HighlEfficiency Microwave Reflector...
Transcript of HighlEfficiency Microwave Reflector...
1470 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
HighlEfficiency Microwave Reflector Antennas A Review
PETER J. B. CLARRICOATS, FELLOW, IEEE, AND GEOFFREY T. POULTON
Invited Paper
AbstmctThe paper provides a review of current reEuch on micro wave refkctor antennas with p.rticulrr regad to tho& antennas which pxoduce penciibem diation patterns. After a pdmimry exrminr tioa of microwave antennas of difft types attention is f d on drculrrty symmetric antennas with miplly symmetric feed systems. This class of antenna s c a m @ for the largest n u m b of applications which indude miaowave pointtupoint communication, satellite communiution, and dio astronomy. The cboice of optimum feed is dered in more detail and it is demonstrated that metrllic feeds with cormgated walls represent the best choice in most cases. Brief reference b dm m d e to dielectric cone feeds which may find a place in future systems The deriga of dualreflecfor systems is undered in depth and different computer optimisation techniques are reviewed. Th incMe the latest methods of ditfrrction optimization in which more than one of the antenna spedhtbns k mvolved. An example is given where efficiency and VSWR are simultaneously optimized.
I . INTRODUCTION HE PURPOSE of this review is to trace recent develop ments in the field of microwave antennas. Many of these developments cover the period from 1970, although for
continuity, references to work prior to 1970 are included. A comprehensive review would produce an unacceptably long paper so the authors have confined their attention mainly to topics with which they and their colleagues have been closely involved.
To set the contents in perspective, we begin by classifying reflector antennas. Fig. 1 identifies reflectors according to a) pattern, b) reflector type, and c) feed type. Let us first consider the antenna pattern. Pencilbeam antennas are most widely used in pointtopoint microwave communication systems since their patterns yield the maximum boresight gain. In terrestrial microwave relay applications the beam direction is fixed at the time of installation of the antenna. In satellite communication systems the uplink beam may be fully steer able by reflector movement as in INTELSAT ground stations, or capable of limited steering as in the Canadian domestic systems. Although spacecraft antennas often produce shaped beams, many systems also require multiplepencilbeam antennas. Some future earth stations may also employ multiple beams such as provided by the torus antenna investi gated by COMSAT. Microwave antennas for satellite and terrestrial broadcasting and for radar demand shapedbeam specifications.
Manuscript received November 15,1976;revistd March 8, 1977. P. J. B. Clarricoats is with the Department of Electrical and El=
tronic Engineering, Queen Mary College. University of London, London El 4NS, England.
Engineering, Queen Mary College, University of London, London, G. T. Poulton waa with the Department of Electrical and Electronic
England. He is now with the Radio' Phydcs Division of CSIRO, Sydney, N.S.W., Australia.
fixed mvoble
slrgle rnultlple 8 l h o r A cIrcuIor cnaxis A offoxis opertue A axial
Fig. 1 . Classification of reflector antennas.
Many microwave communication antennas operate with one sense of polarization at a given frequency and require only reasonable discrimination between orthogonal polarizations. However, many of the next generation of microwave com munication antennas will operate with dual polarization at the same frequency (socalled frequency reuse antennas). Such a requirement imposes stringent demands on the polar ization performance of the antenna and accounts for a quite significant proportion of reflector antenna research effort in recent years. Radioastronomers are also interested in measuring the polarization characteristics of radio sources and they too have an interest in antennas with good polarization discrimination.
Next consider the reflector. The paraboloid is the most commonly used type because this will produce a highgain pencilbeam with quite low sidelobes when fed efficiently from the focal point. Frontfed operation has been' used widely in terrestrial links for reasons of economy although as new shaped dual reflector antennas become available, designs based on the Cassegrainian principle may supersede the front fed design. Slight shaping of the reflector's surfaces leads to substantial gain enhancement and these techniques have been widely used in earth station antennas. Other reflector surfaces have been used to produce shapedbeams and nonparaboloidal reflectors have also been used in special purpose antennas. For example, the sphericdreflector antenna has been investigated for radioastronomy and for small earthstation applications because the beam can be efficiently scanned by movement of the feed system. In beamscanning and multiplebeam systems the aperture of the antenna may be excessively blocked by the feed system. Besides the inherent reduction in gain, sidelobe levels may be raised to an unacceptably high level. For this reason, offset designs have been developed. However, because
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1471
of the inherent geometrical assymetry of the reflector, design techniques are rriore involved; it has a less favorable intrinsic crosspolarization discrimination arid the relatively high cost of producing a large reflector of this type accounts for the limited application of the design to date. However, it is likely to prove a popular choice in future spacecraft applications:
We turn finally to the feed. Until about 10 years ago, horn or waveguide feeds operating in a sirigle pure mode were tHe only types considered for iliuminating reflectors. However, mainly because of radio astronomy and earthstation antenna requirements, considerable efforts have gone into the produc tion of new feeds to efficiently illuminate either tHe main reflector or subreflector of the antenna. Useful t o an under standing of these designs has been the analysis of the fields in the focal region of reflector antennas. This shows that a hybrid mode field (combining TE and TM fields) is required in a feed if theie is to be efficient matching between the feed distribution Pnd the desired focal field distribution. It is also the ideal feed for reducing crosspolarizdtion. The most important class of feed producing a hybrid mode field is that in which the wall of the feed is a corrugated metal surface. When excited with a dominant waveguide mode, such a feed cari generate in its aperture, fields which very closely resemble the focal region field of a reflector antenna. Other structures which support hybrid modes include the dielectric cylinder and cone; feeds of this latter kind also have the advantage that they can be used to support the subreflector in a Cassegrainian system.
In the present paper, we shall concentrate our attention on two topics: feeds and reflectors for fixedbeam onaxis systems. Discussion of reflectors for fixedbeam offaxis sys tems and antennas to provide multiplebeam formation Wil l be dealt with in a forthcomidg review paper by Rudge.' Inevi tably theTe must be interplay between these topics especially since feeds, by definition, are developed specifically for use with reflectors. This is the context in which we present Section 11, those readers wishing to obtain a more general treatment of waveguide and horn radiators are recommended to consult for example, papers appearing in the reprint volume edited by Love.
11. FEEDS A . Introduction
In this section, we consider the radiation characteristics of fixed fkeds illuminating circularly symmetric reflectors which are intended to produce single highefficiency pencil beams. We concentrate, therefore, on feeds for paraboloidal reflectors of both prime focus and secondary focus types. Detailed dis cussion of the design of specially shaped dual reflectors is defeired until Section 111A.
During recent years, two main problems have concerned feed desikners, namely, high antenna efficiency and low cross p o l ~ z a t i o n . Design features of the over+ antenna such as low sidelobes and low input VSWR are also important but, especially in dualreflector systems these q e influenced more by the design of the reflector and subreflector than by the feed. They are therefore discussed in Section 111.
To gain insight into the above problems we shall look at the antenna in two ways. First, by considering it as a radiator which leads us to an examination of the antenna radiation
Note: Review paper on offset reflector antennas by A. Rudge, is to be published in PROCEEDINGS OF THE IEEE.
Y
W Fig. 2. Paraboloid reflector showing coordinates.
pattern and its gain. Secondly by considering it as a receiving antenna when we shall examine the fields in the focal region of the reflector where the feed is located.
Fig. 2 shows a paraboloid reflector and the coordinates of a point on the reflector and a point in the farfield. In order to obtain the electricfield strength at the point R , .$, we may integrate the contributions to the field from currents induced in the reflector by the fields EO, E@ radiated by the feed.
We assume an illumination that possesses a degree of circular symmetry so that the @ integration may be performed analyt icaily, leaving only the integration over the 8 coordinate to be performed numerically.
The field incident upon the reflector is chosen as
Ei = fe sin $8 + f@ COS $& (1)
where fe, f6 are functions of 0 and p . For a paraboloid p = 2f(1 t COS e)'.
n e electric field at R, .$, O is given by
where
e u1 = 2kf tan sin
2
Examination of (2) shows that the conditions for pattern symmetry and zero crosspolarization (according to Ludwig's [21 definition 3) &e both achieved when the feed radiation pattern is symmetrital, i.e., fe = f@. Since pattern symmetry is a necessary f i t step towards achieving maximum antenna gain, we see this as a prime objective in feed design for antennas of this class.
The antenna gain in the boresight direction G, is given by 131, [41
where q, the antenna efficiency can be partitioned as
T) = q p I)i I), r), q b 7)r (4)
1472 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
following Thomas [ 5 1 , we shall now identify the terms in (4). If fe of ( l ) , is represented in the form F E E ~ ~ ~ I P where the function FE is the 8 dependent Eplane radiation pattern of the feed and similarly for f@,
7)p =
(phase error loss) ( 5 )
e qi = 2 cot2
2
currents
diffraction coefflclents
dlffracted \ I fietd i
F%. 3. Feed radiation patterns and techniques for derivation of far field pattern,
(illumination loss) (6) shown in Fig. 3, with
(1  q p ) 100 represents the percentage power loss which arises if the field over the aperture is not everywhere in phase. (1  vi ) 100 represents the percentage power loss due to non uniformity of the aperture amplitude distribution. ( 1  7)$) 100 represents the percentage power loss caused by energy from the feed spilling past the main reflector. (1  q X ) 100 represents the percentage power loss caused if there are crosspolarized fields present in the antenna aperture. (1  Q,) 100 is the percentage power loss due to blockage of the reflec tor by the feed, supporting struts and, in the case of a dual reflector, the subreflector. ( 1  qr) 100 is the percentage power loss due to random errors over the reflector surface. The overall antenna gain is further reduced because of attenua tion in the antenna feed and associated waveguide.
If the feed radiation pattern is symmetric, FE = FH and 1), = 1 while if the feed has identical Eplane and Hplane phase centers and misalignments are avoided during antenna con struction, qp will also approach unity. This leaves illumination and spillover losses as the principal contributors to antenna gain loss and the expressions show they depend on the feed pattern function. The two factors of necessity involve a com promise; near uniform illumination could easily be achieved if spillover were inconsequential and vice versa. Thermal energy entering from the spillover region also contributes to the antenna noise temperature and provides an additional reason for it to be minimized in certain applications. Uniform illumination over, the reflector aperture requires a pattern, as
e sec2  o < e < e o
F E = F H = F ( e ) = {o, 2 ' e >eo. ( 9 )
We note that uniform illumination of a circular aperture leads to a fmt sidelobe level of  17.6 dB which could also be unacceptable. A feed pattern function of the optimum form also shown in Fig. 3, would be generally ideal for highgain pencilbeam antennas and such a form is adopted in earth station antennas. The realization of such a pattern function over a reasonable bandwidth has been the objective of many research groups in recent years.
Because most radiotelescopes are intended for operation over a very wide frequency spectrum, an efficient Cassegrain ian or Gregorian design would require an unacceptably large subreflector at the lower frequencies. Thus in practice it is usual to resort to prime focus operation over most of the frequency range. Where the radiotelescope antenna design makes provision for a subreflector, it is either Gregorian, which leaves access to the prime focus, o r Cassegrainian, when it is arranged that the subreflector can be removed at the lower frequencies. An example of the former design is to be found in the 100m diameter Effelsberg radiotelescope of the Max Planck Institute for Radioastronomy, while the latter design has been chosen by the same organization for a Euro pean 25m diameter millimeterwavelength radiotelescope.
As a result of the above mentioned constraint, it is not possible to use shaped dualreflector antenna designs to achieve high efficiency and the burden of gain optimization in radioastronomy antennas has fallen entirely on the feed designer.
Many present generation microwave relay systems also use frontfed paraboloid reflectors both for reasons of economy and far out sidelobe control. Thus improvements in feed design are of interest there.
With the exception of antennas intended for radioastron omy, nearly all veryhighgain reflector antennas are of the Cassegrainian type, as shown in Fig. 4(a). The prototype comprises a paraboloid main reflector with a hyperbolic subreflector. Within the main beam, the Cassegrainian antenna behaves essentially as if it were a long focallength frontfed paraboloid, as shown in Fig. 4(b) [6]. The prototype Casse
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1473
(b) Fig. 4. C a s e g m h h antenna. (a) Casegrainian reflector configuration.
(b) Equivalent frontfed parabobid reflector.
grainian design permits one to achieve more uniform illumina tion of the main reflector and lower spillover. Typical measured efficiencies for Cassegrainian antennas are of the order 6570 percent which are about 10 percent higher than for frontfed paraboloids with simple feeds. The Cassegrainian design also permits the receiver to be located close to the feed which is an advantage in lownoise applications. However, its greatest advantage is realized when both reflector shapes are deformed. Then, provided the feed pattern is circularly sym metric, an antenna aperture pattern approaching the ideal can be achieved. The Cassegrainian antenna is not without dis advantages but a discussion of these is deferred until Section 111.
We turn next to an inward view of the paraboloid antenna following the investigations of Minnett and Thomas [ 71 who provide references to earlier work. Fig. 5 shows a plane wave incident upon the reflector in a direction parallel to the reflector axis. The field of the plane wave induces currents on the reflector surface and these currents reradiate. By evaluating their contributions to the electric field E at a point rf, $f (in 'the plane of the geometricoptics focal point) it can be shown that provided rf/A 5 10
E = jEokf C0{ Jo(u l ) s in8 i+Jz (u l ) s ine
e 2
* tanZ  [cos Wfi + sin 2$ffy]
2 (10)
U1 = krf sin 8 .
Fig. 5. Paraboloid reflector showing incident plane wave and coordi nates of point located in the focal plane.
1 0
h 0 5
0
Fig. 6. Field distribution in focal plane of paraboloid with B o = 63' (Minnett and Thomas [ 71). (a) Polarization of Efield. (b) Contours of electric field amplitude E(U).
