OUT-OF-BAND RESPONSE OF

REFLECTOR ANTENNAS

David A. Hill

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

April 1985

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1 CAT SID: 09241 AF

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AUTH pn "HILL, DAVID A - " Record 1 of 1 LC/NACO - NAME

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1 010 n 82208918 1

2 040 DGPO *C DLC *d DLC 1

3 005 19870318174613.4 1

4 100 1 Hill, D. A. *q (David A.) [AACR2] 1

5 500 1 Hill, David A. f

6 670 Gates, H.M. Technical sharing issues between broadcast-satellite

and fixed services, 1982 (a.e.) =t=b t.p. ( D . A

Hill) bibl. data sh. (David A.

Hill, Inst, for Telecommunication Sciences, Nat. Telecommunications and

Information Admin., U.S. Dept, of Commerce, Boulder, Colo.) 1

NBSIR 85-3021

OUT-OF-BAND RESPONSE OF

REFLECTOR ANTENNAS

David A. Hill

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

April 1985

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

1. Introduction 1 2. Fields in the Focal Region of a Paraboloid 2 2.1 Physical Optics Integration 3

2.2 Large f/D Approximation 7 2.3 Poynting Vector 8

3. Received Power 9

3.1 Feed Horn Response 9

3.2 Circular Aperture Integration 11

3.3 Effective Aperture 13

4. Extensions to the Model 16

4.1 Distant Si delobes 16

4.2 Transient Fields 18

4.3 Reflector Roughness 21

4.4 Offset Parabolas and Dual Reflectors 22

4.5 Coax-to-Waveguide Adapter 24

5. Conclusions 29

6. References 31

Appendix A Evaluation of the <\> l Integration 34

Appendix B Open-ended Waveguide 3 7

Out-of-Band Response of Reflector Antennas

David A, Hill

Electromagnetic Fields Division

National Bureau of Standards

Boulder, CO 80303

The response of reflector antennas to out-of-band frequencies has

been analyzed using physical optics. A simple approximate ex- pression has been obtained for the effective aperture, and this

expression yields both the receiving pattern and the frequency

dependence of the on-axis gain. The theory has been compared with

published out-of-band measurements, and the pattern agreement is

good, but the measured gain falls below the theory. This discrepancy is caused by mismatch loss in the coax-to-waveguide adapter,

and this mismatch loss has been analyzed theoretical ly. The basic

physical optics model for symmetrical reflectors has been extended

to include offset and dual reflectors, reflector surface roughness, and transient excitation.

Key words: coax-to-waveguide adapter; effective aperture; focal

region; paraboloid; physical optics; Poynting vector; out-of-band

response; receiving pattern; reflector antenna.

1. Introduction

The response of antennas to out-of-band frequencies [1-3] plays an impor

tant role in interference and jamming problems. Reflector antennas are of

particular interest because they are used so frequently and because they have

a strong response to above-band frequencies. The analysis of reflector anten

nas at above-band frequencies is complicated by the presence of higher-order

modes which can propagate in typical waveguide feeds. Frequencies well below

the in-band frequency are not important because they are cut off by typical

waveguide feeds. Consequently, "out-of-band" will refer only to above-band

frequencies throughout this report.

An earlier out-of-band analysis of reflector antennas [4] yielded the

radiation pattern for each propagating mode and performed a statistical analy

sis based on the statistics of the higher-order mode coefficients. The probl

em is that the statistics of the higher-order modes are highly dependent on the specific feed system and are not generally known. Such problems can be

avoided by analyzing reflector antennas in the receiving mode and computing

only the total received power carried by all the propagating modes. This

1
total power is easily related to a generalized effective aperture which appears to be the most convenient receiving characteristic for out-of-band frequencies. Both the receiving pattern and the frequency response are given by

the effective aperture, and the results depend only on the antenna parameters. The power which is coupled from the waveguide to the detector does of course depend on the details of the feed system, but that portion of the problem can be analyzed separately. In any case, the total power in the waveguide

is a useful upper bound for the detected power.

The organization of this report is as follows'. Section 2 contains an analysis of the fields and the Poynting vector in the focal region of a symmetrical paraboloid for plane-wave incidence. In Section 3, the Poynting

vector is integrated over the aperture of the feed horn to yield the total

received power. The integration can be done numerically or analytically, and

a simple analytical approximation is adequate for most cases. The theory is

compared with some earlier out-of-band measurements for frequencies from 3 GHz

(in band) to 10 GHz [5], and the agreement is fairly good. Section 4 contains

a number of extensions to the theoretical model, such as offset and dual re- flector geometries, the effect of surface roughness, and the response of a typical coax-to-waveguide adapter. Transient excitation is also considered.

Section 5 summarizes the results of this study and makes recommendations for

future work

2. Fields in the Focal Region of a Paraboloid

In this section we derive expressions for the electric and magnetic

fields and the Poynting vector in the focal region of a symmetrical parabolic

reflector. Much of the previous work on the focal region fields of paraboloids [6-10] was directed toward design of feed systems, and only on-axis

incidence was considered. More recently, Valentino and Toulios [11] computed

the electric field in the focal region of an offset paraboloid for off-axis

incidence. For simplicity we consider only the symmetrical paraboloid, but we extend the previous derivation [11] to include the magnetic field and the

Poynting vector for off-axis incidence.

