A procedure is described for designing dual reflector antennas. The analysis is developed by taking each reflector to be the envelope of its tangent planes. Rather than specifying the phase distribution in the emitted beam, the slopes of the emitted rays were specified. Thus, both the output wave shape and angular distribution of intensity can be specified. Computed examples include variations from both Cassegrain and Gregorian systems, permitting deviation from uniform source distributions and from parallel beam property of conventional systems

Barger, Raymond L.

English

NASA Technical Reports Server

NASA 
Technical 
Paper 
2797 
1988 
National Aeronautics 
and Space Administration 
Scientific and Technical 
.Information Division 
A Simplified Approach 
to Axisymmetric 
Dual-Reflector 
Antenna Design 
Raymond L. Barger 
Langley Research Center 
Hampton, Virginia 
https://ntrs.nasa.gov/search.jsp?R=19880007279 2020-03-20T07:21:04+00:00Z
Summary 
A procedure is described for designing dual- 
reflector antennas. The analysis is developed by tak- 
ing each reflector to be the envelope of its tangent 
planes. The slopes of the emitted rays are speci- 
fied rather than the phase distribution in the emit- 
ted beam. Thus, both the output wave shape and 
the angular distribution of intensity can be specified. 
Computed examples include variations from both 
Cassegrain and Gregorian systems. These examples 
include deviation from uniform source distributions 
and from the parallel-beam property of conventional 
sys tems. 
Introduction 
In theory it is possible to specify, within lim- 
its, both the emitted amplitude distribution and the 
phase distribution of a dual-reflector system when 
both surfaces are properly shaped. The problem of 
determining these shapes has been treated in refer- 
ences 1 and 2 with a combination of differential and 
algebraic equations. This system of equations tends 
to be unwieldy, and consequently in both references 
the method of solution is only indicated, with the 
specific formulas omitted. 
Although the present approach is an approximate 
procedure, it can, in theory, be made as accurate 
as desired by taking sufficiently small step sizes. 
The simplifying concept is to treat each of the two 
reflectors as the envelope of its tangent planes. This 
technique permits the mathematical problem to be 
reduced to one of solving a set of nonlinear algebraic 
equations. 
Computed examples include both modified 
Cassegrain and modified Gregorian systems. Inas- 
much as the emphasis in the present analysis is on 
simplicity of concept, only axisymmetric systems are 
treated. It should be noted that if the reflector sys- 
tem utilizes only a segment (e.g., a quadrant) of the 
axisymmetric design, then a cross polarization exists 
in the output beam since the antenna normally does 
not emit a circularly polarized wave. 
I 
t 
I 
I Symbols 
I b exponent of cos 80 
E beam energy 
I intensity 
m slope of ray relative to 
system centerline 
coordinates, x taken along 
system centerline and y in 
radial direction 
(XI, Yl), (X2, Y2) midpoint of straight seg- 
ment tangent to reflector 
meridian line 
x3, y3 
(XI, yl), (22, y2) 
point at which ray intersects 
reference plane 
initial point of straight 
segment tangent to reflector , 
meridian line , 
8 angle ray makes with 
system centerline 
Subscripts: 
0 
1 
2 
3 
b 
C 
origin or ray emanating 
from origin 
tangent segment of 
subreflector 
tangent segment of main 
reflector 
vertical plane along which 
output intensity is specified 
ray reflected from main 
.reflector 
ray reflected from subreflec- 
tor to main reflector , 
i index I 
min, max, total minimum, maximum, and 
total 
Analysis 
Reflector and Ray Geometry I 
As was mentioned in the Introduction, each 
curved reflector surface is treated as the envelope 
of its tangent planes. Since each of these surfaces 
is axisymmetric, it can be specified by the meridian 
line cut by a vertical plane. Then the tangent plane 
along this meridian appears simply as a straight line 
segment. (See fig. l(a).) 
