Offset and Syhmetrical Reflector Antennas

San Fernando Valley State College

OFFSET AND SYHMETRICAL REFLECTOR ANTENNAS

H Polarization and Pattern Effects

A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in

Engineering

Daniel Francis DiFonzo

,January, 19 7 2 The thesis of Daniel Francis DiFonzo is approved:

San Fernando Valley State College

March, 1972

------

ii Acknowledgment

I would like to express my deep gratitude to my advisor, Prof. E. s. Gillespie , for his invaluable help and encouragement throughout the course of my graduate studies at SFVSC. I am also grateful to Dr. Geoffrey Hyde for several helpful discussions and for his gentle but persistent encouragement to complete this work and to Mrs . Barbara Scheele who typed the manuscript.

iii Table of Contents

Page No.

•••• 0 • • • • • • • • • • • • • • • • • • • • • • • ••• ABSTRACT . X

. 1.0 INTRODUCTION ...... •...... ••...... 1

2.0 FORMULATION OF THE PAT'rERN INTEGRALS 11 2.1 D�splaced Feeds and Arrays of

- Feeds .•...... •••...... ••.... e: • 2 5

2.2 Integration Technique ...... 29 2.3 Surface Definitions for Parabolic

and Spherical Ref lectors ...... 32

2.- 3.1 Parabolic Re flectors 32

2. 3.2 Spherical Reflectors 35

2.3.3 Discussion ...••...... 37

2.4 Feed Patterns ..•....••••...•.... 39

2.4.1 Generalized Huygen's

Source ...... • . . . . . • . • 39

2.4.2 TE Mode Rectangular 1 0 Waveguide 42

2. 4.3 Dual Mode Circular Wave- guide 44 n 2. 4.4 cos w Pattern .•...... 48

2.4.5 Generalization to Arbitrary

Feed Polarization 48

2.5 Pattern Quantities Obtained from the Computer Program ...... 53

iv Table of Contents (continued) Page No .

2.5.1 Principal and Cross- Polarized Field Com-

ponents ...... 56 2.5.2 Circular Polarization

Components ...... 60 2. 5.3 Pattern Information Pro- vided in the Program

Output- ...... 64

3.0 POLARIZAT ION EFFECTS IN SYMMETRICAL RJ:;-

FLECrrORS . • • • . • • • • • • . . . . . • • . . • • • • . • . . • . • 6 5

3 . 1 Parabola Currents .••...••..•.... 65

3.2 Generalized Huygen's Source Feed 69 3.3 TE Mode Rectangular Waveguide 1 0

Feed e e "' e e "' .., e e "' •••••••••••••• � • 74

3.4 Circular Waveguide Feeds .•...... 76

3.4.1 TE Mode Feed ..••..... 79 1 1 3.4.2 Dual Mode Circular Wave�

guide Feed . . . • . • . • • . . . . 87 3.5 Polarization Degradation Due to Beam

Scanning ...... 9 6

4.0 OFFSET-FED REFLECTORS ....•..•...... •... 102

4.1 Offset Parabola . .' ...... 10 5

4 .1.1 Scanned Beam • . . • ...... 111

v Table of Contents (continued) Page No.

4.2 Offset Spherical Reflector 112

4.3 A Comparison Between the Offset

Sphere and the Offset Parabola .. 118

5.0 CONCLUSIONS • • • � • • • • • • • • • • • • • • ! • • • • • • • • • 121

REFERENCES 123

APPENDIX A. DERIVAT ION OF EQUATION (2)·. 126

. + APPENDIX B. EXPANSION OF F IN RECTANGU-

LAR COORDINATES .•...... 128

APPENDIX C. OFFSET REFLECTOR COMPUTER

• • • • • • • • • • • • • • • • • • • 3 PROGRAM 1 0

. . . . . • • • . . . • . . . . 1 C .1 Discus sion 30 C.2 Description of Input Quantities

Needed by the Program ..•.. 131

vi List of Illustrations

Figure No. Title Page No. 1 Multi-beam Spherical Reflector 5

2 Offset Spherical Reflector •.•..�.. 8 3 Coordinate Defintions for Offset Reflectors ...... 1 2

4 Coordinates for Displaced Feed ••.. 27

5 "Side View" of Offset Parabolic Reflector ...... 34

6 "Side View" of a Spherical Reflector 36

7 Huygen's Source ..•..••..••...... • 41

8 Rectangular Waveguide Feed ...... • 43

9 Coordinate System for Secondary Pattern ...... 55 10 Currents for Parabola with Electric Dipole Feed •.•...... ••••••.•.... 68

l/J vs. f/D for Symmetrical Re- 11 max flectors .•.•..•...... � .... � ....•.. 71

12 Cross�Polarized Lobes for a Symmetrical Parabola with Electric Dipole Feed 73

13 Computed Cross Polarization vs. f/D for Parabola with Electric Dipole

Feed. D/A = 15 . ...••.•....•.•...• 75

14 Patterns for Parabola with D/A = 15 TE Mode Rectangular Waveguide Feed 77 . 1 0 15 Current Distribution in the � = 45° Plane for Symmetrical Parabola with

TE Mode Feed · · �···· ········· ···· 80 1 1 Radius of TE Mode Feed Required to 16 1 1 Provide -10 dB Edge Current for Symmetrical Parabola ··· · ······· ··• 8 2

vii

'· -- List of Illustrations (continued) . Figure No . Title Page No. 17 Peak Leve l of Cross-Polarized Current, Ky ', for TE11 Mode Circular waveguide Feed ...... 84 18 Pat.terns for Parabola with TE Mode Feed. D/A = 15 ·····�·······!2 .... 85 19 Maximum Level of Cross Polarized Lobe vs. f/D for Symmetrical Re flectors with TE Mode Feed ••...• 86 1 1 20 Cross-Polarized Parabola Currents vs . � for Dual Mode Circular Waveguide Feed. (a/A = 0.8) ..•..••.....••.. 90

21 Cross-Polarization Current Distribu-. tion for a Parabola fed with a Dual Mode Circular Waveguide when P�M is · Adjusted to Minimize Ky at Var1ous 0.8 .•... Values of �(�=�M). a/A = 92 22 Cross-Polarization Current for Parab ola with a Dual Mode Circular Wave guide vs . Evaluated at the Angle PT M O .•. where Ky has its Peak for PTM= 93 23 Cross-Polarization Lobes for Sym metrical Parabola with Dual Mode Circular Waveguide Feed ...•.•..... 94 24 Maximum Cross-Polarization Radiation from a Symmetrical Parabola with Dual Mode Circular Waveguide Feed vs . pTM ••• • • • • • • • •••• • • • • • • • • • • • • • • •• � 9 5 25 PTM which Minimizes the Maximum Cross-Polarization Current (ky) for a Dual Mode Circular waveguide Feed 97 26 Effect of Beam Scanning for Symmetri- cal Parabola ...... 98 27 Polarization Purity for Parabola with Scanned Beam . • ...... • . . . . . l·oo 28 Reflector Currents for Offset Reflec tors with "Vertically" Polarized Feed

··············�!""!".!! • ···· · Antenna . 103

viii List of Illustrations {continued} Figure No . Title Page No . 29 H-Plane Pattern for Offset

. Parabola ...... 10 7 30 H-Plane Patterns for Offset Parab ola with Generalized Huygen's Source Feed ...... � ...... 110 31 Offset Parabola with Beam Scanned 8° in the H-Plane ...... •...•... 113 32 Comparison Between Computed and Measured E-Plane Patterns for Offset Spherical Reflector ...... 114 33 Comparison Between Computed and Measured "H-Plane" Patterns for Offset Spherical Reflector .••..... 116 34 Effect of Focusing for Spherical Re flectors ...... 117 35 Comparison Between Offset Sphere and Offset Parabola with Scanned Beam • • . . . • • • . . . . . • . • • • • • • • • • • • • • . . 120

ix ABSTRACT

OFFSET AND SYMMETRICAL REFLECTOR ANTENNAS Polarization and Pattern Effects

by Daniel Francis DiFonzo Master of Science in Engineering January, 1972 An analysis of polarization and pattern effects is presented for parabolic and spherical reflectors which are either symmetrically or offset fed by "point source" antennas such as electric and magnetic dipoles, rectangular waveguides, and dual mode circular waveguides . A computer program is presented which yields the vector far field patterns of a number of feed and reflector combinations based on the numerical integration of induced surface currents . For symmetrical reflectors it is shown that "Huygen 's source" feeds can be. used to eliminate cross-polarized reflector radiation and that the small circular waveguide feed is almost ideal from this viewpoint. It is also shown that the introduction of controlled amounts of TM mode into a cir- 1 1 cular waveguide feed results in cross polarization reduction for a wide range of symmetrical reflector parameters . The scanning and polarization properties of offset spheres and parabolas are compared.

X 1.0 INTRODUCTION This thesis is concerned with polarization and pattern effects in offset fed and symmetrical antennas . In particular , an analysis is pre�ented for parabolic :and spherical reflectors which are either symmetrically f or offset fed by "point source" antennas such as electric and magnetic dipoles, rectangular waveguides , and dual

•mode circular waveguides . A computer program is pre sented which yields the vector far field patterns of a :number of feed and reflector combinations based on the numerical integration of induced surface currents . There is a current need for detailed pattern ;and polarization data for symmetrical and offset fed reflector antennas. For example , an application which has recently become very important relates to the rela tively new technique of "frequency reuse" in communications· i satellite systems . This concept takes advantage of the fact that the communications capacity of a geostationary satellite can be significantly increased when it uses a number of antenna beams which occupy the same frequency band. The increase in capacity is directly proportional to the number of beams transmitting independent informa- . tion at the same carrier frequencies. In order for this ·concept to be practical, the beams must be sufficiently isolated from each other to insure adequate signal�to-

1 2

, interference ratios as measured at the earth terminal 1antennas of the communications system. For two beams , originating from a satellite

• and occupying a common spatial region, such isolation can only come about from their polarization orthogonality . 'For beams which illuminate adjacent earth regions , the . isolation can result from polarization orthogonality and pattern enve lope shape . 1 Typical half-power beam widths for this application are in the range of one to four degrees and a typical goal for beam isolation is

30 dB. Furthermore , since the angle subtended by the earth as seen from synchronous altitude is approximately 1 7° , the "field of view " of a multiple beam satellite antenna may be as wide as +8 .5 degrees. Reflector antennas seem to offer the most practical means for achieving a modest number of narrow beams . Reflectors have been the object of intensive investigation for many years due to their ability to

form narrow beams while maintaining physical �implicity . However, their polarization properties have been studied in detail only for a number of special cases . Further more , th e scanning properties of front fed reflectors have generally been studied experimentally and on the basis of scalar pattern analysis of the symmetrical parabola. The effects of beam scanning on the antenna polarization have not been examined in detail. 3

The cross -polari zation properties of symme trical paraboloids have been studied by several inv�stigators. Condon/ showed th at , for a symmetrical paraboloid excited .by a dipole , there are four symmetrically disposed cross- polarization lobes at 45° to the principal planes of the antenna. These lobes are generally referred to as the

:"Condon lobes ." Jones3. concluded that if a symmetrical ... · reflector is fed with a combination of electric and magnetic dipoles such that the ratio of electric to magnetic fields is equal to the free space characteristic im pedance , the cross-polarized radiation in the far field will vanish. This type of feed is referred to as a

"Huygen •s source ." Koffman4 showed that cross polariza tion in a reflector whose surface is generated by a conic section and which is excited by electric and magnetic dipoles can be made to vanish if the ratio of electric to magnetic field intensities is equal to the eccentricity of the surface . Kerdemelidis 5 formulated and evaluated the pattern integrals for a symmetrical paraboloid excited by electric and magnetic dipoles and he examined the cross-polarized radiation both near and far from the antenna axis as a function of variations in the focal length and diameter of the reflector. Minnett and Thomas6 and Rumsey7 examined the characteristics of an ideal feed which would have no cross polarization and showed that the necessary requirement of a circularly 4

symmetric feed pattern can be satisfied by a hybrid mode 'antenna as exemplified by the corrugated horn . The effects of scanning the beam of reflector antennas have been studied with reference to pattern amplitude properties . Lo8 computed the beam deviation i factor of a paraboloid with a feed displaced from the focal point . Kelleher and Colernan9 experimentally . studied the degradation in gain and sidelobe levels for parabolic and spherical reflectors . Sandler1 0 and Ruze11 . used analytical methods to determine the effects of feed displacement in paraboloidal antennas . Ruze presented a series of graphs which depict the degradation in gain and sidelobe levels with scan as a function of the ratio of focal length to diameter. The pattern quality of paraboloidal antennas degrades due to aberration effects as the beam is scanned away from the reflector axis by displacing the feed. This limitation on the field of view has motivated the use of spherical reflectors for certain wide angle applications . The appeal of this class of reflectors sterns from the fact that, while a spherical reflector has an inherently greater amount of spherical aberration compared to a parabola, this aberration is not dependent on scan angle as long as the feed is constrained to lie on a focal surface which is concentric with the reflector surface as illus trated in Figure 1. For a sphere of 5

Spherical Reflector Surface " " "' Focal Sphere Feeds " " \1

R- f / / � / \ / / R / / / / /

\ \ \

Figure 1. Multi-beam Spherical Reflector 6

radius R, the focal surface is a spherical cap whose :radius is approximately R/2 . Since the spherical re-

i flector has no preferred axis , it "looks " the same to all feeds situated on the focal sphere and the beam direction coincides with an axis passing from the center .of the sphere through the feed. The properties of the symmetrical spherical

; reflector have been extensively studied and much of the , early theory comes from their application to optical sys , terns . Their suitability as microwave scanning antennas ·has been recognized for some time . 1 2 Lil3 determined some of the limitations caused by aperture phase error

! on point fed spheres and he determined the optimum focal parameters based on the allowable phase error . Several investigators have designed feeds which can cor rect for the inherent phase error in a spherical re flector. 14, 1 5 Further insight into the properties of spherical reflectors has been gained by an examination of their focal region fields . Hyde 1 6 used the method of stationary phase to calculate these fields and Ricardi 1 7 used spherical wave expansions to synthesize an optimum feed based on matching the fields at the surface of the reflector. The patterns of symmetrical reflectors , whether spherical or parabolic, are subject to the degrading effects of aperture blockage caused by the feed and 7

associated supporting hardware . This blockage can cause serious degradation in sidelobe levels particularly when several feeds are used to obtain multiple beams from a single antenna. These limitations motivate the use of offset fed reflectors for certain applications . In this case the feed (s) are removed from the reflector aperture and they illuminate it in an unsymmetrical way . Offset fed reflectors have been used for many years , particu larly in radar applications . 1 8 Pagones 19 considered an offset parabola excited by the class of feeds having a n pattern proportional to cos �, where � is the polar angle measured from the feed axis . He determined the gain factor as a function of the feed parameter and re flector parameters . Bem2 0 calculated the focal region fields of an offset paraboloid and he showed that the electric field distribution in the focal region is nearly the same as for a symme trical re flector with an equiva lent focal length . The beam scanning properties of an offset parabola are still limited by the effects of aberration caused by feed displacement. A particularly interesting geometry which may have application to wide angle systems is the offset fed spherical reflector2 1 illus trated schematically in Figure 2. As in the case of the front fed sphere , the feeds are located on the focal surface but they are removed from . the reflecting. 8

I I I Surface I I�\----- R .

Figure 2. Offaet Spherical Reflector 9

ap�rture . This arrangement results in a greater amount :of aberration compared to the front fed case but again ,

• this aberration is independent of scan angle. This arrangement should be useful with point feeds if the ratio of focal length to diameter (f/D ) is relatively high . For small f/D the possibility exists for design- ing a phase correcting feed.22 At any rate , the offset sphere is a relatively new and unexplored antenna geometry which could be useful in certain wide angle applications.

An important characteristic of offset reflectors which has not been investigated in detail is that of cross polarization. The unsymmetrical nature of the illumination for these antennas causes their c�oss-polarized radiation to behave differently than that of symme trical (front fed) reflectors. Because of the importance of offset reflectors in eliminating aperture blockage effects , it is essential that their polarization properties also be characterized for various commonly used feed antennas . The primary objective of this thesis , then, is to formulate the pattern expressions for offset fed reflectors and to present a computer program with which their far field properties may be conveniently determined as a function of the type of feed and the appropriate antenna parameters. In Section 2 of this thesis, the far field integrals are formulated and this forms 10

the basis for a computer program which is included in an appendix. In Section 3 the polarization properties 'of symmetrical reflectors are examined as a function of ·feed type and reflector parameters . It is shown for example that a small circular waveguide feed has nearly ideal polarization characteristics for parabolas with a low f/D. It is also shown that a dual mode circular waveguide feed can be used to provide nearly id�al polarization for a range of values of f/D . In Section 4, several examples of offset reflector patterns are presented in order to illustrate some of their polariza� tion and beam scanning properties. P . 2.0 FORMULA'fiON OF THE ATTERN INTEGRALS The geometry and coordinate system definitions

' · to be used for offset reflectors is illus trated in Figure 3 . The primed coordinates are those of the feed and ; reflector (i.e., the source coordinates). The unprimed coordinates are those of the far field pattern.

By a direct integration of the vector Helm- . holtz equations, Silver2 3 (following Stratton and Chu24)

' 'has shown that the far fields of a current distribution on a surface of infinite conductivity and containing

: only electric currents are

-jkr -+ e [K r �r (l) s

(2)

; where

-+ -+ EP, Hp are the far fields at P -+ r = r r is the vector from the origin to P (r is a unit vector) · p is the radius vector from the origin to an element of surface current K is the vector current distribution on the surface S

k = 2n/A; (A is the wavelength)

� = 376.7 ohms, the characteristic impedance of fr�� $pace (�Z0)

11 12

X � is measured in the x'-y' plane

Figure 3. Coordinate Definitions for Offset Reflectors 13

W]J = kZ0 The calculation of the far fields then involves

determining the current distribution K and then integrating over the reflector surface . The exact deter .mination of K involves the solution of an integral equation . 2 5 In the event a computer solution is being sought, the integral equation method is prohibitive for :all but the smallest surfaces because of its large 'storage and time requirements . Fortunately , it is not always necessary to : resort to an integral equation solution . A commonly i

• used method for approximating the reflector 's current . distribution is to assume that the currents are those which would be generated by an incident field that is reflected optically . That is , K is calculated accord- ing to

= 2n x (3) K H.1

-+ where n is the unit normal to the surface and H. is '1 the incident magnetic field. Equation (3), along with the assumption that the currents are zero in the shadow region of the reflector constitues the physical optics approximation for computing the current distribution . It is known to be a good approximation when the reflector 's radius of 14

curvature is large . FUrthermore , if the reflector is in

:the far field of the feed (which is presumed to be ithe primary source for the reflector currents) then the : :incident magnetic field can be expressed in terms of :the incident electric field as

p X E (4) �.

