OFF SET AND SYHMETRICAL REFLECTOR ANTENNAS H
Polarization and Pattern Effects
A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in
Engineering
by
San Fernando Valley State College
March, 1972
I would like to express my deep gratitude to my
advisor , Prof . E. s. Gillespie , for his invaluable help
and encouragement throughout the course of my graduate
studies at SFVSC . I am also grateful to Dr . Geoffrey
Hyde for several helpful discus sions and for his gentle
but persistent encouragement to complete this work and
to Mrs . Barbara Scheele who typed the manuscript .
iii
1.0
2.0
2.1 Dsplaced Feeds and Arrays of
Page No .
2.3 Surface De finitions for Parabolic
2.4
2.3.2 Spherical Ref lectors 35
2.3.3 Discus sion . . . • • . . . . . . . . 37
Feed P atterns . . • . . . . • • • • . . . • . . . . 39
Waveguide
Generalization to Arbitrary
iv
3.0
Page
vided in the Program
FLECrrORS . • • • . • • • • • • . . . . . • • . . • • • • . • . . • . • 6 5
Feed e e "' e e "' .., e e "' "' "' • • • • • • • • • • • • • • •
Circular Waveguide Feeds . • . . . . . .
65
69
74
76
79
Scanning . . . . . . . . . . . . . . . . . . . . . . . . 9 6
v
No .
4.3 A Comparison Between the Offset
Sphere and the Offset Parabola . . 118
CONCLUSIONS
REFERENCES
121
123
126
PROGRAM • • • • • • • • • • • • • • • • • • • 1 30
vi
3 Coordinate Defintions for Offset Reflectors . . . . . . . . . . . . . . . . .. . . . . . . . 1 2
4 Coordinates for Displaced Feed • • . . 27
5 "Side View" of Offset Parabolic Reflector . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 "Side View" of a Spherical Reflector 36
7 Huygen's Source . . • . . • • . . • • . . . . . . . • 41
8 Rectangular Waveguide Feed . . . . . . . •
4 3
10 Currents for Parabola with Electric Dipole Feed • . • . . . . . . . . • • • • • • . • . . . . 6 8
11 l/J max
vs. f/D for Symmetrical Re- flectors . • . • . . • . . . . . . . . . . . . . . • . . 71
12 CrossPolarized Lobes for a Symmetrical Parabola with Electric Dipole Feed 73
13 Computed Cross Polarization vs. f/D for Parabola with Electric Dipole Feed. D/A = 15 . . . . • • . • . . . . • . • . . . • 75
1 4 Patterns for Parabola with D/A = 1 5 TE Mode Rectangular Waveguide Feed 77
1 0 .
15 Current Distribution in the = 4 5 ° Plane for Symmetrical Parabola with TE Mode Feed · · · · · · · · · · · · · · · · · · · 80
1 1 16 Radius of TE 1 1 Mode Feed Required to
Provide -10 dB Edge Current for Symmetrical Parabola · · · · · · · · · · · · · • 8 2
vii
Figure No . T itle Page No .
17 Peak Leve l of Cros s-Polarized Current, Ky ', for TE 1 1 Mode Circular waveguide Feed . . . . . . . . . . . 84
1 8
1 9
Pat.terns for Parabola with TE Mode Feed. D/A = 15 · · · · · · · · · · · · ! 2 . . . .
Maximum Leve l of Cros s Polarized Lobe vs. f/D for Symmetrical Re­ f lectors with TE Mode Feed • • . . . • 1 1 Cross-Polarized Parabola C urrents vs . for Dual Mode C i rcular Waveguide Fee d . ( a/A = 0.8) . . • . . • • . . . . . • • . .
Cross-Polarization Current Distribu-. tion for a Parabola fed with a Dua l Mode C ircular Waveguide when PM is · Adjusted to Minimize Ky at Var1ous Values of (=M) . a/A = 0.8 . • . . .
Cross-Polarization Current for Parab­ ola with a Dual Mode C i rcular Wave­ guide vs . PTM Evaluated at the Angle where Ky has its Peak for PTM=O . • .
Cross-Polarization Lobes for Sym­ metri cal Parabola with Dual Mode C i rcular Waveguide Feed . . . • . • . . . . .
2 4 Maximum C ross-Polarization Radiation from a Symmetrical Parabola with D ual Mode C ircular Waveguide Feed vs .
85
86
90
p TM • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 9 5
2 5 PTM whi ch Minimizes the Maximum Cros s-Polarization Current (ky) for a Dual Mode C i rcular waveguide Feed 97
2 6 Ef fect o f Beam Scanning for Symmetri- cal Parabola . . . . . . . . . . . . . . . . . . . . . . 9 8
27 Polarization Purity for Parabola with Scanned Beam . • . . . . . . . . . . . . . . • . . . . . l·oo
2 8 Re flector C urrents for Offset Reflec­ tors with "Vertically " Polarized Feed Antenna · · · · · · · · · · · · · · ! " " ! ".!! • .· · · · · 1 0 3
viii
Page No .
Parabola . . . . . . . . . . . . . . . . . . . . . . . . .. . 10 7
30 H-Plane Patterns for Of fset Parab­ ola with Generali zed Huygen ' s Source Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3 1 Offset Parabola with Beam Scanned 8° in the H-P lane . . . . . . . . . • . . . • . . . 113
3 2 Comparison Between Computed and Measured E-P lane Patterns for Offset Spherical Re flector . . . . . . . . 114
3 3 Comparison Between Computed and Measured "H-Plane" P atterns for Offset Spherical Re flector . • • . . . . . 116
34 Effect of Focusing for Spherical Re flectors . . . . . . . . . . . . . . . . . . . . . . . . 117
3 5 Comparison Between O f fset Sphere and Offset Parabola with Scanned Beam • • . . . • • • . . . . . • . • • • • • • • • • • • • • . . 120
ix
ABSTRACT
by
January, 1 9 7 2
An analysis o f polari zation and pattern effects i s
presented for parabolic and spherical reflectors which
are either symmetrical ly or offset fed by "point source "
antennas s uch as electric and magnetic dipoles, rectangu-
lar waveguides, and dual mode circular waveguides . A
computer program is presented which yields the vec tor
far field patterns of a number of feed and reflector
combinations based on the numerical integration of
induced surface currents . For symme trical re flectors
it is shown that "Huygen ' s source" feeds can be. used
to e liminate cross-polari zed ref lector radiation and
that the small circular waveguide feed i s almos t ideal
from this viewpoint . It is also shown that the intro-
duction of control led amounts of TM mode into a cir- 1 1
cular waveguide feed results in cross polari zation re-
duction for a wide range of symmetrical reflector
parameters . The s canning and polari zation properties
of offset spheres and parabolas are compared .
X
This the s i s is concerned with polarization and
pattern ef fects in of fset fed and symmetrical antennas .
In particular , an analys is is preented for parabolic
:and spherical re flectors which are either symmetrically
f or offset fed by "point source" antennas s uch as e lectric
and magnetic dipoles , rectangular waveguides , and dual
•mode circular waveguides . A computer program is pre­
s ented whi ch yie lds the vector far field patterns of a :number of feed and re flector combinations based on the
numerical integration of induced surface currents .
There is a current need for detai led pattern
;and polarization data for symmetri cal and offs et fed
re flector antennas . For example , an application which
has recently become very important re lates to the rela­
tive ly new technique of " frequency reuse" in communications·
i s ate l lite systems . This concept takes advantage of the
fact that the communications capacity of a geostationary
s ate l lite can be s ignificantly increased when it uses a
number of antenna beams which occupy the s ame frequency
band. The increase in capacity is directly proportional
to the number of beams transmitting independent informa- . tion at the s ame carrier frequencies. In order for this
·concept to be practical , the beams must be sufficiently
isolated from each other to insure adequate s ignalto-
1
1antennas of the communications system .
For two beams , originating from a satellite
• and occupying a common spatial region, such isolation can
only come about from their polari zation orthogonality .
' For beams which i l luminate adjacent earth regions , the
. isolation can res ult from polari zation orthogonality
and pattern enve lope shape . 1 Typical hal f-power beam­
widths for thi s appl ication are in the range of one to
four degrees and a typi cal goal for beam isolation is
30 dB. Furthermore , s ince the angle s ubtended by the
earth as seen from synchronous altitude is approximate ly
1 7 ° , the " fi e ld of view " of a multiple beam satellite
antenna may be as wide as +8 . 5 degrees .
Ref lector antennas seem to offer the most
practical means for achieving a modest number of narrow
beams . Ref le ctors have been the object of intensive
investigation for many years due to thei r ability to
form narrow beams whi le maintaining physical implicity .
However , their polari zation properties have been studied
in deta i l only for a number of special cases . Further­
more , the s canning p roperties of front fed re flectors
have general ly been s tudied experimentally and on the
bas is of s calar pattern analys i s of the symmetrical
parabola . The effects of beam s canning on the antenna
polari zation have not been examined in detai l .
2
paraboloids have been s tudied by several invstigators.
Condon/ showed th at , for a symmetrical paraboloid excited
.by a dipole , there are four symmetrically disposed cross-
pol ari zation lobes at 4 5 ° to the principal planes of the
antenna . These lobes are generally referred to as the
:"Condon lobes . " Jones3. conc luded tha t i f a symmetrical · ref lector is fed with a combination of electric and mag-
netic dipoles such that the ratio of electric to magnetic
fie lds is equal to the free space characteristic im­
pedance , the cros s -polari zed radiation in the far field
wi l l vanish. This type of feed is re ferred to as a
" Huygen • s source . " Koffman4 showed that cross polari za­
tion in a reflector whose surface i s generated by a conic
section and whi ch is excited by electric and magnetic
dipoles can be made to vanish if the ratio of e lectric
to magnetic field intens ities is equal to the eccentricity
of the surface . Kerdemelidis 5 formulated and evaluated
the pattern integrals for a symmetrical paraboloid
excited by electric and magnetic dipoles and he examined
the cross -polari zed radiation both near and far from
the antenna axis as a function of variations in the
foca l length and diameter of the ref lector . Minnett and
Thomas6 and Rumsey7 examined the characteris tics of an
ide a l feed which would have no cross polari zation and
showed that the necessary requirement of a circularly
3
symmetric feed pattern can be satisfied by a hybrid mode
'antenna as exempli fied by the corrugated horn .