The geometry for tracing a ray through the sys- 
tem is shown in figure l(b). The reflection condition 
at the subreflector gives 
R .*2 mz-1 
Main reflector 
x 2 ' y 2 \  
Su breflector 
x y (source location) 
0 '  0 
(a) Tangent-plane geometry. 
m 2  = -cot e2 
I 
\mc =tan ec  
(b) Angles and slopes. 
Figure 1. Basic system geometry. 
2 
To find the relation between the slopes, we take the 
tangent of both sides: 
(2) 
- tan Oc + tan 80 
1 - tan8, tan80 
- 
2 tan 81 
1 - tan281 
tan281 = 
Similarly, at the second reflector, 
tan(282) = tan[& + (8, - T ) ]  (3) 
or 
(4) 
- tan 83 + tan 8, 
1 - tan83 tan8, 
- 2 tan 62 
1 - tan282 
Since the slope of the local tangent to the reflector 
is the negative reciprocal of the slope of its normal 
(ml = -cot el), equation (2) yields 
-2m1 - rno+m, 
m 7 - 1  l-mornc (5) 
-- 
and, similarly, equation (4) yields 
(6) 
-2m2 - m3+mc 
mg-1 1-m3rnC 
-- 
Referring to figure l(a),  if we denote the coordi- 
nates of the initial point of a subreflector segment by 
(21, y1) and denote the reflection points by capital 
letters (XI, Yl), then the reflection point is specified 
to be at the midpoint of the segment simply by tak- 
ing the segment length to be twice the distance from 
(21, y1) to (X1,Yl). The main reflector is treated 
similarly. The slopes and point coordinates are re- 
lated linearly as follows: 
y 2  - Yl 
x2 - x1 m, = 
Yl -Yo 
x1- 20 mo = 
y1- Y 1  
x1- 21 ml = 
y2 - Y2 
x2 - 2 2  
m2 = 
(7) 
(9) 
The intensity distributions are determined as fol- 
lows. The intensity distribution to be assigned ar- 
bitrarily as the output of the system is most conve- 
niently specified along a vertical plane, usually taken 
near the aperture plane. Thus, if the output inten- 
sity distribution I(Y3) is assigned as a function of 
Y3 along a vertical plane taken at X3, then the ray 
that intersects this plane at Y3 is related to the ini- 
tial source emission angle 80 through the energy rela- 
tion, which is given in the next section. Furthermore, 
it is appropriate, with the present method, to spec- 
ify the ray slope distribution of the system output 
beam rather than the phase distribution. Thus (see 
fig. 2(a)), in the equation for the ray reflected from 
(X2, Y2) through (X3, Y3), 
and X3, Y3, and mb are all known quantities. 
Energy Relation 
The value of Y3 is determined from the energy 
relation as follows. The intensity distribution Io(80) 
emitted by the source antenna is (see fig. 2(b)) 
and consequently all the energy emitted within this 
segment is 
and the relative amount of energy emitted is 
(13) 
Eo (80) Eo (80) 
EO (00,max) Etotd 
- 
where denotes the edge of that part of the 
source beam that is to be utilized. 
The intensity of the output beam is specified as a 
function of Y3 along the vertical plane at X3. Thus, 
the energy emitted through the annulus at Y3 with 
width dY3 is (see fig. 2(a)) 
The energy emitted through the ring at Y3 is 
where the ray through Y3,min is that emitted at 
8 0 , ~ i ~  by the source. The relative energy is 
Comparing equation (15) with equation (13) enables 
one to determine the Y3 corresponding to a given 80. 
As a rule, the prescribed intensity distribution 13(y) 
in equation (14) is taken as a relatively simple an- 
alytic expression. Consequently, the energy integral 
3 
(a) Vertical reference plane at X3. 
(b) Source rays at unit circle. 
Ray (constructed 
normal to wave front) 7 
(c) Method for determining Y3 and 8b from prescribed wave shape. 