,Then, the surface current can be expressed as

A (p X E.) (S) �

The term [l(- (K.;);J in Equation {1) is jus t the component of K perpendicular to the unit vector r ; in the observation direction . In terms of the polar

unit vectors e, �at the far field point P,

( 6)

where K8 Equation (1) can then be written

(7} 15

•where 1 = ;��:j(j(�X (p X ( 8) s This is the basic expression wh ich must be solved in order to calculate the far fields . It is assumed that the reflector surface can

A -+ be described by a radius vector p � p (�,r,,�0 )n. Also,

the incident fiel. d E., produced by a feed antenna at l the origin of the coordinate systems in Figure 3, can be expressed in terms of its polar coordinates � and � ; as

- p E. = �, e jk l f ( :E o p

= jk E�(�,F,)� + E�(�,ui] e-:- p ( 9) [ p

The expressions for feeds displaced from the origin and for arrays of identical feeds will be described later . The feed pattern is most naturally represented as a function of � and � which are defined relative to the primed coordinate system in Figure 3. However, the definition of the reflector surface will usually be most conveniently defined in terms of the unprimed axes and the observation coordinates e and � are also defined relative to the unprimed axes. Therefore , it is necessary to re late the two coordinate systems in order to 16

carry out pattern calculations. This relationship is easily determined by using standard coordinate trans- ; forrnations .26 For the coordinate representation of 3, Figure it can be observed that the z • axis is simply w , rotated about the y-axis by an amount 0 (the offset

'angle of the reflector) relative to the -z axis. The

coordinate transformation relating these rotated sys- terns is

cosw 0 x' X o 0 -1 0 (10) y = y•

z sinw 0 -cosw ·. ' 0 0 z

·Also, since the transpose of the transformation matrix is itself,

0 sinw x' cosw0 0 X 0 y' = -1 0 y (11)

Z 0 -cosw z I sinw0 0

Expansion of Equation (10) and use of the transformations from spherical to rectangular coordinates yields explicit relationships between the polar coordinates of the two systems: 17

X = sinO cos� x' + sin• ' COS$ = 0 0 z

=cos� 0 sin� cos� + sin� 0 cos� (12a),

y - sine sin¢ = -y' = -sin• sin� (12b):

= coso = sin� x' - cos$ z 0 0 �·

= sin� sin$ cos� - cos$ cos� (12c) · 0 0 . .

The evaluation of the pattern integral re- ;quires the explicit form of the phase factor p·r . A unit radius vector p can be expressed in either of the two polar coordinate systems , where

' ' p = sin� cos� x' + sin$ sin� y + cos$ z ( 13a) :

p = sine cos¢ x + sine sin¢ y + coso z (13b)

Now upon substitution from Equation (12) into Equation (13b) the radius vector can be expressed in the un- . primed system as a function of the primed coordinate variables:

� p = (cos$0 sirt$ cos� + sin�0 cos� )x - sin$ sins y

(14) 18

The phase factor p·r is then given by

r·r = p· (sine cos� x + sine sin� y + coso z)

where p is given in Equation (14) . The result is

r costjJ sintjJ cos� sin� sine cos� P. = [ 0 + sintJJ0 ]

- sintjJ sin� sine sin�

+ (sintJJ0 sintjJ cos� - costjJ0 costjJ] cose {15)

Several other quantities must be established

· in order to complete the formulation of the pattern

expression. A useful formula for the quantity n dS can . be obtained from differential geometry . Given a surface i : defined by the curvilinear coordinates u and u , a l 2 . + + vector r = r(u ,u ) can be be described as 1 2

+ + + ar r = du + � du (16) au 1 au 2 1 2

The element of surface area is

a +r ar dS = __ _ x du du ( 17) au au 1 2 1 2

By the definition of the cross product, the vector dS

+ is normal to the surface r{u ,u ). The unit inward 1 2 normal is therefore given by + +

ar X --ar au au 1 2

n = + {18)

1 -- ar X ar I au au 1 2 , In terms of the coordinates in Figure 3, the expression l

� for n dS is

+ Cl n dS = --Xp (19) CltjJ

·where (as derived in Appendix A) ,

+ a A !e.= p (20) atjJ � p + p tjJ

+ .l.e_ = (21) a�

Expansion of Equation (19) with the use of Equations

(20) and (21) yields

dS 2 sintjJ sintjJ � d {22) � = [-p p + p :� + p :� �] tJi d c;,

+ Now, in order that K dS be evaluated, it is noted that

{23)

where, with Equations (9) and (22), it is found that 20

<�·Ef>� dS = pE sintJ; !e. + p E� !e.ar; 1j; ()tj; ;and

<�-�>E = - 2 sintJ; E + E f p [ tJ; 1j; E; i]

- P2 E sintJ; - p2 sintJ; ;:: 1j; E I; 1j; s

Therefore , Equation {23) may be written as

n x { p x E ) ds = p F � + F ; tJ; + d tJ; d , 1; { 24) f [ P tJ F I; �]

where

(25)

F = E sintJ; {26) tJ; p tJ;

( 27)

The pattern integral of Equation {8) now has the form

a a m ; -jkp rn _e ___ kpp·r = 1 ej d tJ; d {28) i J j F p I; ljl E; min min For numerical computations , it is necessary to express the vector F in rectangular coordinates since 21

'the spherical unit vectors vary over the region of inte-

-+ gration. Furthermore , F should be expressed in terms of \ the unprimed rectangular coordinates since this is the system in which the far fields are to be evaluated . The .sequence of transformations from the primed spherical ! coordinates of F to its unprimed rectangular coordinates

• is

(29)

·where ,

sint/J sin� · costjl sin� (30) ' cos�

and , the transformation relating x',y',z' to x,y,z coordinates is given by Equation (10) . The overall expression to be evaluated is then

F = F F F [sinO cost; sinw sint; coso [ P tJi <] co�tJi cost;, costjl sinE;, -�inljJ] -sl.n� cost;,

[ cos 0 sin oo oo • 0 -1 ( 31) - l sintjl 0 os · � __o . - . . - - 0. . o_. m 22

The details of the above expansion are carried out in

Appendix B. The result is

= F (sintjJ COSF, COSI/1 + COStjJ SintjJ. ) p 0 . 0

F sinl; COStjJ ( 32 ) - l; 0

(33)

+ F (costjJ cos( sin� + sintjJ costjJ ) 1/1 0 0

- F sinE. sini/1 (34) F; 0

The vector F is a function of only the source coordinates and is therefore independent of the observation coordinates , e and �. This means that, in respect to programming considerations , it is necessary to compute F only once and to store it as an array . Since a typical integration grid may contain several thousand elements (i.e. , units of dS) , the impact of this approach on computer time is quite significant . 25 Of course, the exponential term in the integral must be evaluated for every value of the observation coordinates, e and �. It should also be noted that F is permitted to be

,_. -- �-- -- !complex. This situation can arise, for example, when the i feed pattern is complex due to a difference ih the I ; . I ! locations of the feed's principal plane phase centers . The vector F can be conveniently and concisely expressed in terms of the components of the feed radiation :pattern and the surface parameters . Let

I A = A(tjJ , tjJ , F,) = sinl)J + sintjJ (35) 0 COS[, COSl)J0 COSt/! 0 B = B (tjJ , ljJ , E;,} = COSljJ cost;. COStjJ - sinw sintjJ (36) o 0 0 c ::: C = sinl)J sinw - (37) (ljJO,tjJ,t;,) COS(. 0 COSljJ COSljJ0 ::: = sint/! + sinl)J (38) D D(ljJO,tjJ,t;,) COSljJ COS(. 0 COStjJ0

It should be noted that , in regard to the expression of source coordinates in terms of the un iprimed (8,$) coordinate system,

A = sine cosq, (39) and ::: a A (40) B aw .Also, ::: c cose ( 41) 'and ac D = ( 42) aw

Upon substitution of Equations (25) - (27) and (35) - (38) into Equations (32) - (34) the rectangular 24

1 • ->- icomponents of F become !

= A B) Fx E� sin� ( �� + p

+ Ec (A � - p sin�. sin� . cos� ) (43) s a� o

= E + F y � sin� sin� (sin� �� p cos�)

� - E sin� (sin + p cos�) (44) � ��

F E sin� (C ap + D) z = � � p

- � + E� (C �� p sin�. sin sin:tl.J0) (45)

Although Equations (43) - (45) represent the !final form to be used for the computation of the vector I F, ·it is interesting to note that they can be expressed in !a particularly compact way by noting that

()A = - sin1jJ sinr: cosw (46) at: o land

ac = sinljJ sinE, sinljJ 7) a[ - 0 ( 4

Then, Equations (43) - (45) can be written

= E sin� a A a l/J �(p ) + E � af(p A) (48) 25

F = sinljJ sinE.: a sin41) E sinlji () . sinE;) (49) y - E t;j;"( p - �(p tjJ i;

F = E sinljJ d C) + E a C) (50) z tjJ �(p i; �(p

The phase factor p •r given by Equation (15) !Can be expressed concisely by using the definitions of ' ;A and C:

p • r = A sine cos� - sinljJ 13int; sine sin� + c cose (51)

The preceding formulas are subject to the :restriction that the surface be in the far field of :the feed since they do not include the possibility of ia radial component of feed field . The near field case is not included here since most reflector configurations :of practical interest satisfy the far field condition . iHowever , if required, the near field case may easily be incorporated by allowing a radial component of H.l. in Equation (3) . Then, a completely arbitrary feed i field may be generated by using , for example, a spherical wave expansion as described by Ludwig. 27

2.1 DISPLACED FEEDS AND ARRAYS OF FEEDS The pattern integral as given in Equation (28) implies a single feed antenna which is located at the

!Origin of the co.orc:l:i.I'la_t� syste:rn. It: i.s Ci simple matter

-.· -- 26

to generalize the pattern formulas to include the effects of feed displacement and arrays of identical feeds . Consider a feed displaced from the origin :as shown in Figure 4. The radius vector to the phase -+ :center of the feed is given by E:. The vector from the (feed to an element of sur face area dS is p', where

-)>- -+ -+1 (52) p = p - E:

-+ ' The vector p makes new angles �� and �� with respect ito the axis of the displaced feed so that the incident electric field at dS due to this feed is

� -+ -+ . - E. (E 1 � + E 1 �1)e J'kl p-E:1 1 I = � � � (53)

In many practical cases the feed displacement is small compared to the distance between the feed and dS. Then , if lti!IP"I << 1, it may be assumed that

� 1 � � and r,' :::::: t;. Furthermore , if I£"I is small, the "path loss" given by the denominator of Equation (53) is nearly the same as for the case of a feed at the origin. Therefore , the path loss term in Equation (53) may be written

(54) 27

Figure 4. Coordinates for Displaced Feed 28

10f course, the phase term represented by the �xponential in Equation (53) mus t be maintained intact since even ismall feed displacements can cause significant phase

·c' hanges at the surface of the reflector. If a number of identical feeds are arrayed in 1the focal region of the reflector, it is merely neces isary to superimpose their contributions at the surface .

Then , if M feeds are arrayed with equal amplitude of excitation at each feed , and if the above approximations are assumed to hold for each feed, the total incident· ,field at the reflector may be written

M � � = · (E + E F,)e- Jk I+p-e:+m I E. .2: l/1 ]_ l/1 f, m=l p M 1 " = (E + E e -jklp-e:m I (55) p l/J l/1 F, F, ) m=lL: th •where t m is the radius vector to the m feed and ,

2 p· e: 2 + p e: 2 - ) + ( - ) · ( - ) (56) ( p e: 1 IP-t m I x mx ' y my z . mz 1

Equation (55) is simply the incident field :due to a single feed multiplied by the array factor of M feeds . When Equation (5) is used to represe�t the incident field at the reflector, it can be seen that in the pattern integral of Equation (28) , the p term in the numerator cancels that in the denominator , the 29

· -r - k 'vector F is left unchanged , and the e J' p term is re- placed by the array factor term .

2.2 INTEGRATION TECHNIQUE For pattern computations , the array factor �term could be included either in the express ions for

;-+ :F or it could be incorporated into the exponential A A jkpp·r 1phase factor (e ) of Equation (28) . The manner 1in which the array factor is treated depends on the integration method to be used. For example, in one possible technique for ithe numerical evaluation of the pattern integral the 'entire integrand is expressed in terms of its r�al and imaginary parts and Simpson's rule is applied to each . iTests of a numerical integration program using Simpson's

;rule have shown that to achieve errors at least 40 dB l !below the pattern maximum, the integration grid must ,be subdivided into areas (dS) of at most 0.04 square ;wavelengths . This fact may readily be appreciated when ;it is recalled that the phase factor of the integral is proportional to the reflector size and it causes the real and imaginary parts of the integrand to behave as rapidly oscillating functions as the far field ob servation angles are moved away from the reflector 'axis. This rapid variation may be further complicated ,by the inclusion of the feed array factor. At any rate , 30

the use of Simpson 's rule demands that the integrand be "sampled" at enough points to insure that these rapid oscillations are taken into account. The impact

:of this method on the computer time necessary to eval- :uate a moderately large reflector is quite significant . An alternative technique which appears to offer a considerable savings of computer time compared :to Simpson 's rule has been developed by Ludwig. 2 8 The pattern integral to be evaluated can be written in the form

-+ :where F has already been defined and y includes the iphase factor and·the array factor . Ludwig notes that,

:for a fixed observation coordinate (8,�}, the individual

-)>. terms F and y are each well behaved over an increment of surface area , i\S ' whose area is on the order one mn of :square wavelength . He then integrates the contribution

:over i\Smn analytically yielding the incremental contri bution i\Imn to the field integral . The total integral . I (for any of its three vector components) is then obtained by summing the contributions from each incre-

mental surface area. The explicit formula for i\Imn . is given in Reference 28. Ludwig claims that for an incremental surface 31

area approximately 2/3 of a square wavelength in area, the pattern errors are more th an 40 dB below the pattern 'maxima . The reported reduction in computer time for 'this technique as compared to Simpson 's rule is a factor :of 11 to 19 .

Because of its apparent computational advantages over more conventional quadrature methods , Lud- wig's integration technique is used herein. Again , since the basic idea behind this approach is to separate the oscillatory terms of integrand , it is desirable to incorporate the array factor from Equation (55) into 'the phase factor term of the integrand . Let

- kiP ; l -j� e j - m -a + jb = Qe (57) where �m=l Q =·.Ja? +b2 (58)

-1 r; = tan (b/a) (59)

p ; a=L cos (kl - ml> (60) m

= p-; b L sin (kl ml> (61) m

the . Then, the final form for int�gral to be evaluated is 32

max max -+ = j (p•r)+sJ I i / / Q F e [kp d d (6 2) iJ; E.. A !Jimin �':min The computer program to be presented herein

1uses a subroutine (FINT) whic::h was developed by Ludwig. ,A listing of this subroutine is included with the program :and a description of FINT may be found in Reference 25.

2.3 SURFACE DEFINITION S FOR PARABOLIC AND SPHERICAL REFLECTORS

The pattern integral requires explicit defini-

tions for the reflector's radius vector p(iJ;,�) and its

I derivatives ap/aw and ap /oE.. in order that the reflector !surface be completely specified. In this section , these quantities will be defined for the specific cases of :parabolic and spherical reflectors . While the computer program included herein incorporates only these two re-

•fleeter types , the inclusion of others is straightforward. It will be seen that the expressions for the reflector surface can be concisely defined in terms of

the quantities A, B, C, and D which are given in Equa- 'tions (35)- (42) .

2.3.1 Parabolic Reflectors Consider the "side view" of a surface as 33

:shown in Figure 5. The vertex of the reflector is

:located at z = - f. If the surface is that of a parab

:oloid of revolution about the z-axis with vertex at i :z = - f, the polar equation of the surface in terms of the

lunprimed (B,�) coordinates is

2f = ( 63) p 1 - cose

(Note that in this case e is not a far field coordinate ). j But, from Equations (37) and (41) ,

cose = C = sin� cosc sin�0 - cos� cos�0 (6 4)

'SO that,

2f p =

2f = (65) 1 - c

The derivatives are obtained in a straightforward manner:

2f(-ac;ap) �a� = (1 = c) 2

2f D = ( 66) (1 - c) 2 34

X x'

� z \ \

Note: D should not be confused with Equation (38).

Figure 5. "Side View" of Offset Parabolic Reflector 35

2f (-ClC/;)e_:)

(1 - C) 2

2f sin�0 sin� sin� = (6 7) (1 - c>2

,2. 3. 2 Spherical Reflectors

Consider the "side view" of a spherical surface ,as shown in Figure 6. The radius of the sphere is R and

= its center is at z (R - f) . The equation of the surface in the unprimed rectangular coordinates is

(68) iBut i

= x p sine cos (69)

y = p sine sin$ (70)

= (71) z p case

:rf Equation (68) is divided through by R and transformed

�to polar coordinates using Equations (69) - (71) the polar expression for p/R is�

(72)

:where

m = 1 - f/R (73) 36

/ / / ,1 X x' �----

D z L I

-f 0 . (R- f) z \

Note: D is not to be confused with Equation (38) .

Figure 6. "Side View"· of a Sph�rical Reflector

, __ . .- 37

Now, in order that Equation (72} be expressed

in terms of tfJ and � it is noted (from Equation (41)} that

cose = c so. that

sine = � 1 - cz (74) Then,

p /R = .V 1-m2 + m 2 C 2 + m C (75)

It can be easily be shown that

ac m2 C � = R � m + (76) dt/J [ " l-rrt2 +m2 c2 !\ J :and

ap = ac [ m2 R m + C (77) ar IT ..Jl- m2 +m2 C2l

2.3.3 Discussion The preceding equations are sufficient to completely describe the reflector surface in terms of the r�flector parameters and the feed polar angles . For syrnrnetrical.reflectors it is merely necessary to set

= 0. In general, the size of the reflector is deter tfJa mined by the parameters' 1JJ0 , f (and R in the case of the

I .J 38

:spherical reflector) , and the range of � and � over ,which it is desired to integrate . The term (1/A ) pre 'ceding the pattern integral in Equation (62) can be in cluded with f and R so that it is possible to specify ithese parameters directly in terms of wavelength . A comparison of the equations defining the parabolic and spherical surfaces shows that the quantity 2f for the :parabola is analagous to the spherical reflector radius R. For a spherical ref lector the parameter f/R deter mines the "focal length " for a feed located at the origin.