2
2.1 Physical Optics Integration

The perfectly conducting, parabolic reflector is shown in figure 1. The

reflector has a diameter D and a focal length f. The origin of a rectangular

coordinate system (x,y,z) is located at the focus of the paraboloid. Our

derivation and notation follow Valentino and Toulios [11] fairly closely, but we introduce several simplifications. Because we consider the symmetrical

paraboloid, the offset angle 0 Q

in [11] is zero. For the symmetrical paraboloid, without loss of generality we assume the incident plane wave is incident in the xz plane (

4> s = 0 in [11]). Finally, we assume that the field

point is located in the focal plane (z 2 = 0 or 0 2 = tt/2 in [11]).

The physical optics surface current J on the reflector is given by

J = 2 n x H, , ( 1 )

where n is the unit normal to the reflector and H_. is the incident magnetic

field. For plane wave incidence, FL is given by

H.

-i -H

H. e

l

( 2 )

where k is the free space wavenumber ( =2n/X) , X is the wavelength, and the

exp (jwt) time dependence is suppressed. In all cases, y with a subscript

denotes a unit vector. We chose the quantity to be zero at the center of

the reflector, and it is given by

8 = u • r

s -p -s

(3)

where r c is directed from the center of the reflector.

An infinitesimal surface current patch of area dS at a point P on the

reflector produces electric and magnetic fields [8,11] at the point ?

2

:

-jkR

- Ur (J • Ur)] •

ik e"

jkR

‘ 17 u * Hr) —K— dS -

(4)

3
where n is the intrinsic impedance of free space. As indicated in figure 1, R

is the distance between P

1 and P

2

, and u R is directed from P to P

2

. Also it

is assumed that kR is much greater than unity.

In order to evaluate eq (4), we restrict our analysis to the region near

the focus (r 2 «r 1 ). Thus we can assume that u R

- -u^ and R - r

1 except in

the phase term. The required expressions for the unit normal n and the sur- face differential dS have been given by Bern [12]:

and

3 -

r

i n = x 30

3 r 3 r 3 r -1

, -1 -1

/ |— x

3 <J> 30 3 (j)

• (5)

8 El3

-r

l dS = x d8 i d*i • 3 (j)

Substituting eq (5) into eq (4) and carrying out some of the differentiations,

we obtain

dE = -j —- e

J

' k ^s

+R

^(r 1 sin0 1

d<j>

1 d0 x

)

. r L x

1

{[(u • u M ) +

1 3r

(u • u..)] u

-<)>]_ -H r

1 sin0 1 3^ -r 1 -H -0

L

( 6 )

1

" [( V U-H>

+ 7^<V U-H

)] V

and

dH = —— e Jk ^s + R

^(ij sin0 1

dcj> 1 d0 x

) r

L Xn

‘““e. Vtt

3r

^

(u-r/ “h

)] -e/ (d

,

* U H

) } » l “H -<p

^

where Ej = nH-j. The expression for dE agrees with previous results [11,12].

Note that dE and dH are orthogonal and both dE and dH are transverse to u^

This is consistent with the large kR assumption. The expression for R can be

approximated as follows

4
( 7 )

R = r

: [1 - 2 — sine 1 sine 2 cos - 4> 2 )

r 2 '2 1/2

-2 — cos0 1 cos0 2 + 2 ]

1

= r

l - r

2 [si nL sin0 2 cos(^ 1 -cj) 2

) + cos0 1 cos© 2 ] .

In eq (7) we have neglected terms in r 2

2 /r i, and the validity of this approximation has been discussed previously [11]. The phase term ft $

in eq (3) can

be written

ft = (f - r , ) cos0 + r, sin©, sin0 co s<t>

s

1 s

1 1 s

1

( 8 )

Also, r

x is given by

2 f

Pl

1 + COS0 1 (9)

The polarization of the incident magnetic field y H can be written

u,, = a u + a u + au

-H x -x y-y z-z

where a

2 + a

2 + a

2 = 1. x y z

( 10 )

Without loss of generality, we confine the analysis to two orthogonal polarizations. For the incident magnetic field polarized in the xz plane, we have

a = cos0 , a = 0, a = -sin0 . (11)

x s y z s

For the incident magnetic field polarized perpendicular to the xz plane, we have

a x = 0, ay

= 1, a

z = 0 . (12)

In order to evaluate the integration on ^ from 0 to 2n in eq (6), we take advantage of the fact that the integrands are periodic in 4> 1

. The evaluation of the 4

> L integrations is given in Appendix A, and the resultant ex- pressions for E and H are

5
0 -j k M sine,

E = -2 jkE.f / e — — de, . i l + cose, 1 0

1

2 n

[ Z j J (k M ) (C cos n 4

> + S sin nip) u

n = o n

1 xnvn xnvn -x

2 n + Z j J (kM ) (C cos ncp + S sin nip) u

n = o n

1 yn yn -y

2 n + Z j J (kM ) (C cos nip + S sin n<p) u ] n = o n

1 znTM zn -z

and (13)

H =

e jk E f m sine,

i , -jk M

1

J e o de, .

T) o 1 + cose

1

1

2 n

[ Z j J (kM ) (D cos nip + F sin nip) u

n = o n

1 xnvn xnvn -x

2 n + Z j J (kM ) (D cos nc[i + F sin n<^> ) u n = o n

l yn yn -y

2 n + Z j J (kM ) (D cos n 4

> + F sin n<p) u ] n = o n

1 7n 7n - 7 zn zn -z

The upper limit e m of the 0! integration is given by [11]

_i

0m

= tan (D/4 f) . m

(14)

All other quantities in eq (13) are defined in Appendix A.