Fig. 6 , due to Minnett and Thomas [7] shows for an antenna with Bo = 63' (F/D = 0.4), the electricfield distribution in the focal plane. This also represents the field which would be coupled into a dipole located there. In the central region, which includes the geometricoptics focal point the field is almost entirely linearly polarized. However, as Fig. 6(b) and equation (10) shows, the distribution is not completely circularly symmetric and in the region of the first null of the focal field, there are the crosspolar fields which reach their maxima in the $f= f45' planes. Additional regions of cross polarization lie further away from the axis.
If a feed with aperture S is introduced into the focal region, the antenna efficiency is obtained from [ 81
1474 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
where E l , H1 are the electric and magnetic focal fields due to a plane wave with unit power over the reflector aperture and E2 , H z . those fields created at the apeiture of the feed when unit power is incident in the dominant mode of the feed wave guide. We shall now see that for a particular choice of aper ture field in the feed, only the field corresponding to the first term of (1 0) contributes to the efficiency.
Let the electric field E1 defined by (10) be partitioned into @ independent and @ dependent terms. Likewise for the
=  B ( U ) (Sin W f i  COS 24fj) L
2 0 (13b)
where C is a constant and A( V ) and B ( V ) are integrals defined by (1 0).
Then if Ez and Hz are chosen so that
E2 = W f , @I i (14)
and
Under conditions (14) and (1 51, maximum efficiency.occurs when the feed function r dependerice Matches the function A ( V ) . A graph of the function A ( U ) together with the func tion B( V ) is shown in Fig. 7(a). The efficiency integral is then given by
Q = 3 lUo A' ( U ) U d U
VO = krf sin B o (1 7)
which is shown in Fig. 7(b) together with an integral repre senting the time average normalized power flowing through the corresponding surface in the focal region
uo f)o = 3 1 ( A 2 ( V )  B 2 ( U ) } V d U . (18)
We note that r ) > 90 ; it may seem swprising that, in principle, more power can be coupled into a feed than would actually cross an aperture of the same radius in the focal region. How ever, detailed examination of the Poynting vector in the focal plane reveals that there are regions where it is oppositely directed to that of the integrated vector. In these regions QO decreases as rf increases. However, with the above class of
0 9 
0 8 
0 7 
0 6 
7 , 0 5 
__
7 0 . 2 0 1 o w ~ ~ ' ~ l ~ ' ' ~ ~ ' J
1 2 3 L 5 6 7 8 9 1 0 1 1 1 2 U.
(b) Fig. 7. Functions related to focal plane, characterha of paraboloid
reflector. (a) Functions A(U) and B(W for puaboloid subtdnding k d a n g k 8, .at the focus (Mmnett and Thodss [7]). (b)' Integral representihg maximum antehnr efficicdcy (17) () and the time average nbrmdized power flowixig through comspondirig surface in focal retjioh (1 8) ().
feed we fihd that the "B type" of field does not couple and it responds d d y to the "A type" field. For small B o , r ) exceeds unity whdh rf is very large and, neglecting the biocking effect of the feed, we have a super gain phenomenon. As is well knowri it is generally quite impractical to operate antennas under supergain conditiuns but there is experiniental evidence to Suggest that Q may exceed QO when QO < 1.
Recalling that the ideal feed discussed earlier in this section did nof generate crosspolarization when transmitting, we see from (1 6) that the teed just discussed also does not respond to the crdsspdlar componetlts of the focal field. In this section we shall discover that a feed supporting a hybiid mode of prdpagation can satisfy equations (14) and (15) and, if the feed possesses an aperture of circular crosssection, it will also
1475 CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS
d B
1
1
dlslance from (1x1s 1x1
Fig. 8. Eplane focal region fields for various edge illuminations on a frontfed paraboloid, D = look, FID = 0.3  0 dB edge taper; _ 10 dB; 20 dB.
exhibit pattern symmetry, when the condition (Ez/Hz) = Zo is satisfied.
When the F / D ratio of the paraboloid is large, as might be realized in practice in a Cassegrainian configuration, the focal field corresponding to an arbitrary normalized aperture distri bution g(s) = g(e /O0) , reduces to a simple form [ 71
where q = r e o / A and Bo = D/2f. The integration can be performed analytically when g(s) = 1
(uniform illumination) and for a variety of other specific distributions as discussed for example by Hansen [9] Ramsay [ 101 and Sciambi [ 11 ] . In particular, when g(s) = 1
Finally, since g(s) = 0 when s > 1 the integration range in (19) may be extended to infinity reminding us that the normalized focalfield and aperture distributions approxi mately form a Hankel transform pair in the case of very long focal length antennas. Focalfield contours are shown in Fig. 8 for a number of different aperture distributions.
B. PureMode Feeds The simplest form of feed for a paraboloid reflector antenna
comprises a conventional waveguide or horn supporting the dominant mode of propagation. In prime focus applications, reasonable efficiency can be achieved with a waveguide whose diameter approximately matches that of the centrd region of the focal field shown in Fig. 6. When transmitting, the radia tion pattern of such a feed will taper monotonically from 0 dB on boresight to a value which lies between 8 and  10 dB, at an angle corresponding to the reflector rim. In general, the feed radiation pattern will be asymmetric causing a loss in efficiency, increased crosspolarization and increased spillover.
Crosspolarization can be reduced by judicious choice of wave guide radius while improvements in efficiency and spillover can be achieved for a singlemode waveguide feed, by the use of a corrugated flange. For narrowband applications, multi mode waveguides can be used to synthesize the desired aper ture distribution.
Horns of either circular or square crosssection have been extensively used as feeds in Cassegrainian antennas however, they too produce asymmetric patterns unless designed for multimode operation with corresponding bandwidth penalties. The Cassegrahian configuration is generally reserved for larger, hence more expensive, antennas where the slight increase in overall cost arising from the use of a more efficient hybrid mode horn is acceptable. Thus following the advent of hybrid mode horns, rather little effort has been expended on horns of conventional design. For an account of developments in that area the reader is referred to the digest edited by Love [ 1 1 .
As mentioned above, the conventional circular waveguide is extensively used as a feed for the frontfed paraboloid. The radiation pattern of a waveguge of radius r l when excited by a TEll mode is specified by Eo and E @ , the 8 dependent pattern functions [ 31
\x1 J K2 = k2(1  B2). (22)
From (2), we recall that for the secondary pattern of the paraboloid to exhibit symmetry and zero crosspolarization, the condition E8 =E+ should be satisfied. Fig. 9(a), due to Hockham [ 121, .shows however that when the waveguide radius is such that krl = 27rrl/A = 2.25, the equality condition is nearly maintajned over a wide range of 9. Experimental evidence to nipport this conclusion has been obtained. Fig. 9(b), also due to Hockham [ 121, shows the corresponding predicted 45' plane crosspolarization pattern when the above waveguide illuminates a puaboloid with D/X = 26.5 [ 3dB beamwidth = 3'1. In practice, the crosspolar maximum will exceed the predicted value principally because of scattering by the feed and the feed support and due to cunents on the outside of the waveguide and evanescent fields in the aperture. Nevertheless, a singlemode circular waveguide constitutes a suitable feed for a frontfed paraboloid when narrowband width operation i s . required with good crosspolar character istics. I t also has advantages when multiple feeds are required so as to produce multiple beams since, if the feeds must be closely spaced, other larger types of discrete feed cannot be accommodated in the focal region.
Although the crosspolar performance of a paraboloid with a singlemode feed can be excellent, the antenna aperture efficiency is not high . Fig. 10, due to Koch [ 131, shows theo retical aperture efficiency curves for a number of feed systems, including the TEll mode circular waveguide feed described above. With the parameter F/D in the range 0.250.4 the semiangle 8 0 lies in the range 90'40' and corresponding theo retical aperture efficiencies lie between 55 and 67 percent. Before discussing techniques for raising aperture efficiency of
1476 PROCEEDINGS OF THE IEEE, VOL. 6 5 , NO. 10, OCTOBER 1977
f ,c
002 
0
0 02 
001 
006 
*OB
0 10 0 ? 0 2 0 3 0 4 0 5 0 6 0 x ) B o
 30  w Y  b
0 1 2 3 1 5 6
Fig. 9. Characteristics of paraboloid reflector fed by circular waveguide (Hockham [ 1 2 1 ) . (a) fe  f6 as a function of boresight angle 0, parameter 2 m 1 / h . (b) Normalized crosspolarization secondary pattern in 4S0 plane for paraboloids fed by waveguide with near optimum radius. D/h = 2 6 . 5 , 2 m , /A = 2.25, parameter F / D .
feeds excited by the TEll mode, we mention that in narrow bandwidth applications, very high efficiencies can be achieved through the use of multimode feeds. In 1963, Potter [ 141 proposed the use of a dualmode horn utilizing TEll and TMll modes in order to produce a more nearly circularly symmetric radiation pattern. The concept of pattern synthesis using a number of TE and TM modes in a circular aperture was then more fully developed by Ludwig [ 151 who predicted aperture efficiencies of 85 percent for a 3mode horn.
In applications of the pattern synthesis technique the results of the above authors are mainly confined to horns of large aperture suitable for feeding antennas with long focal lengths or their Cassegrainian equivalent. We return now to consider techniques for enhancing the efficiency of antennas with shorter FID in the range 0.250.4.
With reference to Fig. 6, we observe that the focal field corresponding to uniform illumination of the aperture of a paraboloid, has a distribution which is approximately J , (tr)/(v); the 'A type' focal field referred to earlier, ap proaches this form for large FID ratios. Thus in order to produce uniform illumination over the paraboloid aperture, a feed is required with an approximately J1 (W/U distribution, as shown in Fig. 7(a) for different values of FID.
Three approaches to this problem have been made. We consider first that adopted by Koch [ 131 and his coworkers.
O 6 t measured values
0 5 00 zoo LO0 60' 80'
eo  Fig. 10. Maximum obtsiaable aperture efficiency of paraboloid anten nas with either a circular waveguide feed or coaxial feeds as shown in general in Fig. 11 . The number of coaxial regions (rings) included in the feed is denoted. Aperture blocking is neglected in the calculation (Koch [ 131).
They attempt to synthesize the focal field by means of coaxial resonators. Fig. 1 l(a) shows the field distribution in the aperture of a 3ring coaxial feed excited by the TEll mode, while Fig. 10 shows the maximum obtainable aperture efficiency of a paraboloid for coaxial feeds with 0, 1, 2, and 3 rings. Evidently the major improvement over a simple wave guide comes from inclusion of the first ring of the coaxial resonator. Radiation patterns for such a feed see Fig. 1 l(b) are shown in Fig. 1 I(c).
Koch [ 131 has also investigated feeds in which TEll and TM1l modes propagate in the central region and TEll and TElz modes propagate in the coaxial region. It is found that this type of feed gives a higher efficiency for reflectors with F/D  0.4 whereas for deep reflectors (F /D  0.25) a feed excited by the TEll mode is sufficient. More recently Schaefer has reported results for parasitically excited multi mode feeds, see Fig. 12, of which Feed 111 gives very low edge illumination (25 dB) for a reflector with a semiangle of 80'. Such a feed would yield a very low antenna noise temperature which is essential in the radioastronomy applications for which the above feeds were developed.
The second approach is that adopted by Wohlleben, Mattes, and their coworkers [ 161. Their prototype feed is shown in Fig. 13, together with the focal region electric field of the Effelsberg radiotelescope for which it was intended. The recessed corrugated surface causes the backward directed radi ation from the circular waveguide to be reflected in such a way as to synthesize the aperture distribution over the first annular ring. The surface behaves as an anisotropic impedance 2, with 2 = 0 in the Hplane and 2 = 00 in the Eplane. It thus produces a polarization independent reflection of the back warddirected radiation. A theory for feeds of this class is being developed by Hockham [ 171. The optimum dimensions of the feed of Fig. 13 were obtained from experimental observation of the radiation pattern of the feed. Additional discussion of related corrugated feed horns is presented in the next section.
The third approach to the synthesis of a uniform illumina tion over a paraboloid reflector is that adopted by MiMett, Thomas [ 181, and their coworkers [ 191. They utilize a multi mode hybrid waveguide to effect the synthesis. Discussion of
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1477
c
I I I 1 I I I 20. 90' do. 30. 00 300 6 0 0 900 1M*
measured  eo measured calcu!atec]E plane calculated I Piane
(C)
Fig. 11. Characteristics of coaxial feeds (Koch [ 13)). (a) Electricfield distribution in aperture of Jring coaxial feed excited by TE,, mode. (b) Coaxial feed with single ring excited by TE,, mode and experi mentally optimized for 0, = 70' (Koch [ 131). (e) Radiation patterns of coaxhl feed of (b).
their approach is deferred until the properties of corrugated waveguides have been discussed.
We conclude our review of feeds utilizing simple waveguides with a brief account of those shown in Fig. 14. Their origin goes back to World War I1 when a feed similar to that of Fig. 14(a) was utilized in a microwave communication system. Their advantage lies in simplicity and ruggedness. They differ from the standard Cassegrainian design of dualreflector antenna in that the subreflecting plate is mounted in close proximity to the feed waveguide giving rise to the term splash plate feed. The feed of Fig. 14(c) has been designed by workers at the British Post Office using geometricaptics
I 11 I I I 
0  Eplane  Hplane 
/ \
Lo I I I I I I I
90' 60' 30' 0' 30' 60' 90' e
( 4
Fig. 12. Parasitically excited coaxial ring feeds. (a) Feeds and aperture field patterns of modes (Schaefer [ 7 ] ) . (b) Feeds. (c) Radiation patterns (Schaefer [ 171) corresponding to f e d I, 11, and III.
r h
Fig. 13. Sectional view of feed for paraboloid refkctor with F/D = 0.3 [ 161 together with f o d region electric fiilds.
