Design of Circular Apertures For Narrow Beamwidth and Low Sidelobes-fvQ
Design of Circular Apertures For Narrow Beamwidth and Low Sidelobes-fvQ
Design of Circular Apertures For Narrow Beamwidth and Low Sidelobes-fvQ
CONCLL-SIONS
but i t is probable that the sidelobe level in the radiation pattern of a short Yagi antenna ma>- be improved
by this method; the impro\:ement may not be as good
as i n the case of longYagi antennas. This method of
tapering. although applied here to the particular case
of longYagi antennas, ma>- be utilized i n the case of
other long end-fire antennas of traveling-wave type.
Summary-This article extends a method of antenna design described in an earlier article' by the same author.A family of continuous circular aperture distributions is developed in such a way as to
involve only
independent parameters, A , a quantity uniquely rea number controlling the delated to the design sidelobe level and
gree of uniformity of the sidelobes. An asymptotic approach to the
condition of uniform sidelobes thus becomes possible. A companion
article by Robert Hansen contains aperture distribution tables and
examples.
Authorized licensd use limted to: IE Xplore. Downlade on May 10,2 at 19:043 UTC from IE Xplore. Restricon aply.
January
zeros in the total pattern, chosen with due regard for
thegeneralpropertieswhichthearrangementmust
possess, is first proposed and then a method, employed
by Dossier and similar to thatof Woodn-ard, is invoked
for the calculation of the distribution function. I t m a y
beremarkedthatthepracticalrealization
of optical
apodization in the general case is contingent upon the
successful construction of absorbing screens n-hich are
free from phase errors.
The concept of a circular aperture as a source of
radiation is, in the field of antenna theory, an outgrowth
of the analysisof that very commonconfiguration which
consists of a paraboloidal reflector and feed. If such a
device is completely enclosed bs; an imaginar): surface,
then, inprinciple,
it is merelynecessaqto
know
tangential E and H upon this surface in order to calculate the radiated fields. I t is customary to draw the surface in such a m a ~ that
r
it hasa large plane expanse just
in front of the antenna configuration and to suppose
that the tangentialfields are zero except within the circular area which is directly illuminated. This circular
area is regarded as the aperture.
I n this article i t will be assumed that, upon any small
element of the aperture, the fields are identical in character with those which would be found in a n emerging
plane electromagnetic wave. I t is further assunled t h a t
the direction of polarization is the samefor all elements
b u t t h a t t h e field strength varies from place t o place,
perhaps in both amplitude and phase, and is proportional t o a distribution function
I). I t is not necessary, of course, that the device producing the fields actually be a reflector and feed; any device which genera circular areawould
ates fields of the type assumed over
come under the scope of this article.
Let the geometry of the aperture, which has radius a ,
be as illustratedinFig.
1. Theradiationpattern
in
power per unit solid angle is
Fig. 1-Geometry
of circular aperture.
s(e)
2 r k 2 1 u p g ( p ) J o ( k p sinelap.
(2)
p-
(3)
a
2a sin 0
The quantity
will be Itnown as a standard beamwidth; the beamwidth (in radians) between half-power
points for a uniformly illuminated circular aperture is
1.029 standard beamwidths. It is interesting to recall
thecorrespondingfigureforauniformlyilluminated
line source, 0.8859 standard beamwidth.
(2) becomes
\l;ith the substitution of (3) and
pg(p)Jo(pu)dp.
( 3
Here
whichisproportionalto(l+cos
is the
v d be dropped and the integral deobliquity factor .of aHuygenssource6and
is what The constant factors
would norm all^^ be called the element factor. The inte- noted F(
gral is called the space factor and,because of the broadness of the obliquity factor, the relationship betn-een the
F(U) ~ r 9 R ( P ) J a ( P z a ) d P .
(6)
beamwidth and sidelobe level of the space factor alone
will be adopted as the subject for discussion.