In general, the 0! integration must be performed numerically, and a com- puter code has been written to evaluate eq (13). Note that the summations

truncate at n = 2 which is in contrast to the case of the offset parabola

[11,12] where the summations run to n = °° .

6
2.2 Large f/D Approximation

When f/D is large, then <3 m

as given by eq (14) is small, and small argument approximations can be used for the tr igonometric functions of ©i . In

this case, eq (13) simplifies to

- F o

^ a y -x

a x -y^ *0m *

H

“ ~~ (a

x ux - ay

uy

)

I 6m ,

(15)

where E = -2 jkf E.e’

J

' kf

sin 2 (0 /2) 5

I 0m

g cosec

2 (0 /2) m

j j ( k

x P ) s i n

x d0 x , 2 o o

and P = / (x 2

- f sin0^) ? + y 2

2

.

By setting sin0, - 0, and cosec 2 (0 12) - (2/0 ) 2 m m ate I Qm analytical ly:

2 (k© P)

I 0m k © P m

in eq (15), we can evalu-

(16)

Note that E and H are orthogonal, and both are transverse to y . The results

in eqs (15) and (16) are consistent with earlier approximations [8,11].

A fairly simple physical picture can be obtained from eqs (15) and

(16). The maximum of I Q occurs for P = 0, and this occurs for the following

focal plane coordinates:

x 2

= f sin0^ and y 2 = 0. (17)

The point determined by eq (17) is essentially the geometrical optics point

for a ray incident on the center of the reflector. For 0

s equal to zero the

maximum fields occur at the focus, but for 0 not equal to zero the maximum is

s

7
shifted as indicated by eq (17). Away from the maximum, the decay is more

rapid for higher frequencies (larger k) and also for larger reflectors

(larger 0 ) . Also the peak electric field E Qis larger for higher frequencies

and for larger reflectors.

2.3 Poynting Vector

The Poynting vector is of particular interest because in Section 3 we

will integrate it over the aperture of the feed horn in order to determine the

total received power. The real Poynting vector S is given by

S

= i Re (E X H*) , (18)

where Re denotes the real part and * denotes complex conjugate. A computer

code has been written to evaluate eq (18) using the integral expressions for E and H in eq (13)

For large f/D, we can substitute eqs (15) and (16) into (18) to obtain

the following expression

2 n y

+ a. 2

)

2J, (ke P)

1 m

k0 P m

u-z (19)

For the incident magnetic field polarized in the y direction, a = 1 and a

x

= 0. For the magnetic field polarized in the xz plane, a = cosQ and a =0.

x s y

Since we are interested in small scan angles, we can replace cos by unity

and rewrite eq (19) as

S = u S

- -z o

2J, (k0 P)

1 m ¥1 P m

where S

= - —

o 2

E

o'

2

( 20 )

Thus the approximate 2 is independent of the polarization of the incident

field and contains only a z component S

z

.

8
Most reflector antennas do not have large f/D ratios, but the approximation in eq (20) turns out to be surprisingly good even for relatively small

values of f/D. Figure 2 shows S

2 for three different values of f/D for

the on-axis case 0

$ = 0. The curves for f/D = 1.0 and 0.4 were computed from

the general integral expressions in eq (13), and the curve for f/D = 00 was computed from eq (20). We see that the dependence on f/D is quite weak, and

the large f/D approximation in eq (20) is adequate for most cases. For offaxis incidence (8 * 0), eq (20) is also a good approximation, and figure 3

shows some results for f/D = 1. The large f/D curves were computed from eq

(20), and the f/D = 1 curves were computed from (13) for f = 1 OX and for

polarization. The dependence on f/X and the incident polarization is fairly

weak. Note that the agreement with the large f/D approximation is better for

the smaller scan angle 0

$ = 10°, but is not too bad for 0 $ = 20°. We are

not interested in very large 0

$

because the entire physical optics method be- comes questionable for 0 $

too large.

3. Received Power

3.1 Feed Horn Response

Initially we consider a feed horn with an aperture of arbitrary shape.

Normally feed horn aperture dimensions are on the order of a wavelength at the

in-band frequency. Consequently, the aperture dimensions can be assumed to be

electrically large at the higher out-of-band frequencies. Thus we make the

simplifying Kirchhoff approximation that the fields in the aperture are equal

to the incident fields.

If the incident fields are nonuniform, as in the focal region of a paraboloid, then the power passing through the feed horn aperture P r is given by

the integral of the Poynting vector over the aperture.

P = - / S dA . (21)

r

A

2

S

z is the z component of the incident Poynting vector, and the geometry is

shown in figure 4. Since we assume that no power is dissipated in the walls

of the horn and waveguide, the power propagating down the waveguide is also

9
given by P r in eq (21). Because the waveguide will normally be multimoded at

out-of-band frequencies, P r

is the total received power in all the propagating

modes. The simple theory in eq (21) does not give the individual waveguide

mode amplitudes, b

p

, but it gives the sum of the square of the modal amplitudes, £ | b

^

|

2

, which is equal to the total power if the modes are properly

normal ized.

