.. ' 

NASA Technical Memorandum 87162 

I 

I 

I 

J 

I NASA-TM-87162 19860005007 

~ ~--------------~/ 

Secondary Pattern Computation of an 

Arbitrarily Shaped Main Reflector 

Peter T.e. Lam and S.W. Lee 

University of Illinois 

Urbana, Illinois 

and 

R. Acosta 

Lewis Research Center 

Cleveland, Ohio 

November 1985 

NI\S/\ 

-

'.: -... ; . 1.-"':: ., ! _.; ~: • \- ! .:~ :. .._.'.: ... 

IIIII~IIIIII 1111 1111111111111111111111111111· . NF01484 ',-
1. INTRODUCTION 

Reflector antennas are widely used in communication satellite systems 

because of their relatively good radiation characteristics, low cost, and 

light weight. A central problem in the analysis of a reflector antenna is 

the secondary pattern computation. As sketched in Figure 1, for an incident field from a feed located at PI' the problem is to calculate the scat-

+s . + 

tered field E from a known reflector L at a far-field observation point r. 

Several methods exist for calculating the high-frequency asymptotic solu-

+s+ tion of E (r), as explained below. 

(i) Physical Optics Method (PO, Fig. la) [1]-[7]. The induced 

+" 

current on the reflector is approximated by 2n x H1. An integration of this current over the curved reflector L gives the far 

+s field E • 

(ii) Geometrical Theory of Diffraction (GTD, Fig. 1b) [8]-[10]. At a 

+ +s far-field observation point r, the scattered field E consists of 

two terms: the reflected field on ray ~ PIO ,and the edge 

. ~ 

diffracted field on ray PIO • 

(iii) Aperture Integration Method (AI, Fig. 1c) [1]'[11]-[13]. The 

field on aperture plane La is first calculated by tracing a 

r reflected ray PIO P2 using geometrical optics theory and an edge 

d diffracted ray P1

0 P2 using GTD. Next, we integrate the field 

over L via FFT to obtain the scattered far-field ES • a 

The accuracy of the above three methods is discussed below. As a 

reference, let us represent the exact solution of ES by a high-frequency 

asymptotic series, namely, 

1 
Figure 1. 

r4 I 

L~ _____ ....... 

(a) Physical Optics Method 

(b) GTO 

,'P2 L ..... ---_--i ...... I 

, 

, 

I 

I 

I 

"Es (7) 

lc) Aperture Integration Method 

·s Three methods for calcUlating Scattered field £ from a reflector. 

2 
Then, we may summarize the accuracy and limitations of the three methods in 

the following table: 

TABLE 1 

ACCURACY AND LIMITATIONS OF PO, AI, AND,GTD, 

Methods Accuracy 'Limitation 

PO not accurate for 

1.. and partial + recover Al 

AI 0 wide-angle lobes 

GTD A + recover and Al predicts infinite 

0 field in main 

beam direction 

(caustics) 

In this report, we will study the main reflector far-field pattern 

using the aperture integration (AI) method for the following reasons: 

(i) Unlike the other, two methods,' AI gives. the near-field (aperture 

field) as well as the far field. Most of today'~ large r~f1ector 

"measurements are done i!\. a near field range. Thus, only AI 

provides a convenient theoretical check for the near field 

measurements. 

(ii) The accuracy of AI is comparable to the popular PO. The use of 

FFT in AI makes it numerically efficient. Furthermore, as will 

be discussed 1ater,the present AI formulation is most suitable 

for extension to multiple (2 or more) reflectors. 

There exists an extensive list of published literature on AI - notably, 

Silver [1], Kauffman and CroswelL [11], Acosta [12l,and Hwang, Tsao, and 

3 
Han [13J. In comparison with these prior works, the present AI analysis may 

be considered as an extension in one or'more of the following areas: 

(i)' The surface of the reflector is completely arbitrary. It can be a 

numerically specified surface. 

(ii) The edge of the reflector is not restricted to a circular curve. 

It can be an arbitrary curve lying 'on an elliptical cone or 

cylinder. 

(iii) The divergence factor of the GO field is correctly computed.' 

Hence, our analysis' is not restricted to feeds located very close 

to the focal point (in which case the divergence factor is nearly 

unity and is ignored by several researchers). 

(iv) The edge diffracted field is included in the ~perture field 

calculation. Near, the incident and reflected shadow,boun~aries, 

two uniform theories [14J-[15J are used 56 that the aperture 

field is continuous from the lit to the shadow region. 

In short, the present AI analysis represents a' genE~ralized and improved 

version of previous work. In particular~'it is amendable for a convenient 

extension for analyzing multiple reflecto'rs. 

The organization of this report 'is as follows: The description of the 

problem is described in Section 2. Th~ incident field, feed coordinates, 

and power radiated from the feed are covered in Section 3. Sections 4 and 

5 give a step-by-step procedure to compute the various contributions that 

make up the aperture field, namely, the geometrical optics ~nd edgediffracted fields. Section 6 covers the aperture field theory, the application of the FFT, and gain normalization'. Numerical results and concluding 

remarks will be presented in Sections 7 and '8. 

4 
2. DESCRIPTION OF PROBLEM 

The geometry of the problem un~er consideration is sketched in 

~., 

Figures 2 and 3. A reflector E is illuminated by the incident field from 

a point source at Pl. The problem is c (a) to determine the high-frequency 

asymptotic sO,lution' of, the total field at an observation point on'the aperture grid E as shown in Figure 2~ and (b) to determine the secondary a 

'pattern using the FFT as depicted in Figure 3. In this section, we shall 

describe the various elements involved in the problem. 

2.1. Coordinate Systems and Time Convention 

The main coordinate system is'therectangular system (x,y,z), whose 

origin and orientation are arbitrarily chosen. In calculating the edge 

diffracted field which involves the boundary of the reflector, we employ 

a primed rectangular system (x"',y'" ,z"'), whose relation with (x,y,z) is 

explicitly stated later. The feed coordinate 'system (x£,y£,zf) required to 

describe the polarization and incident field is related to the main coordinate system by Eulerian angles [16]. This will be discussed in detail in 

Section 3. The time factor is exp(+jwt) and is suppressed throughout. 

2.2. Source 

We assume that the source has a' well-defined "phase center~' at PI with 

, ti +i coordinates (xI,y1,zl)' and radiates a spherical wave denoted by (H ,E ). 

If the feed is an array, it is necessary to consider each element in the 

array separately and superimpose their final scattered fields. 

2.3. Reflector E 

The perfectly conducting surface is described by the equation 

5 
Yt .. .!. . 

~Z . PI (XI' y"z,) Source '. 

Figure 2. A reflector I with edge r· being 'illuminated by 

the incident field from {l point source at PI •. 

FFT 

Aperture 

Grid La 

Secondary 

Pattern' 

Figure 3. Secondary pattern using the Fast Fourier Transform. 

£> 
z = f(x,y) for a < x <b and c < Y < d (2.1) 

It is not necessary to know the analytical form of the function f(x,y). In 

fact, the present computer program requi!es only a set of discrete data 

points (x. ,y ,f ) with n = 1,2, •• • ,N as the description of 1:. Those points n n n 

are fitted by a cubic spline which gives automatically first and second 

2 2 2 '. 2 2 partial derivatives of f, namely, af/ax, H/ay, a flax , a f/axay, a f/ay .• 

There are .two requirements on. the cubic-spline fit: (i) the data points 

can be distributed over a random grid, but they' must be dense enough' to 

describe the fine details of E; (ii) the domain of the dai~ points 

(a < x < b,c < y < d) must be somewhat greater than the area defined by the. 

boundary r of the reflector. .Thus, we must know surface 1: in the shaded 

region in Figure 4 as well. Typically, "the "width" of the shaded region is 

about 3 to 4 wavelengths. Our present program contains an extrapolation 

subroutine, which automatically extends E outward "smoothly to obtain the 

necessary data points in the shaded. region." Th~ final scattered field, 

for all practi~al purposes, is independent of the surface outside r. 

Consequently, the exact manner in which the extrapolation is done is 

unimportant .. 

2.4. Boundary r 

Two types of reflector boundaries are most frequently used in practice, and they receive our special attention. 

(a) Cylinder Case. In the first case, r is the intersection of surface L and an elliptical cylinder (Figure 5a). The parameters of the 

cylinder are: 

7 
x=a 

Figure 4. Projection of reflector L and its boundary r on x-y plane. 

8 
t

y 

y 

" \ 

r , 

Yc J x' I 

/ 

,.z 

X Xc Z 

a. r on ellipt!cal cylinder • 

• y 

• Z 

X,X • ®-___ --J~ ___ ~>_-__:! ... Z 

r"""---P~ 

b. r on elliptical cone. 

Figure 5. Two examples of boundary r of the reflector. 

9 
center of the ellipse 

semiaxis along (x,y) direction. 

Using a point on the axis of the cylinner as the origin, we introduce the 

second rectangular coordinates system (x~ ,y~ ,z~) such that 

~ 

x x - x 

c 

y~ y - Yc (2.2) 

~ 

z = z 

The curve r may be described by a parametric equation with parameter .~ , 

O~V < 2n, 

r: y~ 

where 

gl (.~) 

g2(·~ ) 

g3(·~) 

p~ cos .~ 

p~ sin .~ 

f(x 

The projection of r on the x-y plane is always an ellipse. 

(2.3) 

(2.4) 

(h) Cone Case. In the second case, r is the intersection of surface 

E !Inn !In elliptical cone (Figure Sb). The axis of the cone lies in the 

y' - z~ plane and its parameters are 

(x O,y = O,z = -p) = lip of cone 

6 3 inclination angle of cone axis measured from z-axis 

(6 1,6 2) = half-cone angles in the x~ - z' ann y~ - z~ plane 

10 
Using the cone tip as the origin, the primed coordinates (x",Y.",z") are 

introduced such that 

x" = x 

y" -y cos 6

3 - (z - p) sin 6

3 

z" = y sin 6 ...; (z p) co.s 6 3 3 (2.5) 

Using parameters. q," , the curve· r is described 'by 

x" = g1 ( q,") = p" cos q," 

r: y" = g2 ( q,") = p" sin <P" 

z" = g3( q,") = . p" [( cot 6

1 cos q, .. )2 + (cot ,62 sin q, .. )2]1/2 (2.6) 

To determine p" as a function of q,",. we must solve the following nonlinear 

equation: 

. f(x,y) - p + y" sin 6

3 + z .. cos ~3 = 0 . (2.7) 

For a given q,", there is a unique root p" fro~ (2.7). The pair (q,",p") 

gives the desired relation p" = p"(q,"), which is fitted by spline functions. 

The projection of r on the x-y plane is, in general, a pear-shaped curve. 

(c) Arbitrary Case. In addition-to the above two fiequently used 

special cases, r may be an arbit~ary cur~e described by 

where (x",y",z") is related to (x,y,z) by either (2.2) or (2.S). The functions (gl,g2,g3) can be specifi~d either analytically or.numerically. 

11 
2.S. Aperture Grid Points 

A general point on the aperture plane ~a is denoted by P2 with coorEach point on ~ as shown in Figure 3 is fed into the 

a 

computer for repeated calculations of the scattered field. It is essential 

that successive observation points are adjacent, because of the following 

fact. In determining the reflection (specular) points on the reflector, we 

make an exhaustive search ~nly for the first observation point in a batch. 

From the second point on, we use the reflection point of the previous 

observation point as the initial guess for the current reflection point. 

It is only when successive observation points are adjacent that such an 

initial guess ensures fast convergence. In the cases that were considered, 

only one iteration was needed to obtain the reflection point for all observat ion points other than the first observation point. 