Eq. (6) expresses the basic relationship which
will be
I n ( l ) ,k is the free space wave number,
and
studied here.
is the angle measured between the observers direction
PROPERTIES
OF THE DISTRIBLTTIOK
FCNCTION
and the radius vector to the element
Since the
scope of this article will be restricted to those cases in
in the corresponding article on line sources, the
which the space factor has rotational sJ-mmetry about
variables p and are imbedded in the complex domains
the axis [an extension to certain othercases involving E and
respectively, such that i = p + i q and =u+iv.
a linear phase shift in
I) is obvious], the samet!-pe T h e visible range of the plane is the real segment
S. A. Schelkunoff. Electromagnetic IVaves, D. Van Nostrand
Co., Inc.,
Kew
York,
Y . , p. 354; 1943.
Authorized licensd use limted to: IE Xplore. Downlade on May 10,2 at 19:043 UTC from IE Xplore. Restricon aply.
19
0 5 ZL 5 2 a j X and a unit distance i n the z plane corresponds t o one standard beamn-idth. An observer who
travelsaroundtheaperturetoexamineitsradiation
pattern will cover all real angles in
the range O < ~ _ < T .
This coverage is translatedtwiceintothepointset
O < Z L <2a/X, = 0 , in the z plane by the relationship
zt=?a(sin 0j/h as shown graphically in Fig. 2 . The profile of F(zj on the visible range is the only part of F(zj
lvhich enters into the formationof the radiation pattern.
I t is evident from Fig. 2 that the space factor attains the
same value a t T - 0 as i t does a t 0 ; the total radiation
pattern is not "double-endedl" hou-e\ler,for the obliquity
factor nullifies the rearward beam.
Consider,now,the
plane.Theintegral in (6) becomes the limit as T tends to zero of the integral along
the path C of Fig. 3 , beginning a t zero and foIloLving the
axis of reals to T
T h e reason for the limitingprocess
will be made evident later. Eq. (6j becomes
~ ( z ) lim
7-0
lq
h(Q(r'
t')";
real and
1.
(8)
Fig. 3-Contour
of integration i n complex
plane.
h(:!
PROPERTIES
o b - T H E S P ~ I CF;.IC-TOI<
E
Given that the distribution functionis in accord with
the mathematical model just described and following a
procedure analogous to thatemplo).ed earlier,' i t is e x \ to prove thatF ( z j is an even entire functionof ivith the
follon-ing asymptotic forms for large 2 :
1.5
a\
TRANSBCTIOSS
IRE
20
0:Y A:YTE:\NAS
AiVD PROPAGATION
January
TABLE I
COMPARISON
OF ASYMPTOTIC FORXS
OF F(z) FOR LISE
SOURCE AXD CIRCULAR
APERTURE
Asymptotic Form of F ( z ) , Re
Line Source
which, for u slightly greater than unity,is a n ideal function with slight horizontal dilation.
rZ set of points on the axis of reals, hereafter designated as the p points, will now be defined. T h e definition of p n ,the nth such point, is simply
Circular Aperture
TABLE I1
3
cos r z
/z(T)(~T))(~)
k(r)(2~).)(6)
COS
T(Z
1/4)
SOLUTIONS OF
JI(T%)
, A
DESIGNOF
PRACTICAL
CIRCULAR
APERTURE
1
2
3
4
5
6
7
8
9
10
0
1.2196699
2.2331306
3.2383154
4.2410628
5.2427643
6.2439216
7.2447598
8.2453948
9.2458927
10.2462933
DISTRIBUTION
FUNCTION
As waspointedoutintheearlierarticle
on line
sources, essentiaI1y two areas of flexibility are available
T h e plan is now to construct an entire function with
in regard to the form of F ( z ) . These are 1) the value of
a and 2) the placement of the central zeros, that is, the the following zeros:
zeros near the origin in the plane. I t is possible to draw
the same conclusions as before, namely, t h a t for highly
directive patterns, a should equal zero (this gives the
distribution a pedestal), and that the centralzeros a s
well as the remote zeros of F ( z ) should be simple zeros This function, normalized so as to have unit value at
on the axis of reals. For choosing the positions of the the origin,is a s follows:
Authorized licensd use limted to: IE Xplore. Downlade on May 10,2 at 19:043 UTC from IE Xplore. Restricon aply.