A more detailed analysis of the feed horn and the junction between the

feed horn and the waveguide would be required in order to determine the individual waveguide mode coefficients. We have stayed away from this additional

complexity because a knowledge of the individual mode coefficients is probably

not very useful. Any waveguide bends, transitions, or irregularities would

produce mode conversion and a change in the mode coefficients. Even in the

uniform waveguide, the field distribution would change along the waveguide

because the individual modes have different phase velocities. In contrast,

the total power remains constant along the waveguide.

For the simple case where the incident field is a uniform plane wave,

then S

z is simply given by

S

z -S coso , o

( 22 )

where S Q

is the incident power density, and 0 is the incidence angle shown in

figure 4. In this case the integral in eq (21) is easily evaluated to yield

P r

= P q cose , (23)

where P^ = AS„ . o o

Thus we have the simple result that the receiving pattern of an electrically

large receiving horn is simply cose and is independent of polarization and

the detailed shape of the aperture. This result only holds when we consider

the total multimode power and even then is a high frequency approximation

which neglects edge diffraction. If we consider the receiving pattern for an individual waveguide mode, then the pattern has a lobe structure as expected

for an electrically large antenna. The specific case of an open-ended, parallel plate waveguide is analyzed in Appendix B in order to illustrate the difference between the mode patterns and the total power pattern.

10
3.2 Circular Aperture Integration

For a feed horn aperture of arbitrary shape, the integral for the total

received power in eq (21) must be evaluated numerically. This numerical evaluation can be rather time consuming, but for the special case of a circular

aperture we can obtain an analytical approximation to eq (21).

For a circular aperture of radius p m , the integral in eq (21) can be

written

2 tt p m

P r

= _

/ j S^p 2

d p 2 d<t>2> (24)

o o

where p 2

= /x 2 + yf, <j> 2 = tan ^(y

2 /x 2 ),

and x 2 and y 2 are the focal plane coordinates as shown in figure 1. For the

general case where S

z must be determined by numerical integration, the p 2 and <t> 2 integrations must also be done numerically, and a computer program has

been written for this case. For large f/D, we can substitute eq (20) into eq

(24) and obtain the following:

2w p m 2 J i

( k O P ) 2

P

r = - S J j [ ke p m

] P 2 dp 2 d* 2 . (25) oo m

First we evaluate eq (25) for axial incidence (0 $ =0). In this case, P

= 92 and the <t>2 integration simply yields a factor of 2 tt:

m

P = - 2tt S r o

2 Ji(ke

m

p 2 ) 2

[ —\7r~z:— J p 2 dp 2 . kem

P2

(26)

The 92 integration can be done to yield the following result:

where P o

P = P ri - J (ke p ) r o

L ov mm'

* (D/2) 2.

(27)

P Q is the total power incident on the reflector, and the factor in brackets

varies from zero to unity as the quantity ke p increases from zero to infinity. Both the power density S

z and the received power P r are shown in figure

5. The ratio P r /P

Q is called aperture efficiency, and typical values for inband reception are on the order of 50 to 90 percent.

11
The result in eq (27) has been given previously by Minnett and Thomas

[8], but they did not treat the case of off-axis incidence (e

s *0). For the

case of 0 s * 0, we lose the 4> symmetry and must resort to an approximate integration method. Consider first the case where f | si n

$ | is less than pm

. Then

the maximum intensity (or geometrical optics point) is located at X

2 = f sine

s and y 2 = 0 inside the circle p 2 = Pm

as shown in figure 6. In order to do the

integration analytically, we change the integration area from a circle of

radius pm centered at the origin to two semicircles of radii p + and p_ cen- tered at the geometrical optics point as shown in figure 6. In this case, the

integration can be evaluated as follows:

p + 2 Ji(k0 p) 2 - ' S o

f / l ~Te/- ) p dp

P - 2 Ji(ke p)

I 1 o m (28)

= —S- [2 - J n (k0m pj - J 1 (k0mP .) - J_(k0 P ) - J i (k0 P )], 2 o m + 1 m +' o m - 1 m -

where p = pm

± f sin0

$

. Note that for 0

S = 0, eq (28) reduces to eq (27).

For the case where p | s i n

S | is greater than pm

, we use a similar strategy

as indicated in figure 7. In this case, the circular area is replaced by an annular sector, and the received power integral can be written

p+ 2 J i

( k e p)

2 P n “ - 2cf> S A / m 1 r

Y c o

J L p-

<f>

“ko p m

p dp

(29)

= P — [J (k0 p ) + J i ( k 0 p ) - J (k0 P J - J i o ir o' m - 1 m - o' m

+' ( k 0m

PH J], +y ’

-1 pm where P+ = f sine

$ + p , p.

- f sine

$ - pm

and *

c

= sin ( rTfwr ).

s

It is easy to see that when f sin0s is equal to pm

, then p_ = 0, 4> c

=

tt/2, and eqs (28) and (29) yield the same result for P

p

. Also, it can be seen from figure 7 that eq (29) provides an upper bound for P

r because the sector

includes all of the circular area and the integrand is always positive.