1478 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
==4 dteleclric suppal
__
Fig. 14. Crosssections through splashplate feeds. (a) Rototype feed (in practice the splashplate win be attached to the mveguide by means of a low permittivity dielectric or fiberglass tube). (b) Spksb plate with cone to reduce input VSWR of feed. (c) Dielectric sup ported splashplate (the dielectric may be shaped co as to improve aperture distribution over paraboloid reflector aperture).
O r
.. G.T. 0.  measured
Fig. 1 5 . SpMplate feed results. (a) SpMplate fsed showing rays considered by Mdik and James (201 in their analysis, 0' and 0" represent Geometricoptics imager of source. Broken lines denote GTD rays. (b) Splashplate configuration investigated experimentally by Mdik and Junes. (c) and (d) Hplane radiation pattern of splash plate feed of (b); (c) L = 2X b = 0; (d) L = 2.Sh b = 0.SX.
(4 Fi. 16. Hybridmode feeds. (a) Corrugatcdhom feed. (b) COM
gatedwaveguide fecd. (c) Dielectric cone.
methods so as to produce reasonably uniform aperture illumination of a paraboloid with F/D = 0.35. It gave a measured efficiency of approximately 62 percent over a narrow although useful frequency band.
A theoretical investigation of the feeds of Figs. 14(a) and (b) has been made by Malik and James [20] using the geo metrical theory of diffraction (GTD) to supplement geometric optics analyses. Fig. 15(a) shows the rays they considered while Figs. 15(b) and (c) shows measured and predicted radiation patterns and VSWR. When the feed is used to illuminate a paraboloid with F/D = 0.38 and D/h= 60, the predicted aperture efficiency is 65 percent for an optimum choice of feed parameters. The inclusion of a cone in the design provides a means to control the input VSWR, which otherwise exceeds 1.4 with a flat plate. The aperture effi ciency is constrained by feed spillover unless the feed and plate are in very close proximity, then the VSWR becomes intrinsically high .
C. HybridMode Feeds 1) Introduction: The term hybridmode feed describes a
feed in which both electric and magnetic field components exist in the direction of propagation of the guided wave. The presence of both of the above components in the ratio of the freespacewave impedance, can then lead to an ideal form of field transverse to the direction of propagation. Such a field when radiating, produces a circularly symmetric radiation pattern with zero crosspolarization.
Fig. 16 shows structures ivhich will support hybrid modes. They are characterized by a boundary surface such as r = r l in Fig. 16(b), where in a first approximation, the wave imped ance of an azimuthally or @ dependent mode is anisotropic when viewed in a direction normal to the surface. Minnett [ 2 1 ] and his coworkers in Australia, were the first to recog nize in 1963 the importance of cylindrical hybrid mode wave guides as feeds for paraboloid reflectors while at about the same time in the U.S., the conical corrugated horn was de
veloped by Kay [22] . From a ray standpoint, the ideal impedance surface treats both polarization equally so Kay originay coined the term scaZarfee4 to describe the conical corrugated horn. This term is less widely used today.
2) Cylindrical HybridMode Waveguides: The propagation and radiation characteristics of cylindrical hybridmode wave guides have been considered extensively by many authors [ 1 ] . To date the form of waveguide in which the impedance boundary is created by transverse metallic corrugations has proved the most popular form for feeds, although the analysis follows closely that for a cylindrical dielectric rod antenna. In both structures, the HEll mode of propagation is the one most widely employed. In the corrugated waveguide, the operating regime is where the phase velocity of the HE11 mode exceeds that of light and conversely for the dielectric rod. Because the corrugated waveguide is a periodic structure, an exact representation of the field in the waveguide interior re quires an infinite set of space harmonics while in the slot region, infinite sets of TE and TM resonant modes are re quired. Clarricoats, Olver, and Chong [ 231 have analysed the behavior of corrugated waveguides in this way, following in the path of those concerned with linear accelerators. They find that for many purposes it is sufficient to neglect higher order space harmonics and to include only the dominant slot mode.
With these assumptions, the field compodents in the interior region of the waveguide ( r < r l ) for modes with unity azi muthal dependance become [ 241 :
where x = K r and the factor ejBz is suppressed throughout the field equations.
To obtain the propagation coefficient, the fields E, and H6 are matched at r = r l while, sub je t to the assumption that only the lowest order mode is supported in the slot, we have
becomes
where
CLARRICOATS A F ~ D POULTON: MICROWAVE REFLECTOR ANTENNAS
E4 = 0 at that boundary. Then the characteristic equation
1479
and
A similar although somewhat more complicated equation is obtained when the assumptions are removed. Typical solu tions are presented in Fig. 17(a). The formulation leading to (3 1) treats the boundary at r = r l as if it were a continuous impedance surface. Then for a given mode, the passband for propagation has lower and upper frequency limits given by
as shown in Figs. 17(b) and (c). The latter condition is an important one because at the
frequency where @ + 00 for a given mode, other propagating modes exhibit special characteristics. For inodes with fl finite when S1 + 0, equation (3 1) reduces to
axid (26) then shows that
x= 71 (35)
where the + sign refers to HEl, modes and conversely for EHI, modes.
The condition expressed by (35) is referred to as the balanced hybrid condition.
From (23) and (24), when 1x1 = 1
thus the longitudinal components of the electric and magnetic fields of the hybrid mode are in the ratio of the freespace wave impedance. As previously mentioned, the radiation pattern of the waveguide then exhibits symmetry and zero crosspolarization. In practice the presence of spaceharmonics and the waveguide flange modifies the condition but their neglect greatly simplifies preliminary discussions of the subject.
By use of Bessel function recurrence relations in (25)(28), the transverse fields can be rephrased, then on making use of the relation
we obtain for the approximate Cartesian components of the transvefse fields
(39)
1480 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
''Or
0.8
'1 ' ro
0.6
0 2 6 8 10 wr1 / c
( 4 Fig: 17. Characteristics of corrugated circular waveguides [ 241. (a)
Propagation characteristics of corrugated circular waveguide with r, = 25mm, d = lOmm, b = 2mm, r, /ro = 0.714. (b) Lowfrequency cutoff mode chart. (c) Highfrequency cutoff mode chart.
0
10 
g 20 
2 w
W 30  A 2171
Fig. 18. Transverse fEld patterns for corrugated Waveguides [24] in region r < r,. (a) Transverw electricfsld pattern of HE,, mode under baianced hybrid conditions for r, /r, = 0.9. (b) Transverse electric and magneticfsld pattern of EH,, qode under balanced hybrid conditions for r, /ro = 0.9. (c) Transverse electricfield pattern
Crosspolarized component of HE,, mode as a function of r/r, in of HE,, mode under balanced hybrid conditions for rl /ro = 0.9. (d)
corrugated pcular waveguide for r,/ri = 0.9 and plane J, = 45O. Parameter x, .
We shall require these expressions when we discuss radiation from a corrugated waveguide.
For waveguides for which rl /X >> 1, + 1 under balanced hybrid conditions and then the transverse fields of HEI, modes are essentially linearry polarized as Fig. 18(a) and (c) show. For the EH modes A =  1 and there is a strong cross polarized component as in Fig. 18(b).
When r / h is large
Sl(Xi, x:) x; cot (x:  x; 1. (40)
The balancedhybrid condition S1 = 0 is then satisfied when x:  x; = n/2 or (ro  r l ) = x/4. The impedance surface then appears as an opencircuit to the longitudinal electric field and as a shortcircuit to the longitudinal magnetic component.
To obtain the radiation pattern of an openended corrugated waveguide, many authors have used a KirchhoffHuyghens
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1481
X
0 1L 15 16 17 18 19 20
kr ,
(e) Fig. 19. Radiation from corrugated waveguide [24]. (a) Coordinate system for plane aperture. (b) HE,, mode radiation
patternsfor r l / r o = 0.85;  kr , = 9.245 (balanced hybrid condition);    kr, = 13.868, x= 1.27 (E plane);   k r , = 13.868 A = 1.27 (H plane). (c) Radiation pattems for HE,, and EH,, modes for r , / r o = 0.9, kr , = 14.47;  mode;    HE,, mode. (d) Radiation patterns of TM,, and TM,, modes r , / ro = 0.9 and kr, = 4.0 for TM,, mode rl/r,, =0.87 and kr, = 10.85 for TM,, modes. (e) Beamwidths of HE,, radiation pattern as a function of k r , for r , Ir, = 0.9  E p l a n e ,       H p b e .
EH,,
1482 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
integration over the aperture field in the region r < rl , assum ing all fields to vanish for r > r l . The procedure leads to pattern functions given by [ 241
Eo = (1 + COS e ) A L ( e ) + (COS e + s> e(e) (41)
 Z + = ( I + ~ c o ~ e ~ ~ ( e ~ + ( c o ~ e + ~ ) ~ ~ ( e ) (42)
where
Fig. 19(a) shows radiation patterns of the HEll mode pre dicted using (41) and (42) while Fig. 19(b) shows the depen dence of beamwidth on normalized frequency.
Fig. 19(e) also shows the beamwidth of a circular waveguide o f radius rl supporting the TEll mode; the pattern of the HE11 mode is considerably broader because its aperture field has a more pronounced taper. A high degree of pattern symmetry for the HE11 mode is maintained over almost a 2 : 1 frequency band thus demonstrating one of the principal advantages of hybridmode waveguides.
Under balancedhybrid conditions, pattern symmetry occurs for HEl, modes for which A = 1 and the Cartesian com ponent of the radiated fields become
E, = c1 [ u e ) (1 + pcos e ) + Qce) (cos e + (43)
E , = 0. (44)
Near boresight where 0 + 0, the functions L@) and @e) simplify, leading to expressions for E, and E,, for arbitrary A
E x = c J l ( x l ) J o ( ~ 1 ) ( 1 + A ) ( 1 + g , (45)
E y = C J 1 ( x t ) J z ( u l ) ( l  x)(1 +p)sin2+. (46)
Equation (46) shows that the crosspolar radiation is a maxi mum in the 45' planes and that it reaches a maximum offaxis, at an angle 0 such that
JI (u l ) = 0. (47)
To a very good approximation, the condition is satisfied
sin e = A/2rl. (48)
when
The result in (48) is borne out in practice but the maximum amplitude is not in close accord with (46) unless the wave guide diameter is very large. This is because the crosspolar pattem is quite strongly influenced both by the first higher order spaceharmonics and by the flange. The effect of a flange on the copolar radiation pattern of a corrugated wave guide was first considered by Knopp and Weisenforth [ 25 ]. They treated the flange as if it was infinite, which is a good approximation because the aperture field decays rapidly with increasing radius beyond r l . Their results show that the flange has little influence on the copolar pattern.
Parini, Clarricoats, and Olver [ 26 ] have extended the above method to the crosspolar field and have also included the first pair of higher spaceharmonics. F i g . 20(a) shows predicted and measured crosspolar patterns for the waveguide shown in the inset to Fig. 20(a). The agreement is seen to be much improved compared to that of the surface impedance model. Also. Fin. 20(b) indicates that the freouencv at which cross
0 2 1 1 3 4 5 6
ww,
(C)
Fig. 20. Radiation characteristics of corrugated waveguides [26 ] including the effects of the m e and spaceharmonics. (a) Measured (curve b) and theoretical (curves u and d) crosspolar radiation pat terns of openended cormgated circular waveguide shown in inset a = 50mm; b = 25mm; c = 11.6mm; d = 10mm; frequency = 9 GHz. Curve ta b based OR surface impidonce model, curved includea effects of spaceharmonics and flange. (b) Maximum value of crosspolar
a is based on surface impedance model, curve b includes space power aa a function of frequency for waveguide of Fig. 20(a). Curve
harmonics, curve c includes .the effect of a flange, curve d includes
minimum crosspolar radiation. both spaceharmonics and the flange. (c) Optimum slot depth for
polar radiation is a minimum, is appreciably shifted between the approximate and more exact models.
The optipwm slot depth for m m u m crosspolarization is shown in Fig. 20(c); because of the flange, the asymptotic value approaches 0.3 rather than 0.25.
To obtain the pattern of a paraboloid illuminated by a corrugated waveguide we may use (2) while the aperture efficiency is obtained from (4). Calculations nerformed bv
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1483
e.,% Fig. 21. Aperture efficiency u function of wmkngular aperture for
parabolic reflector fed with cormgated circukr waveguide fead [?SI. Parameter r, / r ; .
Rudge [27] show that the pattern symmetry and cross polarization of the secondary pattern closely follow that of the prime feed. Under balancedhybrid conditions the cross polarization loss vanishes and if qP = 1 the aperture efficiency is given by
2 cot2 8o ilea F(8) tan 2 2 u
t7a = I)irls = * (49)
J " l F ( e ) i z sinede
Fig. 21 shows qa as a function of eo for different values of x i = krl and rl/rO chosen from Table I so as to satisfy the balancedhybrid condition. The maximum gainfactor a p proaches 84 percent in the limit as Bo becomes small which is precisely the result obtained from consideration of the focal field [ 71. Unless higher order modes are used to synthesize the outer rings of the focal plane distribution of Fig. 6 , the best compromise is obtained when the feed just encloses the central region of the focal field. For the frontfed paraboloid or Cassegrainian equivalent, the aperture efficiency r) is then given by
Qa = 1  r: (UO). (50)
With Uo = 3.83, va = 0.84 in accord with Fig. 21. This is a limiting value of aperture efficiency, for very long focal length antennas. As another example of the optimum performance of a singlehybrid mode feed, for a reflector with F/D = 0.4
To improve on the aperture efficiency, Minnett and Thomas [ 181 have designed multimode corrugated waveguide feeds. If there are m modes, the waveguide radius is chosen equal to the mth zero in the electricfield focal distribution of the reflector. The amplitudes of the modes can be found using an optimiza tion program which compares the predicted radiation pattern of the modes with the desired pattern and adjusts for best match. Vu [28] developed this method using a graphic display computer. Table I1 shows theoretical results taken from a paper by Thomas (51 which compares focal plane matching and matching over the paraboloid aperture. These results are computed for the paraboloidal radio telescope at Parkes, Australia, which has F/D = 0.4. The figures f o r r ) ~ correspond to those of Fig. 7(b).