2.6. Method of Solution 

+i +i For a given incident field (H ,E ) from the source at PI' the asymptotic solution of the total field (ijt,gt) at point P2 is determined by 

Keller's geometrical theory of diffraction (GTD) [17). Explicitly, the 

total magnetic field is asymptotically given by 

k + co (2.8) 

+g 0 Here H , the geometrical optics field, is of order k relative to the incident field and is the dominant term. +d The second term H is the edgediffracted field and is of order k -1/2. It is well-known that tId becomes 

infinite and (2.8) fails if observation point P2 is close to the incident 

or reflected shadow boundary. In the latter case, we will use the uniform 

+d asymptotic theory (UAT) [14), (18)-[2l), which amounts to replacing H in 

12 
+D (2.8) by a new term H (with capital D): 

(2.9) 

Once Ht is found, we calculate Et from it by using the fact that Hg , Hd , 

and HD are all the so-called "ray fields" which are locally plane waves. 

. . +t +t 

Once the tangential fields (E or H ) on ta are obtained, the Fourier 

transform of these fields will essentially produce the secondary pattern. 

This will be discussed in detail in Section 6. 

13 
3. INCIDENT FIELD 

The definition of the incident field is given. The incident power 

from an arbitrarily polarized feed is derived so that the secondary pattern 

has the correct gain level. Finally, the computation of the incident field 

at a point on the reflector in the reflector system is described. 

3.1. Definition of Incident Field 

The surface current at the radiating aperture of the feed element may 

be expressed as 

(3.1) 

where (a,b,$) are real and 

By choosing (a,b,$), one may obtain any feed polarization. Table 2 shows 

the values of a,b,w corresponding to linear and circular polarizations. 

TABLE 2 

VARIOUS FEED POLARIZATIONS 

a b 

linear x 1 0 0 

linear y 0 1 0 

RHCP 1/ .f[ 1/ .f[ 90 0 

LHCP 1/ .f[ 1/ .f[ -900 

14 
The radiated electric field 'due 'to j given by (3.1) is 

-'jkr i - e. 1(6,.) (3.3) ·r 

where f(6,.) is the active element pattern. The function f(6~.) may be 

approximately expressed by 

f(6,.) = aU (6)(aej1jl cos. + b sin.) +.UH(6)(b cos. - aej1jl sin.) (3.4) E ' 

where 

UE(6) = E-plane active pattern 

UH(6) = H-plane active pattern 

Typically, these active patterns may be approximated by (cos 6)Q, i.e., 

(3. Sa) 

(3.Sb) 

3~2. Incident Power Radiated 

From (3.3) and (3.4), the radiated electric field is given by 

+ e-jkr"1jI 'A "1jI A 

E - -r- [UE(6 )(aeJ cos. + b sin .)6 + UH(6)(b cos. - aeJ sin.).] 

The power radiated, assuming. forward radiation only, is 

where 

p 

inc 

Z 

o 

21T . 1T/2 E E* f f _._. r2 sin 6d6d~ 

1»=0 8=0 Zo 

120rr ohms. 

Using (3.2), (3.6), and (3.8), the incident power radiated is 

p" l.nc 

15 

(3.6) 

(3.7) 

(3.8) 

(3.9) 
For (cos a)q type patterns, 

p 

inc 

3.3. Determining the Incident Field on the Reflector 

(3.10) 

The geometry of this problem is illustrated in Figure 6. The feed 

coordinate system (xf'Yf,zf) and the main reflector system (x,y,z) are 

related by Eulerian angles Yl'Y2'Y3. In the ~ost general case, these 

Cartesian systems can be aligned by three rotations. The angles of these 

rotations are known as Eulerian angles. Figure 7 illustrates the Eulerian 

angles Y1'Y2'Y3. The definitions of these angl~s [22] are as follows. 

Angle Y1 describes a counterclockwise (ccw) rotation about the Z axis which 

brings the x axis to the x" axis aligned with the line of nodes (line of 

intersection between xy and x

f Yf planes), angle Y2 defines a rotation about 

the line of nodes in a ccw sense as indicated so that 'this brings the z 

axis to zf' and angle Y3 is another rotation about the zfaxis and aligns 

the x" axis with the xfaxis in accw sense. Typically, xf and zf are 

given. Let these unit vectors be expressed by 

(3.11) 

0.12) 

Then, the Eulerian angles are given by 

1[ z 1 ) tan -z2 0.13) 

_1[/1 - z~ 1 

tan 

Z3 (3.14) 

16 
Y 

x 

z 

Figure 6. Feed and reflector coordinate systems. 

LINE OF NODES 

(INTERSECTION OF xy 

. AND Xt Yt PLANES) 

Figure 7~ . Eulerian~ngles. 

17 
! '." .' .. ~. 

(3.15) 

From [16], it can be shown that 

(3.16) 

where 

cosy 1 cosy 3-siny 1 cosy 2siny 3 

= A sirt( 2cosy 3 

s iny 1 siny 2 -cosy 1 siny 2 CO!I'( 2 

(3.17) 

Thus, a point with coordinates (x,y,z) in the main coordinate system on 

F F F ~ has coordinates (x ,y ,z ) in the feed coordinate system given by 

F 

x - xl 

F 

y A y - Yl 

z z - zl 

The corresponding spherical coordinates are 

<P f 

-1 F F tan (y Ix ) 

18 

(3.18) 

(3.19) 
From the feed furiction given' by (3.6), the incident E-field may be 

obtained. To find the incident H-field, the following equation is used: 

+ r'X E + 6 T inc 

H inc - [Hr He'· H4> ]. = -""Z,---

f f .. fo 

where Z. is given by (3.8). Next, the incident 

0 

field 

spherical ,components to rectangular components using, 

Hx 

f 

sinS fcos4> f cosS fcos4> f -sin4> f Hr 

f 

H. = sinS fSin4>f . cosS fSin4> f cos4> f HS Yf f 

H cosS f -sinS f " 0 H4> zi f 

Finally, the incident H-field in the (xf'Yf,zf) system 

(x,y,z) system using the following equation 

H H 

x xf 

H =T A H 

Y Yf 

H H 

z zf 

where A is given by (3.17)., 

, 19 

(3.20) 

is converted from 

(3.21) 

is converted to the 

(3.22) 
4. GEOMETRICAL OPTICS FIELD 

The geometrical optics field ~ in (2.8) consists of two components: 

the incident field Hi and the reflected field Hr which is calculated in 

this section. It should be noted that the incident field at an observation 

point on the aperture grid is taken to be zero because the incident field 

does not 'contribute to the secondary pattern. 

4.1. Reflection Point 

For a given source point PI and an observation point P2 (F~gure 2), ~ 

reflection point Or may exist on the reflector E, and we denote its coordinates by (x,y,z=f(x,y». The vectors 

A A A 

II = x(x - Xl) + y(y - Y1) + z(f(x;y) - zl) (4.1a) 

A A 

x(X2 - x) + y(Y2 - y) + z(z2 f(x,y» (4.1b) 

r r 

are the connecting vectors between PI and 0 , and 0 and P2, respectively. 

The condition on the reflection poi~t is that the distance (d1 + d2) must 

be stationary, i.e., 

(4.2) 

which is explicitly given by 

1 {(y _ Y1) + [f(x,y) - zl]' ~ft + t- {(y - Y2) + [f(x,y) - z2] ~f t = 0 

d1 Y 2 Y 

(4.3a) 

20 
A root (x,y,z=f) of the two nonlinear equations in (4.3a) gives the location of a reflection point. For a given PI and P2' there may be none, one, 

or more than one reflection point. It may be shown that (4.3a) is equivalent to the.satisfaction of Snell's law. 

The system of equations (4.3a) can be also satisfied if PI' Or, and P

2 

are collinear. Such a spurious root may be eliminated by an additional 

where 15 is a small positive number. ,We set 15 = 0.0001. 

A root of (4.3) mayor may not fall inside the boundary r of the 

reflector (Figure 4). Thus, for each root (x,y,z=f) or its corresponding 

coordinates (x' ,j',z') in the primed system, the following test must be 

performed. If 

(4.4) 

then the root is inside r and it ,is indeed a reflection point on the 

reflector. If (4.4) is not satisfied, ,then the root should be discarded. 

The parameter Roi in (4.4) is given by 

1/2 r[xK'I]2 + [YK'Z]2]-1/2 'I " [(x,)2 + (y,)2] l (4.Sa) 

if r lies on an elliptical cylinder (Figure, Sa); and 

. 1/2 ' ' -1/2 

2.1 = Iz' 1[(x,)2 + (y,)2] , [(x' c:ot 9'1)2 + (y'" cot 9 2)2] (4.Sb) 

if r lies on an elliptical cone (Figure 5b)., 

21 
4.Z. Formula for Reflected Field 

The reflected magnetic field at Pz is given by 

(4.6) 

, +i 

which is given in terms of the incident field H at the reflection point 

r the surface normal N of the reflector at 0 , and a divergence factor 

OF. We choose N pointing toward the source; thus, (Nez) is always greater 

than zero (Figure 8). Explicitly, N is given by 

~ 

N = ~(-f x - f y + z) x y 

(4.7) 

where ~ = +(fZ + fZ + 1)-I/Z and the subscript x of f , for example, means 

x y x 

partial derivative with respect to x. The divergence factor in (4.6) is 

OF (4.8) 

where the square roots take positive real, negative imaginary, or zero 

value (so that OF is positive real, positive imaginary or infinite). 

(R~,R~) are the principal radii. of curvature of the reflected wavefront 

passing through Or. Their computation is given next. 

4.3. Curvatures of Reflected Wavefront 

We use the formulas given in Section IV of [181 for calculating 

The three orthonormal base vectors of the incirlent pencil are 

chosen to be (Figure 8) 

x(z 

[(z -

22 

- x ) 1 (4.9a) 
x 

~----II .. z 

Figure 8. Reflection from reflector L. 

Figure 9. Diffraction frqm boundary r of the reflector. 

23 
Ai Ai 

X3 x Xl 

lijx iii 

X(x 

[(X - 2 xl) + (y -

- z ) 1 

(4.9b) 

(4.9c) 

. r where (x,y,z) are the coordinates of the reflection point o. Those of the 

reflected pencil are chosen to be 

.... r ....i .... 1 ........ 

xl x - 2(x l

o N)N 1 (4.10a) 

.... r Ai Ai A A 

X

2 x - 2(x2

o N)N 2 (4.10b) 

"r x(x2 - x) + Y(Y2 - y) + z(z2 - z) 

x3 2 2 (z - z)2]1/2 [(x2 - x) + (Y2 - Y) + 2 

(4.10c) 

Note that (4.10) chosen above 1s a left-handed system, .... r .... r r i.e., xl x x2 -x3 • 

r r This choice, of course, does not affect the final solutions of (Rl ,R2). 

The three orthonormal base vectors of reflector ~ at Or are chosen to be 

x + zf x 

Y + zf 

Y 

(l + 

N 

From (4.9) and (4.11) the elements 

(4.11a) 

(4.11b) 

(4.11c) 

24 
· ... i "'1: 

= x ·x 

m n m,n = 1,2,3 

can be calculated with the results 

i (z - zl) - fx(x ~ xl) 

P11 =-[-1-+--f~;~]l~/~2--[(~X~--X-1~)~2-+--(-Z~---Z-1-)2-]-1~/2-

i 1 P = -- Mf (x - xl) + f (y - Y1) - (z - zl)] 33 .d1 x Y 

(4.12) 

(4.13a) 

(4.13b) 

(4.13c) 

(4.13d) 

(4.13e) 

The first four elements in (4.13a) through (4.13d) form the 2 x 2 matrix 

=i =r =i P. Because of the particular choice in (4.10), we have P = P. The 

cur~ature mat~ix of the incident pencil is . 

