORD
J.,
3)
1963,
5)
HANNAN P. W.,
Sept.
6)
ELAM E. M.,
ibid,
I.
Antenna;
"Optim=
pp.
"The Open Cassegrain
pp. 1255-1300.
Feeds for
All
Three
Technical
pp.
Modes of
a Monopulse
on Antenna and Prop. ",
Publication
on
Aerial
Feeds",
Aerospace Antennas,, No. 27,
"The Theory of Electromagnetism",
with
Off-set
Note 108, January
S. and TURRIN R. H.,
Reflector
339-345.
The Macmillan
1964.
J. W., "Beam Shift
CLARE J. D. and CROMPTON
Off-set
429
200-205.
JONES D. S.,
CHU T.
ibid,
Characte-
and ZUCKER H.,
"Dual Plane Monopulse Multimode
Microwave Aerials
9)
"The Electrical
Antenna",
I. R. E. Trans.
Theory",
Company, New York,
8)
System
Horn-Reflector
44,1965,
I. E. E. Conference
7)
Bell
1961, pp. 444-454.
BOWESK. G.,
1971,
Echo",
1187-1212.
COOK J. S.,
Antenna"
Proj.
and TURRIN R. H.,
LI T.
N.,
pp.
and Design",
1095-1116.
pp.
of the Conical
ristics
4)
40,1961,
HINES J.
Theor-I
and HUNT L. E. 0 "A Horn-Reflector
Space Communication.
Antenna for
Tech.
Antenna
pp. 454.
McGraw-Hill
2)
"Microwave
Antenna",
Parabolic
Phenomena in
Reflectors",
R. R. D. E.
1954.
I'Depolarization
Properties
I. E. E. E. Trans.
AP-21,1973,39
of
- 91 -
CIHAPTER 4
SYMMETRIC CASSEGP-kIN AERIAL
THE CROSS-POLAIRIZATION OF AXIALLY
Introduction:
The feed system of a CasseZrain
or a feedhorn,
radiator
their
positioned
so that
reflector,
as shown in Fig.
The design
of
components are
main
of a parabolic
into
a spherical
plane
ba3ed
on the
principles
of the components is such that
This wave is in turn
an.1 tranuformed
of the primary
wava emerging
source is
from the
by the
reflected
waves to form a well
defined
beam.
However,
since
system are usually
the
dimensions
appreciable.
who also
calculated
as the diffractive
of this
in axial
in
subreflector
of diffraction
An indication
the loss
of
the
to the operating
comparable
1OX-40X , the effects
of
effects
into
by the subreflector
radiated
quite
is
system
reflector
The arrangement
focia3 of the paraLoloid.
order
the axis
with
wave emerging from the phase centre
a spherical
paraboloid
system these
a hyperboloid
(4.1a).
dual
the
coincide
axis
optics.
of geometrical
converted
Cassegrain
source
a point
comprises
which is usually
and a subreflector
In the standard
of revolution.
aerial
wavelength,
at this
of the
can be
given by Ruschl'ý,.
was originally
in the aperture
usually
surface
gain of the aerial
phase errors
a practical
due to such
distribution
of
the main reflector.
The diffraction
C
process
significant
cross-polarized
cross-polar
performance
at the subreflector
energy,
can also generate
which principally
of the overall
system.
This
determines
follows
the
directly
-92-
FIG
FIG
4.1b:
4-1a:
The
Co-ord. 1nate
Cassegraln
geometry
antenna
of
the
configuration.
hyperbolic
subreflector.
- 93 -
of the axially
polarization
by that
produced
In this
by its
In Part
feeding
properties
to the polarization
II,
front
with
of the simple
Part
I:
The Depolarization
is mainly
carried
of the hyperboloid
are used to compute the diffraction
that
is
analysis
determined
In Part
out.
I,
examined in
are first
of the fields
data obtained
performance
the cross-
system.
property
the numerical
The cross-polar
was shown that
SYMMetriC Daraboloid
a twro part
chapter,
the de-polarization
relation
2, where it
of Chapter
from the conclusion
incident
from the calculations
on it.
of Part
of the main reflector.
patterns
of the Cassegrain
aerial
is then compared
fed aerial.
Properties
Of The Axially
Symmetric
Hyperboloid.
Analysis
4.1
The geometry of the Cassegrain
It
of a hyperboloid
consists
phase centre
coincides
In terms of the polar
of the reflector
with
of revolution
c
and a primary
one of the two foci
can be written
the equation
02)
('1.1)
where
e=
source whose
as
Cos *t
the
(4.1b)
of the hyperboloid.
shown in the diagram,
co-ordinates
surface
feed system is shown in Fig.
distance
eccentricity
between
of
the
the
foci
hyperboloid.
I
- 94 -
An outward
directed
normal to the surface
vector
unit
;pa)
+
(Cos
n=
is given by
ý a^lp
sin
(4.2)
MM
where
m(*)
The differential
(1 t $2 t 20 cos
:
element
Let the radiated
electric
(E
(4-2)
1ý
be written
(4.5)
side
optics
of the reflector,
currents,
is
(4.5)
assumes the
form
ikp
-,
L'
(4.6)
-p
components of
K
are
K
x
+ 0 Cos
Cos
E+
(C-Cs
+
sin
K
y
+ 0 cos
sin
E
(Cos
+
Cos
E
sin
as
(4.4)
xE fý
x (a
-P
-
2K
z0 M(V)
K
(4.3)
-ikp
2ý
of the Physical
and (4.4),
where the Cartesian
of the feedhorn
on the illuminated
assumption
2
;:;.j0
p2 sin 0 m(o) dý dC
tcos V+ 0)
Ef
+E^)
4c%p
density
current
under the usual
Using
IP
field
is
surface
reflector
d* dC
P2 sin
(A. ý
ds
The surface
on the
*
E
(4.7)
- 95 -
when the incident
Note that
the induced
of a Huygen source,
The far-zone
the
(4.3)
vector
and (4.6),
Tr
the
is
this
2'
in this
fields
_r
the
of
focus
phase centre
co-ordinates
(r,
(4.8)
Cassegrain
F1
fields
point.
used to
system,
of the observation
the virtual
optics
ip d* dt
+
are subsequently
complete
to choose the origin
with
Using
currents.
of the observation
manner the phase of the scatter
In terms of the polar
it
co-ordinate
of the hyperboloid.
is referred
of the reflected
to
F1
fields.
measured as shown in
e, ý),
(4.1: ),
aa
Hence, the Cartesian
written
p sin
(Cos
of the reflector
patterns
which is the geonetrical
Fig.
ep
in the direction
vector
system to be co-incident
Also,
source
0
diffraction
found convenient
the
by
given
*m
can now be derived
reflector
with
f 2-7ry ý'DPU
fr
Since the scattered
is
of the
the eccentricity
the
of
associated
potential
is a unit
a
r
compute
have a non-zero
because of the constraints
occurs
field
scattered
o
where
properties
surface.
reflector
from
in the reflector
by the shape or
imposed upon the currents
the polarization
exhibit
currents
This
component.
cross-polarized
fields
sin
0 sin
ý cos
components
(i
'Ti
+ cos 0 cos
= x, Y, z)
(4.8)
of
can be
as
Ir
ps in
(
a
4, o)
ta
tTer
(044004
(4JO)
where
Fi
f21r
0
inb(ý16)
Ki (ý, C) e
COS(C
d4
(4.11)
0
- 96 -
(I - Cos ý Cos
Cos yl t
n=y
a
sin ý sin 0
(Cos lp + X)
.