(eo = 63O), va = 0.69.
TABLE I
r1/r2 h.1 CI in (dB)
0.70 4.0300 0.75
10. 74 5.0728
0.80 12.68
6.6356 0.81 7.0473
14.98 15.44
0.82 7.5050 0.83 8.0167
15.98 16.55
8.5925 17.15
0.86 17.77
9.9916 0. 87
18.43 10.8530
0.88 11.8580 19. 15
0.89 19.90
13.0470 0.90 14.4730
20.75
0.91 2 1.64
16.2160 22.65
0. a4 0. a5 9.2453
~ 0.5.11
bl rl
Fig. 22. Reflection at homogeneouscorrugatedwaveguide junction
genboua circular waveguide of radius rh on a junction with a c o m  [24]. Reflection coefficients of TE,, mode incident from a homo
gated circular waveguide. Parameters r , / ro , see Fig. 16(b) and r , /rh.
The excitation of modes in a corrugated waveguide is a problem of considerable importance. Clarricoats and Saha (241 have used modal matching techniques to predict the excitation of the HElt mode when a TEll mode in a smooth wall circular waveguide is incident and Fig. 22 shows the reflection coefficient as a function of frequency for different values of step height. These authors have also obtained good agreement between their theory and experiment for the particular case of a narrow flare angle corrugated horn for which the theory might also be expected to apply (see Fig. 23). To improve the mismatch at the junction, several workers
have used a slot depth of approximately X/2 in the throat region of the transition tapering to the appropriate value at the far end. If the slot depth is chosen so that the guide wave length of the HEll mode remains approximately constant throughout the transition a rather low level of reflection occurs over a large bandwidth. Takeichi, Hashimata, and Takada [29] have used a ringloaded slot in the throat to improve the match.
In concluding our survey of corrugated waveguides, we note that Clarricoats and Saha [ 241 discovered that these structures
1484 PROCEEDINGS OF THE IEEE, VOL. 65 , NO. 10, OCTOBER 1977
TABLE I1 i
I O
tzt l o
F o\
frcqKncl GHz
Fig. 23. Experimental VSWR aa a function of frequency [24] mea sured at the throat of a narrowflareangle horn with 8, = 12' and theoretical curve for rl lro = 0.55.
can exhibit exceptionally low attenuation under very nearly the same conditions as those which produce optimum per formance when corrugated waveguides are used as antenna feeds. A very detailed study of the attenuation properties of several different types of corrugated waveguide was made and a preliminary account of t h i s work is to be found in the two papers by Clarricoats, Olver, and Chong [ 231.
3) Conical HybridMode Feeds: Singlemodecorrugated waveguide feeds as discussed in the previous section, will not generally prove suitable for Cassegrainian antennas as their beamwidths are too broad. One of the most convenient means to produce a narrowbeamwidth circularly symmetric radiation pattern is to employ a conical horn supporting a hybrid mode. Then, if a narrow flare angle is chosen and the horn is long, an electrically large aperture is possible yielding a radiation pattern of narrow beamwidth. Two main forms of conical hybridmode horns have been investigated. These are the conical corrugated horn and the dielectric cone, see Fig. 16. We shall consider first the corrugated horn as this is the type most widely used today in highperformance antennas.
I
0 5W 0.400 1 0.960
0725 0720 0 . m 0.827 0.SIY 0.927 0.874 0.870 0.946
0.91s 0.918 0.971
The cylindricalmode analysis of corrugated waveguide feeds presented above may be applied directly to horns with flare semiangles not exceeding about 5'. For larger angles, the above theory may be used to predict the amplitude variation over the horn aperture but to accurately predict the radiation pattern the phase variation over the aperture plane must be accounted for. However, it is usually more convenient to pro ceed directly to a design based on spherical hybrid modes. Around 1970 this approach was considered almost simul taneously by Clarricoats and Saha [24] , Viggh [301, Thomas [3 11, Booker [321, and, Jansen, Jeuken, and Lambrechtse [ W .
With reference to Fig. 16(a) it is assumed that the field within the horn may be described in terms of a single spherical hybrid mode analogous to the cylindricalhybridmode analysis described above. However, in that case the field representa tion could be made exact whereas for the corrugated conical horn this does not seem possible except in special cases. One of these is where the wave impedance E R I H ~ is infinite at the mouth of the slots. Otherwise, to provide a match to the field in the interior of the horn the impedance must be a function of position along the horn and we are not strictly justified then in using the method of separation of variables in deriving the characteristic equation. Nevertheless, as excellent agreement is obtained between experimental and theoretical radiation patterns for horns with flare semiangles up to about 70°, the assumption is evidently a good one. The discrepancies which occur for larger flare angles do so mainly for another reason. Fig. 16(a) shows that the horn is con nected to a circular waveguide. It is assumed in the analysis that only one mode is excited in the horn if, as is usual, a single mode is incident in the waveguide. This assumption fails at large flare angles and while Clarricoats and Seng [34] attempted to extend the sphericalmode analysis to larger angles they encountered serious problems in predicting the excitation coefficients of the spherical hybrid modes. An alternative transform technique has been developed by Hock ham [35] and this yields excellent agreement in the case of the 90' horn; mention of his method will be made later in this section.
The modal field in the horn can be obtained directly from electric and magnetic scalar potential functions then the radial components of the electric and magnetic fields are given by
h i Z ) ( ~ R ) P F (case) (51)
HR =  K E R (52)
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1485
h p ) kR and e (cos 8 ) are respectively spherical Hankel and Legendre functions with Y in general noninteger while x is a hybrid coefficient which depends on the boundary conditions.
In a manner analogous to that for cylindrical corrugated waveguides, see Section I IC2 , the other field components EO, E g , H e , and Hg may be obtained from ER and HR . On applying field matching at the boundary, a characteristic equa tion is obtained for the separation constant Y which governs the dependence on 8 of the fields in the aperture. This constant is a function of the semiflare angle of the horn e l and of the wave admittance at the surface of the corrugations. On introducing
20 = &/eo)'/2.
the characteristic equation for v is
(54)
Under balanced hybrid conditions the slot depth d = h/4 and (54) reduces to
ky"(e1)i2 = m 2 (55)
where qv is formed by taking the derivative of Py" (cos 8 ) in (53) with respect to cos 8 . Equation (54) is valid when the horn flare length R is large enough for an asymptotic approx imation to be used for the spherical Hankel function. The dominant spherical hybrid mode is classified as HEll to main tain consistency with the notation for the cylindrical hybrid case then, on placing m = 1 in ( 5 5 ) we obtain a very simple equation for Y. The special case 8 = 60' yields an integer value for Y namely 2. In general the value is noninteger as Fig. 24 shows. Chan [361 has noted that within the range 7' < 8, < 82', (v sin 8 , ) varies almost linearly with leading to the empirical relation
rn
2.4s  0.0128~ sin e l V = (56)
Over the above range, values of v obtained from (56) lie within 1 percent of those obtained from ( 5 5 ) . On comparing ( 5 5 ) with m = 1, with (34) we find that, when 8 is small so that sin 0 * 8 1, the forms am equivalent as x = K r l may be identified with [ v (v+ 1)l1l2 e l .
Under balancedhybrid conditions, the 6 dependent variation of the aperture field is governed b_y functions eg , e@, h e , and hg which are related. With m = 1, A = + 1 then
e g = T e g h g = T Y o e g a n d h g = 7 Y o e 6 (57)
Again when 0 is small
eg =J~([Y(Y+ 1)1'/*e) (59)
as for the HE11 mode in a cylindrical corrugated waveguide. The farfield radiation pattern may be obtained by Kirch
hoffHuygens integration over the spherical cap R = R 1 to
V
Fig. 24. Solutions of characteristic equation for HE,N and EH,N modes in corrugated conical horn.  modes in uncorrugated horn 1241.
at a farfield point P [ 241
and (I = kRo cos 8'
b = kRo sin 8'.
As with the cylindrical corrugated waveguide, pattern sym metry exists about the axis 8' = 0. There a maximum occurs for HE1, modes while EHI, modes exhibit a null, likewise for all modes with m > 1. Also, under balanced hybrid condi tions there is no crosspolarized component in the radiated field if the incident mode in the horn is linearly polarized.
The radiation pattern can also be found [ 291, [ 371 from an expansion of the fields over the spherical cap in terms of free space spherical modes. Under balancedhybrid conditions, the pattern normalized to boresght is then given by
under balancedhybrid conditions, for the electric fields
1486
r
PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
o h ' A ' 10 ' ' 1: ' t . 13 ' ' 1: ' " f r w e c c y ItHz1
15
Fig. 25. Measured and predicted beamwidth as a function of frequency for a conicll cormgated horn with 0 = 12' (Je*en rml Roumen i401).
d e e'deg. f
Fig. 26. Theoretical and experimentally measured radiation patterns
points. theoretical average of E and H plane values obtained by [24] XH plane experimental points. 0E plane experimental
sphericalmode analysis.  obtained by cylindricalmode analysis assuming constant phase over plane aperture.
(a)+) e = 120. (d)(f) (a) F = 8.5 GHz (d) F = 8.5 GHz (b) F = 9.0 GHz (e) F = 9.0 GHz (c) F = 11.0 GHz (f j F = 11.0 GHz.
where
Clarricoats, Olver, and Saha [38] have used this method and have found that convergence to within 1 percent is obtained with N = kRo . For wideangle horns a good approximation to the final pattern is obtained with only six terms. For long narrow flare horns, Aubray and Bitter [ 391 have found that an alternative expansion using LaguerreGaussian functions leads to a considerable computational advantage. Both of the above methods have the advantage that the near field behavior of the radiation pattern can be determined with ease.
Many workers have made satisfactory comparisons between predicted and measured patterns of cormgated conical horns in the range 5' Q e l Q 70'. Fig. 25 shows beamwidth as a
unrfwrnly illuminated , / circular aperture
S 1 I I I I I 1 1 I 1 2 . 3 4 5 6 7 8 9 1 0
nwmdised aperture diameter(%)
Fig. 27. Gain of a conical corrugated horn, under balanced [ 8 ] hybrid conditions, as a function of normalized aperture diameter. Parameter horn semiangle (Chan [ 36 I).
function of frequency for a horn with = 12' as predicted and measured by Jeuken and Roumen [40]. The high degree of pattern symmetry and constancy of beamwidth are notable features. Fig. 26 shows patterns obtained by Clamcoats and Saha [24] for a 30' horn. While Fig. 27 due to Chan [ 361 shows theoretical gain curves for horns operating under balanced hybrid conditions. He has also found that the 3, 10, and 20 dB halfbeamwidths of horns whose length is chosen to give maximum gain, are given by the following expressions to within an error of less than 1.5'
e(3) = 0 .358~
e(1o) = o m 1 e(2o) = 1.14e1.
The formula agree well with the results shown in Fig. 26 whereas the horn of Fig. 25 did not satisfy the maximum gain criterion.
As previously mentioned, the assumption that only the HE11 spherical mode is excited in a corrugated horn when a TEl l mode is inocident in the circular waveguide, fails when 81 exceeds 70 . A feed of particular interest is one with 81 = 90°, for this is a prototype for the optimum feeds described previously. T o predict the performance of such feeds, Hock ham [ 171 uses a transform method. He assumes that the corrugated surface is embedded in an infinite ground plane and then writes the total transverse magnetic field in the radiation space in terms of the tangential electric field in the aperture plane. The electric and magnetic field over the corrugated surface comprising shortcircuited coaxial waveguides and the
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1487
€plane 1
/ c o p o I ~ I
I I ' V I I I I I
*eg
72 36 0 36 72
Fig. 28. Predicted and measured radiation patterns for 90' corrugated horn with three sloe [ 35 1.
exciting circular waveguide are expanded in terms of normal waveguide modes, then, by allowing the radiation point to lie in the corrugated surface the amplitudes of the modes are determined. Thus the electric field over the aperture is determined and the radiatiop pattern also found. Fig. 28 shows the predicted and measured radiation patterns for a corrugated surface with three slots embedded in the ground plane together with those of the case of a plain flange. Evidently the corrugations improve the pattern symmetry as supported by the relatively lowcrosspolar pattem.
We next turn to the dielectric cone feed of Fig. 16(c). The origin of dielectric cones as feeds for Cassegrainian antennas can be traced to the pateqt specifications due to Bartlett, Moseley and Pietsch 1401 covering the perioa 19691971. These workers had discovered earlier that the efficiency Qf a small Cassegrainian antenna employing a subreflector sup ported by a thinwalled Fiberglass radome, was actually enhanced when the radome was filled with low permittivity dielectric. In a 1966 qrtjcle, a quqlitative explanation of the enhancement was presented on the basis that due to total intemal reflection of rqys within the dielectric cone, feed spillover was reduced. However, the investigations of Clam coats and Salema [41] show that there are other advantages which derive in part from the pattern symmetry of the hybrid mode excited on the cone. These authors appear to have been the first t o establish a design theory far feeds of this kind and the associated Cassegrainian antennas. A discussion of the design of the Cassegrainian antenna using a dielectric cone feed is presented in Section 111A; here, we address the primary characteristics of the cone feed excited by different launchers.