The effects of s canning the beam of reflector
antennas have been s tudied with re ference to pattern
amplitude properties . Lo 8 computed the beam deviation
i factor of a paraboloid with a feed displaced from the
focal point . Kelleher and Colernan9 experimentally
. s tudied the degradation in gain and sidelobe levels for
parabolic and spherical reflectors . Sandler1 0 and Ruze 1 1
. used analytical methods to de termine the e ffects of feed
displacement in paraboloidal antennas . Ruze presented
a series of graphs which depict the degradation in gain
and s ide lobe levels with s can as a function of the ratio
of focal length to diameter .
The pattern quality of paraboloidal antennas
degrades due to aberration e f fe cts as the beam is
s canned away from the re flector axi s by displacing the
feed . This limitation on the field of view has motivated
the use of spherical reflectors for certain wide angle
applications . The appeal of this c lass of re flectors
s terns from the fact that , whi le a spherical reflector
has an inherently greater amount of spherical aberration
compared to a parabola , this aberration is not dependent
on s can angle as long as the feed is cons trained to lie
on a focal surface which is concentric with the reflector
surface as i llus trated in Figure 1 . For a sphere of
4
5
radius R , the focal surface is a spherical cap whose
:radius is approximately R/2 . Since the spherical re-
i f lector has no preferred axis , it " looks " the s ame to
a l l feeds s ituated on the focal sphere and the beam
direction coincides with an axis passing from the center
.of the sphere through the feed.
The properties of the symmetrical spherical
; reflector have been extensive ly s tudied and much of the
, early theory comes from their application to optical sys­
, terns . Thei r suitabi lity as microwave scanning antennas
·has been recognized for some time . 1 2 Li l 3 determined
some of the limitations caused by aperture phase error
! on point fed spheres and he determined the optimum
focal parameters based on the allowable phase error .
Several inves tigators have designed feeds which can cor­
rect for the inherent phase error in a spherical re­
flector . 1 4, 1 5 Further ins ight into the properties of
spherical reflectors has been gained by an examination
of their focal region fields . Hyde 1 6 used the method
of s tationary phase to calculate these fields and
Ricardi 1 7 used spherical wave expans ions to synthes ize
an optimum feed base d on matching the fields at the
surface of the reflector .
spherical or parabolic , are subject to the degrading
e ffects of aperture blockage caused by the feed and
6
serious degradation in s idelobe levels particularly when
several feeds are used to obtain multiple beams from a
single antenna . These limitations motivate the use of
offset fed ref le ctors for certain applications . In this
case the feed ( s ) are removed from the re fle ctor aperture
and they i lluminate it in an unsymmetrical way . Offset
fed re flectors have been used for many years , particu­
larly in radar applications . 1 8 Pagones 1 9 cons idered an
of fset parabola excited by the clas s of feeds having a
pattern proportional to cosn , where i s the polar
angle measured from the feed axis . He determined the
gain factor as a function of the feed parameter and re­
flector parameters . Bem2 0 calculated the focal region
fie lds of an offset paraboloid and he showed that the
e lectric field dis tribution in the focal region is
nearly the s ame as for a symme trical re flector with an
equiva lent focal length .
The beam s canning properties of an offset
parabola are sti l l limited by the e f fects of aberration
caused by feed displacement . A particularly interesting
geometry which may have application to wide angle
sys tems is the offset fed spherical reflector2 1 illus­
trated schematically in Figure 2. As in the case of
the front fed sphere , the feeds are located on the
focal surface but they are removed from . the reflecting.
7
I \----- R .
8
:of aberration compared to the front fed case but again ,
• this aberration is independent of scan angle . This
arrangement should be useful with point feeds if the
ratio of focal length to diameter (f/D ) is relatively
high . For small f/D the possibility exists for design-
ing a phase correcting feed . 2 2 At any rate , the offset
sphere is a relative ly new and unexplored antenna geom-
e try which could be use ful in certain wide angle applica-
tions .
An important charac teris tic of offset re flectors
which has not been investigated in detai l is that of
cross polarization. The unsymmetrical nature of the il-
lumination for these antennas causes their coss-polarized
radiation to behave differently than that of symmetrical
(front fed ) reflectors. Because of the importance of
offset ref lectors in e liminating aperture b lockage effects ,
it is essential tha t their polarization proper ties also
be characterized for various commonly used feed antennas .
The primary objective of this thesis , then , is
to formulate the pattern expressions for offset fed
reflectors and to present a computer program with which
their far field properties may be conveniently deter-
mined as a function of the type of feed and the appro-
priate antenna parameters. In Section 2 of this thesis ,
the far field integrals are formulated and this forms
9
the basis for a computer program which is included
in an appendix . I n Section 3 the polari zation properties
'of symmetrical re flectors are examined as a function of
·feed type and reflector parameters . I t is shown for
example that a sma l l circular waveguide feed has nearly
i de a l polari zation characteristics for parabolas with
a low f/D. I t is also shown that a dual mode circular
waveguide feed can be used to provide nearly idal
polari zation for a range of va lues of f/D . In Section
4, s everal examples of offset ref lector patterns are
presented in order to i llus trate some of their polari za
tion and beam scanning properties .
10
. 2 . 0 FORMULA'fiON OF THE PATTERN INTEGRALS
The geometry and coordinate sys tem definitions ' · to be used for offset reflectors is il lus trated in Figure
3 . The primed coordinates are those of the feed and
; reflector ( i . e . , the source coordinates ) . The unprimed
coordinates are those of the far field pattern .
By a direc t integration of the vector Helm- . holtz equations, Silver2 3 (fo llowing Stratton and Chu24) ' 'has shown that the far fields of a current distribution
on a surface of infinite conductivity and containing
: only e lectric currents are
; whe re
-jkrr -+ e [K r s
-+ -+ EP, Hp are the far fie lds at P -+ r = r r is the vector from the origin to P
( r is a unit vector )
p is ·the radius vector from the origin to an
e lement of surface current
K i s the vector current dis tribution on the
surface S
( l )
( 2 )
of fr $pace (Z0)
Figure 3 . Coordinate Definitions for Offset Ref lectors
X
z
determining the current dis tribution K and then inte-
grating over the re flector surface . The exact deter­
.mination of K involves the solution of an integral
equation . 2 5 In the event a computer solution is being
s ought , the integral equation method is prohibitive for
:all but the smalles t s urfaces because of i ts large
's torage and time requirements .
Fortunately , i t is not always necessary to
: resort to an integral equation solution . A commonly i
• used method for approximating the reflector ' s current
. dis tribution is to ass ume that the currents are those
which would be generated by an incident field that is
reflected optically . That is , K is calculated accord-
ing to
K = 2n x H. 1
-+ where n is the uni t normal to the surface and H. is ' 1
the incident magnetic field .
Equation ( 3 ) , along with the assumption that
the currents are zero in the shadow region of the re-
( 3 )
flector cons ti tues the physi cal optics approximation for
computing the current distribution . It i s known to be
a good approximation when the reflector ' s radius of
13
curvature is large . FUrthermore , i f the re flector is in
:the far field of the feed (which is presumed to be
ithe primary source for the re flector currents ) then the : :incident magnetic field can be expressed in terms of
:the incident electric f ie ld as
p X E.
A (p X E.)
The term [l(- (K.;);J in Equation {1) is jus t
the component of K perpendicular to the uni t vector r
; in the observation direction . In terms of the polar A
unit vectors e, at the far field point P,
where K8 written
( 6 )
( 7 }
14
•where
s
solved in order to calculate the far fie lds .
It is assumed that the reflector surface can -+ A
be described by a radius vector p p (,r,, )n. Also , 0
the incident field E., produced by a feed antenna at . l
the origin of the coordinate systems in Figure 3 , can
be expressed in terms of its polar coordinates and ; as
E. = :Ef (, o -jkp e l p
= [E(,F,) + E(,ui] e-:-jkp p
( 8 )
( 9 )
The express ions for feeds displaced from the origin and
for arrays of identical feeds wi ll be described later .
The feed pattern is most natural ly represented
as a function of and whi ch are defined relative to
the primed coordinate system in Figure 3 . However , the
definition of the reflector s urface wi ll usual ly be most
conveniently defined in terms of the unprimed axes and
the observation coordinates e and are also defined
re lative to the unprimed axes . Therefore , it is neces-
sary to re late the two coordinate systems in order to
15
carry out pattern calculations. This re lationship is
eas i ly de termined by us ing s tandard coordinate trans -
; forrnations . 26 For the coordinate representation of
F igure 3 , i t can be observed that the z • axis is s imply
, rotated about the y-axis by an amount w0 ( the offset
'angle of the reflector) relative to the -z axis . The
coordinate trans formation relating these rotated sys-
terns is
( 1 0 )
· Also , s ince the transpose o f the trans formation matrix is
i tse l f ,
x ' cosw 0 sinw0 X 0 y ' = 0 - 1 0 y ( 1 1)
Z I s inw0 0 -cosw0 z
Expansi on of Equation ( 10 ) and use of the trans formations
from spherical to rectangular coordinates y ields ex-
plicit relationships between the polar coordinates of
the two systems:
X = sinO COS$ = cos x' + s in• z ' 0 0
= cos sin cos + sin cos ( 1 2a ) , 0 0
y - sine sin¢ = -y' = -sin• sin
z = coso = sin x' - cos$ · 0 0
= sin sin$ cos - cos$ cos 0 0 . .