=i Q = (4.14) 

25 
The curvature matrix of reflector I: at Or is . 

where 

·2 

II (eG - fF) 2 II (fE - eF) 

=1: Q = 

2 ·ll (fG - gF) 

E 

e = -llf . 

xx 

F =·f f . x y 

f -llf . xy 

ll. is defined just belQw (4.7) 

G 

g = llf . yy 

(4.15) 

., 

=r The desired curvature matrix Q may becalclllated from the following matrix 

equation 

. (4.16) 

=r Let us denote the four elements of Q ·by 

Then the desired radii of curvature of the reflected wavefront at or· are 

given by 

~r = ~ {(Q1l + Q22 )2 ± /(Qll +Q22)2 - 4(QllQZ2 - Q12Q21)} 

2 

26 

(4.18) 
r r Both Ri andR2 are real. Their signs have the following meaning: 

is positive (negative), the corresponding normal section of the reflected 

wavefront is divergent (convergent). r The same convention applies to R

2" 

Two final remarks about the calculation of the reflected field: 

(i) For a given P 1 and P 2' there maY.be more than one reflection' point on 

E. Then the total reflected field is the superposition of the' contributions from each reflection, point •. If there is no reflection point on E, 

the reflected field is zero~ (ii) If the reflection point is close to the 

boundary f, ~e still calc~late its reflected field in the usual manner. We 

-Ki shall adjust the diffracted field H later by ~sing UAT so that the total 

+t field H in this case is correct. 

·27 
5. EDGE-DIFFRACTED FIELD 

5.1. Diffraction Points 

To calculate the edge-diffracted field, the diffraction points on the 

boundary r of the reflector must be located first. Consider a source point 

P1 at (x1'Yl,zl) and an observation point P2 at (xi,yi,zi), with their 

coordinates given in the primed system (Figure 9). A diffraction point 

ad with coordinates (x~,y~,z~) can be determined from the law of diffraction 

(5.1) 

Here t is the unit tangent of r at ad, and 

-+- x~(x ~ - xl) y ~(y ~ - y~) z ~(z ~ - z ~) d

3 = + + 1 1 (5.2a) 

+ d x~(x~ - x~) + y~(y~ - y~) z ~(z ~ - z ~) 

4 = + 2 2 2 (5.2b) 

From Fermat's principle, (5.1) is equivalent to 

(5.3) 

From (5.2), (5.3) may be written as a nonlinear equation for unknown ~~: 

(5.4) 

where (gl,g2,g3) are defined in (2.3) if r lies on a cylinder, and in (2.6) 

if r lies on a cone. A root of ~~ of (5.4) determines a diffraction point 

28 
on r. Depending on the geometry,' there exist examples where as many as 

four diffraction points have been found. 

5.Z. Formula for Diffracted Field 

d Corresponding to each diffraction point a , there is a contribution 

+d to the diffracted field H in (Z.8). Following Equation (5.Z1) of [18], the 

formula for such a contribution reads 

'( 5.5) 

Here g is a cyliridr\cal wave factor 

g(x) = _1 ~xp[-j x + !.ll UZrrx [ , 4 (5.6) 

The other factors used in (5.5) are explained below. 

5.3. Divergence, Factor 

The square root in (5.5) as usual takes positive real, negative imaginary, or zero value. Rl is a radius of curvature of the diffracted 

wavefront passing through ad. It may be ca'lculated from Equation (5.11) of 

[18] which reads 

t- = ~ + _,K,;.."Z:-- [-d

1 d3 - -d1 ]d4 on 

1 "3 sin e 3 ' 4 

(5.7) 

'+ Here e is the angle between tangent t and d

4 (Figure 9). 'The factor sin e 

is given by 

sin e (5.8a) 

29 
where 

r 

a g212 

aq," + 

l ) 

(5.8b) 

(5.8c) 

All the derivatives in (5.8) and in the remainder of this section are evald uated at the diffraction point 0 , whose coordinates are (x" ,y" ,z"). The 

d curvature K and normal n of the curve r at 0 can be calculated from 

Equation (13.9) of [23]. The results are 

R 

K = 3"" (5.9a) 

p 

.. 

x " 

g" 1 

(5.9b) 

g" 2 

where the prime on g~ signifies the partial derivative with respect to q," 

and 

Summarizing the results in (5.7) through (5.10), we obtain the final 

expression for the divergence factor of the diffracted field 

OFD 1 

'1 + G (5.11) 

30' 
where 

+ [y' - y~ y~ 

~ "2] [gl (gl&2' -g28[') 1 + - g~ (g~ g' ~ -g~ g' , ) ] d 3 2 3 3 2 3 4 .. 

+[ z' 

~ z' 

, 1 - gillJ')]} - zl - z 2 .,- .,..-..- - g;g;') - g'i (g;g'i' d 3 

+ d

4 [g2(g2g3 

5.4. Diffraction Coefficients 

The soft and hard Keller's diffraction coefficients DS and Dh are 

defined fn Equation (5.22) of (18), namely, 

Ds,h = Xi + Xr 

= -sec . 'T 1 (~ - ~ iIi ) ± sec 7(~ + ~ ) (5.12) 

The angles ~i and ~ are shown in Figure 10. Because of the fact that 

N-z > 0, it can be shown that vector ~ x N is tangent to E at Or,. and 

points away from (not toward) E. ·We calculate ~i and ~ from the relations 

cos ~i = 

~ A A 

(Proj d 3)- (t x N) 

IProj rl3 1 

(-1) (5.13) 

cos ~ 

+ A 

(Proj d4)- (t x N) 

Iproj d4 1 (5.14) 

+ + 

Here Proj d3 is the projection of d3 on the plane perpendicular to t, 

namely, 

31 
,. ,. ~. 

t x N ~ L 

/ ,. Plane .L t 

P2 

,. 

~z 

,. ,. t . N 

Od) -

d3 

.... Figure 10. Projection of Figure 9 on a plane perpendicular to tangent t. 

32 
... = x~S31 + y~S32 + z~S33 

where. 

Similarly Proj d4 is given by 

+d . Proj 4 

where 

(5.15) 

(5.16) 

The normal ~ of the reflector at ad is given in (4.7). For the present 

application, it is convenient to change its base vectors to those in the 

primed coordinate system. The result is 

N 

where 

A 

x~N1 + y~N2 + z~N '3 

33 

(5.17) 
N 2 -f::.f N 2-f::.f 1 X, 2 y , (5.18) 

if boundary r lies on an elliptical cylinder; and 

(5.19a) 

(S.19b) 

(S.19c) 

if r lies on an elliptical cone. Substitution of (5.15) through (5.19) 

1 into (5.13) and (5.14) gives the final formulas for calculating ~ and ~: 

(5.20) 

(5.21) 

where 

The solutions of ~i and ~ are subject to the following tests: 

(i) i 

</J takes the value in the range (0, IT) if T3 2. 0, and the 

range (IT,2") if T3 < 0, where 

T3 = (-Proj d3)·N = -S31 N1 - S32N2 - S33 N3 (5.22) 

34 
(ii) cjI takes the value in the range (0,11') if T4 ~ .. 0, and the range 

(1f,'21f) if T4 < 0, where 

(5.23) 

(iii) If the observation point P2 is exactly on the incident shadow 

boundary, then 

. i 0 cjI-cj1 -1f=. (5.24) 

(iv) If the observation point P2 is exactly on the reflected sh~dow 

boundary, . then 

- 1f o (5.25) 

When the observation point· is exactly, on the incident shadow boundary, the 

first facto~ Xi in (5.12) becomes infinite, and causes computational difficulty. A simple remedy is to shift P2 slightly whenever (5.24) is 

satisfied. It should be remarked that when P2 is near but not exactly on 

. i 

the incident shadow boundary, X is large but finite. It does noi cause 

any computational difficulty at the moment. Later on, the diffracted field 

tid in this case will be modified to become AD by using the uniform asymp-

+t r totic theory so that the total field H is correct. Similarly, X in 

(5.12) becomes infinite if the ohservation point P2 falls exactly on the 

reflected shadow boundary. Hence, we shift P2 slightly 'when (5.25) is 

satisfied. 

5.5. Spherical Components of Incident Field 

+i +i Fields Ha and Ha in (5.5) are the two spherical components in the 

A. Ai i d 

directions of 13 1 and a of the incident magnetic field § evaluated at 0 • 

35 
"'i "'i The base vectors (e ,a ) are shown in Figure 11 and may be calculated from 

Ai Ai + e = (a x d3)/d3 

"'i + a = (t x d3)/(d3 sin e) 

Then it may be shown that 

Hi -1 =-- [iii(od). ~] e sin e 

-1 ~ agl ag2 = P sin e HI 3,p'" + H2 acp'" + 

Hi 1 r [a g2 , 

a Pd3 sin e l HI acp'" (z 

- z ... ) 

1 

~ag3 , - x"') ag1 ... 

+ H2 acp'" (x 1 - acp'" (z 

t

agl 

+ H3 , , acp'" (y - Yl) 

ag2 ... 

- acp'" (x 

a

g3] H3 acp'" 

ag3 ... 

- acp'" (y 

-Z')] 

"tn 

- Y!~ 

(5.26a) 

(5.26b) 

(5.27a) 

(5.27b) 

+1 A A A 

Here (H1,H2,H3) are the components of H in the directions of (x'" ,y'" ,z'). 

They are given by 

(5.28a) 

if r lies on elliptical cylinder; and 

H = 3 sin - cos (S.2Rb) 

if r lies on elliptical cone. 

36 
,.. 

t 

,.. ~~::--~~ ___ ~:---~a 

Figure ll. Ai Ai A A Spherical base vectors (~ , a ) and (e, a). 

37 
5.6. Rectangular Components of Diffracted Field 

+d The diffracted field H calculated from (5.5) must be eventually 

.. ... .. expressed in terms of rectangular components in the directions of (x,y,z), 

so that it can be conveniently superimposed with the geometrical optics 

field Hg in Section 4. 

.. .. In (5.5), the two spherical base vectors (e,a) 

(Figure 11) may be calculated from 

(5.29a) 

(5.29b) 

The diffracted field expressed in terms of the primed base vectors are 

.. .. A 

x~hl + Y#h2 + z~h3 (5.30) 

where 

h [Dh a H~ + D

S 

a Hi] g(kd4) (DFD) 1 1,2,3 ilia n = 

n n n a 

1 [ ag3 

Y2) 

3g1 ~ 

- Zl)] a 1 Pd4 sin a a4>~ (y' - aljl' (z 

a 2 Pd4 sin [a'l _ a a4>~ (z - z') 

3g3 ~ 

2 - 34>~ (x -Xl)] 

1 [a'2 _ _ 3g1 ~ 

a 3 Pd4 sin e a4>~ (x - x2) - aljl~ (y - Yl~ 

13 1 

= _1 [a (y~ d4 3 - y~) 

2 - a (z' 2 - z')] 2 

13 2 

= _1_ [a (z~ d4 1 - z~) 

2 - a (x~ 

3 - x')] 2 

13 3 

= _1_ [a (x' - x') d

4 2 2 - a (y~ - y~)] 1 2 

38 
The factor g is defined in (5.6), DFD in (5.11), and sin B in (5.8). The 

final expressions for the diffracted field in the umprimed coordinate 

system are as follows: 

Hd(P2) ~d+ "'Hd + ,. d 

= zH 

x Y Y z (5.31a) 

where 

Hd hI Hd = h2 , Hd = h 

x Y z 3 (5.31b) 

iff lies on an elliptical cylinder; and 

Hd hI x 

·d -h2 cos 6 3 - h3 sin 6 3 H 

Y 

·d 

h2 sin 6 3 - h3 cos 6 3 H 

z (5.31c) 

if f lies on an elliptical cone. The corresponding diffracted electric 

field i d at P

2 is calculated f~om 

(5.32) 

5.7. Detour Parameter 

+d The diffracted field H calculated from (5.5) is not valid when observat ion point P2 is near the reflected shadow boundary which is defined by 

(5.25). To detect if P2 is indeed so, we may calculate the so-called 

"detour parameter" of the reflected field (Section VI of [18]). 