The analysis of propagation and radiation from dielectric cones follows closely that of corrugated horns although there is one important difference which is now discussed. With the latter, one princip+l assumption is that all the power in the incident mode is transferreg fo the dominant mode of the horn with no direct radiation occurring at tbe waveguidehorn discontinuity. This assumption works well provided the horn semiangle is less than about 70'. However, for a dielectric cone excited by a horn or waveguide launcher, as in Fig. 29(a) it is an invalid assumption. Instead, with reference to Fig. 29(b) it is found t4at only the energy associated with the spectrum of rays leaving the launcher at angles 8 < 8 ~ , is
Fig. 29. Radiation from a dielectric cone. (a) Diagram showing rays. For 0 < By rays are trapped and radiation pattern corresponds to that of dielectric cone. For 0 > BT the ratiation pattern is essentially that of the feed waveguide or horn. (b) Dielectric cone ShQwing coordinates.
tTansferred to the modal field of the cane, the remainder leaves the launcher in radiation. We note that a ray with 0 = 0T (which we call the trapping angle) impipges on the coneair interface at an angle to the normal just equal to the critical angle for totalinternal reflection. For angles greater than B T , the radiation pattern of the launcher and cone closely ap proximates that of the lavncher alone provided neither the cone permittivity is too high nor the angle of observation too close to 8 ~ . A theoretical analysis of a related excitation problem has been undertaken by Saqmut and Snyder [421 and their work supports the above findings. In order to describe theoretically radiation from the cone, we must first determine the aperture field of the conical mode, which is assumed to possess the same azimuthal dependence as the mode in the launcher.
Consider Figs. 29; to obtain the characteristic equation for the separation constant v, which is required in order to obtain the fields, we assume that the field outside the cone behaves as if the cone were a dielectric rod of radius R 0 1. The charac teristic equation for propagation along a dielectric rod of permittivity E is
1488 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
9
2 'i Fig. 30. Separation constant Y as a function of dielectric cone semi
angle 8, for HE,, HE,, and HE,, modes 1411. Parameta for HE,, mode is normalized cone length kR, . Dotted curve shows Y for corrugated conical horn.
where
The corresponding equation for a dielectric cone with flare semiangle d l is obtained by making the following trans formations in ( 6 2 )
y : = k 2 ( E  1)x :
r l = R I O 1 .
Then, if the cone is very long compared to a wavelength, so that kR1 >> 1, the transformed ( 6 2 ) reduces to ( 5 5 ) the characteristic equation for the corrugated horn with flare semiangle d l , under balanced hybrid conditions. As most dielectric cones used in antennas have 8 < 15' the following approximation can be made
P;(COS e) = J~ ([v(v + 111 e}. Futhermore, the approximate solutions to ( 5 5 ) discussed pre viously can be of value in an initial design. Fig. 30 shows v as a function of for the first three modes of the cone with m = 1. For the HEll mode, normalized cone length kR1 is a parameter and the error in making the corrugated horn approximation is also revealed. For practical cone configura tions, the cone is usually truncated before reaching the vertex, thus values of kR 2 100 are more realistic than might at first be assumed.
Once v is found, the electric and magnetic fields over the spherical cap R = R l are obtainable as for a corrugated horn except that there is an exponentially decaying field in the regime 6 > e 1. The radiation pattern due to the mode is then
an@e from barewght. 8. deg.
(a)
angle from boresight degrees
(b) Fig. 31. Aperture fEld and radiation patterns for dielectric cone
electric field pattern of a 6 O dielectric cone with E / € , , = 1.1 excited excited by means of a corrugated waveguide I41 1. (a) Aperture
by a corrupted waveguide in the HE,, mode. Frequency 9 GHz,
HE,, mode amplitude. (b) Farfield radiation pattern of dielectric  measured amptitude  measured phase xxxxxx predicted
cone of (a).   Radiation pattern of waveguide and cone. xxxxxx predicted
Radiation pattern of corrugated waveguide.
HE,, mode radiation pattern.
determined by a KirchhoffHuygens integration over the spherical cap. Figs. 31 (a) and (b) show the aperture electric field pattern and radiation pattern of a dielectric cone excited from a corrugated waveguide, good agreement exists between measured and predicted results. Fig. 31 (b) shows that within the trapping angle the predicted pattern is in accord with the pattern of the HEll mode of the cone while outside the trapping angle the pattern compares rather closely to that of the launcher except for a small difference which is mainly due to refraction.
Clarricoats and Salema [ 4 1 ] have investigated a wide variety of launchers which included waveguides, conical, and pyra midal horns as well as corrugated horns (Fig. 32). Throughout they found support for the theory just described. The main
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1489
18
16 
1 . 4 
1 2 
X 10
0 8
06
0 4 
0 2 
0.1 t.
I 01 1 I I 0 02 04 06 08 1.0
Y
(b) Fig. 32. (a) Cassegrainian antenna employing a dielectriccone feed [ 411. (b) Subreflector shape for antenna      
(c) Normalized electric field distribution over subreflector (e scale) and main reflector aperture ( B  scale)       dielectric cone absent;  dielectric cone present. F/D = 0.25, D = 24cm., A = 0.67 cm, E; = 1.1, cone angle = loo.
dielectric cone absent ;  dielectric cone present.
application of the dielectric cone feed to date has been in highefficiency Cassegrainian antennas. But, as the above authors have noted, the technique provides a relatively inexpensive means of achieving a symmetrical radiation pattern given a launcher such as a conical horn which intrin sically lacks pattern symmetry. Brooking, Clarricoats, and Olver [43] have studied the use of pyramidal dielectric horns as an inexpensive means of achieving narrow shaped radiation patterns. They have developed a theory based on an assump tion due to Marcatili [44] and have obtained quite good agreement between experiment and theory.
The main disadvantage of the dielectric cone feed appears to lie in its susceptibility to the effects of precipitation if de ployed in an antenna without environmental protection. Dielectric materials are known to be available which are sealed against water penetration but their long term behavior does not appear to have been investigated. Nor has the behavior of foam materials in a vacuum environment been studied but it is generally assumed that they cannot be used in spacecraft applications. Measurements made around 9 GHz by Clarri coats and Salema [4 1 ] show that the attenuation of expanded polystyrene is so small that the added loss in a feed application could not be determined. However, polyurethane foam which has been used in some prototype dielectric waveguide feeds can induce a feed loss of a few tenths of a decibel which would be unacceptable in an application calling for a low antenna noise temperature.
111. REFLECTORS FOR PENCILBEAM APPLICATIONS
A . Introduction We begin by recalling the most important design require
ments that can apply to pencilbeam antennas in various
applications. These are:
a) high efficiency ; b) low crosspolarization; c) satisfactory sidelobe envelope ; d) low feed VSWR.
Of these, efficiency and crosspolarization have been treated in some detail in Section 11. The sidelobe envelope or general pattern shape is a composite requirement; at the most simple, it may imply a restriction on the height of the sidelobes near the main beam; at the most stringent, upper bounds may be placed on radiation in all directions. Fig. 33 shows the current CCIR recommendations for earthstation antennas with regard to farout sidelobe performance. The main reason for such limits on the radiation pattern is to minimize possible inter ference with other antennas. With the continuing expansion in microwave telecommunicationsystems, especially in areas of population concentration, the probability of such inter ference increases and overall pattern control becomes one of the major antenna design problems of the near future. In addition, a h g h level of radiation at large angles means an increase in noise temperature, important for low noise an tennas used in satellite earth stations. Radiation in the rear hemisphere, including spillover past the main reflector, illuminates the earth, which is the largest natural noise source external to the antenna.
Low VSWR is also a necessary requirement for low noise systems. Even a relatively small value of power reflected back from the feed can modify the operation of a parametric amplifier sufficiently to cause a significant increase in noise temperature. It is also essential in microwave terrestrial systems in order that interchannel noise conditions be met.
1490 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
angle, 0, between t h e a x i s o f m l n beam and the direcllon consldered Idegreest
Fig. 33. CCIR recommended sidelobe envelope for antennas of diam eter > IOOA, applicable to earthstation antennas.
Fig. 34. Four reflector configurations for pencilbeam applications. (a)
frontfed reflector. (d) offset tworeflector antenna. onaxis frontfed reflector. .(b) on axis dual reflector. (c) offset
I) Design Considerations: There are four main reflector configurations, using single feeds, which may be considered for the production of pencil beams, and these are illustrated in Fig. 34. They comprise the onaxis fed singlereflector .and dualreflector antennas, and their offsetfed equivalents. All of these geometries are in use; and each can have specific advantages in terms of the design requirements set out above. A general discussion of these requirements will now be pre sented with concentration on the methods necessary for their fulfilment using the four configurations of Fig. 34.
a) High efficiency: Assuming the use of a simple feed, a singIe reflector antenna, unless it is very small in terms of wavelengths, must be paraboloidal in shape. Departures from this shape will cause phase errors in the aperture which carinot be corrected by a simple feed. Given this, the production of high efficiency with a frontfed antenna is almost completely a matter of feed design. Th’is has been adequately treated in Section 11. However, the dualreflector antenna provides an additional degree of freedom. The classical Cassegrairiian antenna, consisting of a paraboloid with a hyperboloidal sub reflector, is designed by the use of geometric optics to produce a uniform phase front in the aperture of the paraboloid. The aperture amplitude distribution is unconstrained, and with a feed ?f normal type will have a taper as shown in Fig. 35. By redesigning both reflectors this situation may be improved,
Fig . 35. Geometricoptics aperture distributions for tworeflector antennas.  classical Caapegrainian.  dualshaped reflector ante.
and w i t h the geometricoptics approximation an aperture may be obtained which is uniform in both phase and ampli tude, the condition needed for highest efficiency. It must also be remembered that the geometric optics technique will only apply approximately for tworeflector antennas, because the subreflector is usually small in terms of wavelengths. To obtain significant efficiency enhancement in all but the largest systems, diffraction techniques must be used. High efficiency designs based on the above considerations will be discussed in detail in Section 111C.
b ) Low crosspolarization: From the discussion in Section I1 it is evident that feed design is once again of prime impor tance for onaxis fed systems. Excluding the feed, the most significant cause of crosspolarization is scattering from the struts used to keep the feed or subreflector in position Me chanical designs must be sought which eliminate or minimize these supports. For the frontfed antenna a solution would be the splashplate feed of Fip. 14, which requires no asym metric supports. For the tworeflector case the dielectric cone feed suggested by Bartlett and Mosely [45] and later de veloped by Clarricoats and Salema [ 4 l ] is a possibility (see Fig. 32(a)). Strut scattering can be minimized by making the subreflector as small as possible, thus allowing physically smaller struts to be used. The diffraction optimization designs to be discussed in Section 111C will be useful for this purpose. Alternatively, strut scattering problems may be avoided entirely by using the offset reflector configurations of Fig. 32(c) and (dl. Although these structures have a high inherent crosspokrization, the effect can be mitigated by careful atten tion to feed design, at least over a restricted frequency band. Two other problems, higher cost and difficulty in alignment have made offset reflectors less popular in the past, other than for specific applications, but as the design problems have been overcome their use has become more widespread.
c) Pattern control: We may consider two effects sepa rately; control of the nearin sidelobes; and control of farout sidelobes (including the effect of feed spillover).
(i) Nearin sidelobesIn an axially symmetric system, the nearin sidelobes depend mainly on central blockage of the main reflector aperture and the amplitude taper at the edge of the main reflector. Fig. 36 illustrates the dependence of the first sidelobe on the first of these effects. In Fig. 36(a) we see an assumed aperture distribution taking edge taper into account. The amplitude is constant up to a fraction al, then decreasing towards the edge according to the law
C), a, < a < 1. (63)
where Cdb = 20 loglo c is the approp@ate edge taper. Figs. 36(b) and (c) then show the variation of sidelobe level and aperture efficiency with these parameters. Fig. 37(a) shows
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1491
=I
0.75
0.5
0.25
0
70 t €0. 0 10 20 30
edge taper ldEl
(b)
15 1  m 20
30  (C)
0 10 2 0 30 edge taper (dB1
Fii. 36. Effects of aperture distribution. (a) Type of aperture distribu tion umrmed. (b) Variation of aperture efficiency with distribution parametem a. and C ~ B . (c) Variation of f i i sidelobe level with a, md CdB.
how central blockage leads to a sidelobe increase. The equiva lent blocked region yields a broad, low amplitude pattern which must be subtracted from the unblocked radiation to give the total. This reduces the mainlobe amplitude while raising the lst, third and other odd sidelobes. Fig. 37(b) graphs the level of the first sidelobe as a function of the rela tive blockage diameter, for different values of aperture edge taper. Obviously, since the efficiency decreases with increas ing blockage, it is advantageous to make the central blocking as small as possible. A tworeflector system may take ad vantage of its inherent design flexibility to effect a compro mise between efficiency and sidelobe level.
(ii) Far out sidelobesWideangle radiation from a two reflector antenna comes from the following sources: feed
25 0 Dl 0.2 0 3
blocklng ratio
Fig. 37. Effect of central blockage on aidelobe level. (a) Effect of aperture blockage (Silver [ 31). (b) Variation of fiw sidelobe lewl with aperture blockage. auniform illumination; blOdB edge taper. aI = O.5;c1OdB edge taper;d, = 0.25.
Fig. 38. Mechanisms affectiag farout sidelobe level. (a) Main reflector edge dction. (b) Strut diffraction. (c) Feed spillover. (d) Sub reflector edge diffraction. (e) Scattering of reflected power by feed.
spillover past the subreflector and subreflector rim diffraction; main reflector spillover and diffraction, energy scattered from the feed after reflection; and energy scattered from struts. These mechanisms are set out in Fig . 38. The means for reducing strut scattering are those discussed in connection with crosspolarization; the problem does not arise with offset systems, which therefore have a distinct advantage. Main and subreflector spillover can be effectively reduced by increasing the edge. tapers on the two reflectors. However, there will be a significant decrease in efficiency unless an optimum two reflector design procedure is used, as previously discussed. Very low subreflector edge tapers may then be accommodated by designing the reflectors so as to redistribute the power in the aperture according to the desired distribution. If, in addition, very small subreflectors are needed to minimize strut scattering, a diffractionoptimized design procedure is indicated.