The evaluation of the pattern integral re-
; quires the explicit form of the phase factor p·r . A
unit radius vector p can be expressed in either of the
two polar coordinate sys tems , where
p = sin cos x' + s in$ sin y' + cos$ z '
p = sine cos¢ x + s ine sin¢ y + coso z
Now upon substi tution from Equation ( 1 2 ) into Equation
( 13b) the radius vector can be expressed in the un-
. primed system as a function of the primed coordinate
variables :
p = ( cos$0 sirt$ cos + sin0 cos ) x - s in$ sins y
( 1 2b) :
The phase factor p·r is then given by
r·r = p· (sine cos x + sine sin y + coso z)
where p is given in Equation (14 ) . The result is
P . r = [ costjJ0 sintjJ cos + sintJJ0 sin] sine cos
- sintjJ sin sine sin
Several other quantities must be estab lished
· in order to complete the formulation of the pattern
expression . A useful formula for the quantity n dS can
. be obtained from d i fferential geometry . Given a surface i : de fined by the curvi linear coordinates u and u , a
l 2 . + + vector r = r (u , u ) can be be des cribed as
1 2
+ + + r = ar
au 1
du 1
+ dS = _a _r_ x au
1
du 1
du 2
By the definition of the cross product , the vector dS + is normal to the surface r {u , u ) . The unit inward
1 2
( 1 6 )
X -- au au
1 ar + ar X --I au au
1 2 , I n terms of the coordinates in Figure 3, the expression l
for n dS is
·where ( as derived in Appendix A ) ,
+
Expansion of Equation ( 19) with the use of Equations
(20) and (21) yields
( 21 )
d S = [-p 2 sintjJ p + p sintjJ : + p : ] d tJi d c;, { 2 2 )
+ Now, in order that K dS be evaluated, it is noted that
{ 23 )
where, with Equations ( 9 ) and (22 ) , it is found that
<·Ef> dS = pE sintJ; !e. + p E !e. 1j; ()tj; ar;
;and
1j;
where
n x { p x E f ) ds = p [ F P + F tJ; tJ; + F I; ] d tJ; d , 1; { 2 4 )
( 2 5 )
( 27)
the form
i = 1 Jrnajma; F
-jkp _e ___ ejkpp·r d tJ; d I; p {28 ) ljlmin E;min
For numerical computations , it is necessary
to express the vector F in rectangular coordinates since
20
'the spherical unit vectors vary over the region of inte- -+ gration . Furthermore , F should be expressed in terms of
\ the unprimed rectangular coordinates s ince this is the
system in which the far fields are to be evaluated . The
.sequence of trans formations from the primed spherical
! coordinates of F to its unprimed rectangular coordinates
• is
·where ,
s int/J sin · costjl sin cos
and , the trans formation relating x ' ,y ' , z ' to x , y , z
coordinates is given by Equation ( 1 0 ) . The overall
expression to be evaluated is then
F = [F P
F tJi cost; F <] [s inO cotJi cost;, -sl.n [ cosoo
• 0 s intjl ·. - . . - - 0.
sino l -oso
( 3 1 )
2 1
The details of the above expans ion are carried out in
Appendix B. The result is
= F ( sintjJ COSF, COSI/1 + COStjJ SintjJ. ) p 0 . 0
- Fl; sinl; COStjJ0
- FF; sinE. s ini/10
( 3 3 )
( 3 4 )
The vector F is a function of only the s ource
coordinates and is therefore independent of the observa-
tion coordinates , e and . This means that , in respect
to programming considerations , it is necessary to compute
F only once and to s tore it as an array . Since a typi cal
integration grid may contain several thousand elements
( i . e . , units of dS ) , the impact of thi s approach on
computer time is quite s igni ficant . 25 Of course , the
exponential term in the integral mus t be evaluated for
every value of the obs ervation coordinates , e and .
,_. -- -- --
22
!complex . This situation can aris e , for example , when the i
feed pattern is complex due to a dif ference ih the I ; ! locations of the feed ' s principal p lane phase centers .
The vector F can be conveniently and concise ly
expres sed in terms of the components of the feed radiation
:pattern and the surface parameters . Let
A A (tjJ , tjJ , F,) sinl)J + COSt/! = = COS[, COSl)J0 0 B = B (tjJo , ljJ , E;,} = COSljJ cost;. COStjJ0 - sinw
c ::: C (ljJO,tjJ,t;,) = sinl)J COS(. sinw - COSljJ 0 D ::: D(ljJO,tjJ,t;,) = COSljJ COS(. sint/!0 + sinl)J
sintjJ0 sintjJ 0
COSljJ 0 COStjJ0
expression of source coordina tes in terms of the un­
iprimed (8,$) coordinate system ,
A = sin e cosq,
Upon substitution of Equations ( 2 5 ) - ( 27 ) and
( 3 5) - ( 38) into Equations ( 3 2) - ( 3 4 ) the rectangular
. I
I
F = y
Fx = E sin (A + p B )
+ Ec (A - p sin sin . cos ) ( 4 3 ) s a . o
E sin sin ( sin + p cos )
- E sin ( s in + p cos) ( 4 4 )
ap F z = E sin (C + p D)
+ E (C - p sin . sin sin:tl.J0 ) ( 45 )
Although Equations ( 4 3 ) - ( 4 5 ) represent the
!final form to be used for the computation of the vector F, I
·it is interesting to note that they can be expressed in
!a particularly compact way by noting that
()A sin1jJ sinr: coswo ( 4 6 ) = -at: land
ac sinljJ sinE, s inljJ0 ( 4 7) a[ = -
Then , Equations ( 4 3 ) - ( 4 5 ) can be written
a a = El/J sin (p A) + E af(p A ) ( 4 8 )
2 4
s inE.: a s in41 ) s inlji () . s inE;) ( 49 ) F = - E sinljJ t;j;" ( p - Ei; (p y tjJ
s inljJ d C ) a C ) ( 50) F z = EtjJ (p + Ei; (p
The phase factor p •r given by Equation ( 1 5)
!Can be expressed concisely by us ing the definitions of ' ;A and C :
p • r = A s ine cos - sinljJ 13int; s ine s in + c cos e (51)
The preceding formulas are subject to the
:restriction that the s urface be in the far field of
:the feed s ince they do not inc lude the poss ibility of
ia radial component of feed field . The near field case
is not inc luded here s ince most reflector configurations
:of practical interest s atis fy the far field condition .
iHowever , i f required , the near field case may easi ly
be incorporated by allowing a radial component of H. l.
in Equation ( 3 ) . Then , a complete ly arbitrary feed
i field may be generated by using , for example , a spherical
wave expans ion as described by Ludwig . 27
2 . 1 D ISPLACED FEEDS AND ARRAYS OF FEEDS
The pattern integral as given in Equation ( 2 8 )
implies a s ingle feed antenna whi ch i s located at the
!Origin of the co.orc:l:i.I'la_t syste:rn . It: i.s Ci simple matter
2 5
-.· --
to generali ze the pattern formulas to include the e ffects
of feed displacement and arrays of identical feeds .
Cons ider a feed displaced from the origin
:as shown in Figure 4 . The radius vector to the phase -+
:center of the feed i s given by E:. The vector from the
(feed to an e lement of sur face area dS is p', where
-+1 -)>- -+ p = p - E:
-+ The vector p' makes new angles and with respect
ito the axi s of the displaced feed so that the incident
e lectric field at dS due to thi s feed is
E. I = 1 'kl-+ -+1 (E 1 . + E 1 1)e- J p-E:
( 52 )
is small compared to the distance between the feed and
dS . Then , i f lti!IP"I << 1 , it may be assumed that
1 and r,' :::::: t;. Furthermore , i f I£" I is smal l , the
" path los s " given by the denominator of Equation ( 5 3 ) is
nearly the same as for the case of a feed at the origin .
Therefore , the path los s term in Equation ( 5 3 ) may be
written
10f course , the phase term represented by the xponential
in Equation ( 5 3 ) mus t be maintained intact since even
ismall feed di splacements can cause s igni ficant phase
·changes at the s ur face of the reflector . '
If a number of identical feeds are arrayed in 1the focal region of the re flector , it is merely neces­
isary to s uperimpose thei r contributions at the surface .
Then , i f M feeds are arrayed with equal ampli tude of
excitation at each feed , and if the above approxima tions
are as sumed to hold for each feed , the total incident·
,field at the reflector may be written
M .2: (El/1 E. = l/1 + E ]_
m=l
f,
" F, )
M
-jk lp-e: I e m
•where t i s the radi us vector to the mth feed and , m
IP-t I m ( p -e: ) 2 + ( p· - e: ) 2 + ( p - e: ) 2 x mx ' y my 1
· z . mz 1
Equation ( 5 5 ) is simply the incident field
:due to a s ingle feed multiplied by the array factor of
M feeds . When Equation ( 5 ) is used to represet the
incident fie ld at the reflector , it can be seen that
in the pattern integral of Equation ( 28 ) , the p term
in the numerator cancel s that in the denominator , the
( 5 5 )
(56 )
28
-r J'kp · 'vector F is le ft unchanged , and the e- term is re-
placed by the array factor term .
2 .2 INTEGRATION TECHNIQUE
For pattern computations , the array factor
term could be included either in the expres s ions for ;-+ :F or i t could be incorporated into the exponential A A
1phase factor (ejkpp·r ) of Equation (28 ) . The manner
1in which the array factor i s treated depends on the
integration method to be used .
For example , in one possible technique for
ithe numerical evaluation of the pattern integral the
'entire integrand is expressed in terms of its ral and
imaginary parts and Simpson's rule i s applied to each .
iTes ts of a numerical integration program using Simpson's
;rule have shown that to achieve errors at least 4 0 dB l !be low the pattern maximum , the integration grid mus t
,be subdivided into areas (dS ) of a t mos t 0 . 0 4 square
;wavelengths . This fact may readi ly be appreciated when
;it is recalled that the phase factor of the integral
is proportional to the reflector s i ze and it causes
the real and imaginary parts of the integrand to behave
as rapidly oscillating functions as the far field
observation angles are moved away from the reflector
'axis . This rapid variation may be further complicated
,by the inclus ion of the feed array factor. At any rate ,
2 9
the use of Simpson ' s rule demands that the integrand
be " sampled" at enough points to insure that these
rapid oscillations are taken into accoun t . The impact
:of this method on the computer time necessary to eval-
:uate a moderately large reflector is qui te s ignificant .
An alternative technique whi ch appears to
offer a cons iderable s avings of computer time compared
:to Simpson ' s rule has been deve loped by Ludwig. 2 8 The
pattern integral to be eva luated can be written in the
form
-+ :where F has a lready been defined and y includes the
iphase factor and·the array factor . Ludwig notes that ,
:for a fixed observation coordinate (8,}, the individual -)>. terms F and y are each we ll behaved over an increment of
surface area , i\Smn' whose area is on the order of one
:square wavelength . He then integrates the contribution
:over i\S analytically yielding the incremental contri­mn bution i\I to the field integral . The total integral mn
. I ( for any of i ts three vector components ) is then ob-
tained by summing the contributions from each incre-
mental surface area . The explicit formula for i\I . mn is given in Reference 28.