Figure 2. Geometry for identifying source ray at 80 with emitted ray through (X3,  Y3). 
4 
in equation (14) can be evaluated in closed form and 
E(Y3) is determined in analytic form. Furthermore, 
the source intensity distribution Io(6') can often be 
described as some power of cos 6' or as a linear com- 
bination of such functions, so that E(6'o) can be de- 
termined in analytic form from equation (12). 
Before proceeding to the solution of the equations 
for the reflector surfaces, we may pursue further the 
significance of the form of the system performance 
specification. Although for some problems it is 
appropriate to specify the intensity and beam 
direction distributions along some vertical plane, it is 
important to observe that these distributions can be 
obtained by specifying the more fundamental quan- 
tities of output wave shape and the intensity dis- 
tribution as a function of ray direction. Thus, in 
reference to figure 2(c), the wave shape and the rela- 
tive energy as a function o f &  are prescribed. Since 
the wave shape is known, normals to the wave shape 
can be constructed and their intersections with the 
vertical plane at X3 determined. These normals, 
which represent rays, have slope mb = tan&. Thus, 
with E(&) prescribed, these intersections determine 
E(Y3). Consequently, both mb and Y3 are deter- 
mined for each ray of the output beam. 
Solution of Equations 
To determine the reflector shapes, the set of seven 
equations (5) to (11) are to be solved for the unknown 
quantities X I ,  Y1, X2,  Y2, ml ,  ma, and m,. The ap- 
proach taken here is to eliminate all unknowns except 
ma, solve the resulting (highly nonlinear) equation 
for m2 numerically, and then substitute back into 
the other equations to determine the remaining un- 
knowns. The details of this procedure follow. 
Define 
-2m2 
R(m2) = 7 
m2 - 1 
and substitute R into equation ( 6 ) ,  which (noting 
that m3 = mb) can then be written as 
(1 - mbm,)R(m2) = mb + m, 
and solved for m, to obtain 
Equations (10) and (11) are each solved for Y2, 
and then Y2 is eliminated by equating the resulting 
expressions: 
This equation is solved for X2 to yield 
This expression may be substituted back into equa- 
tion (11) to obtain an expression for Y2 as a function 
of mg only: 
Eliminating Y1 between equations (7) and (8) yields 
which can be solved for X I  and expressed as a func- 
tion of m2 with substitutions from equations (17), 
(18), and (19): 
(20) 
This result is substituted back into equation (8) to 
obtain 
Equation (9) now becomes 
Substituting from equations (17) and (22) into equa- 
tion ( 5 )  yields 
This equation is solved numerically by a forward 
seeker algorithm that finds the zeros of the function 
on the left side of equation (23). Once the root is 
found, we can find the remaining quantities by re- 
peating the above substitutions with the known value 
of ma. Thus, m, is obtained from equation (17), and 
X2, Y2, X i ,  and Y1 are obtained from equations (18) 
To determine the next pair of points on the two 
reflectors, 6'0 is incremented. Then 51,  y1, z2, and 
y2 are incremented by specifying ( X I ,  Y1) to be the 
midpoint of a segment on the subreflector and sim- 
ilarly for (X2,Y2) on the main reflector. Thus, for 
example, 
to (21). 
5 
The procedure can then be repeated for the new 
value of 00. Inasmuch as equation (23) is highly 
nonlinear and possesses multiple roots, some care 
must be exercised in setting the limits of the range 
of m 2  over which the numerical algorithm seeks a 
solution. Fortunately, the limits are known to a 
close approximation because specifying even sizeable 
variations from a uniform intensity distribution or 
from a parallel-ray output beam does not result 
in large geometry variations from a conventional 
Cassegrain or Gregorian system. 
Computed Examples 
Figure 3 shows a modified Cassegrain system for 
which the source distribution varies as cos8 00 but 
the emitted beam is specified to have a uniform 
distribution. Figure 4 gives a similarly designed 
modification of a Gregorian system. 