now be obtained
fields
components of the scatter
and the cross-polar
The co-polar
can
froir
Es sin
Es cos
c0
where tha spherical
Es=
0xz
Es
cos
(4.12)
Es
+ sin
Es,
components
7r
Es=0x-.
cos
ý+,
cos
fields
of the scattered
Osiný-7r
nycos
are
sine
ce
ýý F
cos ý 7ry
w+
sin
Es
-ik(r-cos
j
(4.13)
The factor
c cos
as consequence
system
co-ordinate
It
of
0
in
appearing
the
can be readily
01 to
F1
shown that
dependence of the cross-polarized
from these
components vanish
at
45*. to
the
Fig.
planes.
the
origin
sources.
to
occur
arises
of
the
4.1).
sin
the contribution
(ý = 0, ý=
in the principal
can be expected
principal
of
owing to the explicit
system is rotationally
Since the scattering
cross-polarization
exponential
(see
(4.13)
term'of
transformation
geometrical
from
the
symmetric,
in
the
c cos ý
(4.11)
lr/)planes.
the maximum
planes
inclined
e)
- 97 -
Under certain
(4-11)
in
integrals
function
class
of primary
technique
point
;
demands that
the
of
however,
half
over
the
the
azimuth
this
and therefore
rrom the computational
here.
Bessel
a more general
the symmetry of the svstem may be utilized
of view,
efficiency
to reduce the
range.
ýqsults
Numerical
to examine the intrinsic
In order
the hyperboloid
ideal
an
essentially
interest
of
are
patterms
is not implemented
integration
4.2
feed
expressions
analysis,
the
to evaluate
involving
be independ3nt
In the present
C-
co-ordinate
This
E(ý, C) and
function
pattern
(Rusch).
nb
argument
of
to yield
analytically
is possible
it
symmetry conditions,
surface,
feed having
the primary
depolarization
feed was initially
assumed to be
pattern
radiation
symmetric
a circularly
of
properties
of the
fon"i
sin
(Tra sin
(jcL
<ý<
IT
m
0ý<
The double
for
integrals
illumination
e=
having
a reflector
1-5;
diameters
angle,
the following
Vm= 166-50;
extremum reflected
of
as to illuminate
in
appearing
16.2X
.
were
then
geometrical
numerically
evaluated
parameters:
maximum
of the hyperboloid
eccentricity
ray angle,
and 48.6X
a paraboloid
(4.9)
0m= 60"s and reflector
The geometry
of semiangular
of
the
diameter
reflector-is
of 60*.
such
- 98 -
For a 9dB illumination
diffraction
in the
7r
The co-polar
diffraction
and
in the region
diffractive
field
not
Beyond this
decay.
to
owing
significant
increasing
seen that
in the forward
fall
the fields
This merely
effects
tend to be less
by the angular
-60"
patterns
Although
region.
depolarization
the interference
high peak close
sharply
pattern
region
of the fields
angle.
which also
fields
the scattered
as defined
region
on the axis
increases
These oscillations
corresponds
It
feed).
a relatively
practiceBeyond this
in an oscillatory
manner with
are more pronounced
to the "decay"
is seen that
creates
(In
optics
of the reflector,
contributions
of the reflector.
by the primary
diffraction
in the geometrical
away from the axis.
between the diffractive
lobe would be blocked
amplitude
behaviour
have a null
the fields
to the axis
the diffractive
the cross-polar
pattern,
oscillatory
increases
of the smaller
0
-*
s
optics"
of
to +60".
to the co-polar
show a highly
x /D
of
the amplitudes
and consequently
"geometrical
the
is
It
of the oscillations
the amplitude
in the limit
are
has the effect
of the reflector
significant
within
sector
In contrast
that
suggests
to be confined
illumination.
edZe
as compared to the fields
more rapidly
which
appear
sidelobes
in the "Olecay" region
Also,
dish.
the
and decreasing
reZion.
monotomic
of an approximately
minor
of
the diameter
the frequency
increasing
tend
tapering
This region
from the reflector.
(450-700)
region,
by Rusch [1].
are caused by the interference
0-45*
contributions
by a region
then followed
of the
characteristic
as described
of the hyperboloid
effects
among various
shows the familiar
pattern
The minor oscillation
this
(4.2)
of Fig.
(4.3).
rig.
is
fields
and the cross-polar
appear as shown in the plots
X+ plane
ý=
fields
of the co-polar
pattern
the
at the edges of the reflector,
taper
regions
peak the
increasing
near the 450-600
fit.
the
of
CO-Dolar
-Ids.
1ý
0
-10
un
u
-30
-40
-60
--I----II
I'
WCJQ
'-
(d2)
&c(-')(lý
II
------
-
.
--
--------
"1
100
I
ci
--
000
T
i1
---
-- -
--0
1
II
- 101 -
Notze that
than its
polarization
(at
'Although
the
of
computation
cross-polar
level
The Effect
It
primary
significant
'ideal'
the
by the reflector
be some -40dB down.
of the reflector
from that
of an ideal
realistic
large
aperture
5X
of geometries
A sample
that
reveals
the
previous
peak
that
section
the solid
energy within
of the hyperboloid.
the
angle
In order
with
feed condition
this
the "worst
by
case"
(or a dipole
feed
feed),
that
can be
is
calculated
geometry
the cross-polarization
to
in the diffracted
feed cannot vary si6mificantly
the same edge illumination.
generally
the
must be
can be satisfied
considering
of the previous
feed providing
horns
It
cross-polarization.
of the feed radiation
Consequently,
of the reflector,
in practice.
feeds.
of 5X -. 10X.
TE10 rectangular
the maximum cross-polarization
illumination
of
For example,
of an open ended
intercepted
in
us examine the case in which the primary
let
result
feedhorns.
most standard
situation
calculation
at the focus by the periphery
that
role
Cross -Pclarization
no cross-polarized
remarked however,
an important
as -28dB.
fields
generates
the
of diameter
the
source itself
that
may not he true
this
a dish
Of The Primary
this
to
demonstrates
of the Cassegrain
of the order
can be as high
contain
The peak levels
quoted in the above examples may be considered
out for
to generalize
realized
play
in
assumed
was
substended
fields
clearly
cross -polarization
reflector'diameters
carried
more cross-
are -33dB and -39dB relative
This
under most circumstances,
involving
significantly
counterpart.
at the subreflector
the peak levels
acceptable
4.3
fields.
effects
determination
diameter
large
co-polar
diffraction
generates
in the two patterns
60" off-axis)
the on axis
dish
the smaller
'ideal'
required
for
condition
efficient
is even better
With
- 102 -
However, the same cannot be said
in the dominant
feedhorn. operating
lines,
TI
compensatory
mode radiates
the level
of the higher
for
mode;
order
a
of cross-
This expedient
owing to the opposite
cross-polarization
owing
a
used, as in the dual mode conical
mode is also
11
that
(In practice,
to equate the E- and 11-plane beamwidths.
lines
this
is high.
polarization
aperture
We recall
mode.