12
3.3 Effective Aperture

The effective aperture A

0 is defined as the received power divided by the

intensity of the incident plane wave [13]

A

e

= P

r /(|Ei |2/2n) = P

p /(-S

0

). (30)

In our case we use the same definition but recognize that P

r is the received

power contained in all the propagating modes. In general, A

0 must be evaluated numerically because P

r must be evaluated numerically as indicated by eq

(21). However, for the circular aperture we can use the approximate expressions for P

p given by eqs (28) and (29). In doing so, it is convenient to

normalize A

e to the physical aperture A

p = tt ( D/ 2

)

2

. Then we can write the

normalized effective aperture as

1 2 2 2 2 4 [2 - J (ke P J - Ji(ke P J - J (ke P ) - J,(ke p )], f sine„ < P 2 o' m

+' iV m+' o' m - iV m - ’ s n

= 4 (31)

<t> r 2 2 2 2 — [J (ke p ) + J i ( k e p ) - J (ke p ) - J L

( k e p,)], f sine„ > p

tt o' m - iV m - o' m

+' m +' s n

where p , ±i= If sine ± p and <t> is defined in eq (29).

s mi Yc ^

It is possible to study the wide angle behavior (f sine

s >> e ) by using

the large argument approximations for the Bessel functions [14] in eq (31).

J (x) « — cos 2 x and Ji(x) ~ — sin 2 x. (32)

From eqs (31) and (32), we obtain the following asymptotic expansion for A

e /A

p A 4 p

2 _e m

A ir 2 ke (f sine ) 3

’ pm s The (sine

$

)“3 dependence in eq (33) is the same as the wide angle dependence

for the radiation pattern of a circular aperture of constant illumination

[13]. The p m

^ dependence is to be expected because it is proportional to the area of the receiving horn.

13
The angular dependence of A_/An for various values of ke p is shown in

figure 8. The limiting case of k0m

Pm

=

00 corresponds to geometrical optics

where the power is focused to a single spot. When (f/pm

) sin0

s is less than

unity, the spot is inside the receiving horn, and A

g

/A

p is unity. When (f/p

m

) sin0

s is greater than unity, the spot is outside the receiving horn, and A

e /A

p is zero. For k0 p = 10, the focused spot is smeared out, and the pattern is mm

widened. For k0 p =3, the pattern is widened further. For (f/p m ) sin0_

m m s much greater than unity, the patterns in figure 8 begin to approach the asymptotic expansion given by eq (33).

An approximate expression for the beamwidth 0 b can be obtained by setting

0^/2 equal to the incidence angle where the first zero in the Bessel function in eq (20) is located at the edge of the receiving horn. Then the

Poynting vector in the receiving aperture will be small, and the integrated

power and effective aperture will be small. This condition can be written

f sin(0

b /2) = P m

+ pi, (34)

where J L

( kp 1 0 ) = 0 and kp^ = a L « 3.832 represents the first zero of Ji

[14].

The above equation is easily solved for 0^:

0, = 2 sin

b

1 P m

+ a i/k 0 -_1 m

(

-m- -m

-

)

p + a 1 /k0 m m

(35)

When k is infinite, we recover the geometrical optics result, sin0^ = 2 P m

/f,

which is shown in figure 8. As k becomes smaller, 0 b

becomes larger as shown

in figure 8. For sufficiently small k, we have

sin0, «

2 <*!

I<Tf

m

4 a]

To

2.44X

D

' (36)

The above result agrees with the beamwidth between the first nulls for a circular aperture with constant illumination [13].

The patterns in figure 8 were determined from the analytical approximation in eq (31). The accuracy of eq (31) has been compared with numerical

14
integration for numerous cases. A typical comparison for f/D = 1 is shown in

figure 9. The numerical integration curve was done for the incident magnetic

field polarized in the y direction, while the analytical approximation is independent of the incident polarization. For the numerical integration

results, there is a slight dependence on polarization. Note that the general

agreement is fairly good, but that the analytical result is somewhat larger

for (f/p ) sin0

s greater than unity. This is primarily a result of the upper

bound feature of the sector area approximation in figure 7. The numerical

integration is time consuming because it involves triple numerical integration, over Gj in eq (13) and over p£ and <j>2 in eq (24). Consequently, it is

simpler and more efficient to use the analytical expression in eq (31), and

this expression is adequate for most cases.

In figures 10-13, we compare the analytical approximation with previously

reported experimental results [5] for frequencies from 3 GHz (the in-band frequency) to 10 GHz. The antenna tested had a diameter D of 1.22 m (4 ft) and

an f/D ratio of 0.32. The small f/D ratio provided a good test for the analytical approximation which is best for large f/D. The feed horn was rectangular, 7.21 cm x 5.00 cm (2.84 in x 1.97 in) and for the theoretical comparison was modeled by a circular aperture with radius p m

= 3.39 cm to yield the

same area. The pattern data in [5] was taken over a range of -40 to +40° in

the H plane and over a smaller range (either -5 to +5° or -10 to +10°) in the

E plane. Consequently, we show only the H plane comparisons in figures 10-

13. The pattern comparisons are for relative power or effective aperture.

Strictly speaking, the theory should not be used at the in-band frequency, 3 GHz, in figure 10 because the feed horn and waveguide are not electrically

large.

A difficulty in comparing the out-of-band measurements with the theory in

figures 11-13 is that the coax-to-wavegui de adapter has an unknown response

for multimode, out-of-band excitation. Our analysis in Section 4.5 indicates

a very complicated response as a function of frequency and modal content.