Figure 5 shows a modified Cassegrain system for 
which the slopes of the rays of the output beam are 
specified to increase gradually up to a value of 0.25: 
2 
mg = 0.25 (") 
Y3,rnax 
Figure 6 shows an offset system consisting of seg- 
ments of a subreflector and a main reflector designed 
so that the emitted rays all have a slope of 0.25 rel- 
ative to the system centerline. In such a system, the 
cross-polarization phenomenon mentioned in the In- 
troduction would exist. 
Concluding Remarks 
A procedure has been described for designing 
dual-reflector antennas. The analysis was developed 
by taking each reflector to be the envelope of its 
tangent planes, so that the reflection condition is 
satisfied on each of these planes. The slopes of the 
emitted rays were specified rather than the phase 
distribution in the emitted beam. 
Computed examples included variations from 
both Cassegrain and Gregorian systems. These ex- 
amples include deviation from uniform source distri- 
bution and from the parallel-beam property of con- 
ventional systems. 
NASA Langley Research Center 
Hampton, VA 23665-5225 
February 2, 1988 
References 
1. Galindo, Victor: Design of Dual-Reflector Antennas 
With Arbitrary Phase and Amplitude Distributions. 
IEEE Trans. Antennas d Propag., vol. 12, no. 4, July 
2. Kinber, B. Ye.: On Two-Reflector Antennas. Radio Eng. 
d Electron. Phys., no. 6, June 1962, pp. 914-921. 
1964, pp. 403-408. 
6 
12 
Y 
10 
8 
6 
4 
2 
0 2 4 6 8 10 
X 
Figure 3. Modified Cassegrain system. 
7 
12 
/” 
8 
6 
4 
2 
8 
I 
6 Y 
4 
2 
0 2 4 6 a 10 
X 
Figure 5. Modified Cassegrain system with ray slopes of output beam specified to increase to 0.25. 
9 
2 4 6 8 10 
X 
Figure 6. Cassegrain-type section designed to emit beam of rays at constant angle O b  # 0. 
10 
I 
Iw\sA National Aeronautics and Report Documentation Page 
1. Report No. 
NASA TP-2797 
2. Government Accession No. 
1. Title and Subtitle 
A Simplified Approach to Axisyrnmetric Dual-Reflector Antenna 
Design 
19. Security Classif.(of this report) 20. Security Classif.(of this page) 
Unclassified Unclassified 
7. Author(s) 
Raymond L. Barger 
21. No. of Pages 22. Price 
11 A02 
3. Performing Organization Name and Address 
NASA Langley Research Center 
Hampton, VA 23665-5225 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546-0001 
15. Supplementary Notes 
3. Recipient’s Catalog No. 
5 .  Report Date 
March 1988 
6. Performing Organization Code 
8. Performing Organization Report No. 
L- 16392 
10. Work Unit No. 
505-61-71-04 
11. Contract or Grant No. 
13. Type of Report and Period Covered 
Technical Paper 
14. Sponsoring Agency Code 
16. Abstract 
A procedure is described for designing dual-reflector antennas. The analysis is developed by taking 
each reflector to be the envelope of its tangent planes. The slopes of the emitted rays were specified 
rather than the phase distribution in the emitted beam. Thus, both the output wave shape and 
the angular distribution of intensity can be specified. Computed examples include variations from 
both Cassegrain and Gregorian systems. These examples include deviation from uniform source 
distributions and from the parallel-beam property of conventional systems. 
17. Key Words (Suggested by Authors(s)) 
Dual-reflector antennas 
18. Distribution Statement 
Unclassified-Unlimited 
lASA FORM 1626 OCT 86 NASA-Langley, 1988 
For sale by the National Technical Information Service, Springfield, Virginia 22161-2171 


A simplified approach to axisymmetric dual-reflector antenna design

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