main beam and consequently
elliptical
markedly
TE
11
field
of the apertura
to the curvature
of horns such as the conical
horn
the
eliminates
of the field
curvature
a discussion
3
of this
see
point
reference
Another
example,
similar
qualitatively
diagonal
4.
horn
The aperture
of the
lobes
cross-polarized
communication
used where suppression
is of interest
Fig.
illuminated
is
tapered
polarization
frequently
to
modes.
is
a horn
the
an ip-
The resultant
of -16dB occurring
Owing to its
ofýthis
feed in
such feeds must not be
is of primary
importance,
of the hyperboloid
properties
type.
close
broadband
used as the primary
of cross-polarization
shows the
horn
the
edges
the
at
feed
diffraction
predicted
a diagonal
-15dB
of
such
is
horn
The expressions
for
the
in
be
found
horn
the
reference
can
of
radiation
with
which exhibits
conical
Although
systems
feedhorns
with
(4.4)
TE
01
of the order
to examine the diffraction
when illuminated
far-field
horn is
the diagonal
the sattelite
of
of the main beam.
to the -15dB power points
performance,
and
TE
the
distribution
TE
10
variety,
Ehom
have
the
and H-Plane beamwidths
equal
of
fields
but contain
to
characteristics
phase superposition
radiation
of the square aperture
radiation
as described.
of
the
reflector
intercepted
fields
of
a
The primary
and the
by the
25X
dish
illumination
maximum cross
reflector
is
it
103-
111-1ý,
-1:
-1
10
%x'>
-)-
I
43
-
H
)
i
0
C)
Q
tit)
OP
-
(Z)
0
IT
- 104 -
approximately
fields
-16dB.
horn were selected
of this
(in
same edge illumination
is seen that
It
localized
observed
in the case of the ideal
co-polar
fields.
patterns
is -14.3dB
(diagonal
between the two values
and -18.4dB
in the two
(conical
feedhorn).
by the fact
can be explained
that
is
of the two feeds,
and the cross-polarization
the illumination,
as also
below the on axis
of cross-polarization
feedhorn)
The
of the hyperboloid.
is some -2ldB
feed,
The maximum level
of both the feeds are
of the reflector,
to the axis
close
horn.
of the diagonal
patterns
effects
the
approximately
as that
the 45*- plane)
the cross-polarized
peak appearing
The difference
The aperture
to provide
by the diffraction
distorted
severely
mode feed.
TE1,
a conical
with
dimensions
Also shown in the diagram are the predicted
different.
Thus it
is seen that
the cross-polarization
of the Cassegrain
is
also
system
dependent upon the amount of cross-polarization
by the primary
feed ite-elf.
of the last
The results
two sections
feed
introduced
may now be summarized as
follows.
1.
With an ideal
radiation
increases
2.
fields
role
in determination
rapidly
in the limit
When the incident
are scattered
in the far-field.
in the scattered
the cross-polarization
radiation
of the
The diffraction
by the hyperboloid.
fields
imoortant
feed,
as a consequence of the depolrization
arises
incident
primary
effects
play
of the degree of depolarization,
as
Ds/.
Me maximum level
which thus
_* 0.
contains
by the reflector
an
fields,
cross-polarized
to form well
defined
in these patterns
these
diffraction
is nearly
equal
peaks
- 105 -
incident
to maximum level
must not be used in applications
is a principal
3.
design
The presence
of
the
reflector
of
the
feed
symmetric
the
the
paraboloid
reflector.
examples
can focus
This
aerial.
systems
employing
in
cases
such
stringent
particularly
which
the
high
relatively
imposes
for
whom suppression
feeds
of cross-polarization
requirement.
system,
subdishq
aperture
of
Thus such primary
on the reflector.
remark
the
is
energy
particularly
primary
an axially
positioning
areas
of
of
the
the
boresight
to
the
Cassegrain
such
axis
accuracy
as the
main
of
diagonal
due to misalignment
cross-polarization
the
on to
relevant
sources
the
into
lobes
these
undesirable
in
tilt
near
alignment
illuminate
when used to
A slight
peaks
on the
requirement
can deflect
unbalanced
on axis
cross-polarized
horn;
can be
serious.
In the following
antenna are predicted
source.
realized
As pointed
in practice
section,
the radiation
patterns
by assuming that
the primary
out in the previous
section,
with
most standard
feedhorns.
from the Cassegrain
horn is an ideal
this
condition
is easily
- 106 -
Part
II:
fields
The far-zone
integrating
section
1)
From The Main
of the complete
on the surface
the currents
by the fields
Patterns
The Diffraction
4.4-
under the following
induced
of the main reflector
is
carried
by
out in this
assumptions:
is located
the main reflector
can be predicted
aerial
The analysis
to the sub -reflector.
Reflector.
in the far-field
of the
region
sub-reflector,
the fields
scattered
sr
Because of the axial
holds
relationship
true
of the assumption
computing
the double integrals
from using
incident
checked for
and comparing
3).
plane
(The
case by
a specific
w/4
at the
directly
the data with
from
that
obtained
the formula
(6,
c
ý)
=
sin
ý cos ý (EI(O)
- HI(O)).
were found to be in very good agreement).
The results
The expressions
paraboloid
(4.13)
in
patterns
(Rusch, ref.
in the
the cross-polarization
(4.14)
radiation
the fields
representation
was also
-ikr
symmetry of the feed-geometry
that
provided
also have a similar
reflector
validity
are the E- and H-plane
HIM
and
of the reflector.
this
ý El (0). j + cos ý Ht (0)_D a
[sin
EI(O)
in
can be expressed
form as
a separable
where
from the sub-reflector
reflector
the analysis
shown in Fig.
of that
(4.1),
for
the diffraction
have already
chapter
been derived
and referring
the far-zone
fields
electric
of an axially
in Chapter
2.
to the co-ordinate
field
Em(0
aa)
symmetric
Following
geometry
scattered
- 107 -
can be expressed
from the main reflector
-ikR
eaM^
ia
Em(()a* ýa)
where
Caitesian
the
q
M
wI
components
0,2'r
'Ba)
the
of
(1-cos
ikr
Maaa
K. e
ItM(O
iaafJI
as
vector
sin 0 sin
0 cos 6
OBO
xrs
are
potential
in 0 dO dý
(1 t Cos 6)
0
cos(ý-ý
(4.15)
where
in
x
=
ý cos ý (EIM
sin
e=
y
sin2
e=
Z
sin
Ht(6))
-
+ COS2
0
e sin
El(0)
and
2f
=+
Cos 0
(R
ýa )
a9e a'
refer
Note that,
fields,
to
the
because of the diffraction
in phase;
also,
since
components of the surface
the E- and H-plane
EIM,
currents
into
account
of the aperture
by choosing
can result
the lower
limit
of the incident
from the differences
as the familiar
e
in
cross-polarized
amplitude
by the subreflector
in the
constant
the cross-polarized
in the E- and H-plane
blockage
far-field.
is not completely
are phasors,
as well
phase patterns,
difference
from
the
component
The effect
HIM
the
phase ripples
from the paraboloid
the wave reflected
in
co-ordinates
observation
patterns.
is taken
integrations
to be
- 108 -
OB'
the angle subtended, by the blocked
OZ defines
where
of the new co-ordinate
at the origin
the aperture
from any struts
of the blockage
portion
of
The effects
system.
the subdish
is ignored
in
were numerically
evaluated
for
supporting
the analysis.