Ludwig c laims that for an incremental s urface
30
area approximate ly 2/3 of a square wavelength in area ,
the pattern errors are more than 4 0 dB below the pattern
'maxima . The reported reduction in computer time for
'this technique as compared to Simpson ' s rule is a factor
:of 11 to 19 .
vantages over more conventiona l quadrature methods , Lud-
wig's integration technique is used herein. Again , since
the basic idea behind this approach is to separate
the oscillatory terms of integrand , it is desirable
to incorporate the array factor from Equation ( 5 5 ) into
'the phase factor term of the integrand .
Let
where m=l
Q =·.Ja? + b2
r; = tan-1 (b/a )
m
m
Then, the final form for the .intgra l to be
evaluated is
( 5 7 )
( 5 8 )
I i /max/max
Q -+ ej [kp (p•r ) +sJd iJ; d (6 2) = F E.. A !Jimin ':min
The computer program to be presented herein
1uses a s ubroutine (FINT ) whic::h was developed by Ludwig .
,A lis ting of this subroutine is included with the program
:and a description of F INT may be found in Reference 25 .
2 . 3 SURFACE DEFINIT IONS FOR PARABOLIC AND SPHERICAL REFLECTORS
The pattern integral requires explici t de fini-
tions for the ref lector's radius vector p(iJ;,) and its
derivatives a p/aw and a p /oE.. in order that the reflector
!surface be completely specified. In this s ection , thes e
quantities wil l be de fined for the specific cases of
:parabolic and s pherical reflectors . Whi le the computer
program included herein incorporates only these two re-
•fleeter types , the inclus ion of others is straight-
forward .
It wi ll be seen that the expressions for the
reflector surface can be concisely defined in terms of
the quantities A, B , C , and D which are given in Equa-
'tions ( 3 5) - ( 42) .
Consider the " side view" of a s urface as
3 2
:shown in Figure 5 . The vertex of the reflector is
:located at z = - f . I f the s urface is tha t of a parab­
:oloid of revolution about the z-axis with vertex at i
: z = - f , the polar equation of the surface in terms of the
lunprimed (B,) coordinates is
2f p = 1 - cos e ( 63)
(Note that in this case e is not a far field coordinate ) .
jBut , from Equations ( 3 7) and ( 4 1 ) ,
cose = C = s in cosc s in0 - cos cos 0 (6 4)
'SO that ,
manner:
2 f D ( 66) = (1 - c) 2
3 3
Note: D should not be confused with Equation ( 3 8 ) .
Figure 5. " Side View" of Offset Parabolic Reflector
34
z
2f sin0 s in sin
(1 - c>2
Spherical Ref lectors
(6 7)
Cons ider the " s ide view" of a spherical s urface
,as shown in Figure 6 . The radius of the sphere is R and
i ts center is at z = (R - f ) . The equation of the sur-
f ace in the unprimed rectangular coordinates is
(68 )
z = p case (71)
:rf Equation (68 ) is divided through by R and transformed
to polar coordinates using Equations (69) - (71) the
polar expres sion for p/R is
(72)
:where
, __ . .-
Now , in order that Equation (72} be expressed
in terms of tfJ and i t i s noted (from Equation (41)}
that
Then ,
p /R = .V 1-m2 + m 2 C 2 + m C (75)
It can be eas i ly be shown that
R ac[ m2 C
:and
ar = R IT m + ..J l-m2 +m2
2.3.3 D i s cuss ion
The preceding equations are sufficient to
complete ly describe the reflector s urface in terms of
(77)
the rflector parameters and the feed polar angles . For
syrnrnetrical.ref lector s it is merely necessary to set
tfJ = 0. In general, the size of the reflector is deter­a
mined by the parameters 1JJ , f (and R in the case of the ' 0
37
: s pherical re flector ) , and the range of and over
,which it is desired to integrate . The term ( 1/A ) pre­
'ceding the pattern integral in Equation ( 6 2 ) can be in­
cluded with f and R so that it is possible to specify
i these parameters directly in terms of wave length . A
comparison of the equations defining the parabolic and
spherical surfaces shows that the quanti ty 2f for the
: parabola is analagous to the spherical re flector radius
R . For a spherical ref lector the parameter f/R deter­
mines the " focal length " for a feed located at the origin .
' For a parabola with a feed at the origin , the equiva lent
" f/R" is equal to 1/2 .
For an offset-fed parabola i t can be shown2 9
', that the intersection of a cone of cons tant with the
. parabolic surface describes an e l l ipse which , when pro­
j ected onto the " aperture plane " ( i . e . , the x-y plane )
de fines a circ le .
The evaluation of the pattern integra l requires
that the partial deriva tives for the ref leCting surface
be continuous . This condition is clear ly not satis fied
at the edges of the ref lector . S ilver 1 8 introduced a
l ine charge at the e dge to insure that boundary condi­
tions are satis f ied the re . However , as Ludwig2 5 points
out , Sancer 3 0 has shown that the pattern integral
intrins ically contains the effect of thi s line charge .
A conceptual way of circumventing the discontinui ty of
3 8
: the derivatives at the edge of the reflector is to as sume
that the surface is continuous but the incident fields
vanish beyond the integration limits .
i 2 . 4 FEED PATTERNS
I t now remains to define the pattern express ions
for the specific feed antennas of interes t . The computer
program incorporates analytical express ions for four
commonly encountered types of feeds : a generali zed
· Huygen ' s source , a rectangular waveguide with TE -mode 1 0
aperture fields , a dual mode circular waveguide , and a
: cosnljJ pattern having the polari zation of a dipole . Of
: course , the program i tself i s not limited to these feeds
and the inclus ion of other analytical feed expressions
or even measured feed data is s traightforward .
The speci fic feed expressions wi ll be pre-
sented first for the case of linearly polari zed feeds
whose aperture polari zation is along the x ' -axis . Later
the expressions wi ll generali zed to account for arbitrary
feed polar i zation . In all cases , the patterns given below
are for a constant dis tance from the antenna ( i . e . , the ' k e-J P IP term wi ll be s uppressed ) .
2 . 4 . 1 Generali zed Huygen ' s Source
For a short electric dipole , polari zed along
the x ' -axis as shown in Figure 7 , the pattern is
39
1
: where P is a cons tant representing the ampli tude of 1
: exci tation . I
A short magnetic dipo le , s ituated orthogona l
: to the e lectric dipole may be cons idered to be a current
: carrying loop in the x ' - z ' plane as shown in Figure 7 .
' r ts pattern is given by
= P ( cos - cos sin i > ej a 2
(79)
of excitation of the loop and the exponentia l term
A generalized Huygen ' s source may be defined
by s uperimposing the patterns of the dipole and the loop .
Then , the polar components of the pattern are
= cos (P cos + P ej a ) { 80 ) 1 2
E = E e + E m
= - sin (P + P cos ej a ) ( 8 1 ) 1 2
40
i
The three parame ters P , P and u are sufficient 1 ;;
to complete ly speci fy the pattern of this antenna . When
P = P and a = 0 the antenna is known as a Huygen ' s 1 2 ·s ource . In this case , the antenna has a " plane wave "
: po lar i zation and , i f used as a feed for a symmetrical
parabola , wi ll produce zero cross -polari zed currents .
. 2 . 4 . 2 TE -Mode Rectangular Waveguide
For a rectangular waveguide in the x ' -y ' plane
and wi th principal polari zation along the x ' -axis
as shown in F igure 8 , Si lver 1 8 gives the radiated field
. components as
where
E ,,, = z _TI __ cos t; 1 + _l.Q_ cost)! V ( !)! , t; ) . a 2b ( B o/ o 2 A 2 k
E l;
u2 - ( TI /2 ) 2 v
1r a s inlJJ s in E; u = A
1Tb s inlJJ cos t; v = A
( 8 2 )
( 8 3 )
( 8 4 )
{ 8 5 )
! and
b is the E-plane of the guide .
8 is the propagation cons tant for the 1 0
TE mode . 1 0
The above adaptation of Si lver' s equations
( 87)
: assumes that the re flection coe fficient at the waveguide
! aperture may be ignored . This is a reasonable approxi- '
mation for moderate ly large waveguide (e . g . , a/>. > 1 }
and , i f the more exact expres sion is required , the
reflection coefficient may eas i ly be incorporated into
• Equations ( 8 2 } and ( 83 } .
The TE waveguide requires only the parameters 1 0 : a/>- and b/>- in order to complete ly speci fy its pattern .
2. 4 . 3 Dual Mode Circular Waveguide
The patterns for circular waveguides having
TE and TM modes are presented in Si lver . 1 8 For mn mn the particular case of m = 1 and n = 1 , the patterns
for a circular waveguide whose aperture is in the
x' -y' plane and whose principal polari zation component
for TE and TM modes is along the x' -axis can be 1 1 1 1 wri tten as follows :
4 4
For the TE mode , 1 1
ka/" [l + e J J 1 (u ) . E = Qe k COSt/! J 1 (Kea ) cos [, !}Je u
E E, e = - Qe
For the TM mode 1 1
ka Km Bm J E '''m = Q 2 . - + cos 1jJ '' m s J.n!}J k
E = 0 r,m
1
s inE,
. where ,
for the TE and TM modes respectively
a is the waveguide radius
u = ka s in!}J
K = 1. 8 4 1/a ( root of J 1 I (K a ) ) e e
K - 3. 8 3 2/a (root of J 1 (Kma ) ) m Wjl = kZ 0
W £ = k/Z 0
J l {K a ) = J l ' ( 3. 8 3 2 ) = - 0. 4 0 2 76 m J l (K a ) = J l { l. 8 4 1 ) = 0. 5 8 1 9 e
The propagation cons tants for the TE and TM modes are . b f3 d f3 • 1 gJ.ven y e an · m respectJ.ve y :
4 5
( 8 8 )
( 8 9 )
( 9 0 )
- K 2 e
- K 2 m
English 3 1 has de termined the coe fficient Q e
: and Q on the basis that the sum of the powers in the m
TE and TM modes b e normali zed to uni ty ; i . e . , 1 1 l 1
where PTE and PTM are the fractional powers in each of
. the two modes . The coe f f icients are then ,
1r l3 w [K 2 a 2 - 1] J 2 ( K a ) e e 1 e
1r t3 w e K 2 a 2 J 2 (K a ) m m o m
The pattern equations can be s impli fied by
, . incorporating all appropriate cons tants . The final
· result for the polar components of the pattern is
4 6
( 9 1 )
( 9 2 )
( 9 3 )
( 9 4 )
( 9 5 )
e = Qe
1 + 1 - ( 1 . 8 4 1/ka )2 cos ka s in•
where
and
J (ka s in. ) / (ka s in•J . s1nt; (ka sin./1 . 8 4 1 } 2
( 1-PTM )
Q = . 0 6 4 6 e I v 1- ( 1 . 8 4 1/k a ) 2
Q . = 14 . 1 9 t.! p TM .
m · f .Vl- ( 3 . 8 3 2/ka } 2
The overa l l pattern i s , o f course ,
cos r: ( 9 6 )
( 9 7 )
( 9 8 )
( 9 9 )
( 10 0 )
( 10 1 )
where the exponen tia l term e j a h as been inc luded to al low
4 7
for the pos s ibility o f a phase difference between the
TE and TM modes with the TE mode as phse re f- 1 1 1 1 1 1
erence .
of the pattern for the dual mode circular waveguide are :
a/A , PTM ' ,
and a .