'For a parabola with a feed at th e origin, the equivalent "f/R" is equal to 1/2.

For an offset-fed parabola it can be shown29 ', that the intersection of a cone of constant � with the .parabolic surface describes an ellipse which , when pro

jected onto the "aperture plane" (i. e., the x-y plane ) defines a circle . The evaluation of the pattern integral requires that the partial derivatives for the refleCting surface be continuous . This condition is clearly not satisfied at the edges of the reflector . Silver1 8 introduced a line charge at the edge to insure that boundary condi tions are satisfied there . However, as Ludwig2 5 points out, Sancer30 has shown that the pattern integral intrins ically contains the effect of this line charge . A conceptual way of circumventing the discontinuity of 39

:the derivatives at the edge of the reflector is to assume that the surface is continuous but the incident fields vanish beyond the integration limits .

i2. 4 FEED PATTERNS It now remains to define the pattern expressions for the specific feed antennas of interest. The computer program incorporates analytical expressions for four commonly encountered types of feeds : a generalized ·Huygen 's source , a rectangular waveguide with TE -mode 1 0 aperture fields , a dual mode circular waveguide , and a :cosnljJ pattern having the polarization of a dipole . Of :course, the program itself is not limited to these feeds and the inclusion of other analytical feed expressions or even measured feed data is straightforward .

The specific feed expressions will be presented first for the case of linearly polarized feeds whose aperture polarization is along the x'-axis. Later the expressions will generalized to account for arbitrary feed polarization . In all cases , the patterns given below are for a constant distance from the antenna (i. e., the ' k e-J P IP term will be suppressed).

2.4.1 Generalized Huygen 's Source For a short electric dipole , polarized along the x'-axis as shown in Figure 7, the pattern is 40

� P {cos � cos � � - sin� �) ( 78) 1

:where P is a constant representing the amplitude of 1 :excitation . I A short magnetic dipole , situated orthogonal :to the electric dipole may be considered to be a current :carrying loop in the x'-z' plane as shown in Figure 7. 'rts pattern is given by

j = P (cos � � - cos� sin� i>e a (79) 2

In Equation {79), P represents the amplitude 2 of excitation of the loop and the exponential term

A generalized Huygen 's source may be defined by superimposing the patterns of the dipole and the loop . Then , the polar components of the pattern are

= cos� (P cos� + P eja ) {80) 1 2

E � = E �e + E �m

= - sin�(P + P cos� ej a ) (81) 1 2 41

z•

x•

Figure 7. Huygen•s Source 42

The three parame ters P , P and u are sufficient 1 ;; to completely specify the pattern of this antenna . Wh en

= = P P and a 0 the antenna is known as a Huygen 's 1 2 i ·source . In this case, the antenna has a "plane wave " :polarization and , if used as a feed for a symmetrical parabola, will produce zero cross-polarized currents .

.2.4.2 TE -Mode Rectangular Waveguide

For a rectangular waveguide in the x'-y' plane and with principal polarization along the x'-axis as shown in Figure 8, Silver 1 8 gives the radiated field . components as

= a 2b B _TI__ (1 + E,,, z cost; _l.Q_ cost)! V(!)! , t; ) . ( 8 2 ) o/ o 2A2 k �

El; (83)

where

cos u sin v V(lJJ ,l;} = ( 84) u2 - ( TI/2) 2 v

1ra u = sinlJJ A sinE; {85)

1Tb v = sinlJJ cost; A (86) 43

Figure 8. Rectangular Waveguide Feed 44

!and a is the H-plane dimension of the g�ide . b is the E-plane of the guide .

8 is the propagation constant for the 1 0 TE mode . 1 0

/k = � 1- ( 2 a } 2 ( 87) B 1 0 >-I

The above adaptation of Silver' s equations :assumes that the reflection coefficient at the waveguide

!a' perture may be ignored. This is a reasonable approxi-

mation for moderate ly large waveguide (e.g. , a/>. > 1} and, if the more exact expression is required , the reflection coefficient may easily be incorporated into

• Equations (82} and (83}. The TE waveguide requires only the parameters 1 0 :a/>- and b/>- in order to completely specify its pattern.

2. 4.3 Dual Mode Circular Waveguide The patterns for circular waveguides having TE and TM modes are presented in Silver . 1 8 For mn mn the particular case of m = 1 and n = 1, the patterns for a circular waveguide whose aperture is in the x' -y' plane and whose principal polarization component for TE and TM modes is along the x' -axis can be 1 1 1 1 written as follows : 45

For the TE mode , 1 1

ka " J = / + �e 1 (u) . E Q l k COS J (K a) cos[, (88) !}Je e t/! 1 e u ! . [ J

k J (K a) J ' (u) = a Wil e + 1 e 1 E - k COSI/J sinE, (89) E,e Qe 2 � ] 2 1 - (�<�a)

For the TM mode 1 1

ka K (u) J ' (K a m Bm J m ) E = Q . -k + cos 1 1 cos[, (90) ''''�'m 2 sJ. 1jJ K a m n!}J� m 2 J - 1-( u )

E = r,m 0

. where, and are the amplitudes of excitation Qe Qm for the TE and TM modes respectively a is the waveguide radius k u = a sin!}J

= I a) ) K e 1. 841/a (root of J 1 (K e K - 3. 832/a of a) ) m (root J 1 (Km

Wjl = kZ 0

= k/Z W£ 0

J K a) = J '(3. 832) = 0. 40276 l { m l - (K a) = = 0.5819 J l e J l {l.841)

The propagation constants for the TE and TM modes are . • gJ.ven b y f3e an d f3·m respectJ.ve 1 y: 46

- K 2 e

(91)

- K 2 m

(9 2)

English3 1 has determined the coefficient Q e :and on the basis that the sum of the powers in the Qm TE and modes be normalized to unity ; i.e. , 1 1 TM l 1

(9 3 )

where P and P are the fractional powers in each of TE TM . the two modes . The coefficients are then,

(94) w� K 2 a2 -1 J 2 (K a) 1rl3e [ e ] 1 e

weK 2 a2 J 2 (K a) 1rt3m m o m (95 )

The pattern equations can be simplified by

,.incorporating all appropriate constants . The final

· result for polar components of the pattern is the 47

J (ka sint� } E 2 = Q + �1 -(1 .841/ka ) cos� cosr: (96) � e e [1 J ka sin•

(ka si J n.)/(ka sin• s1nt;. (97) (ka sin./1.841} 2 J

(9 8)

where

(1-P ) � TM Q = .0646 (99) e 1- ( 841/ka) 2 �Iv 1.

and

. TM . = 14 . 19 p (100) Q m t.! 2 · f .Vl- (3.832/ka}

The overall pattern is , of course ,

(101)

ja where the exponential term e has been included to allow 48

for the possibility of a phase difference between the TE and TM modes with the TE mode as ph�se ref- 1 1 11 1 1 erence. The parameters necessary for the specification of the pattern for the dual mode circular waveguide are :

a/A , P ' and a. TM ,

n 2.4.4 cos 1/1 Pattern A somewhat fictitious but often used antenna pattern is one which has a circularly symmetric pattern 1about the polar axis and it has the polarization of an /electric dipole . The polar components for this pattern , ( . !having its principal polarization along the z'-axis and its beam maximum on the z'-axis is

n+l cos cos[, Elji = (102}

= (103) E f,;

The only parameter required to completely determine the , pattern is the exponent n.

2.4.5 Generalization to Arbitrary Feed Polarization The feed patterns which have been presented all imply an aperture field whose principal polarization 49

is along the x'-axis . It is a simple matter to generalize

the feed expressions to include an arbitrary orientation

of the aperture field in the x'-y' plane or even to

allow for elliptical and circular feed polarization.

It is interesting to note that all of the

feed patterns given here can be expressed as separab le

functions of � and �. That is, for an x'-polarized feed

each of the patterns has the form

{104)

where

( 10 5)

E f (�} sin� �X, = - (10 6) 2

and the subscript x I refers to the aperture plane polar-

ization.

Suppose this same feed is polarized along the

y'-a xis. Then it is necessary to shift only the � i variable by �/2 . That is,

+ sin� COS/;

sin!; + -COS/; 50

Then ; for a y'-polarized feed aperture

E , f sin� 7) y = ( �� ) ( 10 ljl 1

E I f cost; y = ( ljJ ) (108) F, 2 .

Now , considering that the feed may be dual

polarized, let A ' be the amplitude of excitation of the x x'-polarized aperture , A , be the amplitude of the y' y .polarized aperture and a be the relative phase between

the x' and y' excitations . Then , ;

� A (E E t; ) = � ijJX ljJ + t; x' (109)

( 110 )

The total field is then

E A .E I + A ,E ,eja = iji y ijiy ( 111) lji X X a E A I E I A ,E ,ej = + y y (112) t; X t; X t;

Upon substitution of the functional relation-

ships for the component fields given by Equations (105) - (108) the total field may be expressed as 51

a E = f , + ' sins j ] ( 113) !/J (1/J) [A cos t.: Ay e l X

- - E = f 1 sins 1 cost.: e j (114) {t/J)[A Ay cv.J [, / X

Therefore , the use of proper values of A ,,A , X y and a the principal polarization of the feed may be spec ; ified. Several examples are listed below :

,A I A a Polarization X y' 1 0 0 Linear, x'-polarized 0 Linear , y'-polarized •o 1 ;1 1 0 Linear, 45° !1 1 -90° Left hand circular i 1 1 +90° Right hand circular

Equations (113) and (114) can� of course, be

easily normalized on the basis of any suitable criterion

such as unity total power or unit amplitude at the

· peak of the feed • s beam. The above equations cannot be applied exactly

to the TE rectangular waveguide unless it is square 1 0 . (i! e. , unless a/A = b/A for this antenna ). The general-

ization to dual polarization for this antenna is a

straightforward matter and the computer .program incor-

porates the dual polarization feature .

It is interesting to note that the expressions 52

for the patterns of the generalized Huygen 's source . I have the same general form as those of the TE mode 1 0 rectangular waveguide , i.e., for both antennas ,

E cos� (l + x cos�) � a (115)

E a - sin� (x + cos�) (116) � . I

In the case of the generalized Huygen's source , x is the ratio of electric to magnetic dipole amplitudes and in the case of the rectangular waveguide , x is the normalized propagation constant /k . 8 1 0 For a waveguide whose dimensions are small,

/k is appreciably less than unity . The feed's 8 l 0 polariz ation then resembles that of a magnetic dipole .

Since it is known that the condition x = 1 results in zero cross-polarization in a symmetrical paraboloid, this explains the fact that a small rectangular waveguide does not produce low cross polarization in a paraboloid. This fact was observed experimentally by

E.M�T. Jones . 3 Kinber3 2 attempted to explain this phenomenon by observing that the wave in a rectangular waveguide is really two plane waves whose wavefronts a·re not parallel.

An explanation in terms. of the analogy with the generalized Huygcn 's source was given by Kerdemelidis5

. I 53

and also implied by Koffman . 4 In fact Kerdemelidis

takes the viewpoint that the classical Huygen 's source

(x = 1) is just a special case of the horn excitation

problem. This can be appreciated when it is recalled

that the term x = /k for the waveguide approaches f3 1 0 'unity only as the waveguide aperture size becomes much

greater than the wavelength . Therefore , a large wave

:guide may be expected to behave like a Huygen 's source

in terms of its polariz�tion properties.

Koffman4 notes that when using electric and

:magnetic dipole feeds the conditions for zero reflector

;cross-polarization depend on the type of reflector and i he shows that for symmetrical reflectors whose surfaces

:are generated by conic sections , the optimum ratio of

.electric to magnetic dipole intensities is equal to

, the eccentricity of the surface . While he considers

. spherical reflecting surfaces , his results are not

·applicable to the spherical reflectors described herein.

The reason is that Kof fman assumes that the sphere

is fed from its center whereas the feeds for most

• practical spherical reflectors are located in the vi-

· cinity of their paraxial focus which is at one-half

the radius of the sphere .

2.5 PATTERN QUANTITIES OBTAINED· FROM THE COMPUTER PROGRAM

All �nput quantities needed to describe a 54

)variety of feed and reflector configurations have now

r ibeen determined and the general expressions for the

pattern integral have been defined explicitly in terms

of these quantities . :i A computer program which incorporates all

iof the features described above is presented in Appendix

•c. That ap�endix also contains a brief description of the input quantities required by the program. It is

the purpose of this section to describe the pertinent

output quantities and features of the secondary pattern

which are obtained from the program .

The primary result of the pattern integration

! lis the complex vector i (Equation (62) ) from which all

i l?attern and polari zation quantities may be obtained . The observation coordinate system is redrawn

in Figure 9. For a fixed observation distance r, the

· complex spherical components of the far field radiation

patterp are obtained as

(117)

. where

(I x + I y + I z) ·8 (118) X y Z

A (I x +I y +I z) ·� (119) X y Z 55

Figure 9. Coordinate System for Secondary Pattern 56

and

cos o cos ¢ x cosO sin¢ y - sinO o = + z ( 120)

� = - sin� x + cos� y (121)

Therefore ,

= I coso coscp + I coso sin(jJ - I sino (122) E0 X y Z

- I sin� cos� (123) E� = + I 'I' X y

2.5.1 Principal and Cross-Polarized Field Components

The conventional spherical components , E 8 and E , are not particularly convenient for visualizing � the polarization characteristics of the secondary pattern

except along the principal axes of the observation

coordinate system . For example, the far field pattern of

a reflector having only x-directed currents �i ll generally

have both and components ; the identification of e

not convenient in terms of this coordinate system.

One way to circumvent this difficulty is simp ly

to use the far field rectangular components identify to principal and cross-polarized radiation . For example ,

if the reflector currents are directed along the x-axis ,

E may be called the principal polarization and E is x y 57

ithe cross-polarization of the secondary pattern.

A major disadvantage associated with the use

of rectangular components is that their direct measure

ment is not convenient with conventional pattern measure

ment equipment since experimental pattern data usually

consists of the field components as they appear on the

surface of a sphere surrounding the antenna.

A set of spherical coordinates representing

a natural system for the identification of the antenna 's

polarization components is also shown in Figure 9.

These spherical coordinates are designated with a

"Tilde" (""") . For a linearly polarized antenna whose

principal aperture polarization is in the x direction ,

Ee represents the principal polarization and E� repre

sents cross-polarization in the far field . For a

linearly polarized antenna as described, the distinction

between the two systems is readily seen: The polar

axis of the conventional {6 ,$) system is in the direc

tion -of the antenna 's main beam ; the polar axis of the

Tilde system is in the direction of the principal

aperture polarization� {It should be noted that an

identical set of relations holds for the feed coordinate

system) .

The relationships between the conventional

and Tilde systems may easily be determined by noting

that 58

- - x = sine costp = z = cosO (124)

sine sintj> sine cos (125) y = = x = �

e z = cose = y = sin sin� (126)

· Then

....., - cose = sine sintj> (127a)

- 2 2 sine = v 1 sin e sin � (127b)

- sine cos"'cp sincp = (127c) v1 - sin28 sin21

- cos e costj> = (127d) ...}1 - sin2a sin2r

.The 11 reverse11 transformations are

...., cose = sine costj> (128a)

(128b)

. - cos e s1ncp = (128c) 59

,.., = sine sin COS (128d)

The relations between the unit vectors for wo !:t he t coordinate systems can be obtained by using the :above relationships along with the transformations relating spherical to rectangular coordinates (cf. Equation

(A-3) , Appendix A) • The results are :

(129)

'and i - = (130) m [ : ::J[:J

:where

= cose cos� p (131) '\) 1 - sin2 e cos2 <1>

sin� Q = {132) vl - sin2 e cos 2 <1> - cose sJ.n� R = (133) v1 - sin28 sin2 � -

= cos� s (134) v1 - sin 2e sin2� 60

Therefore , given the and $ components of a e

given field quantity , e.g. , E6 and E$ , th e Tilde com ponents can be found with the aid of Equation (130 ) as ,

E� �� = [ -R S] � � (135) E� E . � $ -S -R I I Also, if Ee and E� are given , the conventional com ponents are ,

(136 )

The relations between coordinate angles are

easily ob tained from Equations (127) and (128) . For

example, division of Equation (128b ) by Equation (128a) ·II yields :

,..., _ - 1 1 - sin2 e cos 2p tan -"--r. -=-_.;.;.;--:--__;,--' (137) e - (...}Sl. n 6 COS$ ) I I 1I The remaining coordinate re lations are obtained in a ' similar manner e

Circular Polarization Components I i It should be noted that either the conventional I coordinates or the Tilde coordinates are suitab le for ...... ····-··· --····· ···-············ ··- ··-·····--·· ·····--···- ···-·· ···--- ... - ... JI l - 61

defining the left and right hand circular polarization

components of the secondary pattern . In fact , any two

components spatially orthogonal to each other and also

orthogonal to the direction of propagation of the wave

are suitab le for th is purpose .

The formulas for obtaining circular polar

ization data are well known . 3 3 They will be listed here

for convenience in terms of the complex E8 and E� com ponents although , as stated , the formulas remain un

changed for any other spatially orthogonal waves . A

wave E may be expressed as

� .t; = ( 138)

where the components have been de fined and where o,cp U and U are unit vectors for right and left hand R L circularly polarized waves respectively. Right and

left hand refer to clockwise and counterclockwise

rotation respectively for a receding wave . Now , U R and U are complex unit vectors and are given by L

- U = G j � (139) R 12

8 + � U = j (140) L 12 62

Then , the right and left circular components of the wave

:are

Eo + j E =

'. E j E = e p EL 12 (142)

For a wave whose principal polarization is ,

for example , right hand circular , the left hand com-

ponent represents cross-polarization and vice versa.

The ratio of left to right hand components for such an

elliptically polarized wave is

1-j (E /E ) = p 0 (143} 1 + j (E

Also,

(144}

;The axial ratio is defined in terms of the circular

components as

1 + r = (145) 1 1�1

.and the angle which the major axis of the polarization 63

·r = i (arg (g) ) (146)

The explicit formulas for r and 1 in terms of the linear components E and E are ob tained as fol- 8

( 14 7)

Substi tution of these these terms into

Equation (143) yields

. 8 E - E eJ 8 j g = � ( 148) . 8 E + E eJ 8 :!