4.5.
Results
Numerical
The integrals
appearing
in (4.15)
an antenna geometry whose dimensions
Rusc
were taken
from the paper by
ese are,
Main reflector
D
m
diameter
201.2X
f=
length
focal
87X
diameter,
semi-angular
0m=
60"
subreflector
diameter
D
s
19.6x
E=1.5
eccentricity
diameter,
semi-angular
radiation
The results
in
Fig.
4.5
Fig.
plane
patterns
of this
andFig.
(4.5)
described
computation
by eqn.
feed having
(4.2)
a -circularly
of section
I.
in graphical
form
in the elevation
(H-)
are presented
4.6.
shows the diffraction
of the-antenna.
at a level
13.50.
feed was assumed to be an ideal
The primary
symmetric
ým=
patterns
The peak side-lobe
level
In this
phase occurs
of -20dB relative
to the field
intensity
on the boresight
In practice
the co-polar
side-lobe
power can be expected
of the. aerial.
-loq
-I(
-
-31
q
-,
A,vi91t;,0f(-ü, 'Ký5
Flý 4.5,
)
t--pj
-110-
(T
J'voý(s
e-Y-Y-0vs
-5
LI-I'D
Fu
I
Fiq
(-5
2
lir)CILýAcet
e--KcI Li cI c4i
- ill
higher
to be slightly
blockage
than the predicted
of the struts,
effects
The cross-polarized
inclined
of the axially
symmetric
obtained
(i. e.
values
this
phase
1EIMI
computation
ripples
radiation
the
is principally
integrals
of
was to
ascertain
the
incident
fields
fields
Thus it
the
main
only
the
modulus
purpose
diffractive
structure,
secondary
pattern
and
radiation.
is relatively
on the cross-polarized
level
pattern
in the antenna
as compared to -40dB in presence
can be concluded
cannot be ignored
performance
the
of
the
patterns
The main
side-lobe
of
the effect
of -5ldB
using
effects
on the
at
fields.
the
the main field
at a level
of the sub-reflector
in the
which,
incident
The maximum cross-polar
ncw occurs
determined
of -35dB.
fields
and
significant.
of the cross-Polar
of
1HIMI)
while
the
with
the peak cross-polarization
the reflector,
diffraction
by the phase-errors,
of the phase errors.
effect
phase
cross-polarized
is seen that,
unaffected
is quite
the
on the
particularly
It
of
of -40dB relative
in the diagram are the predicted
the
and evaluating
reflector
of
the
at a level
is consistent
at a level
occurs
Also shown superimposed
I
by ignoring
4.6).
reflector
illuminating
4.2)
(Fig.
in
am shown
was shown that
paraboloid
by the cross-polarization
(Fig.
case
fields
The prediction
2. where it
of Chapter
and
both the co-polar
for
lobe occurs
main field.
and reaches
at 450 to the elevation
patterns
The peak of the cross-polarized
to the on axis
zero on the boresight
in such a plane
fields
and the cross-polar
is
radiation
The diffraction
the azimuth.
present
in such a
which are always present
in the planes
maximum level
analysis
the aperture
since
value
have not been included.
configuration,
its
-
of the aerial.
that
the diffraction
in realistic
predictions
- 112 -
4.6.
Concluding
Remarks
Comparison
of
A useful
consideration
the
example,
an ideal
given
application,
Cassegrain
system
diffraction
from
for
since
in
the
the
front
the
polarization
subreflector
for
the
are
-50dB
TEiO horn4
that,
Taking
consequence,
for
designed
in its
Cassegrain
cross-Polar
to provide
in the overall
-40dB),
a version
level
since
illumination
of the depolarization
the Cassegrain
due to
the
former
geometry.
the
two
configurations,
to antennas
employing
or the pyramidal
designed
for
2
front-feed
of some 15-200
depending
down.
In
on the illumination
On the other
large
aperture
source
configuration
is
are used
The depolarization
of cross-polarization
comparatively
would exhibit
improvement
dimensions
the cross-polarization
effect
hand, a version
substantial
of the subdish.
then becomes the major
since
the
(Cassegrain-fed).
would exhibit
application
system#
to
from the antenna would have maximum
of the reflector.
performance
efficient
by the hyperboloid
result
aperture,
and the
was shown in Chapter
as an exampleo it
radiation
(or the f/D ratio)
in
such as the dipole
of some -20dB to -25dB,
cross-polarization
the
and -40dB
would have peak cross-polar
the overall
in
avoided
that
be superior
would
does not apply
of feeds,
small
front
the
distortion
(front-fed)
the later
because of its
application
angle
types
both
for
For
indicated
section
cross-polarization
peak
feedhorn.
primary
system
is
be made if
can be only
the
the
of
performance
previous
fed
However,, the same conclusion
the more standard
of
suitable
the
feed,
an ideal
nature
source,
primary
figures
the
Antenna:
paraboloid
cross-polar
paraboloids
presented
Cassegrain
(Typical
to
given
results
the
front-fed
and the
is
and Cassegrain-fed
between
comparison
fed
Cassegrain
Front-fed
the
produced
small
far
as a
(-30dB -
superior
- 113 -
performance
of its
to that
front-fed
Howevers a major difficulty
prime focus applications
to have theoretically,
have been recognized
data on their
experimental
in
literature.
appeared
the design
in modifying
patterns
as required,
cross-polar
-
unless
Off-set
Cassegrain
The principal
geometry is that
the subdish
illuminated
In order
Cassegrain
mode
'
to have the low
feed systems
to the problem of improving
reflector
the
antenna.
Aerial
centrally
of the axially
and its
radiated
symmetric
supports
are placed
from the main aperture.
placed
it
blocks
aperture.
power and reduction
to overcome this
aerial
can be expected
can be designed
a solution
area of the secondary
in the side-lobe
have not
knowledge,
difficulties
of the symmetric-paraboloid
the subdish
is
coaxial
quoted above, the Cassegrain
disadvantage
the path of the fields
since
a feedhorn
as providing
performance
feed Q
sum and difference
I
in monopulse tracking
application.
example,
of cross-polarization
can be regarded
conical
of such feeds to provide
for
In conclusion,
further
Moreover,
of
zero cross-polarizations,
to author's
performance,
bandwidth
as the dual-mode
(5 ) and the multimode
horn
of cross-
over the required
such feeds
Although
(3 ), the scalar
horn
of -40dB)
for
a primary-feed
a low level
which would maintain
frequency.
operational
in designing
arises
(say of the order
polarization
4.7.
a similar.
employing
feedhorn.
primary
level
counterpart
in axial
disadvantage,
Cassegrain
in
directly
Moreovers
out the most intensely
The results
increse
an
are
gain of the antenna.
the off-set
or the open-
(6)
has been proposed
In
such a configuration
.
the axis
- 114 -
of
the
is
subreflector
illuminate
a portion
only
from
tilted
of
the
the
main
of
axis
the
so as to
paraboloid
as illustrated
reflectors
in
(4.7).
Fig.
Although
such an arrangement
of a plane of symmetry has the effect
the loss
problems
of the fields
the polarization
blockage
the aperture
eliminates
of distorting
produced by the combination
of the horn
and the subreflector.