A somewhat fi ctitious but often used antenna
pattern is one which has a circularly symmetric pattern
1 about the polar axis and it has the polari zation of an
/ electric d ipo le . The polar components for this pattern , ( . ! having its principal polari zati on along the z ' -axis and
i ts beam maximum on the z ' -axis is
Elji =
determine the , pattern i s the exponent n .
( 10 2 }
( 10 3 )
2 . 4 . 5 Generali z ation to Arbi trary Feed Polari zation
The feed patterns which h ave been presented
all imply an aperture field whose principal polar i za tion
4 8
i s a long the x ' - axis . I t is a s imple matter to generali ze
the feed expres s ions to include an arbi trary orientation
of the aperture field in the x ' -y ' plane or even to
a llow for e l liptical and c ircu lar feed polari zation .
I t i s interes ting to note that all of the
feed patterns g iven here can be expressed as separab le
functions of and . That i s , for an x ' -po lari zed feed
each of the patterns has the form
where
{ 10 4 )
( 10 5 )
(10 6)
I and the s ubscript x refers to the aperture p lane polar-
i zation .
Suppos e this s ame feed is polari zed along the
i y ' - axi s . Then i t i s neces sary to shi f t only the
variab le by /2 . That i s ,
COS/; + s in
s in!; + -COS/;
E , = f ( ) s in ljly 1
E I = f ( ljJ ) cos t; F, y 2 .
Now , cons idering that the feed may be dua l
( 1 0 7 )
( 10 8 )
polari zed , let Ax' be the ampli tude o f exci tation of the
x ' -polari zed aper ture , A , be the amplitude of the y ' ­y
. polari zed aperture and a be the relative phase between
; the x ' and y ' exci tations . Then ,
= A (E
The tota l field i s then
Elji =
A I E t;
Upon s ubstitution o f the f unctional re lation-
ships for the component fie lds given by Equations ( 10 5 ) -
( 1 0 8 ) the total field may be expres sed as
50
f { t/J ) [A 1 sins / X
+ Ay' s ins
There fore , the us e of proper
ej a ] ( 1 1 3 )
e j cv. J ( 1 1 4 )
va lues o f A , , A , X y and a the principal polari zation of the feed may be spec­
; i fied . Several examples are lis ted be low :
, A I A y'
a Polari zation X
• o 1 0 L inear , y ' -polari zed
; 1 1 0 Linear , 4 5 °
! 1 1 - 9 0 ° Left hand circular i
1 1 +9 0 ° Right hand circular
Equations ( 1 1 3 ) and ( 1 1 4 ) can of course , be
eas i ly normal i zed on the basis of any suitab le criterion
s uch as unity tota l power or unit amplitude a t the
· peak of the feed • s beam .
The above equations cannot be applied exactly
to the TE rectangular waveguide unless it i s square 1 0 . ( i ! e . , unless a/A = b/A for thi s antenna ) . The genera l-
i zation to dual polari z a tion for thi s antenna is a
s traightforward matter and the computer .program incor-
porates the dual polari zation feature .
I t is interes ting to note that the expres s ions
5 1
for the patterns o f the genera l i zed Huygen ' s source
have the s ame general form as those of the TE mode 1 0
rectangular waveguide , i . e . , for both antennas ,
E a cos ( l + x cos )
E
( 11 5 )
( 1 1 6 )
I n the case o f the generali zed Huygen ' s source ,
x is the ratio of e lectric to magnetic dipole ampl i tudes
and in the case of the rectangular waveguide , x is the
norma l i zed propaga tion cons tant 8 /k . 1 0
For a waveg ui de whose dime ns ions are sma l l ,
8 /k i s appreci ab ly less than uni ty . The feed ' s l 0
polari z ation then res emb les that o f a magne ti c dipole .
S i nce i t i s known tha t the condition x = 1 results in
zero cros s -polari zati on in a s ymmetrical paraboloid ,
thi s explains the fact that a sma l l rectangu lar wave-
guide does not produce low cross polari zation in a
paraboloi d . Thi s fact was observed experimenta l ly by
E .M T . Jones . 3 Kinber 3 2 attempted to explain this
phenomenon by obs erving tha t the wave i n a rectangular
waveguide is rea l ly two p l ane waves whose wavefronts
a·re not paral l e l .
An explanation i n terms . of the ana logy with
the genera li zed Huygcn ' s s ource was given by Kerdeme lidis 5
5 2
. I
. I
. I
and a lso implied by Ko ffman . 4 I n fac t Kerdeme lidis
takes the viewpoint that the c lass ical Huygen ' s source
(x = 1 ) is j us t a special case of the horn exci tation
problem . This can b e appreciated when i t is recalled
that the term x = f3 /k for the waveguide approaches 1 0
' unity only as the waveguide aperture s i ze becomes much
greater than the wavel ength . There fore , a large wave­
: guide may be expected to behave l ike a Huygen ' s source
in terms of its po lari ztion properties .
Ko ffman4 notes that when us ing e lectric and
: magnetic d ipole feeds the conditions for zero re f lector
; cros s -polari zation depend on the type of re f lector and
i he s hows that for symme trical re f lectors whose surfaces
: are generated by conic sections , the optimum ratio of
. electric to magnetic dipole intensi ties i s equal to
, the eccentricity of the s urface . Whi le he considers
. spherical ref le cting s ur face s , his results are not
· appli cab le to the spherical re f lectors des cribed herein .
The reason is that Kof fman assumes that the sphere
is fed from its center whe reas the feeds for mos t
• practical spherical re flectors are located in the vi-
· cinity of thei r paraxia l focus whi ch is at one-half
the radius of the sphere .
2 . 5 PATTERN QUANTITIES OBTAINED· FROM T HE COMPUTER PROGRAM
A l l nput quantities needed to des cribe a
5 3
) varie ty of feed and re flector configurations have now r i been determined and the general express ions for the
pattern integral have been defined explicitly in terms
i of thes e quantities . :
A computer program whi ch incorporates a l l
i o f the features des cribed above is presented in Appendix
• c . That apendix also contains a brief description of
the input quanti ties requi red by the program . I t is
the purpose of this section to des cribe the pertinent
output quantities and features of the secondary pattern
which are obtained f rom the program .
The primary result of the pattern integra tion ! l i s the comp lex vec tor i ( Equation ( 6 2 ) ) from which all
i l?a ttern and polari zation quantities may be obtained .
The observation coordinate sys tem i s redrawn
in Figure 9 . For a fixed obs ervation dis tance r , the
· comp lex spherical components o f the far field radiation
patterp are obtained as
( I x + I y + I z) · 8 X y Z
A ( I x + I y + I z ) · X y Z
( 1 1 7 )
( 1 1 8 )
( 1 1 9 )
and
o = cos o cos ¢ x + cos O s in¢ y - sinO z
= - s in x + cos y
There fore ,
E 0 = I cos o cos cp + I cos o s i n (jJ - I s ino X y Z
E = - I s in + I cos 'I' X y
( 1 2 0 )
( 1 2 1 )
( 1 2 2 )
( 1 2 3 )
2 . 5 . 1 Principal and C ros s -Polari zed Field Components
The conventional spherical components , E 8 and
E
, are not particularly convenient for visuali z ing
the polar i zati on character i s tics of the s econdary pattern
except along the principal axes of the observa tion
coordinate sys tem . For example , the far fie ld pattern of
a ref lector having on ly x-directed currents i l l gene rally
have both e and <P components ; the identi fication of
" pr incipal polari zation " and " cros s -polari zation" i s
not convenient i n terms of thi s coordinate system .
One way to circumvent thi s diffi culty i s s imply
to use the far f ie ld rectangular components to identi fy
principal and cros s -polari zed radiation . For example ,
i f the ref lector currents are directed a long the x-axis ,
Ex may be cal led the principal polar i zation and Ey i s
5 6
i the cross -polari zation of the secondary pattern .
A maj or dis advantage as s ociated with the use
o f rec tangular components is that their direct measure­
ment i s not convenient with conventiona l pattern measure­
ment equipment s ince experimental patte rn data usual ly
cons i s ts of the field components as they appear on the
s urface of a sphere s urround ing the antenna .
A set of s pherical coordinate s representing
a natural sys tem for the i dentifi cation of the antenna ' s
polari zation components i s also shown in F igure 9 .
Thes e spherical coordinates are designated with a
" T i lde" (""" ) . For a linear ly polari zed antenna whose
principal aperture polarization is in the x direction ,
Ee represents the principal po lari zation and E repre­
sents cross-polari zati on in the far field . For a
l i nearly po lari zed antenna as described , the dis tinction
between the two sys tems i s readi ly seen : The polar
axi s of the conventional { 6 , $ ) sys tem is in the direc ­
tion -of the antenna ' s main beam ; the polar axis of the
T i lde sys tem is in the direction of the principa l
aperture polari z ation { I t should be noted that an
identica l set of relations holds for the feed coordinate
system ) .