Then ,

j g j = (149) and

1 - 1 T = 2 tan (150)

Substi tution of Equation (149) into Equation

(145) yields the axial ratio of the wave . 64

Pattern Information Provided in the Program 2.5.3 Output

The computer program in Appendix C results in a print-out of the following information for a given pattern cut:

IEX I' IEy I' IEZ I

In the following sections the application of the program, to some specific examples of offset-fed and symmetrical reflectors will be discussed �

'• ...... 3.0 POLARIZA'l'ION EFFECTS IN SYMMETRICAL REFLEC'rORS The pattern equations wh ich have been derived

for offset reflector patterns can be directly applied

to the special case of symmetrical reflectors by merely allowing the offset angle , � , to be zero . 0 This section considers the re lationships among the cross polari zation in the secondary pattern , the reflector parameters and the � of feed antenna which is used. In th is regard it will be possible to gain substantial qualitative insight into cross-polarization effects by first considering the nature of the currents generated on the reflector surface by particular types of feed antennas . The computer program wh ich has been developed herein is then utilized to provide quantitative data concerning reflector polarization and also to confirm the trends indicated by the consideration of re- fleeter currents .

The following discussion is confined to the case of paraboloidal reflectors although as will be shown , the polarization qualities of the symmetrical spherical reflector are qualitatively similar to those of the parabola.

PARABOLA CURRENTS 3.1 The currents which are induced on the surface of a reflector by a feed antenna can be determined from

65 66

r I Equation ( 5 ) • When this equation is applied to the I I specific case of a synune trical paraboloid , the rectangular components of th e currents are found to be

K I K = cos 3 cos t,; - E sins (151) X X ] t[E� f.,;

K l = -K = cos 3 t E sint + E cos � (152) y y 2 [ � � ]

K I. = -K = -cos 2 t sin 1 E (153) z z 2 2 t}J

where the 11 path loss 11 term , 1/p , has been included above 'k and the path leng th phase term (e-J P) has been suppressed .:

Furthermore , the above equations have been divided through

by the cons tant term 2/Z • 0 For a feed antenna having its principal aper-

ture polarization along the x'-axis , K ' ( =K ) represents x x

the principal polari z ation current and K ' ( =-K ) repre y y

sents the cros s-polarization current . The "ideal" linear

polari z ation occurs where K I = The longitudinal y o.

component of current given by 1 (=K ) does not contri- K z z

bute substantially to the far field near the antenna

axis and will therefore be ignored for the remainder of

th is discussion

The cross-polarized radiation in the secondary

pattern will depend upon the amplitude of K 1 and its y

distribution over the surface of the reflector . A VI

general feature of the cross-polarized currents in a

symme trical parabola is their quadrant symmetry . That

is , K is zero along the principa l planes , 0 and I y ,j t: =

90° , and maximizes along the diagonal planes , t: = it

+45° . This situation is illustrated in Figure lOa t: =

: which shows for example the general features of the cur-

rent induced by an electric dipole feed. The quadrant

symme try of the cross-polari zed current dis tribution is

shown in Figure lOb .

In general, cross-polari zed re flector currents

result from the curvature of the surface and from cross-

polarized radiation , if any , from the feed . In th e

case of an electric dipole , the cross-polarized cur-

rents are due entire ly to the curvature of the reflector

since the electric dipole feed has only an E� component

of radiated field (cf. Section 2.5.1) .

A magnetic dipole feed with dipole moment

along the y'-axis (i. e. , principal polariz ation along

' the x'-axis ) would produce a cros s-po larized current

distribution having the same quadrant symme try as

shown in Figure lOb but with the "arrows " pointing in

the opposite directions . As wi ll be shown , a combina-

tion of electric and magnetic dipoles causes ' to Ky

vanish everywhere on the reflector surface . 68

a. Re flector Currents for Electric Dipole Feed

x '

b. Array Model for Cros s-Polari zed Distribution

Figure 10. Currents for Parabola with Electric Dipole Feed <' 69

3.2 GENERALIZED HUYGEN 1S SOURCE FEED

Let the feed antenna be a "generalized"

Huyqen 's source with polar field components given by

( 80) (81) It = 0. Equations and with Wh en these ex (151) (152) pressions are substi tuted into Equations and

the explicit form for the reflector currents is as follows :

= + + + K 1 cos3 ! [ P cosl)J P )cos 2 t,: (P P cosl)J) sin 2 �J x ( 1 2 1 2 (154)

K I = t/1 ) (P + p cos 3 sinr; cos r; [ (P COS t/1 + p COS tp 2 / )] y l .. J 1 = 'P ( 155) (P p ) sin2'v cos sin2.; if 1 2 2

where P and P are the excitation amplitudes of the 1 2 electric and magne tic dipoles respectively . (155) An examination of Equation shows that

= 0 t,; = �9 0 K 1 is zero at � and as expected. Further Y

= 1 = 0. more , if P P then K Wh en P + P , the 1 2 y 1 2 cross-polarized currents are maximum in the planes

Since the worst case cross-polarized currents t,; = �45° , occur in the planes it is instructive to consider their variation in these planes as a function = 0) of l)J. For example , for an electric dipole feed (P 2 = 45° the currentI s at � are ·. ! 70

K I ! (156) = p cos 5 X 1 2

p I = _l � (157) K 4 sin2� cos y 2

(157) From Equation it can be observed th at 0 K 1 is zero at � = and increases with increasing � y Now , the ratio of focal length to diame te r for a parab-

, ( ·1• ola (f/D ) is related to the maximum value of , , ) � �max defined by the edge of the re flector . (Note : The reflector diameter, D, should not be confused with the (38) ). quantity D(t/J,�) as defined in Equation The f/D of the reflector is inversely proportional to The t/Jmax · dependence of w on the ref lector f/D is illustrated max 11. for parabolic and spherical reflectors in Figure This (157) relationship , along with Equation implies that as the f/D of the reflector is increased , t/J wi ll de- max crease . Therefore , the cross-polarized currents and hence the secondary cross polariz ation will be inversely proportional to f/D for a dipole feed .

The relationship between f/D and ref lector cross polarization is a we ll known result and it provides a useful check on the accuracy of the computer program C) , deve loped herein . In the computer program (Appendix the reflector diameter (D/A ) and the f/D are determined by specifying R/A (=2f/A ) and For a symme trical t/Jmax · 7 1

Parabola: � -1 . 8f/D = tan max ] [16 (f/D )2 -1 fJ) (lJ 60 (lJ H tn (lJ n

X = f/R 2f/D

m = 1 - f/R 40

-- Parabola = 3 0 --- Sphere , f/R 0.5

0.3 0.2 0.4 o.s 0.6 0.7 0.8 0.9 1.0

f/D � Figure 11. vs . f/D for Symme trical Re flectors . max 72

re flector the pattern is ob tained by integrating from

• = 0° 360°. I}J = t/J to I}J and 1; to For the case when m1n max ,, , 0. there is no aperture blockage , = '�'min In order to provide a quantitative relationship between secondary cross polarization and f/D , several patterns have been computed for a reflector with a 15 diameter of wavelengths which is fed by an electric dipole . Kerdemelidis5 has shown that the first cross polarized lobe has the largest amplitude and the relative leve l of the cross-polarization lobes is nearly independent of the reflector size in wavelengths . How- ever, the position of the lobes depends on the reflector diameter . Kerdemelidis considers a simp lified mode l of the cross-polarized current distribution consisting of four small dipoles at the corners of the reflector +45° at t; = (Figure lOb) . He then shows that based on this model , the first cross-polarization lobes occur 45° at ¢ = and at a polar angle given approximated by

8 � . -1 (158) Sln (A/D )

Several patterns computed by the program of C 12 45° Appendix are shown in Figure for the ¢ = plane .

It is evident from the patterns that the cross-polari zed radiation depicted by IErl decreases with increasing f/D . This data , along with other patterns wh ich have 73

= • I 2 E8 t /D o s

-10

- 20 t:Q '"d

(1) '"d ::::1 +I -30 · r-i = 0.25 .--I E f/D 0.. � � r (1) . ·:> r-i +I -40 = n.4 cO E f/D .--1 r�

= 1.0 E- I f/D

- 60 10 0 5 l!) - e Degrees

12. Figure Cross-Polarized Lobes for a Symmetrical Parabola with Electric Dipole Feed 74

been computed but not shown , has been s ummarized to

depict maximum IE�I vs . f/D as shown in Figure 13. Also

shown in Figure 13 are cross-polarization levels for

= cJ> 22.5°. These calculations were made in order to

compare the results with those of Kerdemelidis . As can

be observed , the agreement is excel lent and, since

the results of Kerdeme lidis were ob tained by a di fferent

technique , this is a good check on the accuracy of the

patterns computed here .

3.3 TE MODE RECTANGULAR WAVEGUIDE FEED l 0

It was pointed out in Section 2.4.5 that the

pattern expressions for a TE mode re ctangular waveguide l 0

feed are analogous to those of the generalized Huygen 's

source in regard to reflector cross polariz ation . This

can be observed if the polar components of this antenna ,

given by Equations (82) and (83) are substituted into

Equation (152) . The result for the cross-polarization

current K ' is y

(159)

1. The cross-polariz ation current wi ll vartish for B /k = l 0

This in turn occurs only as the "a" dimension of the waveguide becomes very large since B /k = 11 - (A/2a) 2 • 1 0

Therefore , a small waveguide may be expected to produce 75

= lm D/A 15 � = 4--1 A cp 45° 0 Computed at - 10 � = m Computed at cp 22.5° Q) � X Computed by Kerdeme lidis 0 5) = +J (Ref. at cp 22.5° Q) !> -.. 20 ·r-i +J m rl Q) � j:Q '0 -30

l () � 4--1 0 ,.....j� -40 Q) H E ::l E .,..; X m -50 �--�-----�----�--�----�-----�----�--� ::E: 0.3 0�4 0.5 0.6 0.7 0.8 0.9 1.0

f/D

Figure 13 . Computed Cross Polari zation vs . f/D for Parabola with Electric· = Dipole Feed . D/A 15 7()

some cross-polarization . Since , for a given edge illumi- -10 nation (e .g. , K ' = dB relative to the illumination x at the vertex of the reflector) , the angle subtended by the reflector is inversely proportional to the waveguide size , the rectangular waveguide feed is expected to produce lower cross polarization as its size is increased and hence as the f/D of the re flector is increased.

An example of the patterns for a parabola 15 with a diame ter of wavelengths and two different 14 . values of f/D.is shown in Figure In both cases a square waveguide (a/A = b/A ) is used and the size of the waveguide is such that the relative edge illumina- -10 tion of the principal polarization current is dB .

3.4 CIRCULAR WAVEGUIDE FEEDS

It is commonly assumed that a way in which to reduce cross polari zation in symme trical ref.lectors is simp ly to increase f/D . Wh ile this is indeed true for the electric dipole and rectangular waveguide feeds , it is definitely not true for all feeds . In fact, it will be demonstrated in this section th at just the opposite trend is true for a TE mode circular 1 1 waveguide feed . That is , for th is type of feed the cros s-polarized radiation shows a decreasing trend as the f/D of the reflector is decreased . 77

-20

-30 .

-40

E-;p /- -50 1 ' I \ I \ I \ I \ /' I \I ' ,

0 5 10 15

- o Degrees

14. 15 Figure Patterns for Parabola wi th D/A = TE Mode Rectangular Waveguide Feed 1 0 78

It. can be shown34 that the pattern of an

antenna having physical circular symmetry and radiating

only TE and TM modes can be written in the form 1,n 1 ,n

= ( ) (160) E !jJ f !jJ cos� 1

( 161)

In this case , the antenna •s aperture polariz ation is

along the x•�axis so th at f ( �) represents its E-plane 1 pattern and f (�) represents its H-plane pattern . An 2 (160) interesting consequence of the form of Equations (161) and is ·that it is only necessary to measure the

E- and H-plane patterns in order to completely specify the pattern for all angles . (160) If Equations and (161) are incorporated into the expressions for parabola current the results are

( 162)

1 K I cos 3 i sin2�;: [ f (lJ!) - f (lJ! )l (163) y 2 2 . 1 2 �-

The above result is interesting since it expresses the cross-polarized currents directly as the difference between the E- and H-plane patterns . If these principal 79

plane patterns can be made equal the reflector cross-

polarization will vanish .

The princ ipal plane patterns for a single

waveguide mode such as the TE mode are not �qual 1 l but, as will be shown , they tend to approximate the

conditions for low cross polarization for small wave-

guide radius . In general, the E- and H- plane patterns

can be made equal in multi-mode structures as exemplified ' : by the corrugated horn . 7 However , a good approximation

to this condition can be obtained with only one or two

modes .

3.4.1 TE Mode Feed 1 1

In order to illustrate the behavior of re-

fleeter cross-polarized currents as a function of off-

axis angle from the feed and the size of the feed , the

pattern express ions for the TE 1 1 mode as given by (88) (89} Equations and have been incorporated into the

equations for reflector currents . K 1 and K 1 have X y 45° been computed in the t;, = plane as a function of 1jJ

for various waveguide radii , a/ A . The results are plotted 15. in Figure It can be observed that for small wave-

' guide radius the cross-polarized current , K , peaks at y relative ly large value of 1/J. As the radius is increased

the peak of K ' moves to smaller values of 1jJ and its . y peak amp li�:ude increases . 80

, - Principal Polarization K X

-10 � >:: 4-l 0 3 2 1.6 1.0 0.8 0.6 0.4 0.3 -{- a/>. E ;::1 , Cross Polarization f; K - .,..., - y co� 20 - ::8 0 +l (!) :> .,...,+> - co 30 r-l (!) p:; r:fl 't) - +> 40 (!),.::: H H ;::1 u H 0 - 5 +> 0 0 (!) .-t 4-l (!) p:;

-60 0 10 20 30 40 50 60 70 80 90

- Degrees 1jJ

Figure 15. Current Distribution in the � = 45° Plane for Symmetrical Parabola with TE Mode Feed 1 1

.- .- 81

Now , also shown in Figure 15 are the principa l

polarization currents , ' . A reflector f/D can be K X established by observing th e value of at wh ich 1jJ K X ' is down from its maximum by a specified amount , say - 10

dB . This value of 1/J is then related to the f/D by the curve of Figure 11. Therefore , the specification of

the edge illumination provides a re lationship between

the reflector f/D and the waveguide radius . Th i� te-

lationship is plotted in Figure 16 for the case of - 10

dB edge illumination . I t can be observed that for f/D >

0.5 the waveguide radius is approximate ly equal to . the

f/D for this edge illumination .

tfuile the currents shown in Figure 15 indicate

generally that ' increases as a/"A is increased , an Ky interesting exception to this trend occurs for a/"A between

0.3 and 0.4. At a/A � .35 the cross-polari zed current

shows a dramatic reduction in overall amplitude and

it literally .vanishes at � � 65° . One viewpoint is

that at this value of a/"A the cross polarization in the

feed's field nearly matches that wh ich tends to be

generated by reflector curvature thereby causing the

resulting ' to be small. K y For a reflector with its f/D defined as above

it is then possible to plot the maximum val ue of cross- polarized current which falls within the re flector edge as a function of the f/D . This plot is illustrated in 82

Note : Edge Illumination Refers to Peak of Principal Polarization Current

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0

f/D

Figure 16 . Radius of TE 1 1 Mode Feed Required to Provide -10 dB Edge Current for Symmetrical Parabola 83

wh ich shows the dir> in ' at l•'ic.Ju re 17 . r) ronounced . 1

a/;.. � 0. 35. It should be pointed out that the edge illumination is not absolutely constant at -10 dB

since , from Figure 16, the lower limit of f/D for a waveguide radius of 0.3 is approximately equal to 0.42.

Therefore , below f/D � 0.5 the edge illumination is

somewhat more .sharply tapered . However, the overall trend is clearly discernible : The cross�polarized currents de�rease with decreasing f/D - a trend which is opposite to that for a dipole feed .

It is to be expected that the secondary patterns would indicate the same general trend as for the reflector currents . This trend has been verified by computing secondary patterns for a reflector with D/�. held constant at 15 wavelengths as shown in Figure 18.

A summary of peak secondary cross-polarized radiation as a funct ion of a/A (�f/D ) is presented in Figure 19 .

This curve tracks very closely with that of Figure 17 thereby indicating the correlation between reflector currents and far-field polarization effects .

A similar set of pattern computations were made for a spherical reflector with D/A = 15 and h f/R = 0.5. T e maximum cross-polarized radiation levels are also shown in Figure 19 and it can be observed that they track very closely with those of the parabola .

This is confirmation of the fact that the parabolic and 84

-10

-20

-30

lj...j 0 -40

0.3··· 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a/A � f/D

Figure 17. Peak Leve l of Cross-Polarized Current , for Mode Circular Waveguide K { TE F�ed 1 1 85

= 15>..

= 45° -10 = 0.3

= 0.4

= p:j 0.5 'U -20 E- Q) 0 'U :::J +J ·n r-l p, -30 a/A ;::; d. Q) 0.3 !> ·n E ...... +J � ttl r-lQ) - 40 p:::

0.37 -50 0.35

-60 0 5 10 15

- e Degrees

TE Figure 18. Patterns for Parabola with 1 1 Mode Feed. D/>.. = 15 86

- · 0

lr;o = 45° � rp 4-l 0 Parabolic Reflector - 10 8 ;::1 X Spherical Re flector 8 ·.-I X cO ::8 0 +J - 20 QJ :> ·.-l +l cO r--1 QJ )( p:: - 30 t:Q '1:) l-e ILl 4-l 0 r--1 -40 QJ !> QJ H I E I 3 y 8 ·.-l X -50 cO ::8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

"A a/ � f/D

19. Figure Maximum Level of Cross Polarized Lobe vs . f/D for Symme trical Reflectors with TE . Mode Feed l l 87

spherical reflectors should be expected to display simi lar polarization properties . This is quite reasonable since the reflector surfaces are qualitatively similar and approach each other very closely for large f/D .

The quantity jE l, wh ich is the rectangular y component of th e far-field cross polarization , is also 19 plotted in Figure for the parabolic reflector . This quantity is very nearly equa l to !E�I near the beam axis but departs from jE�j as either the off-axis angle is increased or the absolute ampl itude of the radiated field approaches zero . This is just a consequence of the coordina te relations between spherical and rectangular components and it indicates that care must be ex- ercised in defining what is meant by cross polarization for each application .