The polarization
been investigated
have already
front-feed
the reflector
off-set
off-set
axially
j
aerial
feed system;
be expected
for
of some -20 to -30dB,
Cassegrain
will
not be affected
the front-feed
the same level
case using
an ideal
of the crossfor
a vertically
on the parameters
produced by the
typically
at a level
radiation
primary
of its
from the aerial
of cross-polarization
source.
of
of the off-set
by the cross-polarization
in consequence the overall
to exhibit
depending
the performance
that
of
to the plane of the
feed system occurs
is evident
effect
the peak cross-
Since the cross-polarization
geometry.
it
to
some -35 -40dB9
Cassegrain
source,
in the plane perpendicular
and at a level
symmetric
In particular,
primary
antenna
the case. of the simple
in determination
antenna.
fed by an ideal
occurs
of the off-set
role
3 for
the depolarization
was shown that
of the overall
reflector,
polarization
in Chapter
the dominant
plays
radiation
polar
It
system.
fed parabcloid
of the off-set
properties
can
as evaluated
-119, -
N
F
on-A
C-ONFIGOP-wr(nm
115a
RUERENCES
1)
Feed SYstem",
Cassegrain
No. 4,
2)
"Scattering
RUSCHW. V. T.,
July
effects
I. E. E. E. Trans.
Ant.
in a
Reflector
Prop.
Vol.
AP-11,
pp. 414-421.
1963,
RUSCHW. V. T.,
from a Hyperboloid
"Phase Errors
in Cassegrain-fed
and Associated
Cross-
Microwave Antenna",
ibid,
Polarization
AP-14,
May 1966, pp. 266-275.
3)
POTTERP. D.,
Beamwidths",
4)
LOVE A. W.,
5)
SIMMONA. J.,
Feed for
"A New Horn with
Microwave Journal,
Suppressed
6,1963,
Horm Antenna",
"The Diagonal
and KAY A. r., "The Scalar
Large Paraboloid
Reflectors"
Construction
Design
and
on
Side-lobes
and Equal
pp. 71-78.
ibid,
L,
reed -a
1962 pp.
High Performance
I. E. E. Conference
of Large Steerable
117-120.
Aerials",
Publication
21,19660
pp. 213.
6)
KOCHG. F.,
"A New Feed for
Low Noise Paraboloid
Antennas",
ibid,
pp. 163-167.
7)
JACOBSE. 9 KING H. E.,
with
High Pointing
"Large-Aperture
Accuracy".
ibid,
Millimeter
pp. 218-222.
Wave Antenna
- 116 -
CHAPTER 5
CROSS-POLARIZATION CHARACTERISTICS OF DOUBLY OFF-SET CASSEGRAIN AERIALS
Introduction
A second
for
method
performance
the adjustment
of the geometry
depolarization
problems
type
(developed
application.
'
Measurements carried
the theoretical
free
of cross-polarizationi
5.1
Antenna
(in
with
of a hyperboloid
The focus
primary
0
radiator,
the axis
improve
demonstrates
how
of a co=ercial
of this
aerial
monopulse tracking
out by the company on an experimental
thatthe
aerial
doubly
off-set
Is substantially
the
Cassegrain
(5.1a).
comprises
the location
the beam axis
a primary
which has its
reflector
determines
radiator
principal
foci
of the phase-centre
of which is directed
and a
at
0
and
of the
to make an angle *0
of the hyperboloid.
The main reflector
the z' axis
p
use of
system.
Company) for
of
The feed system of the aerial
F.
off-set
conclusions
cross-section)
is shown in Fig.
portion
the
Geometry
The geometry
aerial
for
case can be used to overcome the
is presented
by the Marconi-Elliot
confirm
aerial
in this
of the ordinarily
analysis
This chapter
of the aerial.
the Polarization
In particular,
allows
which can be shown to significantly
hyperboloid
an asymmetrical
Cassegrain
the
off-setting
is a section
and focal-length
f
of a paraboloid
of revolution
about
17a-i
SI DE
VI EW
1Z
ZC)",
r- V Clt!
0, 'ý
)i din ' -ýt (7ý
t c-m
r
Xp
V
O'd
FIG
5-1cl
GE0 E-TRY
17b-1
FROINT
--------
VI EVI
--------
.1
I
I
I
I
/I
/
1
I
main
reflector
ile
pro-I
I
subreflector
FIG
5-lb
XP
profile
0
- 118 -
The feed system is positioned
can be rotated
together
The off-set
(r)
centre
between
angle
Two effects
mode, the fields
the
are operative
undergo further
fields
aperture
fields
the principally
radiated
polarized
hyperboloid.
of this
kind.
are initially
In transmit
depolarized
by the
the
main surface,
so that
the eventual
from the depolarization
of both
components of the fields
and the cross-polarized
sections,
out in two stages.
is used to evaluate
In the first
the scatter
In the second stage,
the aperture-field
distribution
integrated
to yield
Calculations
the final
stage,
fields
of the subreflector
data
this
polar
diagrams
and
is used to determine
and the fields
of the aerial.
From 7he Subreflector
forms a portion
of a hyperboloid,
a spherical
wave emerging from the phase centre
a spherical
from
the virtual
wave emerging
5.2).
the method of the induced
in the main reflector.,
Of The Radiation
Since the subreflector
of the antenna geometr-1 is
the analysis
the data stored.
(Fig.
is
where V
oc
by the subreflectori
currents
5.2
-*)
d
oc
by an
from the
the
of
transformation
polarization
is defined
from the off-set
have components arising
In the following
carried
axis
in an arrangment
Upon reflection
subreflector.
off-set
and the
angle of the
ray reflected
by (*
is given
by the feedhorn
radiated
The angle ýd
the off-set
by the principal
ray
central
fee. d and the subdish
of the paraboloid-.
defines
This angle
of the subdish.
the primary
angle of the main reflector
subtended at the focus
angle
the
about the axis
of the two reflectors
between the axis
subreflectbr.
so that
focus
it
transforms
(0) of the feed-horn
(F) of the hyperboloid
into
Ln
cr:
C)
LLJ
C-e
CT)
D
LI)
Lii
UL-
0
Ld
"5,
0
Lb
0
'I
- 120 -
system used to describe
The co-ordinate
systen, (p, ý, C) with
a spherical
to make an angle V0 with
equation
in this
c
7
0
at
and polar
(z
axis
of the hyperboloid.
the axis
of the hyperboloid
cosý Cos*
origin
The
f)
'
(1 -
0-0-
sinp
by
(5.2)
Cos;
0
directed
is given
system of co-ordinates
sin*
is
feed radiation
the off-set
where
distance
c=
0=I,
e
(sin*
siný
denotes
sin*
0
=
cos*o -6Osiný
0+
0
siq
sin*
0
is given
by
cosc) a
--p
Cos* Cos; )
:
sinC aý]
dC
p2 siný
.a
p
=
of the square brackets.
of the quantity
ds
element,
surface
(n
of the hyperboloid.
normal to the surface
vector
the modulus'value
The differential
ds
unit
[(co4
T-nT
A=1
a
of the reflector
is the eccentricity
e
where
An outward directed
Inj
where
between the foci
p2siný
(a
the
on
reflector
,
dj_dC
is
Inj
bcosC)
-
where
a=
coal cos*
siný
The currents
can be obtained
0-a
sinýo
o
in the reflector
under the usual
due to an incident
approximations
field
of physical
from the feed
optics.