1492 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
Fig. 39. Generation of feed mismatch by power reflected from sub reflector.
d ) Low feed V.S.W.R: The feed itself must of course be separately designed to fulfil requirements on reflection coefficient. However, when operating with an onaxis fed reflector an extra mismatch occurs due to power reabsorbed after reflection as Fig. 39 illustrates. This problem arises particularly with small Cassegrainian systems, and it can be solved by correct design of the reflector nearest the feed. As such designs remove power from the feed region they also improve far sidelobe performance due to feed scatter. Once again this problem does not arise with offset systems.
2) Summary: On axis fed single paraboloids will continue to be used for lowcost systems where the performance re quirements are not too stringent, and also in applications like radioastronomy where frontfed operation must always be available, even if a tworeflector system is used. Shaped dual reflector antennas are necessary when pattern control require ments are strict, as for satellite ground station antennas. These must preferably be designed on the basis of diffraction theory, although geometric optics may be used for large antennas. A full discussion of the necessary design procedures will be found in the next section. Offset reflectors offer real ad vantages because of the absence of blocking, now that appro priate feeds are becoming available. We will not treat offset reflectors in detail, and the reader is referred to the forth coming review of this subject by Rudge.'
B. FixedBeam OnAxis Systems I ) GeometricOptics Design: As discussed in the previous
section, a tworeflector antenna must be optimally designed if it is to satisfy increasingly stringent requirements on efficiency and radiation pattern. If the antenna is large, successful design may be undertaken using geometric optics. indeed, these techniques are widespread and now form part of the classical body of antenna theory. A brief summary follows in order that more recent developments can be seen in perspective.
Two approaches to optimum dualreflector design appeared at about the same time. The first was due to Green [46], who produced an aperture of uniform phase and a desired ampli tude distribution in two stages, as shown in Fig. 40. Firstly, the subreflector was designed optically to give a farfield power pattern equal to the design pattern weighted by the space attenuation of the paraboloid. This led to the desired amplitude distribution in the main reflector aperture, but with a phase error due to the subreflector not being a hyperboloid. This phase error was then corrected by a small perturbation of the main reflector, the resulting error due to a change in the main reflector space attenuation could then be corrected by repeating the entire process.
' O p . cit., to be published in PR~CEEDINCS OF THE IEEE.
anplhde desied d r v r d
Pha
1 1 o p t l a l
dwlation design
main refbctu S l b f d l e C t o r profile >
4
1 recdcdatm
onglitude of
pattern
I
Fig. 40. Geometricoptics design of a twomflector antenna for a given p h w and amplitude d d b u t i o n in the aperturemethod of Greeo (461.
Fig. 41. Galindo's method of optical tworeflector design (witliuns
phase, showing the deviation from I chadcd CutcOrunun [48]). (a) Geometry. (b) Typical result for u a i f o ~ mplitude a d
Ute
CLARRICOATS AND FQULTON: MICROWAVE REFLECTOR ANTENNAS 1493
A more thorough analysis was undertaken by Galindo [ 47 I , who derived a set of equations whose solution yielded directly the two reflector profiles to give a specified phase and ampli tude distribution in the main reflector aperture. A concise description of this method may be found in the book by Rusch and Potter [4 ] , and when applied to a uniform aper ture, in a paper by Williams [48]. For this reason, only a very brief account will be given here.
Fig. 41(a) gives the relevant geometry. Essentially three properties of geometric optics are used to derive a sufficient set of equations; these are as follows.
i) Constant ray path length: The path length of any ray r + r' + r", from the feed to the aperture equiphase surface, must be constant.
ii) Power in each small tube of rays must remain constant, even after reflection, and in the aperture this must correspond to the desired aperture illumination Z(x).
iii) Snell's law (equality angles of incidence and reflection) must be satisfied at each reflector surface.
These conditions lend to the following three simultaneous equations, which must be solved to give the required reflector shapes.
Let the feed have a symmetric radiation pattern F(0) . Then for an aperture with plane phase front and amplitude dis tribution Z(x)
In general, an iterative process using a computer is necessary in order to solve the above equations. Fig. 4 l(b) gives a typical example of the subreflector designed by this procedure.
2) DielectricCone Cassegrainian Designs: In Section I1 we have described an approximate theory for the propagation and radiation characteristics of dielectric cone feeds. Here we consider the design of a Cassegrainian antenna in which a dielectric cone links the horn and subreflector as in Fig. 32(a). The antenna leads to increased efficiency for the following reasons.
a) In common with all Cassegrainian antennas, there is an improved illumination of the main reflector, compared to a primefocus feed, especially when F / D is small, e.g., 0.250.30. Also, the shaping of the subreflector which is now necessary for a constantphase front across the mainreflector aperture fortuituously provides near optimum power distribution.
b) When the mainreflector diameter is small, the sub reflector diameter can be reduced to a level where blockage is tolerable without raising spillover to an unacceptable level. This improvement arises because energy is guided between the horn and the subreflector.
c) The feed pattern has both a high degree of symmetry and low sidelobes, these properties being conveyed to the secondary pattern of the antenna.
In their design procedure, Clanicoats and Salema (411 have used ray optics methods to determine the shape of the sub reflector in order that, for a given dielectric cone, the phase across the main reflector is constant. They have also used an iterative procedure to obtain a suitable aperture distribution Fig. 32(b) shows that the greatest departure from hyperbolic
TABLE 111 PARAMETERS OF ANTENNAS
Parameter
Main reflector diameter D Main reflector focal length F Design frequency f Subreflector diameter D, Effective blocked diameter D, Dielectric material: expanded
polystyrene with (approximate) density
€1
subreflector D,
Relative dielectric permittivity
Dielectriccone diameter at the
Dielectriccone semiflare angle Bo Semiflare angle subtended, at the
dielectriccone apex, by the subreflector OM
mouth of the horn Dielectriccone diameter at the
Launching horn
Distance between launchinglmrn apex and dielectriccone apex
Launchinghorn mouth diameter Launchinghorn semiflare angle
Antenna A
1.22 m 305 cm 112 GHz 20.3 cm 20'7 cm
80 kg/m3
1*101*12
183 cm 5.9"
6.52"
11.9 cm conical
horn
34 cm 125' cm 14'

I
I
Antenna B
0.61 m 17.8 cm 17.7 GHz 9.4 cm
10.6 cm
80 kg/m3
1.101.12
8.4 cm 9.5"
10.6"
4.95 cm conical
horn
0
12" 5.2 cm
shape occurs near the reflector vertex. As a consequence, energy is excluded from the blocked portion of the aperture as in Fig. 32(c), an effect which both enhances efficiency and lowers the antenna VSWR. For comparison we show the aperture distribution obtained in the absence of the dielectric cone but with the same HEll mode feed distribution. The subreflector diameter D, and edge illumination E, are deter mined by considering that for the same type of aperture illumination:
a) A large subreflector decreases aperture efficiency (by blockage) and increases relative firstsidelobe levels. However, it minimizes spillover.
b) For the same aperture blockage, a lower value of edge illumination yields lower relative sidelobe levels but also lower aperture efficiency.
To determine the influence of the edge illumination on the frontback ratio an analytic result due to Kritikos [491 may be used:
Gn = 20 loglo ($)  E, + G
where G p and E, are expressed in dB and G is expressed in dB relative to isotropic, E, is the ratio between fields at the edge and the axis of the reflector (and is usually a negative number when expressed in dB). To obtain a high front/back ratio, a deep reflector and low edge illumination should be used. Also, a higher gain antenna, irrespective of size, yields a higher front/back ratio than a lower gain antenna with the same F/D ratio and edge illumination.
Once D , and E, have been chosen the cone semiangle 8 0 is chosen. It must be less than e,, the complement of the critical angle for total internal reflection for the dielectricair interface:
4 2 e,, = cos1 (:) .
1494 PROCEEDINGS OF THE IEEE, VOL. 65. NO. 10, OCTOBER 1977
TABLE IV MEASURED PERFORMANCE OF A 1.22M DIAMETER
CASSEGRAINIAN REFLECTOR I
E plane Average E plahe , H plane Average H plane
The smaller OD
~
Frequency, GHz
62.7 70.2 55.2 652 705 Efficiency, , 60.0
41.30 41.81 40.71 I 4077 41.05 4034 Gain,dBi*
115 I 10.5 I
b i r d sidelobe
aximum value of 54  6 2  sidelobes for angles larger than 110",dB
Frontiback ratio, dB 60 62  63 65  * Calculated by integration of the m e w r e d pattern
the longer the dielectric cone. Although in dure by including the effects of diffraction. These attempts not undesirable it may force the launcher to will now be discussed. principle this is
protrude from the back of the reflector which may be practi cally unacceptable. The larger 8D the shorter the cone but as the subreflector is then brought closer to the launcher there will be an increase in VSWR and reduction in efficiency if rays reenter the feed after reflection at the subreflector. A value of OD between 0.5 e,, and 0.7 e,, has been found to be optimum. Further details of the design procedure are con tained in the paper by Clarricoats and Salema. As stated, they use an iterative procedure in order to obtain the subreflector shape for an optimum aperture distribution. Subsequently Leong and Poulton [ 501 have developed a closedform expres sion for the subreflector profile and Salema and Seng [ 51 ] have constructed a computer program which yields an optimum design.
To verify the design procedure, two different antennas were designed and tested; a 1.22m diameter focalplane reflector for the 10.51 1 .SGHz frequency range and a 0.61m diameter reflector for operation in the range 17.719.7 GHz. The parameters of the two antennas are given in Table 111, while Table IV gives their measured performance. Generally the performance is significantly better than could be achieved for such electrically small antennas without the use of the dielec tric feed cone. However, an investigation of the crosspolariza tion performance of the antenna has still to be made and especially the effect of precipitation on crosspolarization when the antenna is unprotected by a radome.
C. DualReflector Designs Based OR Diffraction 1) Introduction: While the designs discussed in the previous
section offer a better performance than the classical Casse grainian, the limitations of geometric optics prevent theo retical gain f i e s from being attained. For example, the diffraction analysis of a GalindoWilliams antenna with a 400h diameter main reflector and a 40h diameter subreflector pre dicts a gain loss of 0.5 dB over the design value using g e e metric optics. For smaller reflectors this diffraction loss is much greater. It is evident that the assumptions of geometric optics are far from adequate when the reflectors are small in terms of wavelengths, and for this reason several authors have attempted to improve on the geometricoptics design proce
2) Optimum Illumination Design: Conceptually the simplest way to design for high efficiency is to separate the requirements for uniform phase and amplitude in the main reflector aperture. This is the method used by Daveau [ 521, who applies scalardiffraction theory to scattering by the sub reflector in order to design for an optimum illumination pattern. The phase may then be equalized by small changes in the mainreflector profile. This approach is a direct extension of the geometricoptics analysis of Green [46], see Section 111B1. Consequently, it is subject to the same approxima tions, notably that the main reflector is assumed to lie in the farfield of the subreflector. Furthermore, as small changes in the main reflector cause small changes in the amplitude distribution over the aperture several iterations may be required in order to produce satisfactory convergence. Despite these disadvantages, the method of optimization is elegant, and will now be outlined for comparison with later techniques.
The geometry of the problem is given in Fig. 42. The sub reflector shape p ( 8 ) must be found to give a radiation pattern whose power distribution is as near as possible to a specified pattern A($) . For uniform illumination of a paraboloid A ( $ ) would be proportional to sec4 $/2; alternatively, patterns of the form shown in Fig. 35 may be preferred giving lower spill over and lower sidelobes.
For a given initial reflector Po@), a scalar diffraction formulation gives the scattered field in the form
E o ( $ ) =l: a ( 8 , $1 exp [iwt $11 d e . ( 6 5 )
In this equation a@, $) and @(e, $) are, respectively, the amplitude and phase of the contribution to the far field from the point on the reflector p o ( 8 ) . We assume that a small modification to the reflector shape sp(e) causes a change only to the phase termin (65). If this phase change is ( a @ / a p ) s p ( e ) , then the field changes to E ( $ ) = E o ( $ ) + SE($), where
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1495
Fig. 42. Optimum subreflector shaping by the method of Daveau [ 52 ] phase change due to reflector deviation.
and a$/ap may be calculated geometrically. Fig. 42 shows the effect of a small change in reflector shape. For a given point on the reflector ( p , 8 ) and a given farfield angle $, the phase term in (66) is just
$ =  k ( p + R )
where R is the distance from the reflector point to the point in the far field. Changing p ( 0 ) to p + d p causes a change in total path length
 d p (1  cos (e  $11 from which
The power pattern also changes to E*E = E t E o + 6 (E*E), where
Defining vp(e, $1 = 2 Re W * a ( O , 4 ) (a$/ap) exp t i m , $)I 1
w * E ) =l:' vp(e, $ m e ) de.
we may write
6 p ( d ) must now be found which minimizes the meansquare error between the actual power pattern E*E and the desired pattern A ( $ ) . If B ( $ ) = A ($1  E t ( $ ) E ( $ ) , then the quantity to be minimized is
JJI 1
A gradient search procedure may be used for the purpose, but Daveau's method is more efficient, when it is applicable. The reflector shape is expressed as a truncated series of known functions. Following Daveau [ 521, we use Legendre poly nomials P , ( e ) for this expansion, although other orthogonal sets are also possible
Equation (68) may be written more concisely in vector form
p ( e ) = P T m . S
where P ( 8 ) is a vector of the first N Legendre polynomials, evaluated at 0 , and s is the vector of coefficients representing the shape. A small change in the coefficient vector 8s then
0 degrees
1 0 2 0 ~ 4 o 5 o w m m 9 0
Fig. 43. Amplitude radiation pattern of a subreflector OptimdY designed by the method of Daveau [52] .  desired pattern;  actual pattern.
leads to the small change in reflector shape
6 p ( e ) = p T ( e ) 8s .