3.4.2 Dual Mode Circular Waveguide Feed

The preceding results indicate that a small circular waveguide feed radiating the TE mode is nearly 1 1 ideal from the viewpoint of cross polariz ation for reflectors with rel atively low f/D . However, there are situations which require that a high f/D reflector be . I used, as for example, in beam scanning applications when the side1obe degradation of a low f/D ref lector cannot be tolerated. If low cross polarization is a requirement then other types of feeds must be used. One 88

possibility is the use of rectangular horns as described .

However , one limitation of these antennas is the fact E- that their and H-plane beamw idths are unequal. This

limits their usefulness as circularly polari zed antennas .

In th is event , the feed antenna should have a circularly

symmetric pattern in addition to good polarization char-

acteristics .

An antenna which very nearly satisfies these 10%) requirements over moderate (e .g. , bandwidths is

the dual mode circular waveguide feed , sometimes re-

ferred to as a Potter Horn. 3 5 In this antenna , a small

amount of TM 1 1 mode is added to the dominant TE 1 1 mode E- to equalize the and H-plane beamw idths . This also

has the effect of reducing cross polarization if the

antenna is used as a feed for a symmetrical reflector .

The pattern equations for such a dual mode 2.4.3. feed were developed in Section Several examples

of the inclusion of TM1 1 mode will be given here to

illus trate its effect on ref lector polarization .

In practice TM 1 1 mode may be generated in a

circular waveguide by incorporating a step discontinuity

in the diameter of the guide . A gradual change in the guide diameter may also be used to generate this mode .

In order for the TM 1 1 mode to be effective in yielding

the requisite pattern and polarization properties , it TE must be 11 in phase11 with the 1 1 mode at the waveguide 89

radiating aperture . If the TM mode is generated I l

some distance back from the aperture , the di f ferent

phase ve locity between the modes limits the

frequency band over wh ich the proper phase relationships

are obtained . In spite of these limi tations the dual

mode radiator is very useful for many moderate bandwidth

applications .

A ques tion of particular significance in re-

fleeter applications re lates to the proper amount of TM I l

mode necessary to effect a reduction in cross polari za-

tion . An example of the application of the field equa

tions for th is feed to the computation of reflector 20. currents is illus trated in Figure Here , the re-

fleeter cross-polarized currents are plotted in the 45° � = plane as a func tion o£ � for various fractional

TM mode powers , P (Recall that the mode powers 1 1 TM " are normalized to a total power of unity. ) �n Figure 20 0.8 the waveguide radius is a/).. = and it can be ob- I 40° served th at the .maximum of K occurs at � . The y inclusion of progressively greater amounts of P causes TM I 0.138, K to decrease until , for P = K has vanished y TM y 40° at �= . For th is condition it appears that the average level of K is also a minimum . This suggests y that a reasonable criterion for determining the appropriate amount of TM 1 1 mode power is that it be adjusted 90

0 = = a/>.. 0.8 £. 45°

-10

p:j P '0 -20 TM

0.0 � � - 30

Ul ..jJ � Q) -40 H 0.100 H ::I 0.1 50 tJ -50 '0 NQ) ·r-i 0.1 38 H -60 tO r-1 0 AII -70 0.130 Ul Ul 0 H tJ -80

-90 0 10 20 30 40 5 0 60

t!J - Degrees

Figure 20. ·Cross-Polarized Parabola Currents vs . t!J for Dual Mode Circular Waveguide Feed. (a/>.. = 0. 8) 91

to minimize the maximum cross-polari zed current on the

reflector surface .

This idea has been tested for the particular 0.8 21 case of a/A = in Figure wh ere P has been TM successively adj us ted to minimize K at various va lues y of w denoted by w . The amount of P necessary to M TM accomplish this is also shown for each case . It appears that the lowest average leve l of K occurs where P y TM 1 � 40° is adjusted to minimize K at = which confirms y

• the idea of minimi zing the maximum of K 1 Although y not shown here , similar exercises with other va lues of a/A also confirm this approach . 21 The data of Figure are summarized in Figure

22 which illustrates the cross-polarized current as a

•. function of the fractional TM 1 1 mode power The minimum = 0.138 KY occurs for P in this instance . TM 23 Figure illustrates the computed cross-

= 45° polarized pattern in the • plane for a parabola 15 � 0.8. 13.8% with D/A = and f/D This inc lusion of

TM 1 1 mode power reduces the leve l of the cross -polarized -50 lobe from -27 dB to below dB .

The effect of varying P on the level of TM IE�! for these reflector parameters is illustrated in

Figure 2 4 . As P departs from its "optimum" value , TM the leve l of cross-polarized radiation rises sharply . 92

= P 1/J � 45° TM ' M -50 .144, 10° .1431, 20° 0.1408, 30°

p::j -60 40° '0

� 50° �

+IUl I -70 Q)s:: H \ I H ::l \ I () "d Q) l I N -80 I ·r-lH I 0.1345 , 60° m ,.....j I' 0 � I f I I I Ul , , 0Ul H -90 () 0 10 20 30 40 50 60

1JJ - Degrees

Figure 21. Cross-Polarization Current Distribution for a Parabola fed with a Dual Mode Circular Waveguide when P M is Adjusted to Minimi ze K at Various VaI ues of ljJ(lJJ= lJJ ). a/A=0 .8 y M 93

0 •. 8 aft.. = 38° 1jJ = = t;· 45° - 20

� � -30

- 4 0

-so

- 60 0 0.1 0.2 0.3 6.4

p TM

22. Figure Cross-Polarization Current for Parabola wi th a Dual Mode Circular Waveguide vs . K PTM Evaluated at the Angle where y has its Peak for P =O TM 94

Parameters f/"A = 12 = u lj! 0 o = 35° !)!max = a/ "A 0.8

-20

= 0

-30

0 1.(} ""'

f:J- -40 +I ((j

CQ 'd

-50 = 0.138 2-e. P li-1 TM

0 1 2 3 4 5 6

e ::: Degrees

2 3. Figure · Cross-Polarization Lobes for Symmetrical Parabola \

Parameters f/"A = 12 = o 1)!0 o = 35° lJ! max / "A = 0. 8 a

II) = � -20 45° rp II

B- +l m

� - 30 0 ·.-I +lm

· IE I 'tl.-I !�> m t:t: 'tl Q) N - 40 ·.-I $-1m ....-t � 0 \ Pol I \ U) -50 U) \ 0 $-1 \ () \ s IE I ::I \ y s ·.-I I X -60 m � 0 0.1 0.2 0.3 0.4

Fractional Power in ™ Mode , P 11 TM

Figure 24 . Maximum Cross-Polarization Radiation from a Symmetrical Parabola with Dual Mode Circular Waveguide Feed vs . P TM 96

The amount of P required to minimi ze the TM maximum cross-polari zed re flector currents is illustrated 25 in Figure for a range of waveguide radii . The opti- 0.138 mum va lue of P is very nearly constant at P � TM TM for waveguides above one wave length in radi us . As the

waveguide radius is reduced to approach cutoff for the 0.62) , TM mode (a/A � the required P drops sharply 11 TM 0.62 0.65 so th at , for radii in the range of to only

a small amount is needed to minimi ze K '· y

3.5 POLARIZAT ION DEGRADAT ION DUE TO BEAM SCANNING

It is expected that moderate amounts of beam

scanning by lateral feed displacement would not serious-

ly degrade cross-polari zation leve ls since the basic

quadrant symmetry is not radically altered for small

feed displacements . Parame ter studies for symmetrical

reflectors with scanned beams have confirmed th at for 2 3 scan angles on the order of to beamwidths the

cross -po lari zation levels are typically degraded only 2 6 to dB . Wh ile a complete parameter study is beyond

the scope of this thesis , an example wi ll be given here to illus trate the effects of beam scanning and also to illus trate the application of the computer program to circular polarization . 26 Figure illustrates the IEej component of secondary pattern for a parabolic ref lector with a 97

0.16 (]) '0 .....0 ,.-, 0.14 i r-1 r-1 � [-1 0.12 � ·.-!

J...J (]) 0.1 0 :3: 0 AI

r-1 0.08 C{j � 0 ·.-! .jJ 0 rtl J...J 1'4 o.o

�: 0.02 AI8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

· Waveguide Radius - a/A

25. Figure P which Minimizes the Maximum TM Cross-Polarization Current (Ky ) for a Dual Mode Circular Waveguide Feed 98

65° � 2f/A = 17.2, � . = 7 °, � = , = 0 m1n max o f/D D/A TE = = 0.4, = 21.4, Mode Feed , a/ A 0.36 l I

Scanned Beam 0 On-axis Beam = .0465 = .0465

-10

Ill '1j -20

Q) '0 ::l +I ·r-i .-1 E;0.. -30 1'1! Q) :> ·r-l+I ttl ...-! Q) -40 p::;

-50

225° - __. 45° ¢ = I ·� =

Cf\- -�- - u v

10 5 0 5 10 15

26. ff Figure E ect of Beam Scanning for Symme trical Parabola

.-'{ 99

21.4 diame te r of wavelengths wh ich is excited by a cir-

cularly polari zed TE mode waveguide feed having a l l 0.36. radius a/A = Two patterns are shown : one for

an on-axis beam and one for the case where the beam is 6° 45° scanned appioximately off�axis in the ¢ = plane .

This pattern includes the effect of a small amount of

aperture blockage since the integration limits for � 7° 65° . are from � = to � = The increase in sidelobes

(coma ) in the direction of scan is readily apparent

for the scanned beam .

For a circularly polarized antenna , a measure

of the cross�polarization level is the axial ratio of

the secondary pattern . If the antenna has a nominal

polari zation of righthand circular then the cross polar-

ization is given by the amount of lefthand circular

radiation in the pattern. For such an antenna , the

ratio of right to left circular component amplitudes

(E /E ) is a mea�ure 6f its polarization purity . This R L

� 45° ratio is plotted as a function of e in the ¢ 27 plane in Figure for both the on-axis beam and the

scanned beam . ·While the ideal ratio is infinity , it can be observed that within the 3 dB angles of the scanned beam a "cross-polarization" level of at least 40 dB is maintained .

The f�ct that small feed displacements do not drastically alter the polarization purity of . 100

2£/A 17.2, • 7° 1 = �· = 0 = 1/1 = �· 65° 1 r m1n rmax o 0.36 70 TE Mode Feed , a/A = l l

On-Axis Beam

Scanned Beam (Beam Maximum +J:>1 0 ·.-! 5 at 6 () ) 1-1 ;::1 p.. � 0 ·.-! +J rU 40 N ·.-! 1-1 rU r--i 0 p.. 1-1 rU 30 r--i ;::1 () 1-1 ·.-! tJ

CQ '0 20

c.:; l'.il " ...:! l'.il 10

10 0 10 5 5 15

g e - De rees Figure 27. Po larization Purity for Parabola with Scanned Beam 101

ref lector patterns suggests that, as long as the feed yields acceptable cross-polarization levels , displaced feeds may be utilized. Furthermore , arrays of such feeds can be used to provide shaped secondary beams wh ile maintaining low cross polari zation . 4.0 OFFSET-FED REFLECTORS

The primary utility of offset reflectors arises

from the fact that in such antennas the feed is removed

from the radiating aperture . This can offer an advantage

in certain applications since , for example , aperture block

age by the feed structure can impose a fundamental limita

tion on the control of sidelobe levels . Furthermore ,

there may be a physical advantage to offset feeding in

some applications where the support of a feed at the

center of a symmetrical reflector may be costly in terms

of mechanical complexity and/or transmission line losses .

While offset feeding can offer substantial

advantages in certain applications , it also has certain

potential disadvantages. For example, a consideration

of the basic nature of reflector currents for an offset

fed reflector leads to the conclus ion that the cross

polarization levels would be higher than those of a

symmetrical reflector with comparab le parameters . This

fact can be better appreciated by recal ling that for a

symmetrical reflector the currents display a quadrant

symmetry such that cross polarization is eliminated in

the antenna 's principal planes. However, for a� offset

reflector this symmetry is no longef" obtained. The offset reflector cross polarization curr�nts have a

"left-right" symmetry as shown .in Figure 28. This

102 103

x '

K X

·-- --- K y

y y' Feed

Figure 28. Reflector Currents for Offset Reflectors with "Vertically" Polarized Feed Antenna 104

symmetry can be represented by two anti-phase dipoles

as shown in the figure .

For an offset- fed parabola it then is expected

that the cross-polarized radiation will be zero directly

on axis but will experience its maxima at angles away

from the axis in the y-z plane . This represents the

H-plane for a feed antenna which is' "vertically" polar

ized as shown in Figure 28. If the feed antenna is

11 horizontally11 polarized (i.e., along the y-axis) the

cross -polarization lobes still occur in the principal y-z plane which, in this event, is the E-plane of the reflector . Therefore , .whereas a symmetrical reflector will display four cross-polarized lobes occurring in

the diagonal planes , the offset reflector has two cross-polarized lobes occurring in a principal. plane .

It is intuitive ly to be expected that, since the offset reflector experiences cross polarization cancellation only in one plane , its net cross-polarized radiation would be higher than that of a symmetrical reflector . ' Another consideration is that of beam scanning .

If a parabolic reflector is used to obtain scanned beams via feed displacement, it is well known that the field of view is limited by the effects of aberrations . If, for any number of reasons , an offset reflector is used , these aberrations still constitute a lirtlit on the field of view. 105

A relatively new antenna which may prove use-

ful in certain wide angle applications is the offset

spherical reflector . Wh ile the offset sphere will

suffer from a greater amount of spherical aberration

than an equivalent parabola with an on-axis beam , the

sphere 's aberration can be made to be independent of

scan angle. It is to be expected therefore that beyond

a given scan angle , the offset sphere may be competitive

with a parabola�

It is the purpose of this section to present

a few examples of offset reflector patterns which illus-

trate the above points . A complete parameter study would involve many variab les such as reflector size ,

offset angle, type of feed and scan angle . While such

a parameter study is beyond the scope of this thesis

it can be conveniently carried out with the aid of the

computer program presented in Appendix c. The examples to be presented should illus trate some of the basic pattern and polarization properties of offset parabolic

and spherical reflectors .

.A ...... 1 OFFSET PARABOLA The parameters defining an offset parabola and feed have been described in Section 2. The size of the reflector is determined by the quantity R/A = 2f/A , the offset ang le and the integration limits � 0 , 106

• and (cf. Figure 3) . In all cases to be il- I)!ml n I)!max . lustrated, the integration limits for � are from 0° to

These parameters for th is example are : R/ A =

2f/A = 30.5, 33° , � 33° . The ve rtical height , . I)!o = max = D, of the projected aperture as illustrated in Figure 5 can be obtaine d from the equation of a parabola in the

� = 0 plane . The result is

sinu sinv D I A 2(f/A ) - (164) = [ 1 + cosu 1 + cosv J where

u = ( 1/J + 1/J max 16 5 ) o

= lJ! - l)J (166) v o max

For the above example , 0/A = 19 .8. Since the projected aperture of an offset parabola in the x-y plane is a circle, this represents the diameter of the radiating aperture . . Figure 29 illustrates the computed H-plane pat tern ::z: for an offs t parabola which has the E"<�> 17; go� , e above parameters and which is fed by a TE mode circular 1 1 waveguide with radius a/A = 0.846. The cross-polarization lobes are symmetrically disposed about the � = 90° 107

= 30.5, = 3 3° - .1. 2£/A ��' o - '�'max

TE - Mode Feed , a/>. = 0. 846 1 1

-10

P = 0 E1i'' TM P = 0.138 t:Q E TM 'U 1 -20 ,

QJ '"{) :::1 .j..J ·r-1 r-f P.. 8 -30 �

QJ ·r-1:> .jJ \ cu ' rl ' QJ I � - 40 ' I \ I ' I ' I ' I I I I - 50 I ' I I I I I I I I "'8 = 9 0 ° -60 80 85 90 95 100

"' - Degrees cp Figure ·29 . H-Plane Pattern for Offset Parabola 108

axis . Their leve l relative to the peak of E� is approx mate ly -24 dB . Now , an effective f/D for th is parabola can be defined as the ratio of the focal length , f,

to the "diameter" as defined above . Since f/>. = 15.25

for this example , the "f/D " is approximately . 0.78. A comparison of the cross-polarization level for this reflector with that of a symmetrical reflector having a comparable f/D and fed with a circular waveguide can be made with reference to Figure 19. For the symmetrical. reflector, th� cross-polarized radiation for this f/D is approximately -27 dB. Although not shown here , pattern cuts for other off-axis angles confirm the fact that the cross-polarized field does indeed maximize in the H-plane for the specified feed polarization.

It will be recalled that cross-polarized radiation from symmetrical reflectors could be effectively eliminated by employing feeds which have the polarization properties of a Huygen 's source . The Huygen 's source feed is typified by a combination of electric and magnetic dipoles and it was observed that for symmet rical reflectors with low f/D , the small circular wave guide feed provided a good approximation to a Huygen 's source from the viewpoint of polarization . Further ; it was demonstrated that a dual mode circular waveguide feed almost perfectly satisfied this criterion for a wide range of f/D .

. - 109

It is interesting to inquire whether a lluygen 's

sourc� feed can effective ly reduce cross polarization

in offset ref lectors . In order to test this idea , a

dual mode circular waveguide feed was used for the offset

reflector of this example. That is , 13.8% TM mode 1 l power was included in the feed radiation and the secon-

dary H-plane pattern was computed. The results are also

shown in Figure 29 . The principa l polarization pattern

remains virtually unchanged and it can be seen that

there is a negligible reduction in the leve l of cross-

polarized radiation .

In order to further examine th is concept, the

offset reflector patterns have been computed for the

case of a generalized Huygen's source feed . The results

are shown in Figure 30 where it can be observed that

there is virtually no change in the cross-polarization

level when the reflector is fed with an electric dipole , a magnetic dipole , or equal amounts of both wh ich yield

the Huygen 's source

It appears that the criteria for an optimum

feed from the viewpoint of cross polarization must be based on a consideration of the peculiar symmetry of

the currents for an offset reflector . It is conjectured here that, since the reflector surface as viewed from the feed has an elliptical cross section , certain classes of feeds with a prescribed unsymmetrical pattern and 110

2f/A 30 .5, � 33o - � = 0 = max TE Mode Feed , a/A = 0.846 1 l 0

--- Electric Dipole , P =l, P2=0 1 1 - 0 Magnetic Dipole , P1=0 ,P7=1 Huygen 's Source , P1=l,P2=1

'U!:Q -20

<1J 'U ;:1 +J - � r-f -30

�& � •

(]j :,;;.. -� .j.J l'{j ....-! -40 <1J p::

-50

-60 90 95 100 105

'J;' - Degrees Figure 30 . II-Plane Patterns for Offset Parabola with Generali zed Huygen 's Source Feed I I I

having unsymmetrical (in the quadrant sense) polarization properties may yield optimum results . Possibilities include the addition of TM mode to circular wave a 1 guide , which would create an unsymmetrical feed pattern , and the excitation of "difference" modes in rectangular waveguide . Additionally, it may be possible to utilize small arrays of feed elements which would generate com- pensating cross-polarized currents on the re f lector surface to c�ncel those produced by the surface geometry .