; = €[k(d + d - d - d )]1/2 3 4 1 2 (5.33) 

Here € is the shadow indicator of the reflected field defined by 

39 
+1, if P2 is in the shadow side of the reflected field 

e: = 

-1, if P2 is in the lit ~ide of the reflected field 

It may be shown that 

(5.34) 

The square root in (5.33) takes positive real, negative imaginary, or 

zero value. When the caustic of the reflected field falls on the reflected 

shadow boundary, ~ is imaginary; otherwise, ~ is always real. Detour 

parameter ~ in (5.33) becomes zero when P2 is exactly on the reflected 

d shadow boundary, because the diffraction point 0 and reflection point 

or coincide (Figure 2). Following the numerical study in [24], we take 

I~I = 2 (5.35) 

as on the on-set point. Thus, if I~I > 2, P2 is considered to be away from 

the reflected shadow boundary, and the diffracted field ~ in (5.5) is 

valid. If I~I < 2, P2 is considered to be near the reflected shadow boundary, and we must replace ~ by ~ as stated in (2.9). 

When ~ is small, (d3 + d4 ) in (5.33) is nearly equal to (dI + d2). In 

many practical problems, the reflection point Or and diffraction point 

d 

o may not be determined with great precision. Thus. when ~ is small. a 

direct computation of F, from (5.33) can have a numerical accuracy problem. 

To circumvent this possible problem. we have given below an alternative formula 

for F, when its value is small: 

40 
[ 

1 ] _ (OF)I d4 =- cos '2 (ejI + ejl1) sin a l2k (OOF) if I~ I + 0 (5.36) 

where (DF) of the reflected field is defined in (4.8), and (DDF) of the 

diffracted field in (5.11). The derivation of (5.36) is given in 

Appendix 'A of [14]. In all of the following computations, we use (5.33) if 

I~I > 2, and (5.36) if I~I < 2. 

5.8. Uniform Asymptotic Theory 

+D We shall calculate H by the UAT developed in [14], [19], [21]. The formula reads 

(5.37) 

where F'is a Fresnel integral defined by 

... 2 

F(z) = 11'-1/2 e j 11'/4 f e-jt , dt (5.38a) 

z 

and 

F(z) = ~ expl-j (z2 + !.4)] , 2z/11' L (5.38b) , 

The factor (1 - t:.)/2 in (5.37) is one if P2 is in the ,lit region of the 

reflected field, and zero, if P

2 is in the shadow. , +0 As expected, H reduces 

. +d 

to H when P 2 is away from the re'flected shadow boundary. This is because 

of the fact that for I~ I + ... , 

41 
[ F( t;) - F(~) 1 .... ~ (1 - e:) (5.39) 

:to When P2 is near or on the reflected shadow boundary, H in (5.37) is always 

finite, and compensates exactly for the discontinuity in ~ so that the 

total field Ht in (2.9) is everywhere continuous. 

42 
6. SECONDARY PATTERN COMPUTATION 

6.1. Aperture Field Theory 

Refer to Figure 3. From field equivalence principles, solutions for 

the far field may be obtained knowing the tangential fields at E. The 

a 

aperture plane E is taken to be perpendicular to the z-axis. 

a 

Let us denote the tangential electric and magnetic fields at E . by a 

E and H ,respectively. The field may be determined by using vector 

a a 

potentials [25]. However, it is more convenient to express the far field 

directly in terms of the aperture fields. Let us define the following vector quantities: 

where 

+ f(u,v) 

.. g(u,v) II H ( ) jk(ux+vY)d d a X,Y e- x y 

E 

a 

u = sin e cos $ 

v = sin e sin $ 

k 21T fA 

e,$ spherical coordinates of far field point 

Since the aperture fields are tangent to z, let 

+ 

f f x + f y x Y 

+ 

g 

43 

(6.1) 

(6.2) 

(6.3) 

(6.4) 

(6.5) 

(6.6) 
From each of the 3 equivalence principle formulations, three different 

expressions arise for EO and E~ of the far field, namely, 

(1) 

(2) 

(3) 

+ + using E and H 

a a 

[fXCOS~ + fySi~ + ZocosO (gyCOS~ - gxsin~) ] 

(6.7a) 

= jke -jkr 

E~ 41Tr [ cose (f cos~ - f sin~) - Z (g 81n4> + gxcos4>)l y x 0 y ~ 

+ using H 

a 

jk 

Ee = 

jk 

E~ 

using E+ 

a 

Z 

Z 

-jkr 

e cose 0 (g cos4> - g sin~) 21Tr y x 

e 

-jkr 

0 (g sin4> + g cos~) 21Tr y x 

-jkr 

jke cose (f cos4> - f sin4» 

21T r y x 

(6.7b) 

(6.8a) 

(6.8b) 

(6.9a) 

(6.9b) 

This method suits large apertures (in terms" of A) ,because the Fourier 

+ + transforms of the aperture field, f and g, are highly peaked in the frequency "doMain. 

44 
This method is exact if the fields at E were known everywhere. a 

However, one must truncate La to finite dimensions in order to employ the 

FFT. 

6.2 Fast Fourier Transform 

To employ the FFT subroutine, the integrals (6.1) and (6.2) must be 

rearranged so that the form of the integral,matches the definition given by 

the documentation of the FFT subroutine. For this particular 2-dimensional 

FFT subroutine, the function being considered is assumed to be periodic in 

x and y with. period 1 in x and y. Hence, the aperture grid as shown in 

Figure 12 must be scaled accordingly. 

Many manipulations are required in order to use the FFT. These manipulations are carried out for f. ~imilarly, this can be applied to obtain 

x 

f , gx' and g. From (6~1) and (6.5) y .y 

(6.10) 

where E is the x-component of Ea(X,y) in (6.1). Using the substitutions 

ax 

( 6 • 11) becomes 

(6.13) 

where 

45 
" 

Dp 

2 

Y2 

YI 

o 

H 

Y 

(a) 

Y 

• • 

La 

• • • • 

• • 

• • 

I 

I 

X 

0 • • .. X .. XI X2 0 

aperture grid. (b) FFT grid 

Figure 12. Aperture grid Land FFT grid. a ' 

Y 

-----+-----II~ X 

t----f -I 

Figure 13. An offset parabolic reflector. 

46 
Z'II' 

jr-( ux 1 +vy 1 ) 

e (6.14) 

The expression Eax(x1+(xZ-x1)X, Y1+(YZ-Y1)!) under the integral may be 

interpreted as Eax(x,y) scaled to F(a,b) where O<a<l and O<b<l (see Figure 

1Z). F(a,b) may be approximated by 

HZ 

L 

m=H 1 

(6.15) 

where C are the Fourier coefficients obtained using the FFT subroutine • . mn 

For a 3Zx32 FFT, 

From (6.13) and (6.15), 

f (u,v) x 

(6.16) 

Interchanging the summation and integral signs and noting that 

1 j~X sin(!f) I e dx = 

o 'II'C 

-r 

the expression for f is 

x 

HZ 

I 

m=H 1 

C e mn 

e (6.17) 

jfCm+n) sin[fCm).+u(XZ-X 1») ]sin[i{nA+v(yZ-Yl»)] 

1T (m).+u( xZ-x1) ) 'II' (n>..+v(yz-Yl») 

(6.18) 

47 
where 

j~ (u(x1+x2) + v(y1+y2») 

K = (x2-x1)(Y2-Y1)e (6.19) 

In summary., to evaluate the integral (6.10), we first calculate {Cmn} of 

(6.15) by FFT and then calculate f (u,v) via (6.18). x 

6.3 Gain Normalization 

The secondary pattern is usually decomposed into two orthogonal 

polarizations. Following Ludwig's definition 3 [26], the following unitary 

polarization vectors are introduced. 

R = a (aej1jl cos4> + bsio4» + ; (-aej1jl sin4> + bcos4» (6.20) 

(6.21) 

Let the secondary pattern be expressed as 

E -jkr = _e __ _ 

r (6.22) 

The reference-polarization and cross-polarization expressions of ~ are 

Reference-pol of .. E = (E. ..'" (R*)*) (6.23a) 

Cross-pol of .. E = (Eo ..'" (C*)*) (6.23b) 

The second conjugate operation in (6.23) results from the change in direction of the field after being reflected by the reflector. 

The directivity for the reference polarization is defined by 

(6.24) 

48 
Similarly, the directivity for the cross polarization is defined by 

where Pi is the incident power radiated from the feed. Noting that 

nc 

(6.25) 

-J"kr e /r factor is common in both the secondary pattern (6.22) and the incident radiated field (3.6) the directivity formulas are 

41r Ius (aej$ cos~ + bsin~) + U (-aej$ sin<jl + bcos<jl) 12 

DR(B,<jI) = ______ --;:;---;:;: __ -'-P _______ _ 

ZO Pinc 

(6.26) 

(6.27) 

Thus, for any feed polarization (a,b,$), (6.26) and (6.27) give the 

reference and cross-polarization directivities. 

49 
7. NUMERICAL EXAMPLES 

The present aperture integration method for calculating the secondary 

pattern of a reflector entails the following steps: 

(i) Input the reflector geometry, eq. (2.1), and the boundary 

description. 

(ii) Input the feed pattern as described in eq. (3.4). 

(iii) Calculate GO contribution to the aperture field. 

(iv) Calculate edge-diffraction contribution to the aperture field. 

Use either UAT [14] or UTn [15] for aperture grid points near the 

shadow boundary. 

(v) Use the FFT, eq. (6.15), to obtain the far field. 

(vi) Decompose the far field pattern into reference-pol and cross-pol 

components using eq. (6.23). 

(vii) Use eqs. (6.26) and (6.27) to obtain the reference and crosspolarization directivities. 

In this section, we shall present some numerical results to establish 

the numerical accuracy of the present method. Near field, far field and 

scan data for a large reflector are presented. 

7.1. Effects of Aperture Grid Size and Location on Secondary Pattern 

The aperture field theory used to determine the secondary pattern is exact 

if the tangential fields are known everywhere on the aperture plane L • 

a 

When employing the FFT, L is truncated. To minimize the amount of com- o a 

puter time spent, the size of La should be as small as possible while capturing almost all of the field. To this end, a study of varying the aperture grid size and location of E was performed to determine their effects on 

a 

the secondary pattern. 