The relationships between the conventiona l
and T i lde sy stems may easi ly b e determined by noting
that
57
· Then
y = s ine s intj> = x = s ine cos
z = cos e = y = s ine s in
s i ne
= v 1 - s in2 e s i n2
- "' s in e cos cp =
v1 - s in 28 s in 21
- cos e =
...., cos e = sine cos tj>
. - s 1n cp = cos e
5 8
: the two coordinate systems can be obtained by using the !
: above relationships along with the trans formations
relating spherica l to rectangular coordinates ( c f . Equation
(A- 3 ) , Appendix A ) • The results are :
' and i
'\) 1 - sin 2 e cos 2 <1>
= sin { 1 3 2 ) vl - sin 2 e cos 2 <1>
cose - = sJ.n ( 1 3 3 )
v1 - sin 28 sin2
- = cos ( 1 3 4 )
5 9
Therefore , given the e and $ components of a
given field quanti ty , e . g . , E6 and E$ , the Ti lde com­
ponents can be found with the aid of Equation ( 1 30 ) as ,
E = [ -R S]
E -S -R E . $
Also , i f Ee and E are given , the conventional com­
ponents are ,
The re lations between coordinate angles are
eas i ly obtained from Equations ( 1 27) and ( 1 28) . For
( 1 3 5 )
( 1 36 )
I I
example , division of Equation ( 1 28b ) by Equation ( 1 28a ) · I
I I I
yie lds :
,..., _ - 1 (...} 1 - s in2 e cos 2 p) e - tan -"--r. -=-_.;.;.;--:--__;,--'­ Sl.n 6 COS$
I 1 The remaining coordinate re lations are obtained in a '
simil ar manner e
( 1 37 )
i I t should be noted that either the conventional I coordinates or the Ti lde coordinates are suitab le for
l .... ..... ... ..... · ···-··· --······ · · - ············ · ·- ··-·····--·· · · · ··--···- ···-·· ···--- ... - . .. -
I
I
I J
6 0
de fining the left and righ t hand circular polari zation
components of the secondary pattern . I n fact , any two
components spati a l ly orthogonal to each other and also
orthogona l to the d i rection of propagation of the wave
are suitab le for th is purpose .
The formulas for ob taining circular polar­
i zation data are we l l known . 3 3 They wi l l be l i s ted here
for convenience i n terms of the complex E8 and E com­
ponents although , as s tated , the formulas remain un­
changed for any other spati a l ly orthogonal waves . A
wave E may be expressed as
.t; =
where the o , cp components have been de fined and where
UR and UL are unit vectors for right and left hand
circularly polari zed waves respective ly . Righ t and
left hand re fer to c lockwise and counterc lockwise
rotation respectively for a receding wave . Now , UR
and UL are comp lex uni t vectors and are given by
G - j UR
61
Then , the right and left ci rcular components o f the wave
: are
12 ( 1 4 1 )
Ee - j Ep ' . EL =
for example , right hand ci rcular , the left hand com-
ponent represents cros s -polari z ation and vice vers a .
The ratio of left to righ t hand components for s uch an
e l l iptically polari zed wave i s
A ls o ,
; The axial ratio is defined in terms of the circular
components as
(1 4 3 }
( 1 4 4 }
( 1 4 5 )
. and the angle which the maj or axi s o f the polari zation
6 2
The explicit formulas for r and 1 in te rms
of the linear components E 8 and E <P
are ob tained as fol-
lows : Let the time phase di f ference between E 8 and E <P
be given by 8 . Then
Sub s ti tution o f these these terms into
Equation ( 1 4 3 )
. 8 - j E
( 1 4 7 )
( 1 4 8 )
( 1 4 9 )
( 1 5 0 )
S ub s ti tution o f Equation ( 1 4 9 ) into Equation
( 1 4 5 ) yie lds the axi a l ratio of the wave .
6 3
2 . 5 . 3 Pattern Information Provided in the Program Output
The computer program in Appendix C resul ts i n
a print-out o f the fol lowing in formation for a given
pa ttern cut :
I E I ' I E I ' I E I X y Z
In the following sec tions the appl i cation
of the program , to some speci f ic examp les of o f fset -fed
and symmetrical re flectors will be di scus sed
6 4
The pa ttern equations wh ich have been derived
for o f fset re f lector patterns can be directly applied
to the special case of symme trical re f lectors by mere ly
a l lowing the of fset angle , , to be zero . 0
This section cons iders the re lationships among
the cros s polari zation in the secondary pattern , the
re f lector parameters and the of feed antenna which
is used . I n th is regard i t wi l l be poss ib le to gain
s ubs tantial qua litative ins ight into cros s -polari zation
e f fects by f irst cons idering the nature of the currents
generated on the re f lector surface by particular types
of feed antennas . The compu ter program wh ich has been
deve loped herein is then uti li zed to provide quantitative
data concerning re f lector po lari zation and a lso to con-
f i rm the trends indicated by the cons ideration of re-
f leeter c urrents .
The fol lowing discus s ion is confined to the
case of parabo loidal re flectors although as wil l be
shown , the polar i zation quali ties of the symmetrical
spherical re flector are quali ta tive ly s imilar to thos e
o f the parabo la .
3 . 1 PARABOLA CURRENT S
The currents which are induced on the s urface of
a re f lector by a f eed antenna can be determined from
6 5
r I
I Equa ti on ( 5 ) • When th i s equation is app lied to the
s peci fic case of a synune tri ca l parabo loid , the rec tangular
c omponents o f th e c urrents are found to be
K I K = cos 3 t [E cos t,; - E s in s] X X f.,; ( 1 5 1 )
K l = - K = cos 3 t [E s i n t + E
cos ] y y 2 (1 5 2 )
K I . = -K = - cos 2 t s in 1 z z 2 2 E t}J ( 1 5 3 )
where the 11 pa th l o s s 11 term , 1/p , h as b een included above
' k and the p a th l eng th phase term ( e - J P ) has been s uppre s s ed . :
F urthermore , the above equations have been divi ded through
by the cons tant term 2/Z • 0
For a feed antenna h aving i ts principal aper-
ture polari zation a long the x ' -axis , Kx ' ( =Kx ) repres ents
the prin c ipal pol ari z ation c urrent and Ky ' ( =-Ky ) repre­
s ents the c ros s -polari z a ti on current . The " idea l " l inear
po l ari z ation occurs where K I = o . The longi tudinal y
compone nt o f current given by K 1 ( =K ) does not contr i -z z
b u te s ub s tanti a l ly to the far f i e l d near the an tenna
axi s and wi l l therefore be ignored for the remainder o f
thi s d i s c us s i on
The c ros s -polar i zed radiation in the s econdary
patte rn wi l l depend upon the amp l i tude of Ky 1 and i ts
d i s t ribution over the s ur face o f the re f l ec tor . A
6 6
gene r a l feature o f the c ro s s -po lari zed currents in a
symme trical parab o l a i s the i r quadrant symmetry . Th at
i s , I K ,j i s zero a l ong the principa l p l ane s , t: = 0 and y
t: = 9 0 ° , and i t maximi zes a l ong the di agonal p l anes ,
t: = + 4 5 ° . Th i s s i tuation i s i l lus trated in F i gure l O a
: whi ch shows for e xamp le the general feature s o f the cur-
rent induced by an e l e c tric dipole fee d . The quadrant
symme try of the cros s -po l ari zed current di s tribution is
shown in Figure l O b .
I n genera l , c ros s -polari zed re f l e c tor c urrents
res u l t f rom the c urvature of the s u r face and from c ros s -
polar i zed radi ation , i f any , from the feed . I n th e
c a s e o f an e le c tr i c dipol e , the cros s - po lari zed cur-
rents are due enti re ly to the c urvature of the re f le c tor
s ince the e le c tric dipole feed h as only an E component
of radia te d fie ld ( c f . Sec tion 2 . 5 . 1 ) .
A magne tic dipole feed w i th dipole moment
a long the y ' -axis ( i . e . , principal po l ari z ation a long
' the x ' - a x i s ) wou ld produce a cros s -po lari zed current
d i s tribution having the s ame quadrant s ymme try as
shown in F igure lOb but w i th the " arrows " pointing in
the oppo s i te directions . As wi l l be shown , a comb ina-
t i on o f e le c tr i c and magne t i c dipoles caus e s K ' to y
vani sh e verywhere on the re f le c tor s ur face .
V I
f
y '
a . Re f le c tor C urrents for E lectric Dipole Feed
x '
D
b . Array Mode l for C ros s - P o l ari zed D i s tribution
F i gure 1 0 . C urrents for P arab o l a wi th E lectric D i po le Feed <'
6 8
3 . 2 GENERALI Z ED HUYGEN 1 S SOURCE FEED
Let the feed antenna be a " genera lized "
Huyqen ' s s o urce with polar field components given by
Equations ( 8 0 ) and ( 8 1 ) with It = 0 . When these ex­
press ions are s ub s ti tuted into Equations ( 1 5 1 ) and ( 1 5 2 )
the exp licit form for the reflector currents i s as
follows :
cos l)J + P 2
2 cos l)J ) s in 2 J
( 1 5 4 )
K I = cos 3 t/1 s i n r; cos r; [ (P l
COS t/1 + p ) (P + p COS tp )] y 2 .. J /
1 (P p ) s i n 2 'v 'P s i n 2 .; = if cos 2 1 2
where P and P are the exci tation amp l i t udes o f the 1 2
e l e c tric and magne tic dipoles respective ly .