This prob lem is complicated by the fact that the re- fleeter currents depend on whether the feed is "vertically" or "horizontally" polarized . At any rate , this topic has not been fully explored and it constitutes an area in which further research is needed .

4 . 1 . 1 Scanned Beam

It is of interest to examine the cross-polarized radiation from an offset parabola which is used for beam scanning. It is intuitive ly expected that the maximum level of cross polarization would not be drastically changed as a result of moderate amounts of scan since, in effect, the required feed motion has the effect of merely altering the phase destribution across the aperture . The distribution of cross- polarized radiation would, of course, change as a result of this phase distribution . 112

This effect has been confirmed by computing

the ve ctor patterns of offset re flectors as a function

of scan an9le . An example of such a pattern is shown

in Figure 31 where the waveguide feed has been displaced

laterally (i. e., in the -y ' direction) to cause the

beam to scan by approximately 8.5Q in the H-plane . The

peak relative level of Er is now -23 dB which is a degradation of only 1 dB from that of the on-axis beam.

The shape of Er has changed, however, and it can be seen that the cross-polarized lobe has shifted by an

amount proportional to the shift of the main beam .

4.2 OFF�)E'l' SPHERICAL REFLECrOR The patterns of offset-fed spherical re f lectors would be expected to show higher sidelobes and some beam broadening compared to a parabola with similar parameters . The reason for this is the fact that a sphere does not perfectly collimate the feed radiation and hence , the pattern is degraded due to spherical aberrations or "aperture phase error'' . The E-plane pattern for an offset sphere , having the same parameters as the parabola discussed in the preceding section , is shown in Figure 32. In order to verify the accuracy of the computer program , a model of a spherical re- fleeter was constructed and experimentally evaluated .

The measured pattern is also shown in Figure 28 and it

.- 113

2f/A 30 .5, 33° = �0 = = � max

Mode Feed , a/A = 0.846 TE l I E.: = 0, E = l =: 0 X I y I -0 .085 , Z I 0

E...., -10 e

-20 E 'l" 'I ' ,.- ,.- ... ', / \.- / '\ I \ I \ - 30 I \ I I I I /' I -40 I \ I I \./ I I I -50 I /' = 90° I \ o \ I I \ I

90 100 105 110

......

Figure 31. Offset Parabola with Beam Scanned 8° in the H-Plane 114

R/A 30.5, 33° , f/R 0.43 = w o = = w max =

TE Mode Feed , a/ A = 0.846 1 1

'\ -- Computed \ \ --- Me asured \ \ -10 \ \ \ \ \ E"' 8 \ \ a:1 -20 '"d \ \ I G) I '"d I :::l \ +l ·r-i I H -30 v 0.. �

G) :> ·r-i +l ttl r-1 -40 G) �

-50

-60 80 85 90 95 100

8 - Degrees

Figure 32. Comparison Between Computed and Measured E-Plane Patterns for Offset Spherical Reflector 115

can be seen that there is reasonably good agreement between computed and measured results . In fact this

agreement is quite good when it is considered th at the experimental feed antenna was not a straight circular waveguide radiating pure TE mode but was instead a l l flared conical horn having the sped. radius. ficd '!'he flare produces a phase error across the feed aperture and destroys the coincidence of the feed 's E- and H- plane phase centers . One manifestation of this effect is the slightly higher sidelobe levels which are observed in the measured pattern.

The computed and measured H-plane patterns for this antenna are shown in Figure 33. Again , the patterns agree quite closely attesting to the validity of the computer program . The peak level of E� for this pattern is again -24 dB which is just the leve l obtained from the parabolic reflector . From the viewpoint of cross polarization, it appears that there are few major differences between the sphere and the parabola.

A� additional variab le with the offset spherical reflector is the location of the feed denoted 1 3 by f/R . It is known that for a symmetrical spherical reflector , the optimum f/R is a function of the allowable phase errors across the reflector aperture .

Figure 34 illustrates the effect of focusing on beam 116

R/A = 30.5, f/R = 0.43 33° tjJo = = tjJmax TE Mode Feed , a/A = 0.846 1 1 0

- 10

1=0 "d -20 Q) "d \ ::l ..j.J \ ·r-1 ,.....f \ 0.. -30 \ � \ \ Q) � ', ·r-1 ' ', ' ..j.JI'd ,.....f Q) ', \ p:; -40 \ \ \ E�

-50

-60�------��------� 90 95 100

'¢ - Degrees Figure 33. Comparison Between Computed and Measured " H-Plane� Patterns for Offset Spherical Reflector 117

R/A = 30. 5 ' I� = 33° = vo 1/! max

TE Mode Feed , a/A = 0.846 I I

0 ...... -....···· '.. \ .. •·. / , \ , / ,• \ ·. ' I • \ •. '• I \ I I \ � -10 I : \ ' , . I .' \ '•,, f/R = 0.5 I / \ ,• \ � I ·. \ , I :. �-..... � I ,. ' � . \ � -20 : •. I ' /,. f/R = 0 . 43 \ '• . I : \ .... I / \ , I : - . \..,./'f/R=O .tl7S 1 - . 3 0 I ,• I / : • •�· · • •: ...... • • • • • •• . . - •• 40 •• •

-50 r = 90 °

-60 80�------�------�------�------� 85 90 95 100

"' 8 - Degrees

Figure 34. Effect of Focusing for Spherical Reflectors 118

shape and the location of the beam peak for the offset

sphere . In order to determine the "best" feed position

it is necessary to consider the antenna gain or aper-

ture efficiency . (This feature has not been incorporated

herein since the primary objects of investigation are

pattern shape and cross-polarization leve ls .) It can

be observed that the beam shifts slightly as the feed

is moved. Since the feed motion is along the z'-axis

this corresponds to the introduction of an unsymmetrical

phase error by "axial"defocusing . The best pattern

shape from the viewpoint of sidelobe enve lope seems

to occur when f/R = 0.43. In general , the "optimum" f/R will depend on the reflector size , the effective

f/D and the offset angle.

4.3 A COMPARISON BETWEEN THE OFFSE'l' SPHERE AND THE OFFSET PARABOLA

It was stated that , while the pattern of an offset sphere suffers from aberrations , these aberrations are independent of scan angle as long as the feed is moved along the surface of a prescribed "focal sphere". Since the pattern of a parabola with a scanned beam degrades continuously as scan angle is increased , it is interesting to inquire whether a "break-even" point exists in regard to scan beyond which the offset sphere has a net advantage . While this is a complex 119

question and its answer depends on many parameters ,

it is instructive to consider a simple example . In

Figure are plotted the patterns of an offset parabo la 35 and an offset sphere having the same parameters . The

pattern for an on-axis beam from an offset parabola

is shown as is the pattern for a beam which is scanned

in the E-plane by approximately The pattern of the 8 ° . scanned beam displays the expected increase in sidelobe

level (coma ) compared to the on-axis beam . Now , the

pattern of the offset spherical re flector is shown super

imposed on the scanned beam of the parabola. Wh ile the

sphere 's sidelobe structure is decidedly inferior to

the on-axis parabola beam, it is comparable in quality

to the parabola 's scanned beam . If scan angle is

increased , the parabola's beam will degrade further while that of the sphere will not .

This example illus trates the possibility

that, for some wide angle applications , the offset

spherical reflector can be a useful antenna and can

display a net advantage over parabolic reflectors . 120

J, R/A = 30 .5, � ' ' Mode = 33° = I' E Feed 0 �'max ' I l Feed , a/A . = 8 46 ··· Parabola with beam on-axis --- Parabola with beam scanned by 8.5° .. • Offset Sphere . .• 0 ...... · . . . . . ,: . • . . • 0 · . . • •

•.. \• . . . .· . . • • . • . . • • • . . \ • . .• . . • . . -� •. r' . .. • . • I ' • I \ Q) • • \ '"d • • ::;j . \ • • ' � -30 • . . r-i • ' . . • 0.. • . .. E ...... � � . . .· . . : '· ...... Q) . • • :> . . • e : . . . . ·rl . .. ,: • � -40 • • . . •• •• •• . r-i • Q) p::;

-so

-60�------+------�------�------��------� 75 90 100 80 85

Degrees e -

Figure •. Comparison Between Offset Sphere 35 and Offset Parabola with Scanned Beam

-- 5.0 CONCLUSIONS

An analysis has been conducted for offset-

fed and symme trical ref lector antennas . A computer

program has been deve loped which can be used to compute

the far-field vector patterns of spherical or parabolic

reflectors illuminated by a variety of feed antennas ,

including arrays of feeds .

For symmetrical reflectors , the class of

feed radiators having the polari zation properties of

a Huygen 's source can eliminate cross-polarized radia-

tion . In particular , the small TE mode circular wave- 1 1 guide feed has been shown to be nearly ideal from th is viewpoint. In genera l, the introduction of a specified amount of TM mode to a circular waveguide feed can 1 1 result in the control of cross polarization.

The cross-polarized radiation of offset reflectors is degraded when compared with that of symmetrical reflectors. This is a consequence of the fact that certain geometrical symmetries have been destroyed when offset feeding is used . The deg�adation in cross- polarization level with scan is not great provided that the beam is scanned a moderate amount, e.g. , a few beamwidths .

The cross-polarization leve ls of spherical reflectors , both symmetrical and offset, have been found

121 122

to be very similar to those of parabolic reflectors having similar parameters .

A comparison of the patterns for an offset

spherical reflector with those of an offset parabola having a scanned beam suggests that the offset sphere could find usefulness in certain wide angle applications which require offset feeding . 123

REFERENCES

1. D. F. DiFonzo and R. W. Kreutel, "Communications Satellite Antennas for Frequency Reuse," 1971 IEEE Symposium on Antennas and Propagation (Technical Digest) , September 1971, 287

2. E. U. Condon , "Theory of Radiation from ·Paraboloidal Reflectors ," Westinghouse R. Report SR-105, Septem ber 1941

3. E. M. T. Jones , "Paraboloidal Reflector and Hyper boloid L�ns Antennas ," IRE Trans . on Antennas and Propagation , 1954, 119

4. I. Koffman , "Feed Polarization for Parallel Currents in Reflectors Generated by Conic Sections ," IEEE Trans . on Antennas and Propagation , January 1966 , 37

5. V. Kerdemelidis , "A Study of Cross Polarization Effects in Paraboloidal Antennas ," California Institute of Technology Antenna Laboratory Tech nical Report No . 36, May 1966

6. H. C. Minnett and B. MacA . Thomas , "A Method of Synthesizing Radiation Patterns with Axial Symmetry ," IEEE Trans . on Antennas and Propagation , September 1966 ' 654 7. V. H. Rumsey , "Horn Antennas with Uniform Power Patterms Around Their Axes ," IEEE Trans ·. on Antennas and Propagation , September 1966, 656

8. Y. T. Lo ,· "On the Beam Deviation Factor of a Para bolic Reflector," IRE Trans . on Antennas and Prop agation, May 1960, 347

9. K. S. Kelleher and H. P. Coleman , "Off-Axis Char acteristics of Paraboloidal Reflectors ," Naval Research Lab. , Washington , D. c. , Report 4088, December 1952

10 . S. Sandler, "Paraboloidal Reflector Patterns s. for Off-Axis Feed," IRE Trans . on Antenna� and Propagation , July 1960 , 368

11. J. Ruze , "Lateral Feed Displacement in a Parq.boloid ," IRE Trans . on Antennas and Propagation , July 1960 , 368 124

12. J. Ashmead and A. B. Pippard , "The Use of Spherical

Reflectors as Microwave Scanning Aerials ," J. IEE , Vol. 93, pt. IliA, 1946, 627

13. T. Li, "A Study of Spherical Reflectors as Wide Angle Scanning Antennas ," IRE Trans . on Antennas and Propagation, July 1959 , 223

14. G. C. McCormick , "A Line Feed for a Spherical Reflector," IEEE Trans . on Antennas and Propagation , AP-15, September 1967, 639

15. V. H. Rumsey , "On the Design and Performance of Feeds for Correcting Spherical Aberration," IEEE Trans . on Antennas and Propagation , AP-18, May 1970 , 343

16. G. Hyde , "Studies of the Focal Region of a Spherical Reflector : Stationary Phase Evaluation ," IEEE Trans . on Antennas and Propagation , AP-16 , November 1968, 646

17. L. J. Ricardi , "Synthesis of the Fields of a Trans verse Feed for a Spherical Reflector ," IEEE Trans actions on Antennas and Propagation , AP-19 , No . 3, May 1971, 310

18. Silver, Microwave Antenna Theory and Desian , s. McGraw Hill, New York , 1949

19 . M. J. Pagones , "Gain Factor of an Offset-Fed Parab oloidal Reflector ," IEEE Trans . on Anterinas and Propagation, AP-16 , No . 5, September 1968, 536

20 . D. J. Bern , "Electric-Field Distribution in the Focal Region of an O:ffset Paraboloid," Proc . IEE , Vol. 116 , No . 5, May 1969 , 679

21. D. F. DiFonzo, "The Offset Fed Spherical Re flector ," (paper presented at the 1970 IEEE International Conference on Communications), June 1970

22. T. Pratt, "Offset Spherical Reflector Aerial with a Line Feed ," Proc . IEE , Vol . 115, No . 5, May 1968 , 633

23. Silver , Microwave Antenna Theory and Design , s. McGraw Hill, New York , 1949

24. J. A. Stratton and L. J. Chu , Physical Review , Vol. 56 , 1939, 99 125

25. A. Ludwig , "Calculation of Scattered Patterns from Asymme trical Re flectors ," Ph .D. Dissertation , University of Southern California , June 1969

26 . M. Spiegal, Theory and Prob lems of Vector R. Analys is , New York , Schaum Publishing Co. , 1959

27. A. Ludwig, "Near-Field Far-Fie ld Trans formations using Spherical Wave Expans ions ," IEEE Trans . on Antennas and Propagation, March 1971, 214

28. A. Ludwig·, "Computation of Radiation Patterns Involving Numerical Double Integration ," IEEE Trans . on Antennas and Propagation , Vol . AP- 16 , No. 6, November 1968, 767

29 . J. Cook , et al , "The Open Cassegrain Antenna : s. Pt. Electromagnetic Design and Analysis," Bell I System Technical Journal , September 1965, 12sg-- 30 .. M. Sancer, "An Analysis of the Vector Kirchoff Equations and the Associated Boundary-Line Charge ," Radio Science , Vol . 3 (new series ), No . 2, February 1968, 141

31. W. J. English, "E- and H-Plane Beamwidth Parameters for a Multimode Circular Waveguide Aperture ," COMSAT Technical Memorandum CL-8�71, February 1971

32. B. E. Kinber, "The Space Structure of the Radiation Pattern and Polarization of Radiation from an Axially Symmetrical Reflector Antenna ," Radiotek hnika i Electronika , Vol . 5, 1960, 720 33. V. H. Rumsey , et al , "Techniques for Handling Elliptically Polarized Waves with Special Refer ence to Antennas ," Proc. IRE , May 1951, 533

34. Ludwig , "Gain Computations from Pattern Integra tion," IEEE Trans . on Antennas and Propagation, March 1967, 309

35. P. Potter, "A New Horn Antenna with Suppressed Sidelobes and Equal Beamwidths ," Microwave Journal, Vol . VI� No. 6, June 1963, 71

...... APPENDIX A. DERIVAT ION OF EQUATION (20)

Given a radius vector

-7 y z o = P sinw cos s x + p sinw sins + P cosw (A-1)

A + + + p y cos s �r cos w sinw Cl�() J x sins r� cosw sinw � () "] + + [ - p sinw cosw ��] ; (A-2)

The transformation from rectangular to spherical coordi- by nate is given

s �n�p COSs COSip COSs [s 1 n 1p sins cos w sinE; (A-3) cosw -sinw

Then , upon substitution of the spherical components from

(A-3) into (A-2) it is found that

-7

= + + � A p A,, W A� t; (A-4) () w p 'I' <-,

where

= + A cost; [P cosw sinw ��Jsinw cost; P

+ sinE;[P cosw + sinw �� sinw] sinE;

126 f _ .;..... 127

cos lj! sinlj! cosl); + [-p + ��]

A (A-5) p

A = cos2 sin!JJ coslJ; l/J t; [P COSl/J + �(lp] sin coslJ; sinl/J p + 2 t; [P + a7j; COSt/J (l ] - sinlJ; sin1/J + coslJ; [- p ��]

A = (A-6) lJ; p

A = 0 (A- 7) E,

There fore ,

.?...e._ A (20) (A-8) + P l/1 a t/J P

-+ a The expression for p is derived in a similar · manner. a .; ->- APPENDIX B. EXPANSION OF F IN RECTANGULAR COORDINATES

-� F F + F + F = n � � p � �

= (si cos � x' + sin� sin� y ' F n � p J · + COS� Z 1)

+ F (cos� cos � ' + cos� sin� y' � x - sin� z')

F (-sin� ' + cos � y') (B�1) + � x

Equa tion (11) gives

== COS!p + sin1J; z x' O X 0

' y = -y ( B-2)

z' = sin� - COSl/1 z 0 X 0

Sill)stitution of Equation (B-2) into (B-1) and rearrangement of terms gives

F = (F sin � cos e + F cosl/J cos � -F L sin� ) p w s

· (cosw + sin� z) 0 x 0

+ (F simp sin� + F cosw sin� + F cos� ) (-y) P l/J �

(B- 3)

128 129

Rearrangement of the terms in equation (B-3) yields

the rectangular components of F wh ich are given by equations (32) - (34) in Section 2. APPENDIX C. OFFSET REFLEC'I'OR COMPU'I'ER PHOGRAM

C.l Discussion

This section contains a listing of the computer program which has been developed yield the far-field to vector patterns of offset and symmetrical reflectors .