50 
The reflector used for this study is" an offset parabolic reflector 

(see Figure 13) with typical flD and flD values: " " p 

f 

--D 

f 0.40, 0 - 1.00 (7.1) 

P 

The corresponding values of °

1, "~, and Hare 

(7.28) 

(7.2b) 

H - 0~25D (7.2c) 

The reflector is being illuminated by a Yf-po1arized feed at focus with 

10 dB feed taper. Assuming (cos e)q type patterns, eq. (3.5), the E- and 

H-p1ane feed patterns are given by 

(7.3a) 

(7.3b) 

Refer to Figure 14. Let the aperture plane L be located a distance L 

a 

away from the focal plane. The focal plane is located at z - f. Plots of 

the tangential components of the aperture field along two cuts of La are 

obtained as the distance L is varied. These two cuts are 

x-cut: y = H + 0.5D, z - f + L 

y-cut: x - a z - f + L 

The diameter of this test ref1ector"is SOA. Due to the choice of feed 

polarization, only the y-component of the electric field, E , is plotted. y 

Figures IS(a)-(c) are plots of IE I for a x-cut for L - 0, 10 A, and 20 A, Y 

51 
focal 

i/p,ane l2:a I I I I I I I~y-cut I I (x = 0) 

I 

I I 

I I 

I I I I I I I 

-I z 

f ~. L 

(a) y-cut 

- --- x-cut 

(y = H + 0/2) 

H 

----~-L------------------~·x 

(b) x-cut 

Figure 14. Two cuts of aperture plane La at distance L away from focal plane. 

52 
0.020~--~----~----~--------~----~--~ 

--Total field 

--- GO field 

0.016 

-E ·0.012 ~ 

-

0 

.-J 

W 

u.. 0.008 I 

w 

0.004 

O.OOO~--~----~----~--~----~----~--~ 

O. 5. 10. 15. 20. 25. 30. 35. 

xl)" 

Figure lSa. I Ey I for· a x-cut, L -.0 

53 
0.020 r-----r----r-----r---r---~-__r--.., 

- Total field 

---GO field 

0.016 

-E 0.012 ~ 

-

0 

-l 

W 

IJ.. 0.008 I 

w 

0.004 

O.OOO~--~----~----~--~----~----~--~ 

o. 5. 10. 15. 20. 25. 30. 35. 

xl).. 

Figure lSb. I Ey I for a x-cut, L • lOA 

54 
0.020 ~----r---'---T---""--"""'T""--"T'""""--' 

- Total field 

---GO field 

. 0.016 

-

E 0.012 > -

0 

...J 

W 

lJ.. 0.008 I 

w 

0.004 

0.000 o...-_'-" __ ~ __ '--_~ __ ~ __ ..I--._--' 

O. 5. 10. 15. 20. 30. 35. 

xiX 

Figure lSc. I Ey I for ax-cut, L - 20), 

55 
respectively. Figures l6(a)-(c) are plots of IE I for a y-cut for L - 0, Y 

lOA, and 20A, respectively. 

From Figures l5(a)-(c) and l6(a)-(c), note that the aperture field 

does not spread as L increases. This is so because the feed is at the 

focus. To demonstrate that the aperture field does spread as L increases, 

let's move the feed toward the reflector along ~f by an amount df (see 

Figure 17). Figures l8(a)-(c) are plots of IEyl for a x-cut for d

f - 2A 

at L = 0, lOA, and 20A, respectively. Figures 19(a)-(c) are for the y-cut. 

Figures l8(a)-(c) and 19(a)-(c) show a slight increase of IE I for grid y 

points near the edge of the aperture grid. Also note that the width of the 

region where the GO field is nonzero increased when L is increased. Thus, 

to minimize the field strength for a fixed size of L , LaO was chosen. 

a 

Let the aperture grid be W by W. Choosing the criterion that the field at 

the edge of L is at least 20 dB below the maximum field value on E , W ~ a a 

1.14D centered at the midpoint of the projected aperture was chosen. 

7.2 DBS Antenna 

A direct broadcast satellite (DBS) antenna was designed by Lee et ale 

[27]. Pattern computation programs using the Jacobi-Bessel series technique [41 were developed by Y. Rahmat-Samii. A parabolic reflector was 

used and it is described by 

D = dish diameter = l08.l48A 

f = focal length = 94.867A 

H = offset height" = l6.865A 

(7.4a) 

(7.4b) 

(7.4c) 

The feed is located at the focus and the primary pattern, eq. (34), is 

described by 

56 
0.020~~----~--~----~----r---~----~ 

0.016 

-E > . 0.012 

-

o 

.-J 

W 

lL. 0.008 I 

w 

0.004 

- Total field 

--- GO field 

O.OOO~~~--~----~--~~--~----~~--~ 

5. 10. 20. 30. 40.. 50. 60. 70 . 

. y/'A'· 

'. Figure 16a. I Ey,1 for a y-cut, L .. 0 . 

57 
0.020~~----r---~----~----~--~----~ 

- Total field 

--- GO field 

0.016 

-E 0.012 .~ 

-

0 

-' W 

l.L. 0.008 I 

w 

0.004 

O.OOO~~~--~--~~--~----~~--~~~ 

5. 10. 20. 30. 40. 50. 60. 70. 

y/'A 

Figure 16b. I Ey I for a y-cut t L • 10), 

58 
0.020--~----~~~~--~----~----~--~ 

- Total field 

--- GO field 

0.016 

-

E 0.012 ~ 

-

0 

-1 

W 

u. 0.008 I 

w 

0.004 

0.000 I..-...Io.-J"--_"--_--'-__ -L-___ .l--__ ---'---'-_ ..... 

5. 10. 20. 30. 40.' 50. 60. 70. 

y/'A 

Figure 16c. I Eyl for a y-cut, L • 20), 

59 
y 

I 

I 

------~----------------------~------------~----~z 

~--------- f ---------· ... 1··--- L.:---i~-

... Figure 17. Feed located df away from focus along zf. 

60 
0.020~--~--~~--~--~~--~--~~--~ 

- Totcl field 

--:--GO field 

0.016 

-

E 0.012 "'-

> 

-

0 

...J 

W 

l.I- 0.008 I 

w 

0.004 

0.000 '---__ -'-____ L.-__ -'-__ ----'L.-__ -'-__ ----'L--_-'" 

O. 5. 10. . 15. 20. 30. 35. 

x/x 

Figure 18a. IEyl for a x-cut, df ~ 2A, L - O. 

61 
0.020 r----r---r-----r---r----r--~r__-_, 

- Totcl field 

--- GO field 

0.016 

-E 

........ 0.012 

> 

-

0 

-I 

W 

LL 0.008 I 

w 

0.004 

O.OOO~--~----~--~----~--~~--~--~ 

O. 5. 10. 15. 20. 25. 30. 35. 

x/'A 

Figure lSb. IEyl for a x-cut, df • 2A, L • lOA. 

62 
-

E 

........ 

> 

-

0 

....J 

W 

lII 

w 

0.020~--~--~~--~--~~--~--~~--~ 

- Total field 

---GO field 

0.016 

0.012 . 

0.008 

0.004 

, 

I 

O.OOO~--~----~--~----~----~'--~----~ o. 5. 10. 15. 20. 25. 30. 35. 

xl').. 

Figure l8c. lEy I for ax~cut, df ,. 2 A, L - 20 A. 

63 
0.020~~----~--~----~----~---,----~ 

- Total field 

--- GO field 

0.016 

-E 0.012 ........ 

> 

-

Q 

....J 

W 

u. 

I 0.008 

w 

0.004 

O.OOO~~----~--~~--~----~----~~~ 

5. 10. 20. 30. 40. 50. 60. 70. 

y/"A 

Figure 19a. lEy I for a y-cut, df - V., L - O. 

64 
0.020~~----~--~~--~----~----~--~ 

- Total field 

--- GO field 

0.016 

.-. 

E 

....... 0.012 > 

-

0 

....J 

W 

lL.. 0.008 I 

w 

0.004 

O.OOO~~----~--~~~~----~----~~~ 

5. 20. 30~ 40. 50. 60. 70. 

y/'A 

Figure 19b. IEyl for ay-cut, df .. 2A, L ,. lOA. 

65 
0.020~~----~--~~--~----~----~--~ 

- Total field 

- -- GO field 

0.016 

-

E 

0.012 ...... 

> 

-

Q 

-l 

W 

IJ.. 0.008 I 

w 

0.004 

O.OOO~~----~--~~--~----~----~~~ 

S. 20. 30. 40. SO. 60. 70. 

y/'A 

Figure 19c. IEyl for a y-cut, df .a 2A, L .. 20A. 

66 
a = 1/12 , b D 1/12, 1/1. 90° (RRCP) 

U

E

( a) = (cos a) 3. 6 

uR(a) - (cos 6)2.8 

(7.Sa) 

(7.5b) 

(7.Sc) 

Using this aperture integration technique, the secondary pattern for ~ = 0° 

was computed using GO and GTD constructions. As shown in Figure 20, two 

patterns are superimposed with Rahmat-Samii's results. One pattern used GO 

fields only and in computing the other pattern, the edge-diffracted field 

was included in the aperture field calculation. The gain and sidelobe 

levels of these two patterns are tabulated versus Rahmat-Samii's results in 

Table 3 below. 

TABLE 3 

COMPARISON WITH RAHMAT-SAMII'S RESULTS· 

Rahmat-Samli GO + GTD GO 

main beam 48.28 48.33 48.32 

1st sidelobe 28.42 28.29 28.42 

2nd sidelobe 22.29 22.18 22.93 

3rd sidelobe 18.05 18.02 18.12 

4th sidelobe 14.95 14.96 13.40 

5th sidelobe 12.39 11.85 11.14 

6th sidelobe 10.31 9.05 8.41 

Due to the limited amount of computer working space, a 48 x 48 FFT was 

used. Despite the coarse sampling, the results in Table 3 are in good 

agreement. 

The scan performance of this reflector has been studied by Hung [28]. 

The Fourier-Bessel series technique is used to compute the secondary pattern. Using the same feed, eq. (7.5), the feed was displaced 5.8 in the 

61 
-(I) 

"C 

-

>-

I-

> 

IU 

w 

a:: 

0 

-I 

0 

a. 

I 

ll.. 

w 

a:: 

50.--------~----~------~------~ 

40. 

30. 

20. 

10. 

O. 

- RAHMAT-SAMII 

_.- GO+GTD 

---GO 

, 

, 

, 

I -IO.~------~--~--~------~~--~u 

0.0 1.0 2.0 3.0 4.0 

8 .(degrees) 

Figure 20. Secondary pattern of a DBS antenna using Jacobi-Bessel series, 

GO, and GTD. . 

68 
-xf direction to produce a pattern with peak directivity at a = 3° (see 

Figure 21). The H-plane scan pattern was also computed using the aperture 

integration technique with and without the edge-diffracted field. Figure 

22 depicts the H-plane scan pattern using the aperture integration and 

Fourier-Bessel series techniques. The patterns are in good agreement. 

7.3. TRW Antenna 

A dual reflector antenna system was desig~ed by TRW for NASA-Lewis 

Research Center. The dual reflector is an offset Cassegrain reflector. 

The main reflector is parabolic and it is described by 

D = dish diameter = 257.89A 

f = focal length = 318.74A 

H = offset length = 135.51A 

(7.6a) 

(7.6b) 

(7.6c) 

The reflector is being illuminated by a Yf-polari~ed feed at focus with 

18 dB edge taper. 

Two cuts of I: at the focal plane were taken, a 

x-cut: y H + 0.5D, z = f 

y-cut: x = 0 z = f. 

The magnitude and phase of the y-component of the electric field, E , are 

y, 

plotted in Figures 23(a)-(b) for the x-cut and in Figures 24(a)-(b) for the 

y-cut. 

The secondary pattern for ~ = 0° of this main reflector is shown in 

Figure 25. The key features of the reference-pol directivity plot are 

69 
y 

Zf 

/,fOCUS 

o z 

Yf 

(a) E-plane scan 

z -X, 

(b) H-plane scan 

Figure 21. Feed position for scanning. 