Thus,
- 121 -
for
an incident
field
electric
[E*(*,
Ef
it
ýtE
0 2ýp
:--
can be readily
E of the form
=f
Z Ini
(5.4)
AdT
is given
distribution
the current
shown that
K
-r
c
e-'kp
(*, C)
by
jikr,
X,
the
of
components
where
(5.5a)
along the xf,
resolved
yf,
and zf direction,
are
[a
sinC E; - COSCE* (Cosý
0-
x=
1(
Ky=
)
E;
acos;
b-
-E*
[sin;
siný
z=
The fields
from the subdish
radiated
from the expressions
evaluated
with
w associated
far-field
region
the source
relating
currents
of the reflector
ocosý
Ocosý)
-
(Osiný
E4-E
cos*
0
(co#0
I
+ siný
due to these
the fields
given
the general
sinC]
0
Cos; )l
(5.5b)
can be
currents
to the vector
in (5.5a).
relationship-is
(5,6a)
r
where
as measured'from
7r
the origin
je
0
ik(p.
to express
the diffraction
the scatter
of the co-ordinate
r)
fields
Since the scattered
to evaluate
in the direction
vector
fields
of the observation
system,
point
and
ds
(5.6b)
of the subreflector
effects
in the
For points
-ikr
is the radial
potential
are subsequently
of the main reflectors
in terms
of a spherical
it
used
is preferable
co-ordinate
system
- 122 -
(aligned
rq 0,0
r,
the
with
with
from the centre
axis
(z
axis
polar
directed
(ý
principal
ray
) between this
focus
virtual
reflected
ray and the
angle ý0 by
-U
=$2 - 20cosV
0
1+
& (5.5)
Using (5.1), (5.3)
system to F, it
of the vector
the
the
at
oc
to the feed off-set
is related
co#0
situated
along
The angle
of the dish.
(1 + 02)
co-ordinate
rig. 5.2
S)
hyperboloid
the
of
Cos voc
in
xs9 YS9 zs)
can be shown that
of the observation
the origin
the componopnts ir i (i
= X, Y, Z)
assume the form
potential
BI
ir i
and transforming
ikL(e
eKi
ff
SO
ý)
ikp(l-".
(*, C) ep--
sinýb d* dý
(a - bcost)2
(5.7)
where
I
sinO cosý
Siný
+ co# sin(* +ý
cosý cos(* +ý
0 oc
0 OC
[sin*
sin6
-
siný
sinC
sin*
- cosO
cosC sin(*
0
+ý
)
oc
)
+ý
- Cos* Cos(* 0 ocý
and BI
= c(l
a2)
-
.
The phase factor
integral
centre
sign arises
from 0 to F.
L(8, ý) apperring
from the geometircal
It
is given
A^
c, r
The spherical
term outside
transformation
of the phase
= c[cosO cos*oc - sin%c
by
the
by
components E0, E
the dish are then given
in the exponential
of the electric
sinO coso]
field
radiated
from
- 123 -
(, TI cose
xzr
(7r I
cosý
cosý
-
yr
tw
cose
e-ikr
)
- wt sine
siný
(5.8b)
7rxl siný )e
Is
I
denote the components of the potential
Irl
Ir
ir
9
xyz
where
xS, ys and zs axis
Irl
x0
tw+
7r I-
I
ITz0
oc
wx0
cos(ý
the required
field
+ lpoc
sin(*
the scatter
radiation
radiation
incident
field
+)
(5.9a)
z
due to linearly
oc
(5.9c)
7r
z
equations,
it
from the reflector
from the feedhorn.
illumination
consider
along
(5.9b)
Having formulated
prescribed
x0
+)w
Ir
yy
to calculate
oc
sin(ý
resolved
is
The transformation
respectively.
cosOp
(5.8a)
polarized
is now possible
due to any
In the following
illumination
we shall
so that
the
components have the form
siný
E(ý, O
cosý
cosý
-sinC
corresponding
respectively.
to principal
Here
E
linear
and
H
polarization
along
denote the pattern
the y. and xf axis
functions
of the feed.
- 124 -
5.3
Geometry Of The Feed System In The Elliotts-Antenna
(j)
Primary
Feed
One type of feed used in conjunction
horn with
pyramidal
multimode
12.5 X (H plane).
signals
antenna
is a
of 11.36 X (E-plane)
dimensions
aperture
The sum-channel
the Elliotts
with
are propagated
x
in the H
019
1103 and the EH12 mode.
The aperture
phase errors
propagating
and varying
Ef
:,-
In the
by the horn at the position
radiated
of
by
are closely-approximated
(siný
of the horn
of the modes is compensated by means
velocities
the fields
condition
the subdish
size
lens mounted at the mouth of the horn.
of a bi-cylindYical
focussed
from the large
arising
e-1
p
+ cosý a
--c
where
cosu
)
(12.
U2
3a cosu
sinv
v
(3
7
Y27r)2_u
+a
cosu
(IL
2)2-u2
vsinv
R2 _ v2
where
cL(= 0.35),
fi,,
ir . (12.5)
-,, siný
cosC
Tr . (11.5)
. sin*
sint
0(=0.608)
are the modal amplitudes
to the dominant
mode.
The E- and the H-plane
rig.
relative
(5.3).
radiation
patterns
of the feed are shown in
-125-
PRT"'o'ATRYRADIA'FICINIIPATTEPN
AJ
li -- PLAm iF,
----
E PLA
24b
10
FIG
5-3
12
Cýngt.
-
. 11,
d c,gr
- 126 -
of interest
is also
It
aerial
for
polar
axis
polarized
to consider
the radiation
is rotated
the case where the feedhorn
so that
the principal
(this
in the Xf-direction
beam through
from (5.10)
operation
through
90* about the
in the aperture
vector
function
The pattern
90*).
can be obtained
electric
of the
properties
also
rotates
is
the radiated
of the horn in this
polarization
by the transformation
Ir
i
(ii)
Antenna Dimensions
of both the sub- and the main reflector
The peripheries
design
Elliotts
(20 dB taper)
level
'shaped'
ao as to maintain
at the boundaries
as viewed down'the
the reflectors
Fig.
are
the same illumination
of the reflectors.
beam-axis
in the
of the aerial
The shape of
is shown in
(5.1b).
The dimensions
of the feed system are as
to the parameters
given
follows:
feed off-set
angle,
semi-angular
width
of the subreflector:
9*
H-plane'illumination:
E-plane
illumination:
eccentricity
distance
18*
V0=
7"
of the hyperboloid,
between
the
With these parameters,
focal
points
c=1.8
c=
the projected
the maximum and the minimum dimensions
(S = 0.5545)
127 X.
shape of the subreflector
of 40 A and 30.5 X respectively.
has
- 127 -
of the diffraction
In calculations
to assume that
found convenient
in
9*
V for
an angle of
leads to an increased
but it
dish,
5.4
Numerical
integrations
into
discrete
enough elements
are small. ý
are chosen so that
for
each field
time,
output
storage
size
data.
apprximately
Note,
Thus it
must also
since
point,
be given
a-large
is
essential
to the computation
number of points
in the computation
errors
of thG
in phase and amplitude
variations
however,
it
given to the subreflector,
(37
x 37) elements
of
the grid
of
must
of the diffraction
from the main reflector.