Because our theory yields the total power in the waveguide and the adapter has

a small response for some modes and some frequencies, we can expect the theory

to provide an upper bound or envelope for the detected power. This seems to

be the case in figures 11-13.

15
In figure 14 we show the measured gain for the same antenna [5], To convert our effective aperture A

e to gain, we divide by \

2 /4ti, the effective

aperture for an isotropic antenna [13], Also shown is the gain result using

the physical aperture Ap. The effective aperture curve is seen to gradually

approach the physical aperture curve as the frequency is increased. In con- trast, the measured gain becomes highly variable as the frequency is increased, and this is thought to be due to the irregular frequency response of

the adapter. These measured results point out the importance of the adapter

in determining what portion of the total waveguide power is actually coupled

into the system.

4. Extensions to the Model

4.1 Distant Sidelobes

Because the results in the previous section are based on physical optics,

the patterns are not expected to be valid for large values of 0

$

. Some

interesting comparisons of measured patterns and physical optics calculations

for in-band frequencies have been made by Stubenrauch and Yaghjian [15].

Typically, they found that physical optics gave good accuracy for the main

lobe, less accuracy for the first few sidelobes, and poor results for the distant sidelobes.

For out-of-band frequencies we are not able to calculate the details of

the pattern accurately as indicated in figures 11-13, but we find that physical optics gives a reasonable estimate of the envelope pattern over the entire

forward hemisphere (0

$

< n/2)

To illustrate this point, we use the geometrical theory of diffraction (GTD) to provide an estimate of the edge-diffracted

field. In figure 15, a plane wave is incident on the top edge of the reflector an an angle 0

S from the axis, and we are interested in the diffracted

field near the focus. If the magnetic field incident on the edge is , then

the diffracted field Ha at the focus is [16]

-jkr

H = H, ^ £. D , (37)

“ 1 / r e

e-J*/4

6

where D = C e

2 / 2nk

16
and C

1 1 cos 1/2 (9 -0. ) cos V2T0 . + 9. )*

c scat l nc

c scat i nc

In C, the minus sign applies to the case where the incident magnetic field is

polarized parallel to the plane of incidence, and the positive sign applies to

perpendi cul ar polarization. For completeness, we could add the ray diffracted

from the lower edge, but we neglect it because we are just obtaining the order

of magnitude of the diffracted field. Also, we are neglecting the curvature

of the diffracting edge. The angles and distance required in eq (37) are given by

0 scat

and r e

, e

. _JH a 2 2* w

i nc

2f

1 + COS0 * m

(38)

If we assume that p m

is much less than r g

, then the power density incident on the aperture is constant, and the z component S

z is given by

s

z = - \r\ I H

d | 2 cose m

. (39)

The effective aperture A

0

A

e

is then given by

- S up 2 z m

| H | 2

2 D2 /r . me e (40)

If we normalize l\ e by the physical aperture Ap = u(D/2)^, and use eqs (37) and

(38) in (40), we obtain

A p

3 C2 cose (1 + cos0 ) 3 e _ m m m

A kf 3 64u sin 2 0

p m

(41)

It is interesting that the asymptotic physical optics expression (eq (33)) has

the same factor p ^/(kf^), but a different angular dependence. The p m

^ dependence is simply a result of A

0 being proportional to the feed aperture area.

The k

1 dependence is typical of diffracted power, and the f dependence is a result of the Ap normalization (f dependence) and the inverse distance (f“M

dependence of the edge-diffracted power.

There is a range3 of incidence angle, 3 n/2 <0 < n - 0 , where the feed * s m’ horn is also directly illuminated by the incident field. The geometry for

17
this case is shown in figure 16. In this case it is easy to show that the

normalized effective aperture due to direct illumination of the feed horn is

A 2 p

2 7T

=

( —[p ) cos (» - e

), f

< e $ < * - e (42)

P

In figure 17, we show the various wide angle results for effective aperture for the reflector parameters given in figure 11. Strictly speaking, the

physical optics result is not valid beyond e m

/2 (= 38°) because shadowing of

the reflector takes place, but in practice it is still probably the best estimate out to 0

$ = 90°. Beyond 90°, direct illumination and edge diffraction

can yield a larger value of A

g

. The direct illumination only extends to the

shadow boundary at 0

s = n - 6 m

= 104°. In the vicinity of the shadow boundary, the Keller diffraction coefficients as given by eq (37) are not valid,

but can be replaced by an integral form [16]. However, the maximum diffracted

field is only one-half the incident field even at the shadow boundary. Note

that the GTD results depend on the polarization of the incident field.

Although the diffraction coefficients decrease as 0 $

increases beyond 120°,

there is the possibility of a backlobe at 0

$ = 180° because the entire reflector rim can contribute an in-phase diffracted field.

In summary, it is probably most convenient to use the physical optics

expression for e

s < 90° and to recognize that the effective aperture can in- crease somewhat for e

$ > 90° because of edge diffraction or direct illumination of the feed horn. To try to obtain an accurate result for large 0

$

is

probably unrealistic because the results depend on the construction details

(struts, reflector edge, etc.) of the antenna. Based on a large number of

radiation patterns, the CCIR [17] uses the isotropic value (A

0

= /4tt ) for

reflector antennas at large angles for interference calculations.