The program is written in FOR'l, RAN IV G Level and has been adapted for use with the IBM 360 TSO System

(Time Sharing Option ) . The modifications to the program necessary to adapt it for a batch environment are ob vious . They include , for example , the elimination of the "prompting" statements , i.e. , lines 170 , 180 , 210 ,

220 , 250, 260 , 280, 290 , 970 , 980 . Another modification which may be required is the inclus ion of specific FORMAT statements for input (cf. line 190) .

The integration grid (w ,C ) is dimensioned as

16 x 91 for this program . The absolute accuracy of the patterns depends upon the size of the reflector in wavelengths and the off-axis angle. TherefOre , for studies of 'the main beam region and the near-in side lobes , the allowable integration grid size may be as high as 10 to 20 wavelengths . In general, the computed patterns have been shown to have a useful "dynamic range" , in regard to accuracy , of 30 to 40 dB for re flectors which are sized such that the incremental areas for the integration grid correspond to about one

130 131

square wavelength or less . If greater accuracy is

desired or larger reflectors are to be analy zed , it is

mere ly necessary to change the appropriate DIMENSION

statements . Alternatively, if storage capacity is a

problem , the reflector can be divided into sub-areas

and the integratiop .P,e rformed separate ly over each area .

Then , the total fieTd,.. 11s obtained by superimposing the scattered fields from all the sub-areas .

C.2 Description of Input Quantities Needed by the Program

The program requires 5 lines of input. The

input quantities are listed below. Refer to the text for their meaning .

INPUT QUANTITIES

Line 1 Reflector Description

I REFL 1 For a Parabolic Reflector

2 For a Spherical Reflector

2f/A for a parabola RL R/A for a sphere

FR f/R for a sphere

0.5 for a parabola SIO 1/J o SIMIN 1/J (lower limit of integral) min tJ; SIMAX 1/J (upper limit of integral) max tJ; 132

XIMIN 1/J (lowe r limit of integral) min r, XIMAX 1/J (upper limit of integral) max (

Note : SIMIN , SIMAX , XIMIN , XIMAX are in degrees SIMIN < SIMAX ; XIMIN < XIMAX

Line 2 Feed Description

NFEEDS is the number of feeds ; NFEEDS < 10

I TYPE Feed Pl P 2 P3

Gen. Huygen's Elect. Mag . Re lative 1 Source Dipole Dipole Phase Amplitude Amplitude Between Dipoles (deg.)

2 E Mode a/A. b/A. N/A 'l' 1 a Rectangular (enter 0) Waveguide

3 Dual Mode a/A. P Rel. TM Circular Phase Waveguide (deg. ) Feed n ::.. 4 cos n N/A N/A 'P lJ! (enter 0) (enter 0)

Line 3 Feed Polarization

P4 is A '' amplitude of the "x'-port" excitation x of the Feed

PS is A ' ' amplitude of the excitation y "v..1.' -nort l..- -- - " of the Feed

P6 is the relative phase in degrees between

the port excitations .

(cf. Section 2.4.5) 133

Line 4 Feed Positions

£ XR (N ) I /R for the nth feed X '/R for the nth feed YR (N ) t:y /R for the nth feed ZR {N ) t: z '

Line 5 Definitions for Pattern Cut

(Constant) - - - - - (Variable) - - - - - KORSYS ANGCUT VARMIN VARMAX VARINC --

{1.6 1 rp e min e max l1 2 e <�>min cpmax tP

...., _ ...., ""' 3 8 r cpmin max l'.rp

- '""' "' ...... 4 rp e min e max {1.8 .·

00010 OFFSET REFlECTOR PROGRM-1 D. D I. FONZO c 00020 PROGRAM COMPUTES VECTOR PATTERNS OF OFFSET OR SYMMETR ICAl c 00030 REFLECTORS BASED ON INTEGRAT ION O F SURFACE CURRENTS . c 00040 COMPLEX ES I(1 6,9l),EXI(l6,91 ),CMPL X,ARFACT,F(l6,91,3) ,SUM ( 3) 00050 COMPLEX EX,EV, EZ, ETH,EPH, ETHBR,EPHBR 00060 DIMENS ION R0(16,9l) ,GAM ( 16, 91) 00070 DIMENSION XR(l0),VR(l0),ZR(l0),A(l6,9l),C(16,9l),PHFEED(16,91) 00080 COMMON/GR ID/ SIR(16),XIR (91) 00090 COMMON/TR IG/SSI (l6),CSI(16),SXI(9l),CXI(91) 00100 COMMON/FEED/ ITYPE,Pl,P2, P3,P4, P5, P6 00110 COMMON/GRI NT/MM AX,NMAX 00120 COMMON Pf ,OTR 00130 P1 = 3.1415926535897 93 00140 DTR=Pf/180.0 00150 RTD= 180.0/PI 00160 READ REFLECTOR TYPE , INTEGRATION GRID LIMITS, AND FEED TYPE c 00170 9999 HRITEC6,10) 00180 10 FORMAT( ' INPUT IREFl,R/LAMBDA,F/R,SIO,SIMIN ,SIMAX,XIM!N,XIMAX' ) 00190 READ(5, *) IREFL,Rl,FR,SIO,SIMIN ,SIMAX,XfMIN,XIMAX 00200 FEED DATA- NO . OF FEEDS,TYPE, AND PARAMETERS c 00210 WR ITE(6, 20) 00220 20 FORMAT( ' INPUT NO . OF FEED S, ITYPE,P l,P2, P3 1 ) 00230 R�AD(S, *) NFEEnS, ITYPE,Pl, P2, P3 00240 FEED PO LAR IZAT ION AND POSITIONS ( POS. NORM . TO R) c 00250 R I T E ( 6 , 2 5 ) �'I 00260 25 FORMAT( ' INPUT FEED PO LAR !ZAT IO"l PARM.-tETERS AX " ,AY " ,REL. PHAS E ' ) 00270 READ(S, •) P4, P5, P6 00280 WR ITE(6,30) 00290 30 FORMAT( ' INPUT FEED PO S ITIONS : X 11/R,Y''IR,Z ''/R FOR FEED' ) EACH

;.... w ,::,. 00300 READ(S, *) (XR(N),YR(N),ZR(N),N=l,NFEEOS ) 00310 ESTA BLISH TR IG FUNCTIONS ON INTEGRATION GRID c 0032 0 MMAX =l6 00330 NMAX=91 00340 E��MAX =M��AX -1 00350 ENNAX=NMAX-1 00360 DELSI=(SIMAX-SIMI N)/EMMAX 00370 DEL XJ=(XIMAX-XIMIN) /ENMAX 00380 CAll SETUP (SIMIN,D�LSI,XIMIN,DELXI ) 00390 ESTABLISH FEED PATTERN c 00400 CAll EFEED(ESI,E X!) 00410 SET UP GRID EXCITATI ONS AND VECTOR F c 00420 SSIO= SIN(SIO•DTR ) 00430 CS IO= COS(SI O•DTR ) 00440 DO 90 J=1,NMAX 00450 DO 90 I =l,r-1MAX 00460 A( I ,J)=SSI( I )•CX! (J)•CS IO+CS I(I )•SS IO 00470 B=CSI(I )•C XI(J)•rS IO-SSI(I)•SS IO 00480 C( I,J )=S SI(I)*C XI(J)* SS I O-CSI(I)•r.S IO 00490 D=CSI(I )•C X! (J)*SSI O+SSI (I)•C SIO 00500 DCDX I=-SS I (f )*SXI(J)•SS IO 00510 SET UP RAD IUS VECTOR DERIVAT I VES c AND 00520 c 00530 H1 = 1. O-FI1 00540 c 00550 GO T0 ( 50,60),1REFL 00560 c 00570 PA RABO LA c 00580 c 00590 50 RO ( I,J)=Rl/(1.0-C( I,J))

I-' w (..., 00600 DROOS I=RL•D/(1.0-C( I,J))**2 00610 DROOX I=Rl•DCOX I/(1.0-C( I ,J))*•2 00620 GO TO 70 00630 r, 00640 SPHERE c 00650 c 00660 60 TERMl=SQRT( l.O-EM••2•(1.0-C( I,J ) **2)) 00670 TERM2=EM+ EM••2•C( I,J)/TERMI 00680 DROOS I=R l•D•TERM2 00690 DRO DX I =RL•OCOXI •TEqM2 00700 RO ( I,J)=Rl•( TERMl+EM•C( I,J)) 00710 c 00720 ARRAY FACTOR FOR FF.E D R R Y c A � 00730 c 00740 70 ARFACT=CM PLX(O.O,O.O) 0075 0 DO 80 N=l,NFEEDS 00760 ROX=RO( I,J)•S SI(I)•C XI(J) 00770 ROY=RO( I,J)•S SI(I)•S XI(J) 00780 ROZ=RO( I,J)•C SI(I) 00790 ROP=SQRT( (ROX-RL•XR( N))••2+(ROY-RL•YR( N) )••2+(ROZ-RL•ZR( �)l••2 ) 00800 ERGF=2.•PI•ROP 00810 80 ARFACT=ARFACT+CM PLX(COS ( ERGF),-SIN(ER�F )) 00820 AMFACT=CARS(ARFACT) 00830 PH FEED( I ,J)=ATA N2(AIMAG (ARFACT),REAL(ARFACT) ) 00840 I NTEGRA N D VECTOR F c 00850 GXS I=SSI (I)•(A( I,J)•DRODS I+R O(! ,J)•B) 00860 GXX I =A( I,J )•DRODXI-RO( I ,J)•S SI(I)*S XI(J)•C SIO 00870 GYS I=S SI(I) •SXI(J)•(SSI(I)•DRODS I+RO ( I,J )•C SI(I)) 00880 GYX I=S SI(i)*(SXI(J)•DRODXI+RO( t,J )•C XI (J)) 00890 GZS I=S SI(I)•(C( I,J)•DROOS I+RO( I,J)•D)

i--' w 0"1 00900 GZX I=C( I,J)*DROOX I+RO ( I,J)•OCOX I 00910 F(i,J,1)= AMFACT•(ES!(I, J)•CMPLX(GX SI,O. ) +EXI(I,J)*CMPLX(GX XI,O. )) 00920 F( I ,J,2)= AMFACT•(-E SI(I, J)•CMPlX(GYSI,O. )-E XI(I,J)•CMPLX(GYX!,O. )) 00930 90 F( I ,J,3)=AM FACT*{ESI(I, J)•CP1P lX(GZS I,O.) +EXI{I,J)*CMPLX(GZXI,O. )) 0094 0 c 00950 READ FAR FIElD DATA · c oo96 o c 00970 100 WR ITE(6,40) ' 1 00980 40 FORt·1AT( INPUT KORSYS , ANGCUT, VARM IN,VARMAX,VA R INC, I HEAD ) 00990 REA D(S,•) KORSYS,ANGCUT, VARM IN,VARMAX,VAR INC, I HEAD 01000 PR INT REF. AND FEED PA RAMETERS c 01010 c 01020 IF( IHEAD.EQ. O} GO TO 1119 01030 �I RITE(6, 1000) 010 40 GO TO (111,112),1R EFl 01050 111 WR ITE(6, 1001 ) 01060 GO TO 113 01070 112 WR ITEC6,1002 ) 01080 113 WR ITE(6, 1003 ) Rl,FR,SIO,SIM IN,SIMAX,XIMIN,XIMAX 01090 GO TO ( l14,115,116,117),1TYPE 01100 114 WR ITE(6, 1004) Pl ,P2, P3 01110 GO TO 119 01120 115 WR ITE(6,1005 ) Pl, P2 01130 GO TO 119 .01140 116 WR ITE(6,1006 ) Pl, P2,P3 01150 GO TO 119 01160 117 WR ITE(6, 1017) Pl 01170 119 WR ITE(6, 1007) P4,P5,P6 01180 WR ITE(6, 1008 ) (N,XR(N),YR(N),ZR(N),N=l,NFEEDS ) 01190 WR ITE(6 , 1009 )

I-' w -.j 01200 1119 WR ITE(6, 1010) 01210 c 01220 BEG IN FAR FIELD COMPUTAT ION c 01230 c 01240 PATMAX=O. O 01250 VA R=VARM IN 01260 102 '. GO TO (l038104,106, 107),KORSYS 01270 CONVENT IONAl CUT -- THETA VA R IABLE c 0128 0 103 PH =ANGCUT 01290 TH=VAR 01300 GO TO 105 01310 CONVENT IONAl CUT--PH I VA R IABLE c 01320 104 TH =ANGCUT 01330 PH=VAR 01340 105 STHCPH=S IN(TH•DTR )•COS (PH•DTR) 01350 THBR=RTO•ATAN2 (SQRT(l.O-STHCPH**2),STHCPH ) 01360 PH BR=RTO•ATAN2(1.0,TAN(TH•DTR)•SIN(PH•nTR)) 01370 GO TO 120 01380 TILDE CUT -- PH IBAR VA R IABlE c 01390 106 THBR=ANGCUT 01400 PH BR=VAR 01410 GO TO 108 01420 TI LDE CUT -- THETABAR VAR IABlE c 01430 107 PHBR=ANGCUT 01440 TH BR=VAR 01450 108 STHSPH=S IN(THBR•DT R)•SIN( PHBR•DTR} 0146 0 TH =RTD•ATAN2( SQRT(l,O-STHSPH••2),STHS PH ) 01470 PH =RTD •ATAN2(SIN(THBR•DTR )•COS( PHBR•DTR),COS(THBR•DTR )) 01480 c 01490 ES TABU SH PATH LENGTH c GA�1

� w co .I

01500 c 01510 120 STH=S IN(TH•OTR ) 01520 CTH=COS(TH•nTR ) 01530 SPH=S IN( PH•DTR ) 0154 0 CPH=COS(PH•DTR ) 01550 DO 130 J=l,NMAX 01560 130 I =l, M�1AX 00 01570 RODOTR=A( I,J)•STH•CPY-SS I(I)•S Xt (J)•STH•SPH +C( I,J)•CTH 01580 130 GAM( I,J )=2. 0•PI •RO( I,J)•RODOTR+PHFEEDC I,J) 01590 INTEGRATE c 01600 CALL FIN T(SIR,X!R,F,GAM ,MMAX·l, NMAX-l,SUM ) 01610 EX=SUM(1) 01620 EY=SUtH 2) 01630 EZ=SUM( 3) 01640 ETH=CTH•(EX•CPH+EV•SPH )-EZ•STH 01650 EPH=EY•CPH-EX•SPH 0166 0 c 01670 COMPUTE TI LDE (RAR ) COMPONENTS AND AX IAl c RAT IO 0168 0 DEN=SQRT(l.O-(STH•CPH )••2 ) 01690 RT I LDE=CTH•CPH/DEN 01700 STI LDE=S PH/ DEN 01710 ETHBR=-ETH•RTILDE+EPH•STILDE 01720 EPHBR=-ETH•STI LDE-EPY•RTILOE 01730 AETH=CABS(ETH ) 01740 PETH=ATAN2 (AIMAG (ETH},REAL(ETH)) 01750 AEPH=CABS(EPH) 01760 PEPH=ATAN2(AIMAG ( EPH),RFAL{ EPH)) 01770 PO I FF=PETH-PEPH 01780 HYP=AETH••2+AEPH••2 01790 HYP1=2.0•AETH•AEPH•S IN(POIFF)

1-' w \.0

\ 01800 X=SQRT(HYP+HYPl ) 01810 IF(HYP1 .GE.HYP) Y=O .O 01820 IF(HYPl. LT .HYP) Y=SQRT(HYP-HYPl ) 01830 IFCX.GE.Y•l. OE05 ) RAT=l.OEOS 01840 IF(X.LT.Y* l.OE05 ) R�T=X/ Y 01850 ARNm�=l . O+RAT 01860 AROEN=l. O-RAT 01870 IF(RAT. LE.l.OE-05 ) ELEqDB=-99 .99 01 880 IF(RAT .GT.l.OE-05 ) ElEROB=20. •ALOGlO(RAT) 01890 IF(RAT .GE.l.OEOS ) ELEROB=99 .99 01900 IF(ABS(ARNUM) .GE.ABS(ARDEN)•l.OE05 ) AR=l.OEOS 01910 IF(ABS(ARNUM). LT. ABS( ARDEN)•1.0E05 ) AR=ARNUM/ARD EN 01920 AETHBR=C ABS( ETHBR) 01930 AEPHBR=CABS( EPHBR) 01940 IF(PATMAX.LT.AETHBR) PATMAX=AETH BR 01950 IF(PATMAX. EQ. AETHBR) ANGMAX=VAR 01960 PETHBR=ATAN2(AIMAG( ETHBR),REAL( ETHBR)) 01970 PE PHBR=ATAN2 (AIMAG( EPHBR),REAL(EPHBR)) 01980 PD IFBR=PETHBR-PEPHBR 01990 ANG=PD IFF *RTO 02000 ANGBR=PD IFB R•RTO 02010 IAR=AR/ ABS(AR) 02020 SENSE=IAR 02030 AROB=SENS£•99.99

02040 IF(ABS( AR).LT.l.OEOS ) AqnB=S ENSE•20.0•ALOr.lO(AB S(AR) ) · 02050 TAU=O. S •ATAN2 (2.•AETHBR•AEPHBR•COS( P OIFBR), (AETHBR••2-AEPHBR••2 )) 02060 TAU=TAU•RTO 02070 PTHOB=-99 .99 02080 PPHOB=-99.99 02090 PTHRDB=-99 .9tJ

1-' ..,. 0 .·

\ 02100 PPHRDB=- 99 .99 02110 iF( AETH.GT. l.OE-05 ) PTHDB=20. 0*AlO r,lO(AETH ) 02120 IF(AEPH.GT.l.OE-05 ) PPHDR=20.0*ALO nlO(AEPH) 02130 IF(AETHBR.GT. l.OE-05 ) PTH�OR=20. 0*ALOnl O(�ETHRR } 02140 IF(AEPHBR.GT.l.OE-05 ) PPHROR=20.0* AlO GlO(�EPHRR) 02150 AEX=CARS(EX) 02160 AEY=CABS(EY) 02170 AEZ=CABS(EZ) 02180 WR ITE(6,1011 ) TH,PH, PTH DB, PPHDB,ANG, THBR,PHBR, PTHRDR, PPHRDB,TAU, 02190 lAROB,E lEROB,AEX,AEY, AEZ