Substituting in (67), remembering that 8s is independent of 0
6 @*E) = GT( $) * 8 s (69) where
The problem may now be restated as follows. Find the small change 8s in Legendre coefficients which will minimize
JJ, I
The solution is straightforward, and can be represented as
(70) Note that the first integral in (70) is an N X N matrix.
The above solution assumes that a linear relationship exists between 6 (E*E) and 8s. This relation, expressed in (69), will only be true for small reflector deviations. Consequently, several iterations of the above procedure will usually be necessary for convergence.
The main disadvantage of Daveau's method is that con vergence is not assured in all cases. In particular, increasing the number of terms in the representation of the reflector shape makes divergence more likely. This places an upper limit on the accuracy with which a reflector shape may be specified.
Fig. 43 illustrates the application of the technique to the design of a subreflector of diameter 40?, intended to give a uniform amplitude distribution for a main reflector of semi angle 60'. (Since it is the farfield pattern which is being shaped the diameter of the main reflector cannot be specified.) Evidently a f e l y good power pattern is obtained with a minimum in the feed direction. This latter effect eases the burden of feed matching which is so important in antennas for microwave communication applications.
3) Phase Matching on the Subreflector: Instead of designing the subreflector for a desired amplitude distribution, an
1496 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
into the design. From (71) and (72), amplitude matching is accomplished if
7 15
l o   d 0
dB 5 \
\ \  synthesized horn pattern \
O t  \ optlmurn farmode pattern
5t \ .a1 1 I I 1 I I I I I 0 2 I 6 8 10 12 1L 1 6 1 8
synthesized horn radiation patterns
6) Fg. 44. (a) Optimum subreflector design by phase matching. Phase
front of wave required to optimally illuminate main reflector. (b) Feed radiation patterns for the optimum design (Potter [ 531).
alternative is to use a phasematching criterion in order to obtain the subreflector shape. For this t o be possible we require knowledge of the fields in the vicinity of the sub reflector which, traveling outwards, will form the desired uniform fields in the mahi reflector aperture. In Fig. 44(a) sd represents an equiphase surface of this desired field. The reflector shape is then chosen, using geometric optics, so as to transform the feed phase front into the desired equiphase surface. Following the analysis of Potter [531, let the feed radiation pattern be
(71)
Assume also that the desired field just after reflection from the subreflector is known in the following form
E J P , V , 4) = [ s i n & + COS $ 4 1 F J P , V )
. exp {  j [kp  6 ( p , v ) l l (72)
where 6(p, v ) represents the deviation from a spherical phase front. To perform the required phase transformation, the sub reflector shape must satisfy the following relation
where 2c is the distance between feed and focus, and a is an arbitrary parameter specifying the subreflector position.
The amplitudes should also be matched, and this may only be done in Potter’s analysis by incorporating the feed pattern
and
This restriction on feed radiation pattern is a disadvantage. Fig. 44(b) gives the pattern required to illuminate a sub reflector of 51X diameter, for a main reflector subtending a semiangle of 70’. The production of such a pattern is a con siderable design problem, and any multimode solution as suggested by Ludwig [ 541, and shown in Fig. 44(b), will be narrow band.
To carry out the above design method, it remains to cal culate F , ( p , v ) and 6 ( p , v ) in (72). Potter, assuming the main reflector to be in the far field, expands the desired radiation pattern in terms of TE and TM spherical harmonics. Once the modal coefficients are known, the field in the vicinity of the subreflector may be found. Excellent treatments of spherical harmonics appear in many sources [ 551, [ 561, but since the results will also be needed for the next section, a brief sum mary of the important formulas is given in Appendix I.
For low cross polarization, the subreflector far field should be circularly symmetric
vo < v < Y o
elsewhere.
The pattern function F(v) may be chosen to be sec’ v/2 for high efficiency, or one of the patterns of Fig. 36 if other criteria are important. The circular symmetry implies that only modes with unity azimuthal dependence will appear in the expansion. This may be accomplished using standard techniques [ 551, with a result of the form
where E?: and E?? are modal Efields as discussed in Appendix. The reader is referred to the paper by Potter 1531 for details of the calculation. The above is reproduced only to discuss the accuracy of such expansions.
Note that, since outgoing fields are involved, the radial dependence of (75) involves functions h p ) (kp). Such func tions are very large for kp < n , becoming infinite at the origin. If the number of terms in the expansion is too large, or if the field is evaluated too near the origin, inaccuracies may result, and in any case the phasematching procedure will break down. Such effects are common in farfield to nearfield transformations of this t y p e , and represent a basic limitation on the applicability of the method. Fortunately, an alterna tive use of spherical modes avoiding this difficulty has been developed by Wood, and this will be discussed in the next section.
4) Field Correkltion on the Subreflector: A significant advance, enabling direct optimization of efficiency to be
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1497
Fig. 4 5 . Calculation of efficiency by field correlation on the subreflec tor surface.
undertaken, was made by Wood [ 5714 591. By using a method based on reciprocity, a concise expression for efficiency is obtained which only involves an integration over the subre flector surface. This expression may also be derived directly using the coupling theorem of Robieux, [ 8, equation ( lob) ] . Referring to Fig. 45, let (Ei, H i ) be the fields generated at the subreflector surface when an axial plane wave containing unit power impinges on the main reflector aperture; let (Et , H t ) be the fields of the feed transmitting unit power, also evaluated at the subreflector. Then, subject to an assumption equiva lent to the physicaloptics current approximation at the sub reflector surface, the following may be derived for the overall efficiency of the tworeflector system
7) = J*J (76)
where J is given by either of the following expressions, the integration being over the subreflector surface
or
J = l Et X Hi *dS. (77b)
The two expressions (77a) and (77b) are the result of the choice involved in the application of the coupling theorem. One may either calculate the coupling between the fields of the feed and the currents generated by the incoming wave from the main reflector, or the currents may be assumed to be those generated by the feed. In the general case the two results will not be the same, which at first may seem sur prising. It is, however, a consequence of the fact that the physicaloptics approximation does not obey reciprocity. A good discussion of this phenomenon is given by Harrington [ 601, who also describes an alternative derivation of the above formulas using the imageinduction approximation.
Wood [58] concludes that (77a) is the only correct form, and uses this throughout his analysis. In fact, either (77a) or (77b) may be used within the accuracy of the physicalptics approximation. The actual difference in the solution is very small for most cases which have been considered, and indeed it reduces to zero for a circularly symmetric primary feed.
Thus far, no approximations have been made regarding the calculation of the main reflector fields (Ef , Hi) . Ideally, these will be calculated taking into account all blockage effects as well as surface deformation. Wood [ 591 ignores surface perturbation and strut blockage, but includes subreflector blockage in his analysis. In terms of the various expressions
discussed in Section 11, equations (4) to (8), the efficiency given by (76) is
I
t7 = T)p vi V s r)x 7)b
where T& is the blockage efficiency due to the subreflector
Efficiency of a tworeflector antenna: The above formula tion leads to a simple expression for efficiency when the reflectors have circular symmetry. Referring to Fig. 44(a), let the feed produce the following fields in the region of the subreflector
E , = ~ E f e ( B , r ) c o s q 5  6 E ~ e ( 0 , r ) s i n q 5 + ; 6 f , ( B , r ) c o s q 5 .
The fields in the subreflector region caused by an incident
only.
linearly polarized plane wave are as follows:
Hi = ;H,,,, sin q5 + 6Hme cos q5 + pHrnp sin 9. Then, with the use of (77), the efficiency is
r) = J*J
where
+ r sin OH,,,, {rEfe  rEf,} dB. (78)
This expression simplifies even further if it can be assumed that the subreflector is in the far field of the feed as follows:
J =  n f ( 8 ) exp ( jkr) (tan BH,,,,[b sin v + p cos v 4” ;sin81 +  s i n ~ [ p H ~ , + p H ~ ~ ] } d ~ . P (79)
Calculation o f fields from main reflector: Equation (79) will apply irrespective of how the calculation of the incoming main reflector fields is carried out. The GTD may be used, and this will be discussed in section 111CS. Wood [581 employs a spherical harmonic expansion which, unlike Potter’s method, is convergent since it calculates the incoming fields generated by the reflector in a sourcefree region.
Referring to Fig. 46, s’ represents a general reflector surface, on which currents K = 2A X H are generated by an incident plane wave. We require the scattered fields at a point P in the interior region in terms of spherical harmonics centered at 0. To obtain both TE and TM modes, it is convenient to find the radial field components at P, E,, and H,. E, then generates the TM modes, and H, the TE modes. The fields due to the currents on sf are given as follows [ 561 :
r
(gob)
The expansions may be obtained by expressing exp (jklr  r’l/lr  r‘l in terms of the spherical mode functions n:, which themselves are proportional to the radial field components of the individual modes. Thus using the addition theorems for
1498 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
Fig. 46. Spherical modes generated by a current distribution. (1) interior region; (2) intermediate region; (3) exterior region.
 spherical modes [ 561
exp (jklr  r'l) Ir  r'l
@ n
n=O m=n = k ( 1 1 ~ r 1 f ( ~ , e, I $ ) ~ I ; ~ ( ~ ' , e', 9'). (81)
In this equation
n:: = (2n + 1)  z, (kr)P;: (cos e ) exp ( im9) [ (n  m)! 1/2
(n + m)! 1 where P:: (cos 0 ) is an associated Legendre function, and zn(kr) is a general spherical Bessel function, which becomes jn(kr) in rI?(r, 8 , I$), but h$')(kr') in I I r ( r ' , e', I$'). This choice of Bessel function assumes finite fields at the origin. By slibstituting (81) in (8Oa) and (80b) the radial fields are expressed as sums of the individual modal components. This allows the entire fields to be expressed as follows:
E = CanmEi," ( r , e,4) + brim ~ii (r, 8 , I $ ) } . n
n=O m=n
(82a)
H = {anmH:," ( r , e,I$) + b n r n E 3 r , 8 , 9))  n
n=O m=n
(82b)
where E;,", E:;, Hi,", H : i are the modal fields given in Appendix I , with radial functions #'I and the coefficients are
For a circularly symmetric system, the problem simplifies since only modes with unity azimuthal dependence need be considered. Then, in (82a) and (82b) only values of m of f 1 appear. The same method may be applied to find the fields in region
3, external to the reflector. Eqautions (82) and (83) are st i l l valid, with an interchange of the type of Bessel function used. For region 3, the modal fields must involve outgoing spherical Bessel functions hi2) (kr), while j n ( k r ' ) appears in the calcula tion of coefficients.
selected local sufoce
mdl f l ca i ton to sub dlsh shape
t secondary dlogram by K H S melhod
rnodtflcnltan to mom
reflector
Fig. 47. Generation of optimum reflector profiles by diffraction optimizationmethod of Wood [ 591.
Iteration procedure: Having calculated the mainreflector fields, and with known feed radiation, the subreflector shape may be chosen to optimize the efficiency expression in (78). Wood [ 5 9 ] uses an iterative method based on geometric optics to converge on the required shape. Unfortunately, precise details of the technique have not been published. By using a second spherical mode expansion for the radiated subreflector fields, constructive deformation of the mainreflector profile may also be implemented. The final tworeflector system then results from successive iterations on the main and subreflec tors. Fig. 47 gives a block diagram of the entire tworeflector computation.
Fig. 48(a) shows the results of the above procedure applied to reflector systems of various diameters, all with a semiangle of 80'. The feed is assumed to be a balanced hybrid plane aperture situated halfway between the reflector vertex and focus. The subreflector diameter is chosen to satisfy the geo metric optimum blocking condition.
Two designs are shown; DPwhere only the subreflector is optimized for a paraboloidal main reflector; and DDwhere optimization of both reflectors is carried out. For compari son, the diffractioncalculated efficiencies of a conventional Cassegrainian HP and a modified GalindoWilliams reflector MWW are also shown. Perhaps the most remarkable result is that diffraction optimization of the subreflector alone gives results better than the dualreflector geometricoptics design, up to a diameter of 400X. It is evident that shaping the sub reflector produces the most significant effect, only a few percent in efficiency being added by the simultaneous main reflector optimization. The optimum designs have the add itional advantage of lower sidelobe levels than the Galindo Williams reflector, as can be seen from the graph of radiation patterns in Fig. 48(b). The only disadvantage of the optimum designs is that they are inherently frequency sensitive, a con sequence of the design being performed at one frequency only. The bandwidth curves for a diameter of 375X can be seen in Fig. 48(c), for constant feed pattern beamwidth. If the reflec tor is required to operate at widely differing frequencies, the optimization may be modified to apply over the required band, see Section 111CS. In this case, for large reflectors, the geometric optics design may be preferred, although the better sidelobe performance of the optimum reflectors must be borne in mind.