4.2 Transient Fields

When a transient wave is incident on the reflector antenna, the waveform

undergoes dispersion. If the frequency spectrum of the incident waveform is

known, then the problem can be analyzed in the frequency domain, and the final

result can be transformed to the time domain. Another approach is to obtain

the impulse response of the reflector and to convolve the incident waveform

with the impulse response. This method has the advantage that the impulse

18
response can often yield a useful physical interpretation. Also, the duration

of the impulse response provides an estimate of the pulse stretching which

occurs when the impulse response is convolved with an incident pulse. In this

section we obtain a simple approximate expression for the impulse response of

the reflector, but do not attempt to analyze the effect of the feed horn or the waveguide feed.

In eqs (15) and (16), the approximate expressions are given for _E_ and H_

in the focal region. For the approximations made in eqs (15) and (16), the

pol arizations of _E_ and H_ are the same as their incident polarizations and the

ratio of E to H is the free space impedance n. Thus it is sufficient to treat

the scalar problem for either E or H, and the polarization factors in eq (15) can be ignored. Thus we can write the electric field spectrum as

E(u>)

2 J i ( k 0 P) 1 v m

' ke P m

(43)

where E q = -2 jkf E^w) e ^ s i

2

( 6^/2 ) and P is defined in eq (15) as the

distance from the geometrical optics point (x 2

= f S1

’ n8

s

> yz

= 0). E^w) is

the spectrum of the incident waveform, and for impulse excitation E^w) is

unity. By setting E^w) = 1, making the small 0 m approximation, and writing

in terms of the modified Bessel function Ij_ [3.2], we can rewrite eq (43)

D e E(«)

2P 1 1( j“b) (44)

where b = P0 /c. m

The impulse response E(t) is the inverse Fourier transform of eq (44)

E(t) = F

_1

[E(u.)] = \ E ( oi) e

Jajt d u . (45)

The inverse transform of I^(jwb) can be found in Laplace transform (s = ju>)

tables [18]

F“ [I^jwb)] =

- t

, It I < b

it b / b

2 - t2

0 , It I > b.

(46)

19
The exponential factor in eq (44) introduces a time delay, f/c, because the

phase reference was taken as the center of the reflector. Consequently,

E(t) can be written

!(t) =

2^n

< T >’ (47)

x/b

/ (x/b) 2 -1

where E ( x) =

n

cl < b

, |t| > 0

and x = t - f/c.

Thus the approximate impulse response has a finite time width w given by

2

w = 2b =

2 P 0 2 0 / (x 9 - f sin0 ) 2

- y9

: m m ^ s (48)

The shape of the normalized impulse response E (x) is shown in figure 18. The

singularities at x = +b are integrable square root singularities and present

no difficulty in convolving E(t) with any realistic incident waveform.

When P is zero, both eqs (44) and (47) are indeterminate. However, if we take the limit of P approaching zero, eq (44) becomes

EM “

- TT1 j“ e" J “f/c, P = 0. (49)

The inverse transform of jw [19] is the unit doublet, 6

' ( t ) , and the inverse

transform of eq (49) is

D0

E(t) - - 6’(t), P = 0. (50)

When the doublet eq (50) is convolved with an incident waveform, the incident

waveform is differentiated [19].

From eq (48) we see that the width of the impulse response is proportional to P, the distance from the geometrical optics point. For on-axis

incidence (0

$ = 0), the most distant points of interest are at the edge of the

20
circular aperture, P = p m

. As an example, consider the parameters in figures

10-14: 13 = 1.326 and Pm

= 3.39 cm. In this case, eq (48) yields a pulse

width w = 0.3 ns. However, if we consider off-axis incidence, then P can be

much larger. If we again consider the antenna parameters from figures 10-15

(f = 39 cm and 9 m

= 1.326), an off-axis incidence angle of 0

S = 30° yields a pulse width of w = 1.73 ns at the center of the aperture (x 2

= y2 = 0).

It should be stressed that the results in this section are based on physical optics and are not valid at low frequencies. This means that the

transient results in eqs (47) and (50) are not valid for large times.

However, if the incident waveform has very little low frequency content, the

convolution results should still be fairly accurate. The problem of dispersion caused by the feed horn and the waveguide has not been addressed, and

such results would depend on the detailed geometry of the horn and guide.

Dispersion in waveguides has been studied by inverse transforms of the modal

series [20] for the acoustic problem, and the same technique would be valid

for the electromagnetic problem.

4.3 Reflector Roughness

The analysis in this report has assumed a smooth parabolic reflector, and

this assumption is valid when the tolerance of the reflector surface is sufficiently small in terms of the wavelength. The effect of surface roughness

has been studied in the classic paper of Ruze [21] and in more recent theoretical papers [22,23]. In general, the problem is very complicated, but we can make use of the simple expression from Ruze [21]. By a statistical analysis

of the physical optics integral, Ruze derived the following expression for the

radiation pattern G(0,<j>):

G(0,<|>) - G ( 0 , 4>) e + G ( 0 , 4>) , (51)

o s

where G

Q

is the radiation pattern of the smooth reflector, 6

2 is the mean square phase error over the reflector aperture, and G

s is a scatter term which

is normally negligible in the vicinity of the main beam. The mean square

phase error is given by

= (4tt£/ A) 2

, ( 52 )

21
where e is defined as the effective reflector tolerance. For large f/D, e is

approximately equal to the reflector tolerance measured normal to the surface,

and for smaller f/D the effective tolerance e is somewhat less than the normal

surface tolerance [21],

For cases where the second term in eq (51) is negligible, the effect of

reflector roughness is simply given by the exponential factor in eq (51).