70 
m 

"0 

-

>-

l-

> 

lu 

W 

0:: -..J 0 

-.J 

0 

0.. 

I 

IJ.... 

w 

0:: 

50 

40 

30 

20 

10 

0 

I V I 

-10 

1

0 

-20 

0.0 

0 

1.0 2.0 3.0 

8 "(degrees) 

4.0 

--HUNG 

)( )( )( GO + GTD 

oooGO 

5.0 

Figure 22. "-plane scan pattern of a DBS antenna using Fourier-Bessel 

series, GO, and GTD. 

6.0 
0.004 

- Totol field 

--- GO field 

0.003 

-

E 

...... 

> 

-

0 0.002 

.....J 

w 

lL. 

I 

w 

0.001 

0.000 '---__ --'--__ -1 ___ ...l-__ ---.1.---.l._~ 

O. 30. 60. 90. I 20. I 50. 

X/A 

Figure 23a. Amplitude of E for a x-cut of TRW main reflector. 

y 

72 
360.~--~~------~----~------~----~ 

--Total field 

300. ---GO field 

-C\ 

Q) 240. 

"C 

->0-

W 

lL. 180. 

0 

w 

en 

<t 

J: 120. 

a.. 

60. 

O.~------~------~------~------~~--~ O. 30. 60. 90. I 20. I 50. 

x/x 

Figure 23b. Phase of E for a x~cut of TRW main reflector. 

y 

73 
-

E 

....... 

> 

-

.0 

...J 

w 0u. I 

W 

0.004 

-Total field 

---GO field 

0.003 

0.002 

0.00 r 

0.000 L--'_......L-_--' __ ...L.-_--'-__ "'--_--'-_--'-r-...;! 

120. 160. 200. 240.280. 320. 360. 400. 

'Y / ).0 

Figure 24a. Amplitude of E for a y,:"cut of TRW main° reflector. Y ° ° 

74 
- Total field 

300. ---GO field 

-C\ 

Q) 240. 

"'0 

->-

W 

lJ.. 180. 

0 

w 

(J) 

<t. 

l: 120. 

a. 

60. 

O.~--~----~----~----~----~--~----~ 120. 160. 200. 240. 280. 320. 360. 400. 

y/'A 

Figure 24.b. Phase of E for a y-cut·of TRW main reflector. 

y . . 

75 
60. 

TRW Reflector 

-GO+GTD 

50. ---GO 

-

CD 

"0 

-

>- 40. 

f-

> 

fU 

w 30. 

a:: 

o· 

-.J 

0 20. a.. 

I 

IJ.. 

W 

a:: 

10. 

O.~------~----~~--~--~--~~--LL--~U-~ 

O. .4 .8 1.2 1.6 2.0 

8 (degrees) 

Figure 25. Reference-pol. directivity plot of TRW main reflector, ~ a 0°. 

I 

76 
GO + GTD GO , 

Gain (dB) 56.95 56.95 

HPBW (deg) 0.276 0.277 

SLL (dB) -32.00 -32.26 

Figure 26 is a plot of the cross':"pol directivity. 

The hyperboloid subreflector has a magnification factor of 2. In 

approximating the performance of this dual reflector system, we employ the 

equivalent paraboloid [29]. The equivalent paraboloid is described by 

D': 257.89 (7.7a) 

f 2 2(318.74) = 637.48 

H = 135.51 

(7.7b) 

(7.7c) 

The corresponding secondary pattern for = 00 is shown in Figure 27. 

The gain, HPBW. and sidelobe level (SLL) are 

GO + GTD GO 

Gain (dB) 56.87 56.88 

HPBW (deg) 0.279 0.280 

SLL (dB) -33.87 .. -35.15 

; , 

Figure 28 is a plot of the cross-pol di~ectivi~y. 

Scan performances of the TRW main and equivalent reflector were 

studied. The scan plane chosen is the xfYf plane and E- and H-plane scans 

were performed (see Figure 21). 

For E-plane scan, the coordinates of the feed are 

17 
60. 

TRW Reflector 

-GO+GTD 

- 50. ....--GO 

co 

~ 

-

~ 

I- 40. 

> 

IU 

W 

0: 30. 

Q 

-I 

0 

DI 20. 

en 

en 

0 

0: 

U 10. 

O.~ ______ ~~~L-~~D-__ ~ ________ ~ ______ ~ 

O. .4 .8 1.2 1.6 2.0 

8 (degrees) 

Figure 26. Cross-pol. directivity plot of TRW main reflector, ~ ~ 0°. 

78 
60. 

TRW Reflector 

-GO+GTD 

50. --.:.. GO 

-

CD 

"'0 

-

>- 40. 

I-

> 

lt) 

w 30. 

a: 

0 

..J 

0 20. 0-

I 

IJ.. 

W 

a: 

10. 

O.~-------~-----~~--~---~---~~---~---~U-~ O. .4 .8 1.2 2.0 

8 (degrees) 

Figure 27. Reference-pol. directivity plot of TRW equivalent reflector. ~ ~ 0°. 

79 
60. 

TRW Reflector 

-GO+GTO· 

- 50. ---GO 

m 

"'0 

-

~ 

I- 40. 

> 

IU 

W 

a::: 30. 

a 

. 

..J 

0 

a. 20. I 

en 

en 

·0 

a::: 

u 10. 

o.~------~~~~~~~--~--------~------~ 

O. .4 .8 1.2 1.6 2.0 

8 (degrees) 

Figur~ 28. Cross-pol. directivity plot of TRW equivalent reflector, ~. 0°. 

80 
where 

where 

x .. 0 

y .. -f tan (N·HPBW) cos 6 

c 

z .. f - f tan (NoHPBW) sin B 

c 

f = focal length 

Bc .. 2· tan -t (H \~.5D) 

N ... number of beamwidth's scan 

(7.8a) 

(7.8b) 

(7.8c) 

(7.9) 

HPBW "" beamwidth of secondary pattern (.4> .. 90°) with feed at focus. 

For H-plane scan, the feed coordinates are given by 

x ... -f tan (N·HPBW) 

y ... 0 

z .. f 

f ... focal length 

N "" number of beamwidth's scan 

(7.l0a) 

(7.l0b) 

(7.l0c) 

HPBW" beamwidth of secondary· pattern (4) .. 0°) with feed at focus. 

As the feed moves away from the focus, the secondary pattern degrades. 

The degradation may be characterized by peak gain loss and half-power beamwidth. These 2 figures-of-merit are plotted versus number of beamwidths 

scanned for the following four cases: 

(1) TRW main reflector, E-plane scan (see Figures 29-30) 

(2) TRW equivalent reflector, E-plane scan (see Figures 31-32) 

(3) TRW main reflector, H-plane scan (see Figures 33-34) 

(4) TRW equivalent reflector~ H-plane scan (see Figures 35-36). 

As expected, the scan performance of the equivalent reflector is much 

better than for the main reflector due to its larger f/D value. For the 

same number of beamwidths scanned, the peak gain loss and amount of beam 

81 
O~~--~----r-----.-----'----' 

-GO+GTD 

---GO 

-

m -I 

"'C 

-

en 

en 

0 

-.J 

2 

<! 

(!) 

~ 

<! 

w -2 a.. 

" 

-3L-----~----~----~----~----~ o 2 4 6 8 10 

BEAMWIDTHS SCANNED 

Figure 29. Peak gain loss versus beamwidths scanned for E-plane scan of TRW 

main reflector. 

82 
0.270 __ --....----...-----r-----'r-----, 

-GO+GTD 

--- GO 

0.280 

- 0.290 

C'I 

Q) 

"'C 

-

J: 

~ 0.300 

Q 

3: 

~ 

<t 

w 0.310 

CD 

0:: 

W 

3: 

0 0.320 

0-

I 

lJ.. 

..J 

<t 

J: 0.330 

0.340 

0.350 1--__ ...1....-__ --'--__ --'-......-._--.1.-----' 

o 2 4 6 8 10 

BEAMWIDTHS SCANNED 

Figure 30. Half-power beamwidth versus beamwidths scanned for E-plane scan 

of TRW main reflector. 

83 
O.O~~--~----~----~----~----~ 

-GO+GTD 

---GO 

-0.1 

-0.2 -

m 

"C 

-

en 

en -0.3 

0 

...J 

2 

<! 

t!) -0.4 

~ 

<l: 

w 

a. -0.5 

-0.6 

-0.7~----~----~------~----~----~ 

o 246 8 10 

BEAMWIDTHS SCANNED 

Figure 31. Peak gain loss versus beamwidths scanned for E-plane scan of TRW 

equivalent reflector. 

94 
0.270 .----~--,.......;.--r__--r__-___, 

-GO+GTD 

--- GO 

0.280 F-=-=-~~-::-::_=-___ 

- 0.290 ........ ... Cl .... .. Q.) ., ~ .. - .. " .. J: 0.300 Ia 

:r: 

:2 

<{ 

w 0.310 

CD 

a:: 

w 

:r: 

0 0.320 

0-

I 

l1.. 

-.J 

<{ 

J: 0.330 

0.340 

0.350 1...-__ ~ __ ..l-__ ..1-__ -'--__ ....I 

o 2 4 6 8 10 

BEAMWIDTHS SCANNED 

Figure 32. Half-power beamwidth versus beamwidths scanned for E-plane scan 

of TRW equivalent reflector. 

85 
O~~--r-----r-----r-----r----. 

-GO+GTD 

--- GO 

-

m -I 

"0 

-

en 

en 

0 

...J 

Z 

« 

C,!) 

~ 

« 

w -2 a.. 

-3~-----L------~----~----~~----~ o 2 4 6 8 10 

8EAMWIDTHS SCANNED 

Figure 33. Peak gain loss versus beamwidths scanned for H-plane scan of TRW 

main reflector. 

86 
0.270.----r---...,-----r-----r----, 

-GO+GTD 

--- GO 

0.280 

- 0.290 

C' 

Q) 

"0 

-

J: 0.300 b 

3; 

:E 

<t 

w 0.310 

CD 

a:: 

w 

3; 

0 0.320 

a.. 

I 

I.L. 

-l 

<t 

J: 0.330 

0.340 

0.350 L..-__ ..L..-__ ...i...-__ ....L...-__ """"--~-..." 

o 2 4 6 8 10 

BEAMWIDTHS SCANNED 

Figure 34. Half-power beamwidth versus beamwidths scanned for H-plane scan 

of TRW main reflector. 

87 
0.0 r--=:!!!:::o;;::r----r--.,---,---~ 

- GO+GTD 

--- GO 

-0.1 

- -0.2 

m 

"C 

-

en 

en -0.3 

0 

...J 

Z 

<X: 

C!) -0.4 

~ 

<X: 

w 

a. 

-0.5 

-0.6 

-0.7~----~----~----~----~----~ 

o 2 4 6 8 10 

BEAM WIDTHS SCANNED 

Figure 35. Peak gain loss versus beamwidths scanned for H-plane scan of TRW 

equivalent reflector. 

88 
0.270.-----....-----r----,------r-----, 

-GO+GTD 

---GO 

0.280 r------.,_=__~ 

- 0.290 

Cl 

Q) 

"'0 

-::I: 0.300 I0 

3 

~ 

<:{ 

w 0.310 

CD 

cr: 

w 

3 

0 0~320 a.. 

I 

IJ... 

..J 

<t 

::I: 0.330 

0.340 

0.350 '-__ ~ __ ........ __ ~_---L.-----' 

o 246 8 10 

BEAMWIDTHS SCANNED 

Figure 36. Half-power beamwidth versus beamwidths scanned for H-plane scan 

of TRW equivalent reflector. 