For the dimensions
grid
of the
on the reflector
relies
equations
over each element.
be computed to ensure minimal
patterns
illumination
the phase and amplitude
elements,
However, consideration
time required
This assumption
the prediction
not affect
of the field
being assumed constant
fields
power over
of the feedhorn.
will
was
from the aerial.
Computations
that
this
that
Numerical
being divided
intercepts
(> 30 dB) in the E-plane
taper
it
of the subsish
the subreflector
both orientations
is expected
radiation
overall
fields
was optimal
space and accuracy
from the view point
A further
requirements.
showed no change in the fourth
The time required
was found that
to yield
each field
of computation
increase
figure
significant
point
a
in
of the
(E09 E)
was
22 seconds and the program used about 14K words of store.
that
these
figures
computer on which the calculations
are only
typical
were made.
of the ICL 1905F
128
-
5.5
Results
Nwnerical
field
The e and ý components of the electric
due to the
subdish
fields
incident
in rig. (S. 4)shOws the radiation
The plot
the 0=I
planes
2
for
as predicted
oriented
to be polarized
relative
to the field
increases
its
in the plane perpendicular
containing
in this
rig. (5.5)shows
value
to that'in
Note that
the E-plane,
illuminated
of this
since
the primary
can be seen by canparing
patterns
reaching
of the off-set
and
The maximum value
in the vertical
the case where
higher
by the fact
illumination
due to this
that
is
in the y-direction.
has a stronger
in the elevation
The peak
plane.
can be explained
illumination
for
at a slightly
which the feed is polarized
is
feed
the
polarized
when
of the feedhorn.
to the plane
now occurs
level
the edges of the dish
of the radiation
of the symmetry plane,
in the reflector
distribution
the current
different
fields
The change in this
dB.
of -23
do not
However,
plane.
in the two planes
to be polarized
of the cross-polarized
fields
of -28 dB.
the distributions
the feed is orientated
are
= 0.
the radiated
from the dish.
ray reflected
at a level
plane occurs
outside
and
feed is
values
0=0,0
components in the 0=0
depolarization
the central
The decibel
by sy=etr)r,
rapidly
in the ý=0
the case where the primary
in the direction
intensity
any cross-polarized
peak value
and the data
patterns
in the y-direction.
Owing to the cancellation
exhibit
of (52 x 74) elements
tapes.
on magnetic
stored
in the x and y directions
polarized
grid
ware computed over an output
from the
radiated
plane
in the y-direction.
taper
in
are krightly
The effect
Figs -(S. 6) & (S. 7) which show the phase
in the plane
of symmetry for
the two orientations
6
0
N
0
-0
aj
c
in-
V-
Oj
-iC3 c
(.)
0
U)
Lb
(D
1-4
LL
I-
I-
'.
.01
0
CL
i
0
10!
U)
0
ki
(D
(D
LO
-130C*ý4
-Q
0
(Y)
0
0
C
C"
cl)
>
C
r
a,
C
S.-
C
-9--0
ci
ILI-0
C
c
cl,
I
ai
N.
r..J
LJC)
(D
LL
L-(114
C)
04
(x
0
CL
Lr)
Lf)
0
C)
C)
T-
%L
00
C4
LO
C)
e
-131-
61
Q)
0
a)
11
cn
0
0-
40
-C
73
ID
Qj
(D
Mp
0
0
(J)
Co
(_O
Lb,
C)
lkz,
(5; z-)P) Or-]
0
U
Q
-0
C)
0.,
ý.
(D
.ý
ýý-qb
@GiDL4d
LL
-132
-
0)
Q)
-v
ui
4.0
CL
CD
(Y)
'
":
C)
>
D
1)
CZ)
C'4
(3)
0
L)
ýD
C)
Oo
C)
LL
Q)
0
()
-
C?
)
- 133 -
5.6
Radiation
In dealing
with
from the main reflector,
In this
axis of the aerial.
'ongitudinal
the field
the reflector
Under this
close
in the aperture
in the aperture
distribtuion.
in the far-field
the phase centre
to the focal
constant
phase spherical
incident
field
is mainly
(i. e.
the
r,
wave is incident
it
is assumed
origin
of the subdish.
fields
scattered
of*the
point
so that-an
from the
approximately
This
on the paraboloid.
is given by
I
(0,0
where E.,
region,
electric-field
is located
assumption,
dish is placed
the radiation
can be neglected).
currents
To evaluate
we are primarily
a small cone of angles about the
contained within
determined by the reflected
that
The Main Reflector
radiation
in the fields
interested
optical
rrom
Patterms
E0 are the spherical
il
-Lkr
er
fields
components of the subreflector
as computed.
Referring
it
to Fig. (S. la)
which 0 is measured is
is
inclined
seen that
the plar
axis
zsj,
from
at an angle
ePýa-
to the axis
r,
The equation
of the paraboloid.
0, ý co-ordinate'system
of the paraboloid
is then given by
2f
+ cose cose
pp
sinO
sinO
goc;
in the
- 134 -
between the points
The relationship
and their
is
onto the xp. y plane
pi-ojection
on the reflector
[Cose
01
r
+
Cos
cosý
sinO
=.
sinO
XP
p
p
sinO siný
=r
yp
For an incident
field
.
in the form (5.13),
laws at the paraboloid
reflection
in the projected
surface
of the
an application
gives
the field
distribution
aperture
J(bl
XA
2f
- a' cosý) Et
at cosý) E
cl sinO E
cl sino El)
5*14
where
1+ cosO cosO
p
bl
=
cl
= coso + cose
p0-
sinO sinO
The expression
subdish
in(5.14)indicates
have undergone further
from the off-set
cross-polarized
in the contour
polarization
plots
for
of Figs. (S. a)
of *d equal to 4.9*(i.
illumination
angle
is evident
polarization
distribtuion
on reflection
of the
of the feedhorn
and (5.9).
e. 0P=
by the
radiated
transformation
the two orientations
to the value
It
the fields
The aperture
main reflector.
fields
that
These plots
55.3")
is shown
correspond
and aperture
(20m) of 54".
from this
of the incident
that
fields
the main reflector
has rotated
in such a manner that
the
the aperture
135-
CROSS POLAR
CONTOURS
APE..RTURE
MAIN
FEED
IN
THE
HORZ. POLARIZED
xp
//
CL
Pi
Is
e
6
*FI
5-9
5 dB
a =-42b=-45
47.5
-cz
50
d=
52-5
e=
-136-
CROSS POLAP
CONTOURS IN THE MAIN
APERTURE: FEED VERT. POLARIZED
xp
Yp
Fl G 5-9
az: --47.5 dB
b=- 50
c= -52.5
d:: - 55
e-- - 57.5
- 137 -
fields
have very small
energy is
distributed
expected,
the
symmetrically
The radiation
points
lying
at small
diffraction
In the far-field
fields
ff
E (e 10 )x =iN
raaXR
E, (O, O) eik
sinoa
cosý
pap
(shown in Fig.