An examination of Equation ( 15 5 ) shows that
( 1 5 5 )
K 1 i s zero at = 0 and t,; = 9 0 as expe c ted . Further­Y more , i f P = P then K 1 = 0 . When P + P , the
1 2 y 1 2 cros s -polari zed currents are maximum in the planes
S ince the wors t case cro s s -po lari zed currents
occur in the planes t,; = 4 5 ° , it is ins tructive to
cons ider thei r variation in thes e planes as a function
of l)J . For example , for an e le ctric dipole feed (P = 0 ) 2 the currents at = 4 5 ° are
I ·. !
y 4 2
From Equation ( 1 5 7 ) i t can be observed that
K 1 i s zero at = 0 and increases with increas ing ­ y
( 1 5 6 )
( 1 5 7 )
Now , the ratio o f focal length to diame te r for a parab-
ola ( f/D ) is related to the maximum value of , , , ( · 1 • ) max de fined by the edge of the re flector . (Note : The
re fle c tor diameter , D , should not be confus ed with the
quanti ty D ( t/J , ) as de fined in Equation ( 3 8 ) ) . The f/D
of the reflector is inve rs e ly proportional to t/Jmax · The
dependence of w on the re f lector f/D is i l lus trated max for parabo l i c and spherical re flectors in Figure 1 1 . Thi s
relationship , a long with Equation ( 1 5 7 ) implies that
as the f/D of the reflector i s increased , t/J wi ll de-max crease . There fore , the cro s s -polari zed currents and
hence the s econdary cro s s polari z ation wi l l be invers e ly
proportiona l to f/D for a dipole feed .
The re l a ti onsh ip between f/D and ref lector
cros s po lari za ti on i s a we l l known res u l t and it provides
a use ful check on the accuracy of the computer program
deve loped herein . I n the computer program (Appendix C ) ,
the reflector diameter (D/A ) and the f/D are determined
by speci fying R/A ( = 2 f/ A ) and t/Jmax · For a symmetri cal
70
n
Parabola :
= tan- 1 [ . 8 f/D ] max 1 6 ( f/D )2 - 1
X = f/R 2 f/D
m = 1 - f/R
0 . 2 0 . 3 0 . 4 o . s 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0
f/D
F i gure 1 1 . vs . f/D for Symme trical Re flectors . max
7 1
re flec tor the patte rn is ob tained by integrating from
I}J = t/J • to I}J and 1; = 0 ° to 3 6 0 ° . For the case when m1n max there is no aper ture b lockage , ,, , = 0 . ''min
In order to provide a quanti tative re l ationship
between s econdary cross po lari zation and f/D , several
patte rns h ave been computed for a re flector wi th a
diameter of 1 5 wave length s which i s fed by an e lectric
dipole . Kerdeme l i di s 5 has shown that the firs t cros s ­
polari zed lobe has the large s t amp l i tude and the rela-
tive leve l of the c ros s-po lari zation l obes i s nearly
i ndependent of the reflector s i ze in wave lengths . How-
ever , the pos i tion of the lobes depends on the ref lector
diameter . Kerdeme l idis cons i ders a s impl i f ied mode l
o f the c ros s -polari zed current dis tribution cons i s ting
of four sma l l dipoles a t the corners o f the reflector
at t; = + 4 5 ° (Figure lOb ) . He then shows that based
on thi s mode l , the fir s t cros s-polari zation lobes occur
a t ¢ = 4 5 ° and a t a polar ang le given approximated by
. - 1 8 S l n ( A /D ) ( 15 8 )
Several patterns computed by the program of
Appendix C are shown in Figure 12 for the ¢ = 4 5 ° p lane .
I t is evident from the pa tterns tha t the cross -po l ari zed
radiation depi cted by I Er l decreas e s with increasing
f/D . This data , a long with other patterns whi ch have
7 2
.--I 0.. (1) :> ·r-i .
- 5 0
- 6 0
0 5
Er f/D
Er f/D
l !)
F igure 1 2 . C ro s s -Polari zed Lobes for a Symmetrical Parabola wi th E lectric D ipole Feed
7 3
been computed but not shown , has been s ummari zed to
dep i c t maximum I EI vs . f/D as shown i n Figure 1 3 . Also
shown i n Figure 13 are cros s -polari zation leve l s for
cJ> = 2 2 . 5 ° . These calculations were made in order to
compare the results w i th those of Kerdeme l idis . As can
be observed , the agreement is exce l lent and , s ince
the res ul ts of Kerdeme l idis were ob tai ned by a di fferent
technique , this is a good check on the accuracy of the
patterns computed here .
3 . 3 TE MODE RECTANGULAR WAVEGUIDE FEED l 0
I t was pointed out in Section 2 . 4 . 5 that the
pattern expre s s ions for a TE mode re c tangu lar waveguide l 0
feed are ana logous to those o f the genera l i zed Huygen ' s
s ource in regard to reflector cros s po lari z ation . This
can be observed if the po lar components of this antenna ,
given by Equations ( 8 2 ) and ( 8 3 ) are s ubsti tuted into
Equation ( 1 52 ) . The res u l t for the c ros s -po lari z ation
current K ' i s y
( 15 9 )
The cros s - polari z ation c urren t wi l l vartish for B /k = 1 . l 0
This in turn occurs only as the " a " dimension o f the
waveguide becomes very large s i nce B /k = 11- ( A / 2 a ) 2 • 1 0
Therefore , a sma l l waveguide may be expec ted to produce
7 4
0
- 10
l ()­ 4--1 0
D/A = 1 5
A Computed at cp = 4 5 °
Computed at cp = 2 2 . 5 °
X C omputed by Kerdeme l i dis ( Re f . 5 ) at cp = 2 2 . 5 °
m - 5 0---------------------------- ::E: 0 . 3 0 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0
f/D
·
7 5
s ome c ross -polari zation . Since , for a given edge i l lumi-
nation (e . g . , Kx ' = - 1 0 dB rel ative to the i l l umination
at the ver tex of the reflector ) , the angle s ubtended
by the reflector is inversely proportional to the
waveguide s i ze , the rectangular waveguide feed is
expec ted to produce l ower c ros s polar i z a ti on as i ts
s i ze is increased and hence as the f/D of the re flector
i s increased .
An example o f the patterns for a parabola
with a diame ter of 15 wavelengths and two d i f ferent
va lues of f/D . is shown in Figure 1 4 . I n both cases
a square waveguide ( a/ A = b/A ) i s used and the s i ze of
the waveguide i s s uch tha t the re l ative edge i l l umina-
tion of the principal polari zation current is - 1 0 dB .
3 . 4 C IRCULAR WAVEGUIDE FEED S
I t i s commonly a s s umed that a way in whi ch to
reduce cross polari zation in symme trical re f.lectors
is s imply to increase f/D . Whi l e thi s i s indeed
true for the e lectric d ipole and rectangular waveguide
feeds , i t i s de f i n i te ly not true for a l l feeds . In
fac t , it wi l l be demons trated in thi s section that j us t
the oppos i te trend i s true for a TE mode c i rcular 1 1
waveguide feed . That i s , for th i s type of feed the
cros s -polar i zed rad i a ti on shows a decreas i ng trend
as the f/D of the reflector is dec reased .
7 ()
o - Degrees
1 5
Figure 1 4 . Patterns f o r Parabola wi th D/A = 1 5 TE Mode Rec tangular Waveguide Feed 1 0
7 7
I t. c an be shown 3 4 that the pattern o f an
antenna having phys ical c i rcular symmetry and radiating
only TE and TM modes can be wri tten in the form 1 , n 1 , n
E !jJ = f 1 ( !jJ ) cos
I n this c ase , the antenna • s aperture po lari z ation is
( 1 6 0 )
( 1 6 1 )
along the x • axis s o that f ( ) represents i ts E-plane 1
pattern and f ( ) represents i ts H-plane pattern . An 2
interes ting c onsequence o f the form of Equations ( 1 6 0 )
and ( 1 6 1 ) i s · that i t is only necess ary to measure the
E - and H-pl ane patterns in order to complete ly spec i fy
the pattern for all angles .
I f Equations ( 1 6 0 ) and ( 161 ) are incorporated
into the expre s s i ons for parabola current the results
are
K I y 2 1 cos 3 i s in 2 ;: [ f ( lJ! ) - f ( lJ! )l 2 . 1 2 -
The above result i s interes ting s ince i t expre s s e s
( 1 6 2 )
the cross -polarized currents directly as the dif ference
between the E - and H-plane patte rns . I f these principal
7 8
'
p lane patterns c an be made equal the re flector cro s s -
polari z a tion wi l l vani sh .
The princ ipal plane patterns for a s ing le
waveguide mode s uch as the TE mode are not qual 1 l
b ut , as wi l l be shown , they tend to approximate the
conditions for low cros s polari zation for sma l l wave-
guide radius . I n genera l , the E- and H- plane patterns
can be made equal in multi-mode s tructures as exemplified
: by the corrugated horn . 7 However , a good approximation
to thi s condi t ion can be obtained wi th only one or two
mode s .
3 . 4 . 1 TE Mode Feed 1 1
I n order to i l lus trate the b ehavior o f re-
f l e e ter cros s -polari zed currents as a func tion o f off-
axis angle from the feed and the s i ze o f the feed , the
pattern expres s ions for the TE 1 1 mode as given by
Equations ( 8 8 ) and ( 8 9 } have been incorporated into the
equations for re f lec tor currents . K 1 and K 1 have X y been computed i n the t;, = 4 5 ° plane as a func tion o f 1jJ
for various wavegui de radii , a/ A . The res u l ts are plotted
in F igure 1 5 . I t can be observed tha t for sma l l wave-
guide radius the cross -polari zed current , K ' , peaks at y re lative ly large value o f 1/J . As the radius is increased
the peak o f K ' moves to sma l ler val ues o f 1jJ and its . y peak ampl i :ude increas e s .
79
>:: - 1 0 4-l 0 3 2 E ;::1 f; .,..., - 2 0 -co ::8 0 +l (!) :> .,..., +> - 3 0 co r-l (!) p:; r:fl 't)
+> - 4 0 ,.::: (!) H H ;::1 u H 0 - 5 0 +> 0 (!) .-t 4-l (!) p:;
- 6 0
0 10
K , - P rincipa l Polar i zation X
1 . 6 1 . 0 0 . 8 0 . 6 0 . 4 0 . 3 -{- a/ >.