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS
95
1499
Fig. 48. (a) Antenna efficiency versus mainreflector diameter for various reflector profdesWood [59l).HPclassical Cassegrainian; DD
Williams geometricoptics design; MWP, MW’Galindodesigned subreflector, paraboloid main reflector. (b) Secondary Eplane radiation diffractionoptimized main reflector, and subreflector; DPdiffractionoptimized subreflector, paraboloidal main reflector; MWWGalindo
patterns for the reflector profdes of (a), for a main reflector diameter of 375A (Wood [ 591). (c) Antenna efficiency versus frequency for the reflector protiles of (a), assuming a constant feed beamwidth; main reflector diameter 37%
5 ) General Reflector Optimization: The expression for Thus far we have examined various techniques for finding efficiency given by (76) may be written in the following optimum reflector profiles, all of which involve somewhat general form, once again referring to Fig . 45, heuristic methods. As set out above, however, the calculation
of p(v) to maximize 11 is close to the classical problem in the calculus of variations [ 6 11. Successful application of th@
v) . If the incoming fields are found by GTD this requirement is fulfilled. An advantage of this general approach is that, with
where only slight modifications, it can be applied to a variety of
F ( p , p, v ) = lrf(8) exp (jkr) {tan OH,, [ p sin v reflector problems other than efficiency maximization. Thus Poulton [62] has simultaneously optimized efficiency and
+ p cos v  sin 81 + @ / r ) sin v [pH,, + & , p ] } . (85) VSWR performance in a tworeflector system. Broadband
r / = P J J = l V O F ( p , b , v ) d v (84) technique requires a simple enough analytical form for F ( p , p,
1500 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
h
(b) F%. 49. (a) Calculation of incoming fwldr in the vicinity of a sub
reflector, using GTD. (bj Section through the plaue OQ, P.
results are also easily found. This general technique will now be discussed in some detail.
GTD calculation of main reflector fields: Let a plane wave polarized in they direction be incident parallel to the axis of a reflector, as in Fig. 49. The field at any point P ( p , V , @) may then be obtained, using GTD, as the sum of three components; the geometricptics term; and two edgediffracted contribu tions from the reflector rim
H = H c o +HD,+HDz. The geometricoptics term may be obtained by conventional
methods. The GTD techniques of Keller [63], [64] and Kouyoumjian et al. [ 651, [ 661 must be modified for the calculation of Ho and HD ) to take account of the focussing properties of the reflector. This has been done by several authors [67] [69] and the results may be summarized with reference to Fig. 49, if the incident H field amplitude is Ho
HD1 (P, v, $1 = HO d z e x p  (jkp) {D, cos $ sin @
DhCOS#D,Sin$COS@~}
where
D, = V ( P , x  x o ) T V(P, x + x01 h
I cbl
In the above equation
s= ( 1 + cos a) P 1  P
and 3 { z } is the Fresnel integral Jr exp (  i t2 ) d t . The second edge contribution H D ~ may be obtained simi
larly. The advantage of this formulation is the extreme rapidity with which fields in the vicinity of the subreflector may be calculated.
Fig. 50. Deviation of refkctor profile.  optimum profils p(u) ; _ _ _ _ _ _ perturbed prof& p(u) + dp(u).
Modified EulerLagrange equations: Fig. 50 shows a small deviation d p ( v ) away from the optimum profile p ( ~ ) . Sub stituting in (84)) we obtain an expression for the correspon ding change in efficiency
61) = luo 2 Re {J'A} d p ( v ) d v
where A =    
aF ap d v (aF) ab is the classical EulerLagrange expression.
For the efficiency to be maximum it is necessary that 67 vanishes for any deformation d p ( v ) . This condition is satisfied if a reflector shape p ( v ) can be found such that
V p ( v ) = 2 R e { J * A } = O , forO<v<vo. (86)
Usually this equation is too complicated for analytical solu tion, and numerical search procedures must be implemented. A conventional steepest ascent method may be used, and Poulton [62] has obtained improved convergence with a con jugate gradients algorithm.
Broadband design: A significant advantage of the above method is its generality. Optimum performance over a band of frequencies may be accomplished merely by integrating the efficiency expression over the band. Thus we can define a new performance index Q, a weighted integral of the efficiency over the frequency band (ol , wz)
W t
Q = 1 1)(o) W ( o ) d o w1
where "(a), the weighting function, may be included if some parts of the band are more important than others. The equa tion for the reflector shape is a simple modification of (86)
w2
V p W = [ 2 R e {J*A} W ( w ) d w = O , O < v < v o .
A gradient search procedure may be used as in the previous instance.
VSWR minimization: Another important extension is to demand simultaneous optimization for efficiency and VSWR in the feed. In microwave communication systems where stringent matching requirements exist, the power reabsorbed by the feed after reflection can be a significant part of the total mismatch. Such power may be represented by an inte gration over the subreflector surfaces, with reference to Fig. 45 [70]
P = J t J 2 Jz = d E f X H f  d s
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1501
hyperbolad
I I t I I
8 3 8 5 8 7 8 9 9 1
frequency (GHzI
(a)
1 5
1 2
1 1 5
1 05
1 02
opt ettwency and VSWR
hyperbolold
I d I I I 8 3 8.5 87 8.9 91 9 3
frequency IGHz)
(3)
WSEGRAIN GEc*ETRy
(4
Fii. 5 1. (a) Geometry for optimum reflector shaping [a2 1. (b) An te efficiency veraua frequency for the ante purrmeters given in Table V. i) d.aicll Clgeg. inkn. ii) Subrefkctor optimized for high efficiency. iii) Subrefkctor optimized for high efficiency and low VSWR. (c) Cdcuhted VSWR VCT(NB frequency. i) Classical Case prinhn. ii) Subrefkctor designed for high efficiincy and low VSWR.
TABLE V
Main reflector diameter 2.0 rn F/D ratio 0.375 Subreflector diameter 23 cm Feedaperture diameter 5 cm 1 &dB beamwidth goo Design frequency 8.7 GHz
E f , iYf are the fields of the feed radiating unit power. If the system is circularly symmetric, J z takes on the form of (84), and is amenable to the same optimization process as was used for efficiency maximization
A very simple form for G ( p , P , v ) results when the feed has circular symmetry with radiation pattern f ( 8 ) , and the sub reflector is in its far field
+ p c o s v  ; s i n e } .
The two criteria to be satisfied are the maximization of q and the minimization of P. This may be accomplished most simply by maximizing a composite criterion
T = q  K.P
where K is an arbitrary weighting factor, its value depending on the relative importance of efficiency and VSWR. Other formulations are possible. For example, the reflector profile may be required which will maximize efficiency subject to the constraint that reabsorbed power is less than some specified level. Conversely, minimization of VSWR may be sought subject to efficiency remaining higher than a given value. Although both these formulations are probably closer to the physical requirements than the weighted composite, they involve inequality constraints which make computation diffi cult. For this reason the weighted composite criterion is pre ferred in practice.
The reflector profile may then be obtained by solving the following equation:
V p ( v ) = 2 Re (JYAI }  2K Re {J?A* } 0 (86)
where
Again, a simple averaging procedure allows broadband designs to be accomplished.
Tworeflector optimization: The above discussion has con centrated entirely on subreflector shaping in a dual reflector system. The reason for this emphasis is that the subreflector is by far the most important factor in highefficiency systems, and the only factor in VSWR minimization. The same tech niques may of course be applied to the main reflector. An alternative to the method of successive optimization of main and subreflector profiles described by Wood [591 is possible. If a GTD analysis is used to obtain the mainreflector fields,
1502 PROCEEDINGS OF THE IEEE, VOL. 65, NO. 10, OCTOBER 1977
I IE I
I I 1 I I I J 8.3 8.5 0.7 0 9 9.1
freqwncylGHz1
1 I I I I 1 1 1 2 0 90 60 30 0 30 60 90 120
azimuth 1')
(b)
Fu. 52. (a) Measured VSWR results [62]. i) Claasicd Caswpdaf8n;ii) Optimum ef€iciency/VSWR subreflector. iii) Feed only. (b) Mea m d rubreflector radiation patterns for optimum effiiency/VSWR subreflector and cksaical hyperboloid, at the design frequency of 8.7 GHz.
the possibility exists for simultaneous optimization of main and subreflector. This can be done only because the main reflector profiie appears explicitly as a parameter of the fields, and hence also is explicit in (85).
Applications: The above theory has been applied by Poulton [621 to the subreflector design of a dualreflector system at a single frequency, the main reflector being a paraboloid. The system is shown in Fig. 5 l(a), the parameters being those given in Table V.
The aim is to produce a subreflector profiie so that, at the design frequency, a low total VSWR is obtained with no decrease in overall efficiency, compared with the original hyperboloidal subreflector. The feed pattern used in the design was measured in the far field, and it was assumed that the subreflector would be always in this region. The design was carried out using two different values of the weighting factor K. 1) K = 0, corresponding to a pure efficiency optimization; 2) K = 20, corresponding to a fairly heavy weighting on VSWR. Figs. 51(b) and (c) give the predicted results for both cases over a frequency band surrounding the
1
0 6 rnax
Fig. 53. Variation of reccivcd fields for different dbtg~ces p from paraboloidal focus, p , > Pn > p , .
design value. Fig. 51(b) shows a substantial increase in effi ciency for K = 0, with only a small reduction from this value when VSWR is also taken into account. Fig. 51(c) gives the VSWR over the same frequency band, predicting considerable reduction from the classical Cassegrainian value. Practical verification of the design has been obtained, and is shown in Fig. 52(a). It can be seen that the composite performance of the optimum system is almost entirely limited by the feed, and represents a significant improvement over the unmodified reflector. A similar improvement in gain is promised by the subreflector radiation patterns given in Fig. 52(b). Compari son with the patterns of a hyperboloid shows a greater EH plane symmetry as well as a shape more nearly approaching the sec' $12 pattern needed to efficiently illuminate the paraboloid. Broadband designs for the same problem, as well as application to prime focus systems, have also been reported [711,[721.
The optimum subreflector paradox: One unexpected result emerges from optimization studies on tworeflector antennas namely, that substantially a l l of the efficiency in crease may be obtained by shaping only one reflector. This result is unexpected because it is known that an aperture field which is uniform in both phase and amplitude is necessary for maximum efficiency. Indeed, geometricoptics designs require two variables, the tworeflector surfaces, to satisfy this require ment. Even the early work on diffraction optimization implicitly assumed that two variables were necessary. How then is it possible to obtain substantial matching with
only one reflector capable of variation? The explanation is that there is a hidden variable, namely the position of the sub reflector. It is still necessary to match both phase and ampli tude, even if this is done on the reflector surface. Fig. 53 giving the amplitude profiles of the focussing field at different distances from the focus, shows how this matching is accom plished. As the focus is approached the amplitude pattern changes from the sharp, geometricaloptics pattern near to sec' v/2 to the Jl (u) /u pattern typical of the focal region. The feed amplitude distribution, lying between these two extremes, is best matched at some intermediate value of sub reflector position.
IV. CONCLUSIONS The paper presents a review of techniques which are cur
rently employed to achieve high performance in reflector antennas intended to produce a single fxed main beam. For frontfed configurations the paraboloid main reflector is optimum and a hybrid mode feed should be employed for best performance. For reflectors with F/D larger than about 0.3 satisfactory corrugated feed designs exist and their behavior can be accurately predicted. For deep reflectors, with F/D around 0.25, feed designs have so far been produced largely by
CLARRICOATS AND POULTON: MICROWAVE REFLECTOR ANTENNAS 1503
empirical methods although these have yielded encouraging performance figures. New theoretical techniques are emerging which show promise for the synthesis of optimum feeds for use with short focal length reflectors and it seems possible that efficiencies approaching those of dualreflector configurations will be achieved, at least over a narrowfrequency band.
The realization of low crosspolarization in the radiation pattern has become an important goal in, certain apolications and again the hybridmode corrugated feed emerges as the main contender especially when only a single beam is required. However, when the reflector F / D is smah the feed diameter is also small and the contribution of the feed flange to cross polar radiation pattern becomes significant. It is expected that by the use of choked flanges the performance of such feeds will be improved so as to yield maximdm offaxis crosspolar ization levels approachirig  40 dB.
The dielectric cone feed has been little utilized although when it also supports a subreflector, it offers attractive per formance figures. The main reason for its lack of widespread acceptance, appears to lie in problems of fabrication, dielectric consistency and weather protection. It is believed that these problems will be solved eventually and for small axttenna applications where weight carries a premium, a plastic antenna with deposited metal surfaces, may emerge from commercial antenna product lines. It could be specially useful with the ultimate development of the higher frequency microwave bands for pointtopoint communication.
Cassegrainian antennas are almost universally employed where highgain highefficiency and low noise are essential requirements, as in earthstation applications. Initial designs were based on geometricoptics procedures and many antennas have been produced following the technique first described in detail by Galindo. Since then, improvements have been made using diffraction procedures even for antennas with aperture diameters of several hundred wavelengths. More recently it has been recognized that very h g h performance can be achieved in dualreflector antennas when the subreflector is optimally shaped even when the main reflector is maintained as a paraboloid. It is anticipated that this discovery will bring the Cassegrainian antenna into much wider use for microwave terrestrial communication applications. Looking to the future, it is recognized that stringent sidelobe performance consistent with the maintenance of other specifications will,feature as a target for design initiative. Ingenuity in both feed and re flector design will be called for if the most exactkg of future systems requirements are to be met. The increasing use of computer optimization procedures will feature widely in these developments.
APPENDIX: SPHERICAL WAVES In a homogeneous sourcefree region, the electromagnetic
fields are completely described by the vector wave equation
V 2 E t k 2 E = 0
V 2 H + k 2 H = 0 where k is the propagation constant of the medium.
Solutions to the above equatioh in spherical coordinates are called spherical waves. Two families of soiutions are of interest, characterised by their field components in the radial direction. These are the TE and the TM waves. The former have no radial electric field, whilst the latter have no radial magnetic field. Together, these families constitute a complete set for any sourcefree region bounded by two concentric spheres.
The field components of the spherical waves may be repre sented as follows. TE waves:
EiZ = jkV X
TM waves:
‘ 0
where nr is a solution of the scalar wave equation in spherical coordinates.
n; = (2n + 1) (n  m)! 112
(n t m)! z n ( k r ) P r (cos 0) exp (imq5).
In this expression Pr (cos 0) is an associated Legendre func tion of the first kind, and zn(kr) is a spherical Bessel function, of a type to be chosen in accordance with the boundary con ditions of a given problem.
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