Numerical calculations
dimensions, as provided,
(5.1b)).
at points
have been carried
1381
minor dimension
of the apeture
=
105X
.0
of the subreflector,
angle of the main reflector,
out for reflector
ýd=4.90
0p=
fields
fields.
f- = 127A
=
the upper
where the subdish
are as follows:
of the apeture
point
over the actual
In principle,
major dimension
off-set
a)
has been written
to be performed
are some 20 dB below the maximum of the radiated
angle
siný
of the field
(5.15)
to evaluate
is truncated
in the 0 integration
off-set
+y
(5.15)
the integrations
in a form which permits
length,
of the aerial.
dO dý
The computer program developed
focal
by
are
)
(R,
denote the spherical
0
ýa
co-ordinates
where
at
as measured from the centre of the aperture.
limit
of the aperture.
A
r2 sine
shape of the aperture
in the anti-
which is valid'for
from the axis
the radiated
region,
as
can now be obtained
theory
distance
angular
contained
of the two halves
from the reflector
of the scalar
an application
about the plane of the off-set,
near the centres
patterns
The cross-polarized
component.
of the energy being
portion
mjor
localized
phase lobes
cross-polarized
55.30'.
whose
- 138 -
diffraction
The predicted
components in the elevation
imposed on the provided
The results
of the principally
patterns
and the azimuth
results
experimental
planes
are shown super(5.10a)
in Figs.
to the case where the feedhorn
correspond
polarized
and (5.10b).
is polarized
in
the y-direction.
is made between the predicted
In rigs - (S. 11a) & (s. 111) comparison
The diagrams
feed polarization.
in the azimuth
plane
respectively.
of the feedhorn
are
45
It
the
of partially
-52 dB
figures
vector
The improvement
restoring
the two polarizations
dB
-49
plane:
electric
in the elevation
for
xedirection
- plane:
the radiation
position
taper
in
that
of up to 7 dB can be achieved
the principal
of symmetry.
at 45" to the elevation
-46 dB
can be seen from these
polarization
that
0
radiation
-42 dB
plane:
1
reed polarized
azimuth
the two cases of
to the cross-polar
in these two planes
454' - plane:
(2)
for
in the y-direction
reed polarized
azimuth
correspond
inclined
and a plane
The peak levels
(1)
fields
of the cross-polarized
patterns
radiation
by rotating
than in the azimuth.
the overall
by the fact
the main reflector
in cross-
the feedhorn
of the horn is polarized
can be explained
illuminating
plane
an improvement
so
in the plane
that
in this
has a sharper
This has the effect
symmetry of the aerial.
-139IG 5-1Oci
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PATTERN
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rn r, ut c-cl
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F.
C.,
r
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cl
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(by ElLiotis)
0
I'
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4
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3
-.
-2
cl
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2
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7IJ
x
LLJ
P-1
0
0
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0
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0
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Lf)
0
Z
0
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4ý
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cr
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0)
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0)
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- 143 -
5.7
Discussion
is seen from these results
It
maintains
aerial
involved
in the geometry,
a high level
of polarization
reflectors
off-set
can be attributed
performance
the two reflect.
subreflectors
the plane
in one sense;
upon reflection
is rotated
depolarizes
successive
the incident
and
main reflector,
the plane of polarization
In other
fields
words,
so that
the main
in such a manner that
produced by the subreflector
cross-polarization
is rotated
fields
produced by the subdish
to that
from
at the off-set
of the incident
curvature,
the
reflection
of the depolarization
is restored.
polarization
depolarization
by considering
from the parabolic
in a sense opposite
the original
after
of polarization
has opposite
because the surface
Such a favourable
purity.
to the opposing
vector
As a result
Drs.
from the
the radiation
elements.
of the polarization
orientations
from the two
scattering
of mechanism can be explained
The principle
reflector
directly
of the reflecting
properties
despite
that
is essentially
the
cancelled
out.
The degree of correction
clearly
and thus the amount of cancellation,
depend upon the curvature
feed system illuminates,
of the
is on the off-set
that
the feed system about the paraboloid
to select
those portions
paraboloid
axis
of the reflector
angle
(varying
*d
will
surface
which the
d*
By rotating
it
is thus possible
at which maximum cancellation
occurso
Numerical
computations
of the peak cross-polar
Results
illustrates
of these
were carried
radiation
computations
how the peak level
out to determine
on the off-set
are shown graphically
varies
as the axis
the dependence
angle of the subdish.
in Fig.
(5.13),
of the hyperboloid
which
is
-144--
0ý
m
0Lf)
(f)
0
sub reftector
'V- po
dominant
ci
I
67
0125
OF
THE
%'ArliVi'lON
5.12
IIJG
F
PEAK LEVEL
WITH
CROSS POLAR
THE
SU13RIEFLECTOR OFF-SET
C
UcP-'cs!;
-j
ion
Of
sub. rcflcCtor
Mil-: Aý, URED
I
clxis.
tj
degrees
["ORESIGHT
JITTER
ANGLE
-Td, deg
-144a-
(>Eb-lZrz t-Er-«r 0R
0
o L-oioA,
4- ý-x
/.§--ý-
4Ls
FIG513 MODIFIED ARRANGEMENT
- 145 -
depressed
gradually
for
the parameter
from the axis
reached when *d=4.9*
The validity
-
of the above conclusions
by the E12iotts
the axis
about the theoretical
optim=
at the successive
shift
the baresight
in this
5.8
error
is less
experimentally
through
and measuring
value,
diagram in Fig.
is indeed minimized
the error
position
the
of their
results
(5.14).
It
when *d=4.9",
is seen
and that
1
than yC)6-oth of the beam-width.
Conclusions
It
has been shown in this
of the off-set
fed parabolic
been obtained
for
a method described
can be varied
off-set
surface.
In doubly
been demonstrated
depolarization
polarization
that
at this
how the parameters
to illuminate
furface
is essentially
produced by the feed system itself.
cancellation
antenna,
cannot be achieved
in this
and
of the geometry
aerial,
use is
geometries
a single
asymmetrical
it
has
produced as a result
of
antenna
the cross-polarization
have
from the aerial.
because in these
off-set
by
results
such as the open Cassegrain
properti4s
feed radiation
Numerical
geometry of the Elliotts
aerialss
performance
can be improved significantly
the best performance
made of near ideal
reflecting
the cross-polar
feed systems.
which illustrated
poor polarization
exhibit
reflectors
the particular
to achieve
Conventional
chapter that
Cassegrain
the use of asymmetrical
total
position
of the subreflector
Cl)The
positions.
inset
shown
as
an
are
measurements
that
in this
radiation
was evaluated
company by depressing
a range of angles
boresight
The peak cross-polar
is
condition
of -49 dB.
at a level
occurs
the balanced
of the geometry,
values
is seen that,
It
of the paraboloid.
configuration,
cancelled
Note,
out by the crosshowever,
geometry since
that
the axis
of
- 146 -
the feedhom
equivalent
is effectively
surface
tilted
This suggests
of the aerial.
may be achieved
in the antenna cross-polarization
in
such a manner that
system.
axis
of the main reflector.
Fig.
(5.16).
It
is seen that
at an angle to the axis
the axis
of the
further
that
improvement
the feed
by rearranging
of the horn remains parallel
An arrangement
of this
such an arrangement
kind
is
also allows
to the
depicted
in
the
geometry to be more compact.
REFERENCES
1)
Fraham,
Aerials"
Future.
R.,
: "The Polarization
I. E. E. Conference
pp. 134-136.
Characteristics
Publication
No. 105.
of
off
set
Cassegrain
Radar-Pres6nt
and