K , - C ros s Po lari zation y
2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
1jJ - Degrees
C urrent Dis tribution in the = 4 5 ° P lane for Symme trical Parabola wi th TE Mode Feed
1 1
8 0
Now , also shown in F i gure 1 5 are the principa l
polari zation currents , K ' . A re f lector f/D can be X estab l i shed by observing th e va l ue of 1jJ at which K ' X i s down from its maximum by a s peci fied amount , say - 1 0
dB . This val ue of 1/J i s then re lated to the f/D by the
curve of Figure 1 1 . There fore , the spec i fi cation o f
the edge i l lumination provides a re lationship between
the re flector f/D and the waveguide radius . Th i te-
lati onship is plotted in F igure 16 for the case of - 1 0
dB edge i l lumina tion . I t can be observed that for f/D >
0 . 5 the waveguide radius i s approximate ly equa l to . the
f/D for thi s edge i l lumination .
tfui le the currents shown in Figure 1 5 indi cate
genera l ly tha t K ' increases as a/"A is increased , an y
i nteres ti ng excep tion to thi s trend occurs for a/"A between
0 . 3 and 0 . 4 . At a/ A . 35 the cros s -polari zed current
shows a dramati c reduc tion in overa l l ampl i tude and
it litera l ly . vanishes at 6 5 ° . One viewpoint is
that a t this va lue of a/"A the cros s po l ari z ation in the
feed ' s field ne arly ma tches that wh i ch tends to be
genera ted by re f lector curvature thereby caus ing the
resul ting K ' to b e smal l . y
For a re flector wi th its f/D de fined as above
i t i s then pos s ib le to p lot the maximum val ue of cross -
polari zed current which falls within the re flector edge
as a function of the f/D . Th i s plot is i l lustrated in
8 1
Note : Edge I l lumination Re fers to Peak of Principal Polari z ation Current
0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 1 . 0
f/D
Fi gure 1 6 . Radius o f TE 1 1 Mode Feed Requi red to P rovide - 1 0 dB Edge C urrent for Symmetri cal P arabola
8 2
l•' i cJ ure 1 7 wh ich shows the r)ronounced d ir> in 1< ' at . . . y
a/;.. 0 . 3 5 . I t shou ld be pointed out tha t the edge
i l lumination is not ab so lute ly cons tant a t -10 dB
s ince , from Figure 1 6 , the lower limit of f/D for a
wavegui de radius of 0 . 3 i s approxima te ly equal to 0 . 4 2 .
There fore , be low f/D 0 . 5 the edge i l lumination is
s omewhat more . sharply tapered . However , the overa l l
trend i s clearly discernib le : The cros s po lari zed
currents derease with decreasing f/D - a trend
whi ch i s oppos i te to tha t for a dipole feed .
I t is to be expected that the s econdary
patterns would indicate the s ame general trend as for
the reflector c urrents . This trend has been veri fied by
computing secondary patterns for a reflector with D/.
held cons tant a t 1 5 wave lengths as shown in Figure 1 8 .
A s ummary of peak secondary cros s -polari zed radiation
a s a func t ion of a/ A (f/D ) i s presented in F igure 1 9 .
Th i s curve tracks very closely with that of F igure 1 7
the reby indicati ng the corre lation between re flector
currents and far- f i e ld po lari zation e f fects .
A s imi lar set of pattern computations were
made for a spherical re f lector with D/A = 1 5 and
f/R = 0 . 5 . The maximum c ros s -polari zed radiation leve ls
are also shown in F igure 19 and it can be observed
that they track very c lose ly with those of the parabola .
This i s con firmation of the fact that the parabo lic and
8 3
lj...j 0
- 4 0
0 . 3 · ·· 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0
a/A f/D
F igure 1 7 . Peak Leve l of C ross -Polari zed Current , K { for TE Mode C i rcular Waveguide Fed 1 1
8 4
p:j - 2 0 'U
Q) 'U :::J +J ·n r-l - 3 0 p, ;::; d. Q) !> ·n +J ttl
r-l - 4 0 Q) p:::
- 5 0
0 . 3
0 . 4
0 . 5
1 0
e - Degrees
1 5
E ......
8 5
lr;o rp = 4 5 °
4-l Parabol i c Re flector 0 8 - 1 0 ;::1 X S phe rical Re flector 8 ·.-I X cO ::8 0 +J - 2 0 QJ :> ·.-l +l cO r--1 )( QJ p:: t:Q - 3 0 '1:) l -e ILl
4-l 0
- 4 0
- 5 0
0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0
a/ "A f/D
F igure 1 9 . Maximum Leve l of C ros s Polari zed Lobe vs . f/D for Symme tri cal Reflectors with TE Mode Feed . l l
8 6
s phe rical re f l ec tors should be expected to d i s p lay s imi lar
polari zation properties . Th is is qui te reasonab le s ince
the reflector s ur faces are qua l i ta tive ly s imi lar and
approach e ach other very c losely for l arge f/D .
The quanti ty j Ey l , whi ch is the rectangular
component of the far- f i e ld cros s polar i z ation , i s a l s o
plotted i n Figure 1 9 f o r the parab o l i c re f l e c tor . This
quanti ty is very nearly equa l to ! E I near the beam
axis but departs from j E j as e i ther the o f f-axis ang le
is increased or the abso lute ampl i t ude o f the radiated
f i e ld approaches zero . This is j us t a consequence of
the coordina te re lations between spherical and rectangu-
lar components and it i ndi cate s tha t care mus t be ex-
e rc i s ed in de f ining wha t is meant by cro s s polari zation
for each app l i cation .
3 . 4 . 2 Dua l Mode C i rcular Waveguide Feed
The preceding results indicate that a sma l l
c i rcular waveguide feed rad i a ti ng the T E mode is nearly 1 1
i de a l f rom the viewpoint o f c ross pol ari z ation for re-
f le c tors with rel ative ly low f/D . Howeve r , the re are
s i tua tions whi ch require tha t a h i gh f/D reflector b e
used , a s for examp l e , in beam s c anning applications when
the s ide1obe degradation of a low f/D re f lector cannot
be tolerate d . I f low c ro s s polari zati on i s a requi re -
ment then o ther type s o f feeds mus t b e used . One
8 7
. I
pos s ib i l i ty i s the use of rec tang ular horns as des cribed .
However , one limi tation of these antennas is the fac t
that the i r E - and H-p lane beamwidths are unequal . Thi s
limits the ir us e fu lnes s as circularly polari zed antennas .
I n this event , the feed antenna should have a c i rcularly
symmetric pattern in addition to good polari zati on char-
acteris tics .
An antenna whi ch very nearly s atis fies these
requirements over moderate ( e . g . , 1 0 % ) b andwidths is
the dual mode c i rcular wavegui de feed , sometimes re-
ferred to as a Potter Horn . 3 5 In thi s antenna , a sma l l
amount of TM mode i s added to the dominant TE mode 1 1 1 1
to equa l i z e the E- and H-plane beamwidths . Thi s also
has the e f fect of reduci ng c ros s polari zation if the
antenna i s used as a feed for a symmetri c a l re f lector .
The patte rn equations for s uch a dua l mode
feed were deve loped in Sec tion 2 . 4 . 3 . Several examples
o f the inc lus i on o f TM mode w i l l be given here to 1 1
i l lus trate i ts e f fect on ref lector polari zation .
I n pra c tice TM mode may be generated i n a 1 1
c ircular waveguide by incorporating a s tep discontinui ty
in the diameter of the guide . A gradual change in the
guide diameter may a l s o be used to generate this mode .
I n order for the TM mode to be e f fective in y i e lding 1 1
the req ui s i te pattern and po lar i zation properties , i t
mus t be 11 i n phas e 11 w i th the T E mode at the waveguide 1 1
8 8
radi ating aperture . I f the TM mode i s generated I l
some dis tance back from the aperture , the di f ferent
phase ve locity between the modes l imits the
frequency band over wh i ch the proper phase re lationships
are obtained . I n s pi te of these limi tations the dual
mode radiator is very useful for many moderate bandwidth
app l i cations .
A ques tion o f particular s igni f i cance in re-
fleeter app l ications re lates to the proper amount of TM
mode nece s s ary to e f fect a reduction in cros s polari za-
tion . An example o f the application o f the field equa­
tions for th i s f eed to the computation of ref l ector
currents is i l lus trated in Figure 2 0 . Here , the re-
fleeter cross-polari zed currents are plotted in the
= 4 5 ° plane as a func tion o£ for various fractional
TM mode powers , PTM " (Re ca l l that the mode powers 1 1
are norma l i zed to a total power of uni ty . ) n F igure
2 0 the waveguide radius i s a/).. = 0 . 8 and it can be ob-
s erved th at the .maximum of K y I occurs at 4 0 ° . The
I l
inclusion o f progres s ive ly greater amounts o f PTM caus es
K I to dec rease until , for PTM = 0 . 1 3 8 , K has vanished y y a t = 4 0 ° . For thi s condi tion i t appears that the
average level o f K i s a l s o a minimum . This sugge s ts y that a reas onable criterion for determining the appro-
priate amount of TM mode power is that it be adj us ted 1 1
8 9
- 1 0
0 . 0 - 30
Ul ..jJ - 4 0 Q) 0 . 1 0 0 H H 0 . 1 5 0 ::I - 5 0 tJ
'0 Q) N - 6 0 0 . 1 3 8
·r-i H tO
r-1 0 AI - 7 0 0 . 1 3 0 I Ul Ul 0 - 8 0 H tJ
- 9 0 0 1 0 2 0 3 0 4 0 5 0 6 0
t!J - Degrees
F igure 2 0 . · C ros s -Polari zed P arabola C urrents vs . t!J for Dual Mode C i rcular Waveguide Feed . ( a/>.. = 0 . 8 )
9 0
to minimi ze the maximum cros s -po lari zed current on the
r e f l e c tor surface .
This idea has been tes ted for the particular
case of a/A = 0 . 8 in F i gure 21 where PTM has been
s uccess ive ly adj us ted to minimi ze K at various va lues y o f w denoted by wM . The amount o f PTM necess ary to
accompl i sh this is a l s o shown for each case . I t appears
tha t the lowe s t average leve l of Ky occurs where PTM is adj us ted to minimize K 1 at = 4 0 ° whi ch confirms y the idea o f minimi zing the maximum o f K 1 • Al though y not shown here , s imi lar exercises wi th other va lues of
a/A a l s o confirm thi s approach .
The data of Figure 2 1 are s ummari zed in Figure
2 2 which i l lus trates the cros s -polari zed current as a
func tion of the fractiona l TM mode power • . The minimum 1 1
KY occurs for PTM = 0 . 1 3 8 in thi s ins tance .
F igure 2 3 i l lus trate s the computed cros s -
polari zed pa ttern i