89 . 
> • 

broadening were much less for the equivalent reflector for both E- and Hplane scans. Also, the cross-pol directivity was much lower for the equivalent reflector. Figures 37(a)-(b) ~nd .38(a)-(b) are th~ reference-pol and 

cross-pol directivttyplots for 6 beamwidthscan in the H-plane for the 

TRW main and equ'i valent reflectors, respectively. With respect to "the peak 

directivity, the maximum cross-pol directivity value is -39.15 dB for the 

equivalent reflector and only -27.95 dB for the main reflector. 

90 
60~----~----~------~-----r------~----~ 

50 

-

m 

"'C 

-

>-. 40 

.-

> 

.-

U 

W 30 

a: 

a 

...J 

o 20 0.. 

I 

lJ... 

W 

a: 

10 

Main Reflector 

6 BWs scan 

-GO+GTD 

--- GO 

I' 

~ • O~----~----~----~------~~--~~--~ 0.2 0.6 1.0 1.4 1.8 2.2 2.6 

·S{degrees) 

Figure 37a. Reference-pol. directivity plot of TRW main reflector for 6 

beamwidth scan in theH-plane. 

91 
60 

Main Reflector 

6 BWs scan 

- 50 

CD -GO+GTD 

"C 

- --- GO 

>-

.... 40 

> 

.... 

U 

w 

a:: 30 a 

. 

--I 

0 

a. 20 I 

en 

en 

0 

a::: 

u 10 

O~~~L-~~~----~----~~~~~--~ 

0.2 0.6 1.4 1.8 2.2 2.6 

9 (degrees) 

Figure 37b. Cross-pol. directivity plot of TRW main reflector for 6 beamwidth scan in the H-plane. 

92 
60 

Equiv. Reflector 

50 6 BWs scan 

,- -GO+GTD 

aJ 

"'C --- GO 

-

>- 40 

t: 

> 

.-

() 

w 30 a:: 

a 

-1 

0 20 a.. 

I 

L1.. 

W 

a:: 

10 

O~----~----~~----~------~----~----~ 0.2 0.6 1.0 1.4 1.8 2.2 2.6 

8 (degrees) 

Figure '38a. Reference-pol. directivity plot of TRW equivalent reflector for 

6 beamwidth scan in the H-plane. 

93 
60 

Equiv. Reflector 

50 .6 BWs scan 

-

CD -GO+GTD 

"C 

- --- GO 

>-

... 40 

> 

~. 

U 

w 

a:: 30 

a 

...J 

0 

a... 20 I 

en 

en 

0 

a:: 

u 10 

O~----~----~--~--~----~--~--L-----~ 0.2 0.6 1.0 1.4 1.8 2.2 2.6 

8 (degrees) 

Figure 38b. Cross-pol. directivity plot of TRW eq~ivalent reflector for 6 

beamwidth scan in the H-plane. 

94 
8. CONCLUDING REMARKS 

We have developed a method of calculating the secondary pattern of an 

arbitrarily shaped reflector illuminated by a feed with arbitrary polarization. An edge-diffracted field was added to the geometrical optics field in 

the aperture field calculation. By employing the FFT, the secondary pattern is. computed very efficiently.' The results for the secondary' pattern 

are in good agreement with those obtained by the physical optics integral. 

Furthermore, this method can be conveniently extended to secondary pattern 

computation of multiple reflector systems, whichwlll be done in the next 

phase of this project. 

95 
[1] 

[2] 

[3] 

[4] 

[5] 

[6 ] 

[7] 

[8 ] 

[9] 

[10 ] 

[ 11 ] 

[12] 

[13] 

REFERENCES 

S. Silver, Microwave Antenna Theory and Design, New York: 

McGraw-Hill, 1949. 

w. V. T. Rusch and P. Potter, Analysis of Reflector Antennas, 

New York: Academic, 1970. 

C. C. Hung and R. Mittra, "Secondary pattern and focal region 

distribution of reflector antennas under wide-angle scanning," 

IEEE Transactions on Antennas and Propagation, vol. AP-31, 

pp. 756-763, 1983. 

Y. Rahmat-Samii and V. Galindo-Israel, "Shaped reflector antenna 

analysis using the Jacobi-Bessel series," IEEE Transactions on 

Antennas and Propagation, vol. AP-28, pp. 425-435, 1980. 

C. C. Hung, "Fourier-Bessel series representation for the far field 

pattern of arbitrary reflector antennas," Lockheed Missiles and 

Space Company, Sunnyvale, CA, LMSC Rep. 844955, 1982. 

V. Galindo-Israel and R. Mittra, "A new series representation for 

the radiation integral wirh application to reflector antennas," 

IEEE Transactions on Antennas and Propagation, vol. AP-25, 

pp. 631-641, 1977. 

R. Mittra, Y. Rahmat-Samii, V. Galindo-Israel, and R. Norman, 

"An efficient technique for the computation of vector secondary 

patterns of offset paraboloid reflectors," IEF.E Transactions on 

Antennas and Propagation, vol. AP-27, pp. 294-304, 1979. 

P. A. J. Ratnasiri,· R.· G. Kouyoumjian, and P. H. Pathak, "The 

wide angle side lobes of reflector antennas," ElectroScience 

Laboratory, The Ohio State University, Columbus, Ohio, Report 

2 183-1, 1970. 

W. V. T. Rusch, "Geometrical scattering from inflected surfaces 

of revolution," Tech. Rept. Jet Propulsion Lab., Pasadena, CA, 

August 1975. 

W. V. T. Rusch and o. Sorensen, "The geometrical theory of 

diffraction for axially symmetric reflectors," IEEE Transactions 

on Antennas and Propagation, vol. AP-23, pp. 414-419, 1975. 

J. F. Kauffman, W. F. Croswell, and L. J. Jowers, "Analysis of 

the radiation patterns of reflector antennas," IEEE Transactions 

on Antennas and Propagation, vol. AP-24, pp. 53-65, January 1976. 

R. Acosta, private communication. 

Y. Hwang, C. H. Tsao, and C. C. Han, AP Digest, p. 88, May 1983, 

Houston, TX. 

96 
[14] S. W. Lee and G. A. Deschamps, "A uniform asymptotic theory of 

electromagnetic diffraction by a curved wedge," IEEE Transactions 

on Ari.tennas and Propagation, vol. AP-24, pp. 25-34, 1976. 

[15] R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical theory 

of diffraction for an edge in a perfectly conducting surface," 

Proc. IEEE, vol. 62, pp. 1448-1461, 1974. 

[16] Y. Rahmat-Samii, "Useful coordinate transformations for antenna 

applications," IEEE Transactions on Antennas and Propagation, 

vol. AP-27, pp. 571-574, 1979. 

[17] J. B. Keller, "Geometrical theory of diffraction," J. Opt. Soc. 

Amer., vol. 52, pp. 116-130, 1962. 

[18] S. W. Lee, "Uniform asymptotic theory on electromagnetic edge 

diffraction: A review," EM Lab. Rep. 77-1, University of Illinois 

at Urbana-Champaign, Jan •. 1977. 

[19] D. S. Ahluwalia, R. M. Lewis, and J. Boersma, "Uniform asymptotic 

theory of diffraction by a plane screen," SIAM J. Apple Math., 

vol. 16, pp. 783-807, 1968. . 

[20] S. W. Lee, "Uniform asymptotic theory on electromagnetic edge 

diffraction: a review," in Electromagnetic Scattering, 

P. L. E. Uslenghi, Ed., New York: Academic, pp. 67-119, 1978. 

[21] R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge 

diffraction," J. Math. Phys., vol. 10, pp. 2291-2305, 1969. 

[22] H. Goldstein, Classical Mechanics, Reading, MA: Addison-Wesley, 

pp. 107-109) 1953. 

[23] S. W. Lee, "Differential geometry for GTD applications," EM Lab. 

Rep. 77-21, University of Illinois at Urbana-Champaign, 

October 1977. 

[24] R. C. Menendez and S. W. Lee, "On the role of the geometrical optics 

field in aperture diffraction," IEEE Transactions on Antennas and 

Propagation, vol. AP-25, pp. 688-695, 1977. 

[25] R. E. Collin and F. J. Zucker, Antenna Theory Part 1, New York: 

McGraw-Hill, 1969. 

[26] A. C. Ludwig, "The definition of cross polarization," IEEE 

Transactions on Antennas and Propagation, vol. AP-21, PP:-116-119, 

1973. 

[27] S. W. Lee, Y. Rahmat-Samii, V. Galindo-Israel, W. A. Imbriale, and 

R. Mittra, "A baseline design of STC-DBS antenna version II," Lee and 

Mittra Consultants, Champaign, Illinois. 

[28] C. C. Hung, unpublished work and private communication. 

[29] P. W.Hannan, "Microwave antennas derived from the Cassegrain 

telescope," IEEE Transactions on Antennas and Propagation, vol. AP-9, 

pp. 140-153, 1961. 

97 
APPENDIX A 

INSTRUCTIONS TO RUN COMPUTER PROGRAMS 

To run the computer programs on the University of Illinois CYBER 175 

system to compute the secondary pattern of an offset parabolic reflector, the 

user should perform the following 

(1) create input file TAPE2 

(2) run program BALI 

(3) run programs BPT, BFFT 

(4) for plot, use BPLOT3 

The computer commands corresponding to steps (1)-(4) are 

(first create TAPE2) 

R 

BALI 

R 

P.LOAD(BPT,BFFT)j EXECUTE 

R 

BPLOT3 

The secondary pattern is stored in TAPE13. 

\ 

\ 

The input file TAPEZ consists of 14 lines. These inputs are described 

below. 

1. H,D,f,zo -- offset parabolic reflector parameters as depicted in 

Figure 13 

2. 1 -- cylinder boundary or Z--cone boundary 

3. xc' Y c' K1, K2 -- cylinder boundary parameters or P, 9

1, 9z, 93 --

cone boundary parameters as shown in FigureS 

4. Y1'YZ'Y3 -- Eulerian angles (see Figure 7) 

98 
5. x1 'Y1,zl -- source point coordinates in main reflector coordinate 

system. 

6. a,b,~ -- polarization of feed 

7. qE,qH -- feed taper param~ter for E- and H-plane primary patterns 

'8. L -- location of aperture plane beyond the focal plane as shown 

in Figure 14a (taken to be zero) 

9. type in 1 or 2 

10. 

11. 

X1'Xz,N} 

Y1'Y2,N 

x1 ,X2 'Y1'Y2 are bounds of Ea (see Figure 12a), N is the 

number of points in sampling. 

12. 0 -- use UAT or 1 use UTD, 0-- compute GO and edge-diffracted 

field or 1 compute GO field only, 

13. 6

1, 6

2,NFFP For a theta cut, 6

1 is the first value and 6

2 is the 

last value of theta. NFFP is the number of 6 values for the 

secondary pattern 

14. 1 -- use E-fields only for the aperture field calculation or 2 --

use H~fields only, PHI -- this is the constant ~ value when 

computing the secondary pattern. 

For an arbitrarily shaped reflector, the user should use the binary version of MAIN instead of BALI. However,lines 97 to 107 must be replaced with 

either the coordinates of the points that define the reflector or the new 

equations that describes the reflector. For an arbitrary reflector, line 1 of 

input file TAPE2 has no meaning. Thus, delete line 41 of MAIN and the input 

file is lines 2 through 14 of TAPE2 described above. 