\ 02200 VAR=VAR+VAR INC 02210 IF( VAR-VARMAX) 102,102,140 02220 140 WR ITE(6,1012 ) PATMAX,ANGMA X 02230 WR ITE(6,1014} 02240 REA0( 5,*) LCUT 02250 IF(lCUT) 100,150,100 02260 150 WR ITE(6,1015) 02270 READ(S,•) LCASE 02280 IF(lCASE) 9999,160,9999 02290 160 WR ITE(6,1013) 02300 1000 FORMAT( 1Hl,53X, ' OFFSET REFLECTOR PATTERNS ' ,30X, ' n.OtFONZO � ) 02310 1001 FORMAT( ' PA �AB OLIC REFLECTO R (R IS 2*FOCAL LENGTH)') \ 02320 1002 FORMAT( ' SPHERICAL REFlECTOR REFL ECTOR RAD IUS)1 ) (R IS 02330 1003 FORMAT( ' R/LAMBOA= ' ,F7.3, ' FIR= ' ,F7.3 , ' SIO= ' ,F7.3, ' Sl� 02340 liN= ' ,F7.3,' SIMAX= 0 ,F7.3, ' XH·H n= ' ,F7.3, ' XIMAX= 1 ,F7.3/ ) 02350 1004 FORMAT{ ' Fr.ED TYP E.IS GEN. HUYGEtJS SOURf'E. ELECT. M1P= 1 ,F7 DI P . 02360 1.3, ' DI P. M1 P .=' ,F7.3, 1 REL. PHASE=1 ,F7.3, ' DEGREES ' ) �� AG . 02370 1�05 FORMAT( ' FEED TYPF IS RECTANGULARCTE-10) WAVEGU IOE. A/ LAMBDA= ' 02380 l ,F7.3, 1 B/ LAMROA=' ,F7.3) 02390 1006 FORt·1J\T( 1 FEED TYPE IS OU/I.l-t�OOE CIRt.UlAR A/ LM-1BOA= �JA VEC?U fOE.

f-' � f-' 1 1 1 ' 02400 1 ,F7.3, 1 REL. Tt-1 11 MOflE PO\�ER= ,F7.3, REL. Pl-lAS E =' ,F7.3, ' DEG ) ' 02410 1017 FORMAT( FEED TYPE IS COS(SI )**N (SIRAq PO LAR IZED) 02420 1 N= ' ,F7.3) 02430 1007 FOR��AT( ' FEED PO LAR IZAT ION: AX " = ' ,F7.3, 9 AY " = ' ,F7.3, ' 02440 R F.L. PH As E = , F 7 • 3 , I D R E s 1 I1 " F. G E '" I 02450 �! FEED NO . X /R Y /R Z"/R1 ) 02460 1008 FORMAT( 7X, I2,5X,Fl0.5,3X,Fl0.5,3X,F l0.5) 02470 1009 FORMAT(//40X, 1 RF.FLECTOR PATTERNS-AMPLI TUDES I� DB REL. UN ITY ' ) TO 02480 1010 FORMAT( ' THETA PH I ETH-DB-EPH AN�-DEG THBAR PHBAR E 02490 lTHBR-DB-EPHBR TAU-OEG AR--DB EL/ER-OB IEXI VO LTS IFYI VO LTS ' I 02500 �� EZ I > __ 02510 1011 FORMAT(4(1X,F6.2),1X,F7.2,4X,F6.2,1X,FG.2, 2X,2(F6. 2,1X),F7.2,4X,F6 02520 1.2,1X,F7. 2, 2 X ,3(E10.4,1X)) ' I 1 02530 1012 FORMAT( END OF CUT. MAX MUM VAlUE OF ETHBAR IS , El 0.4, 1 �� AX OCCU 0254 0 lRS AT ' ,FlO.S,' DEGREES FOR TH fS CUT' /// ) 02550 1013 FORMAT( ' ENfl OF LAST CAS E ' ) 02560 1014 FORMAT( ' TYPE 1 FOR ANOTHER CUT; ElSE TYPE 0 1 ) 02570 1015 FORMAT( ' TYPE 1 FOR ANOTHER RUN; OTH ERWISE TYPE 0 END PROGRAM ' ) TO 02580 STOP 02590 END

'-' ..,. "-l 0 2600 SUBROUT INE EFEEn(ESI,E XI) 02610 COMPlEX ES I(l 6,9 l),EXI(16,91 ),CMPlX, PHASE, PHPOl,FlSI,F2S I 0262 0 COMMON/ TR IG/SSf(l6},CSI{l6),SXI(9l),CXI (91) 02630 COMMON/GR I NT/MMAX, NMAX 02640 COMMON/ F�EO/ ITYPE,Pl, P2, P3, P4, P5, P6 0 2650 COMMON PI ,DTR 02660 REAl•4 KEA, KMA,KA, NUM , NUM2 02670 PHASE=CMPLX(COS( P3•0TR),SIN(P3*DTR)) 02680 P4 IS AMPL ITUDE OF X'-POLAR IZED EXCITATI ON c 02690 PS IS AM P LITUDE OF Y' -POlAR IZED EXC ITATION c 02700 P6 IS RElA TIVE PHASE BETWEEN ORTHOGONAl EXCITATI ONS c 02710 PH POl=CMPlX (COS( P6•DTR),SIN(P6•DTR )) 02720 ANORM=SQRT(P4**2+P5**2 ) 02730 GO T0 (1,2,3,4), 1TYPE 02740 c 02750 HUYGEN 1S SOURCE FEED c 02760 P1 IS ELECTR IC DIPOLE AMPLITUDE (VOLTAGE) c 02770 P2 IS MAGNETIC DIPOLE AMP liTUDE (VOLTAGE) c 02780 P3 IS TH E PHASE DIFF. fN DEGREES BETWEEN ELECT. AND c MAG . 02790 DIPOL ES(REL TO PH AS E FOR liNEAR PO LAR IZAT ION) c 02800 WITH PHASE OF ELECTR IC DIPOLE AS REFERENCE c 02810 1 O!NORM=SQRT(P1**2+P2••2 ) 02820 DO 110 J=l,NMAX 02830 DO 110 l=l,MMAX 02840 FlSI=(Pl•C SI(I)+P2•PHASE)/DI NORM 02850 F2SI=( Pl+P2•CSI(I)• PHA SE)/OI NORM 02860 ES I(I,J )=FlSI•(P4•CX I(J )+PS*S XI(J)•PHPOl)/ANORM 02870 110 EXI (I, J)=-F2SI*(P4*SX I(J)-P5•C X!(J)*PHPOL)/ANORM 02880 RETURN 02890 c

� .;::. w 02900 TE-10 MODE RECTANGULAR HORN c 02910 PI IS A/ LAMBDA�DI MENS ION ALONG Y' AX IS c 02920 P2 IS B/ LAMBDA-D IMENS ION ALONG X' AX IS c 02930 c 02940 2 AL=Pl 02950 Bl=P2 02960 BKX =SQRT( l.0-0.25/Al**2) 02970 BKY=SQR T(l.0-0.25/BL••?.) 02980 ENORMX=-4 .0• {l.O +BK X)/PI**2 02990 ENORMY=-4 .0•(l.O+BKY)/PI **2 03000 DO 210 J=l,NMAX 03010 DO 210 l=l,MMAX 03020 ARG=PI•S SI(I) 03030 ARGl=ARG•Al•SX I(J) 03040 ARG2=ARG•Bl•CX I(J) 03050 IF(ABS(ARG2 ).LT. 0.0001 ) TX=l.O 03060 IF(ABS(ARG2 ).GE. 0.0001 ) TX=S IN(ARG2 ) /ARG2 03070 IF(ABSCARGl ).l T.0.0001 ) TY=l .O 03080 IF(ABS(ARGl ).GE.O.OOOl ) TY=S IN(ARGl )/ARGl 03090 DENl=ARGl**2-PI ••2/4.0 03100 IF(ABS(DENl ).l T. O.OOOl ) VX=-TX/PI 03110 IF(ABS(DENl).GE.O.OOOl ) VX=TX•COS(AnGl )/DENl 03120 DEN2=ARG2••2-PI**2/4 ,0 03130 IF(ABS(DEN2).LT.0.0001 ) VY=-TY/ Pr 03140 IF(ABS(DEN2).GE.0.0001 ) VY=TY•COS (ARG2 )/DEN2 03150 ESX=CXI(J)•( l.O+BKX•C SI(I))• VX/ ENORMX 03160 EXX=-SXI(J)•(CSI(I)+ BKX) •VX/ENORMX 03170 ESY=SX I(J)w(l.O+BKY•CSf(I ))• VY/ ENORMY 03180 EXY=CXI(J)•(CSI(I)+ BKY) *VY/ ENORMY 03190 ES I(I,J)=( P4• ESX+PS *ESY•P4POL)/ANORM

.·

f-' ..;::::. ..!:>. 03200 210 EX I(t ,J)=( P4•EXX+PS•EXY•PHPOL)/ANORM 03210 RETURN 03220 c 03230 DUAl MOOF. CIRCULAR WAVEGU IDE c .\ 03240 Pl IS A/ LAMBOA - GU tOE RAO IUS IN WAVELENGTHS c 03250 P2 IS THE FRACTIONAl POWER IN TM ll MODE (O<=PTM <=l .O) c 03260 P3 IS PHASE DI F F. BETWEEN TEll ANO TM ll MODES c 03270 WITH PHASE OF TEll AS REFERENCE c MODE 03280 3 IERE=O 03290 IERM=O 03300 Al=Pl 03310 PTM=P2 03320 Z0=376. 7 03330 KEA=l . 84118 03340 KMA=3.8317 03 350 KA =2 .0•PI•Al 03360 PROPE=l.O- KEA**2/KA**2 03370 IF( PROPE.LT. O.O) IF.RE=l 03380 IF(PROPE .GT.O.O) BEK=SQRT(PROPE) 03390 IF(PROPE.l T. O.O)BEK=O.O 03400 PROPM =l.O-KMA**2/KA••2 03410 IF(PROPM. LT. O.O.AND . PTM . NE. O.O)IERR=2 03420 IF( PROPM. LT. O.O.ANn .PTM.NE.O .O) BMK=l .O 03430 IF( PROPM.GE.O.O) BMK=SQRT(PROPM ) 03440 JF( I ERE.GT. O.OR.I ERM.GT. O) WR ITE(6,320) IERE, I ERM 03450 ENORM =0.0323•(l.O+REK)*SQRT( l.O/BEK) 03460 DO 310 J=l,N�-1AX 03470 310 l=l,MMAX no 03480 AI

;...,; .::>. (..'1 ···----�-- . ---�--�---

03500 BJ COMPUTES TH E BESSEL FUNCT ION c 03510 BJ USES SURROUTINE BESJ FOR COMPUT ING BESSEl FUNCTIO�! c 03520 BESJ IS FR0f'.1 IBM SYS. 360 SCI ENTI FIC PA CKAGE c SUBqnUT INE 03530 IF (ABS(AR G).GE.0.0001 ) TERM1=RJ(1,ARG ) /ARG 03540 NW-1=BJ( 0, ARG ) -TEfUH 03550 DEN=l. O-ARG•*2/ K EA**2 03560 IF(AB S(I1E N).LT. 0.0001 ) TERM2=0. 37754 03570 IF(AB S(I1EN).GE.0.0001 ) TERM2=NlH·1/ DEN 03580 NUM2=ARG•BJ(l,ARG) 03590 DEN2=ARG••2-KMA**2 03600 IF( ABS(DEN2).LT. 0.0001 ) TERM3=-0.20138 03610 fF(ABS(DEN2 ).GE.0.0001 ) TERM3=NUM2/ DEN2 03620 C1=0. 0646•SQR T( (1.-PTM )/BEK) 03630 C2= -0. 0998976•SQRT( PTM/ BMK) 03640 ES I E=C l•(1.+BEK*CSI(I))• TERM1 03650 F1SI=(ESIE+C2•PHASE•(RMK+CS I(I))* TERM3 )/ENO RM 03660 F2SI =Cl•(REK+CSf(I))• TERM2/ENORM 03670 ESI {I, J)=FlSI•(P4•C XI(J)+PS•S XI(J)•PH POl)/ANORM 03680 310 EX I (I,J)=-F2S I•(P4•S XI(J)-P5•C XI(J)•PHPOL) /ANnRM 03690 320 FO!U1AT( ' I CUTO F F, I ERE(TE11 )=',15,11ERM (H111 )=',15) t40 11E S 03700 RETURN 03710 c 03720 COS(SI)• *N (SIBAR PO LAR fZEn ) FF.E� c 03730 P1 IS TH E EXPONENT N c 03740 c 03750 4 DO 410 J=l,NMAX 03760 410 l=l,MMAX no 03770 DEN=SORT(l.O-(SSI(I )•C XI(J))**2 ) 03780 Tl=CS I (I)* *Pl 03790 F1S I =Tl•CSI(I )/DF.N

I-' ,j::,. 0"1

.I 03800 F2S I =Tl/DEN 03810 ES I (I, J)=FlSI•(P4•CXI(J )+PS•SX I{J)•PHPOL)/ANORM 03820 410 EX I(I,J)=-F2 SI•(P4•SXI(J)-PS•C XI (J)•PHPOL)/ANORM 03830 RETURN 03840 END

f-1 � -...! 03850 SUBROUT IHE SETUP(SfMIN,DELSI,XIMIN,DELXI ) 03860 TH IS SUBROUT INE COMPUTES THE INTEGRATION GR ID INTERVALS c 03870 AND TH E TR IG FUNCTIONS OVER THE GR in c 03880 COMMON/G RID/ SIR(l6),XIR(91) 03890 COMMON/TR I G/SSI(1 6),CSI(l6),SXI(91),CXf(91) 03900 COMMON/ GR I NT/MMAX, NMAX 03910 COMMON PI ,DTR 03920 DO 10 I =l,I-1MAX 03930 Al=l-1 03940 ARG=DTR•(SIMIN+A I •DElSI) 03950 SIR( I)=A RG 03960 SS I (I)=SIN(ARG) 03970 10 CS I (I)= COS(ARG) 03980 DO 20 J=l,NMAX 03990 AJ=J-1 04000 ARG=DTR• (XfMIN+AJ•DELX I) 04010 XIR(J)=ARG 04020 SXI(J)=S I N(ARG ) 04030 20 CXI(J)=COS(ARG) 04040 RETURN 04050 END

.J::>, � � 04060 FUNCT ION BJ(N,ARG) 04070 IF(N.E Q. O.OR.ABS (ARG).lT. O.OOOl ) GO TO 10 04080 40 CAll BESJ(ARG, N, BJ, O.OOOI, IER) 04090 IF( IER.GT.O) WR ITE(6,1) IER,N,ARG 04100 1 FORMAT( ' ERROR IN BJ ROUT INE, fER= 1,15,' N= '.,15,1/\RG= ',FlO.S) 04110 GO TO 30 04 120 10 IF(N.E Q. O) GO TO 20 04130 BJ= O.O 04140 GO TO 30 04150 20 IF(ABS(ARG).lT. O .OOOl ) BJ=l.O

04160 IF(ABS (ARG).GE •• 0001 ) GO TO 40 04170 30 CONT INUE 04180 RETURN 04190 END

1-' .!:>. \.0 04200 SUBROUT INE FIN T(X,Y,F,R,MMAX,NMAX,STOT ) 04210 TH IS SUBROUT INE NUMERI CALLY INTEGRATES c 04220 THE RAD IATION INTEGRA L c 04 230 LU IG JUNE 1968 c A. 04240 c 04 250 DIMENS ION XC16),Y(91 ),R( 16,91) 04260 DI��ENSION DUM( 2) 04270 COf1PLEX F(l6,91,3),STOT(3),E,Tl,T2, T3,A, B,C,C MPL X,Fl2,F23,Fl4,SUN , 04280 !TOT 04290 Q U I VAL E N C E ( E , U ( 1 ) , C ) 2 ) , S I ) E D t � 0 , ( D W1 ( 04300 DO 10 l=1,3 04310 10 STOT( L)=( O.O, O.O) 04320 2 00 N=l, N�1AX DO 04 330 OY=0.5�(Y(N+1)-Y(N)) 04340 DO 200 �-1=1,MMAX 04350 S = Y * X ( + 1 ) ( ) ) D D ( �1 - X r·1 04360 R l = { M+1,N+ 1 - R ( , ) R ) M N 04370 R2 =R(M, N+1 }-R{M+1,N) 04380 R3=R(M,N)+R(M +1,N) 04 390 BE=(R1+R2 )•0.5 04400 CE=(R1-R2 )•0.5 04410 AL=(R3-CE)•O.S 04420 IF(AB S(BE)-0.01 )100, 100, 110 04430 100 F31 =RE•0. 33333333 04440 F1 1=BE•0.5 04450 FlR=l. 0-Fl l •F3 1 04460 F3R=0.5-BE•BE/8.0 04470 GiO TO 14 0 04480 110 E=CMPLX(CO S(BE),SIN(BE)) 04490 FlR=S I/BE

1-' Ul 0 04500 F1 1=(1.0-CO }/BE 04510 F3R=F1R-Fl i/BE 04520 F3 1=( FlR-CO }/BE 04530 140 IF( ABS(CE)-0.01 )150, 150, 160 04540 150 F41 =CE*0.33333333 .04550 F2 1 =CE*0.5 04560 F2R=l .O-F21*F4 1 04570 F4R=0. 5-CE*CE/ 8.0 04580 GO TO 170 04590 160 E=CMPLX(CO S(CE),SIN(CE}) 04600 IF 2R=S I/CE 04610 IF 2 1 =( 1. 0-CO }/CE 04620 F4R=F2R- F21/CE 04630 F4 1=( F2R-CO }/CE 04640 170 E=CMPLX(CO S(AL},SIN(AL)) 04650 F12=CMPLX(Fl R*F2R-Fli *F2 1,Fl i *F2R+FlR•F21) 04660 F23=CMPLX(F2R*F3R-F21*F31,F21 •F3R+F2R• F3 1) 04670 F14=CMPLX(F1R•F4R-Fl i*F 41,Fl i*F4R+F1R*F4 1) 04 68 0 DO 200 L=l,3 04690 Tl=F(M+l,N +l, L)-F(M, N, L) 04700 T2 = F(M,N+1,L)-F(�+ l,N, L) 04710 T3 =F (M, N, U+F 01+ 1, N, U 04720 B=Tl+T2 04 730 C=Tl-T2 04740 t\ =T3-0.5* C 04750 SUM=A*Fl2+B*F23+C*F l4 04760 TOT=E*SW-1*0S 04770 200 STO T(L)=STO T{ L)+TOT 04780 RETURN 04790 END

I--' lJ1 I--'