?2
.,I, r?j
2J1(az) *--l
G2[.d2
(JZ
1/2,]
(17)
Pn?
This expression.
a t w o - p a r a m e t e r f a m i l y of entire
functions whose menzbers a.pproach the (nornzalized) ideal
space factor, cos T ~ / z ~ - ~ ~ ? /TcAo, sarbitrarily
~
exactly
a s is increased. The value of
which is finite in any
practicalcase, is of considerable significance. In rela$-pn divide theregion
tionship to the pattern, the points
of uniform sidelobes 14 < p e j from the region of deca).ing sidelobes
where here, as before, z =
As is increased, theregion of unifornl sidelobes extends
farther out from the main beam. In the region of u n form sidelobes the lobes actuallJ- decrease slightll; asz f ,
increasesandtheirinitial
level is veryslightlJ- lovver
( t h a t is: better) than thedesign level 7. This design sidelobe ratio is relatedtoas
follows:
cosh T-4,
(18)
7T
which is the same asin the case of line sources since the
same ideal space factor is approached i n either case.
T h e chief disparit!. between the practical pattern and
the ideal is i n the beamwidth which is greater than the
THE:INVERSETRANSFORM
ideal by a factor almost exactly equal to u. However, n
I n calculating the distribution function g ( p j to give
does not have to be very large to make
u only a few
the space factor F ( z , .-I, -iij, the method of Dossier will
per cent greater than unity.
To sumnlarize.themembers
of thepattern famil?- be used. Let the distribution function g ( p j be built out
of functions of the form Jo(p,p). I n other words,
F ( z : -4,ti) have two independent characteristics:
T h e design sidelobe ratio, 7.
2) The boundary of the region of approximately uniform sidelobes, p e , which depends directly upon
R , an integer.
Once these two parameters have been chosen, the pattern, the distribution function, and
all other relevant
data may be calculated. In selecting A , it is essential to
avoidvalues
thataretoosmall;thechangefrom
F l ( e , -411 t o Fie, -4,
inwhich
thetrans+% zeros The integrali n (22) is vie11 known8 and given by
migrate to the p-points! must be such that the spacing
never increases betweenany of these zeros. In order that
this condition be fulfilled, A must be chosen such
that
a unitincreases i n
doesnotincrease u. Practically,
thisselectionmeansforaspacefactorwith
a design
sidelobe ratio of 30 d b , must be a t least 3 and that
for a designsideloberatio
of 40 db. i t must be a t
least 4. Apart fromthislimitation,onehasconsiderable liberty in the choice of
large values make u more
nearlyequaltounity,therebysharpeningthebeam:
The above expression is an even function of
with a
i n thisway
however,theadditionalbenefitobtained
zero
a
t
every
p
point
except
a
t
i
p
m
,
a
t
which
points
it
soon becomes negligible as increases. If ti is increased
attains
the
value,
so a s t o place p n \\-ell beJrond the endpoint of the visible
range ( S a j X j , the effect is to make all the visible sidelobes uniform. Further increase of ti has little influence
E. Jahnke and P. Emde, Tables of Functions, Dover Publicaupon the visible pattern but has the effect of increasing tions, Kew York. N. p.
Authorized licensd use limted to: IE Xplore. Downlade on May 10,2 at 19:043 UTC from IE Xplore. Restricon aply.
IRE TRANSACTIO;YS
22
lim
7ruJo(7rpm)Jl(7rU)
242
O A V
7r2
[Jo(PpM)12.
F(21)
m=o
nuJl(nu)
D~JO(W~J
212
(25)
is
(26)
pm2
January
F ( p m ) means F ( p m , -4, as
In the present situation,
given by (17). T h e process of setting e equal to p m in
this equation, however, is complicated by the presence
of removable singularities at all the points for which
m
A t m = 0 , the singular factoris J l ( s r z ) / m and for
O<nz<s, it is J l ( n z ) / ( l - z 2 j p n z 2 )These
.
can be evaluated by well-known methods and the results are
Ilt
In
#m
Authorized licensd use limted to: IE Xplore. Downlade on May 10,2 at 19:043 UTC from IE Xplore. Restricon aply.
72