99 
APPENDIX B 

VIEWGRAPHS FOR A PAPER ENTITLED "CALCULATION OF NEAR-FIELD OF A REFLECTOR BY 

GTD" BY P. T. LAM, S. W" LEE, AND R. ACOSTA PRESENTED IN INTERNATIONAL 

IEEE/AP-S SYMPOSIUM NATIONAL RADIO SCIENCE MEETING, JUNE 1984 I G TD AnoJ~sis of Ret lector Antenno.s I 

f. T. Lllm and S. W. Le.(: 

E I ec-rromo.jneObcs LAb 

U nive.y-si~ of 11 \inois 

OvYld 

R! Ac.os-t~ 

No.so.- Lew,s 'Resettrch Center 

Outline 

l. -Problem description 

2. Moti v 0.. tl'on 

3. Approo..ch 

4. Res(tl ts 

5, Co t\cJ LLS \'0 n. 

100 
I frob\emDescriphon I 

~~ sec.ondo..r~ 

pattern 

~s 7 E . = . 

G-iven : 

. De.s UI ptl"O n 0+ re. t I e..c-torFee d r oco.:tion o.l'1d orien to..tion 

Feed po..ttern a.nd polan"za:b"on 

De t e.,r YY\l'n e. : 

Secondo..G pattern 

. 

101 
I Motiv~:\;ioY\ l * gene-raj ret lec.tor .sha.pe.. 

plAra. bol i c.. -4e YMIJ.\ di<;torhoVls shaped 

*" m!Alhple.. re.tlec:tor 5jstems 

~ 

* e+tI'cleVlt se(oY\do.~ paHerl'l COVYIpufation 

1 po. iient L 2. C-lA.ts ) ~ $10 

DB') M-lbIM deSign rql.lires '" 5000 -rtbs . ~ 

$50,000 

102 
THREE METHODS OF CALCLAL~TING 

SECONDARYPATTERNEs . 

(a) Physical Optics Method 

(b) 'GTD 

(c) Aperture Integration Method 

103 
I Compo-rison of -the -three. methodsi 

As 0. re t e r~ nee.) let s 1,.\.5 rep' re.s e.yr!; -th~ exact sollA:t, 0\'1 ot E. b~ -the. foHowins 

kigh -fre.OfeV"\c~ a.s"~rYlptot i c. ser' es : 

E s(-r) "" ~jkS'(1lAo(1)+- k-~ A, (-r)+ k'Air)t,,·l 

-tor k~ co 

Methods Accu.r(Ac~ Lim\tatlon 

'PO ...,.. 

recover Ao o.nd Y"\ot acc.u.rate forAI po..rt-io..l AI . wlde.-aY)s'e. lobes 

&TD ~ . 

recover Ao OJ\~ predicts infinite.-

AI t:-eld in ma;", beam 

di v-ec ·bot'\ (cau-sties) 

104 
I Advantages ot a.pertu.re iYlte3 ra.-hon (A I) I 

t. Mo.~ be. c.onveni entl ~ extended +0 

'YI1(,tltiple re~ector s~s-/;ems 

2. 'Provides 0.. t~eore1xcal check -For neax· +i e ld mea) Vlve W\e~ft~ . 

3. Accuro..c~ i~ CDVllplXY'a.b\e -/:-0 Ph~~lco.l 

Op-tICS Cro). -rhe use of if.e-l=i=T 

makes H: n~merlcoJl~ etticl'~t. 

105 
I APPROACJ-II 

PI (XI' YI' ZI) 

Source 

FFT., ? 

Aperture 

Secondary 

Pattern 

P Grid La 

I 

106 
I Who.:\: 's new in -th is Al o..pproo.c.h? I 

I. Edge - d iffra.cted -Held I'S included in 

-the. operitAre. ,~'e(d co.lcu..llttl'on. 

2" SlArf~ce.. 0+ reflector lOS COMpletel j 

ar b Itro.r'(f-"· ..... 

3" . tog e of retlectot"" IS Ylot- rerh',"c.ted 

+0 cA Ct'rc.u(~Y' ClAy-ve, r t YV\o.~ be. an 

arbitr-o.rj Cf)XVe. I j'"YlS OYI o.h eUiptiud 

. COV\€. or- Cj\\(\der.· 

107 
'Re t {eGtor' SurtlAce : 

• A~o.I1·h·c 

• N rtMtn-coJ 

'Retlec to I" bOIAY1 darj; 

· f II i P ·b· od C j l i Y1 d €-v-

• E\l'ftrc.~1 COVle. 

x=a 

108 
I 

Y 

IfwD -types of retfedor boW\dllr~ I 

" \ 

, 

J 

I 

/ 

Xl 

X 

109 

Y 

Yc 

z 
I Ret I edor and -Fe.e.d coordinate sgstemsl 

y 

x 

z 

'!fer/ector and .feed c.tJord,nat-e .f!fstems 

0; b 1'"1 nus x DY7 x II ahout z 

~ bY'I'nffs Z on Z.; a/',ut X II 

0; b y,'n15 X I( on Xf ~&ut Z.; 

LINE OF NODES 

(INTERSECTION OF xy 

AND xf Yf PLANES) 

110 
IlAvtiform {heovlesJ 

fi 

f ::. {4- I ;.f. 171.. is" ~ -tt.e. S h(i\ dow )IJ e. of 4te. re.({g~fed ,('e/J 

- I i+ 1'2. IS i~ --flu- I'lt >,~d e. "f & veJ~{,fet{ ~-eld 

l~ I > 2. J no correc;/;.on re%~I'red 

\~ \ ~ 2. J H d(~) must be modi-h'ed 

UAT UTD 

osu 

111 

) 
-S e (ond ~Y"j p aHern : 

~ - j kr [ ,.] E = e. r e Ug + ¢ U", 

Iwo UrlitGl.Y'"j po rO.r"iz.o..hoY1 '1ectol'"s: 

J.. ""r . " ''I' R = e (o.eJ CD~ ~ + b~iV\¢) + ¢ (-o.eJ 5C'Ylf+hc.os4» 

C : e (aejo/ 51'rtcp - beos cp) + fp (o..e)t CtJsf+ b~;Vl cp) 

Re+e(eVlce. -jA'Jrtl.r"izat,oVl ot E = E ·(R"')* 

Cross-pOIMiw.t,oVl of E == f· ([I'-t 

r,'nc 

Cro$S - polariz.o..tl'OVi direc.-t-il/ittt: 

Dc (elt?) = 4--n-if· C. r

2

/zo 

~nc. 

112 
\lY\cidtflt power (ltd io.,te.d I 

z 11" . 

7 =: f1l" (.-">. -> * 

inc.. }. E · E \,"'2. sin e ded rp ~ 20 . . 'r~O 9::0 ... ... 

Ee ... = e j:,. [e U~)a.iCos 4>-1- b~ih 4> ) + ¢ Uu,J().e! ~i ~~ + bCos 4> )] 

A ss ~oY\e . a. 

( COS e ) O,fm 

U£M· ~ 

UH~:::' (co'S&)~HM 

Por .ttrbitnlxt po I an"zatlin'l, 

1? ~. be. eXf(e~<;ed ivt fe(WtS of. I~L . a .. 

113 

Se55.e I . GCt »1 W'tl fun( (,dnS . 

. LPmme{ 
... ~ 

....... -.P- .-----.. --.. - ~ 

<V .- / 

./ ~ 

! ~ 

'-J 

:c 

! ~ 

; 

.~ 

, 

-e /\ 

() J \ 

~ 

'-\- ,\ ~ , , 

I 

/ 

.0 

_._-. _ .. _--_. __ . ..-----'-/ 

-:::. / 

Q 

-0 I 

I 

~ 

I 

~ I ~ 

I 

I 

~ 

I ~ 

I 

In 

q) 

I 

\J'"" 

~ I 

\4-

I 

a I 

f 

I 

I I 

I 

I 

I 

I 

I 

-

- l 

I~ r< 

00 

0 + 

.-< 

. - T" 

. 

114 
50.~----~~------------~------~ 

- RAHMAT -SAMII 

-.- GO+GTO 

---GO 

40. 

-

':r~J< aJ 

'"C 30. -

>-

I-

> 17~ 

I- ~ 

U 

w 20. 0:::: 

0 

-1 

0 

11. 

I 

IJ.. 

w 10. 

0:::: , 

I 

0; I 

, 

f 

-10. 

0.0 LO 2~0 3.0 4.0 

(J (degrees) 

115 
z 

-Xf 

50 

- 40 -HUNG 

en 

"C x x x GO+GTD -

>- 30 oooGO 

I-

> 

I- 20 u 

w 

a:: 

0 10 

-1 

0 0 a.. 

I 

lJ... 0 

w -10 a:: 

0 

-20 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 

8 (degrees) 

116 
Ie ONCLUSI ON I 

* With {his AI o..pproO-ch) -the.. mo.in bea.m tlnd 

sld°e.lobe. reqioY\$ of the . sec.on.d o.r.'j po..tte.rV\ 

is ! tA(c.u.n"-tel~C,oyV\p{).ted. . ! 

I * ,He. reflector sha.pe is c.oW)pletelj a.rbitro.r~. 

* /his tec,hnl'tte is nu.rner'ic.(1..llj et-h-Cle.rtt; 

6~ eX'()plo~ inj the FFT.. 

* Nelkr field melkSvtre.menU ma.!j be. c,hec,ke( 

* This Al formlNlttflon ,Os ~uit-ab/e for 

ex ten S Ion -to mul tipf e. ref! e do rs. 

'17 . 
\ Future..' wor k I 

* ex-tend techntqu.e -to CL 

dual ref/utor tr;-{:ern 

* arrfLj feed 

* mult'-ple re.flee-tors 

118 
1. Report No. 2. Government Accession No. 

NASA TH-87162 

4. Title and Subtitle 

Secondary Pattern Computation of an Arbitrarily 

Shaped Main Reflector 

7. Author(s) 

Peter T.C. Lam, S.W. Lee, and R. Acosta 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135 

12. Sponsoring Agency Name and Address 

National Aeronautics and· Space Administration 

Washington, D.C. 20546 

15. Supplementary Notes 

3. Recipient's Catalog No. 

5. Report Date 

November 1985 

6. Performing Organization Code 

506-62-52 

8. Performing Organization Report No. 

E-2196 

10. Work Unit No. 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

Final report. Peter T.C. Lam and S.W. Lee, University of Illinois, Urbana, 

Illinois; and R. Acosta, NASA Lewis Research Center. 

16. Abstract 

This report studies the secondary pattern of a perfectly conducting offset main 

reflector being illuminated by a point feed at an arbitrary location. The method 

of analysis is based upon the application of the Fast Fourier Transform (FFT) to 

the aperture fields obtained using geometrical optics (GO) and geometrical thoery 

of diffraction (GTD). Key features of the present work are (1) the reflector 

surface is completely arbitrary, (11) the incident field from the feed is most 

general with arbitrary' polarization and 10cat1oni and (111) the edge diffraction 

is calculated by either UAT (Lee and Deschamps) or by UTD (Kouyoumj1an and 

Pathak). Comparison of this technique for an offset parabolic reflector with 

the Jacobi-Bessel and Fourier-Bessel techniques shows good agreement. Near 

field, far field, and scan data of a large reflector are presented. 

17. Key Words (Suggested by Author(s)) 18. Distribution Statement 

Antenna; E1ectromagnet1cs; Physics Unclassified - unlimited 

STAR Category 32 

19. Security Classif. (of this report) 

Unc1ass1f1ed 

20. Security Classif. (of this page) 

. Unclass1f1ed 

21. No. of pages 

"For sale by the National Technical Information Service, Springfield, Virginia 22161 

22. Price" 
End of Document