Thus we can define a roughness loss L (the ratio of G

Q

to G)

6

2

( 4ne/X ) 2 L = e = e (53)

Normally, L will be small for in-band frequencies because the reflector toler- ance e will be small compared to the in-band wavelength X. Figure 19 shows curves of roughness loss as a function of frequency for various values

of e/\ where X

Q is the in-band wavelength. It is seen that roughness loss

can increase rapidly with increasing frequency because of the exponential dependence of eq (52). The key parameter is the surface tolerance e which is

not normally known, but has been inferred by Ruze [21] for some antennas from

the frequency dependence of the antenna gain. Typical values of e are on the

order of a millimeter or less. For e = 1 mm and a frequency of 3 GHz (x

Q

= 10 cm), the ratio e/\ is 0.01, and the center curve in figure 19 would

apply. Thus the roughness loss would be negligible at 3 GHz, but would be

approximately 7 dB at 30 GHz.

For large values of e/X, the second term in eq (50) becomes dominant, and

a very broad pattern results. This term depends on a number of parameters

including the correlation distance of the surface roughness [21] and will not

be considered here. However, it is worth noting that this scatter term can actually raise the pattern level in the distant sidelobes. Strictly speaking,

the derivations of Ruze [21] apply to the in-band case, but his results should

also apply to the effective aperture at out-of-band frequencies because both cases are described by physical optics integrals.

4.4 Offset Parabolas and Dual Reflectors

The analysis in Sections 2 and 3 was done for a symmetrical paraboloid

with a prime focus feed, but other reflector antenna configurations are also

22
of interest. In general, other configurations are more complicated to analyze

rigorously, but in many cases can be described by an equivalent symmetrical

paraboloid.

Consider first the offset paraboloid shown in figure 20. The offset

geometry has the advantage of eliminating blockage by the feed horn. The

focal region fields of this geometry have been analyzed by Bern [12] for onaxis incidence (e

s = 0) and by Valentino and Toulios [11] for arbitrary incidence. Typically, the focal region fields of the offset paraboloid are approximately equal to those of the symnmetrical paraboloid if the equivalent

focal length f

1 and equivalent diameter D' are used in place of f and D. Thus

the formulas in Sections 2 and 3 are directly applicable to the offset paraboloid if the equivalent parameters are used. For an offset angle 0 Q

, the

equivalent focal length is given by [12]

f'

2 f

1 + COS0

* 0(54)

The equivalent diameter D' is simply the projected diameter in the plane perpendicular to the axis of the parabola as shown in figure 20. The equivalent

parameter method is most accurate for small values of 6 but Bern [12] has

found good results for 0 Q as large as 45°. Also, Valentino and Toulios [11]

have shown that the geometrical optics shift as predicted by eq (17) holds for

incidence angles of 0

S = 5° and 10° for an offset parabola with 0

$ = 44°.

Symmetrical dual reflector antennas can also be analyzed by replacing

them with single reflectors of equivalent focal length as shown by Hannan

[24]. Consider the Cassegrain antenna in figure 21. The main reflector is a paraboloid of diameter D and focal length f, and the subreflector is a hyperboloid. The real focal point is shown between the reflectors, and the equivalent single paraboloid (dashed) has a focal length f' and a diameter D. The

equivalent focal length can be written as [24]

tan (0/2)

fl = f

ta-nTe;/^'*

(55)

For a Gregorian antenna, the subreflector is a concave ellipse as shown

in figure 22. The equivalent paraboloid [24] again has a diameter D and a focal length f' given by eq (55).

23
Dual reflector antennas have been analyzed by using physical optics on both the main reflector and the subreflector [25]. It was found that the

equivalent paraboloid approach worked well for both Cassegrain and Gregorian

antennas.

4.5 Coax-to-Wavegu ide Adapter

In this section we analyze the out-of-band response of a typical coax-towaveguide adapter. A knowledge of this response is helpful in evaluating the

comparison between the reflector antenna theory and the experimental results

in Section 3.3. Such adapters are also used in many other microwave antennas

which use waveguide feeds, and the adapter response is an important part of

the total antenna system response.

The probe type of adapter which we consider is shown in figure 23.

Collin [26] has analyzed this structure using a variational technique, and he

has presented some numerical results for in-band frequencies where the wave- guide supports only a single propagating mode. Here we follow Collin's notation and formulation, but we avoid some of his approximations which were intended only for in-band frequencies.

The main task is to compute the input impedance Z^

n of the probe as seen

by the coaxial cable at the junction (y = 0). Collin [26] has derived the

following double integral expression for Z,- n

:

where S is the probe surface, _J_ is the surface current on the probe, Q is a dyadic Green's function for the waveguide, and 1^. is the total probe current

at the junction. Collin has shown that eq (56) is a variational expression

for Z.j n

; thus a first-order approximation to J yields a second-order

approximation to Z^. Consequently, a sinusoidal approximation to the probe

current is adequate, and J can be written [26]

Z.in jz 11 11 J ( r ) * G ( r | r

1 ) • J (r ' ) da da',

t S S

(56)

J = a J sink (d - y), -y o (57)

24
where a is a y-directed unit vector and J Q

is an arbitrary constant. We as- sume that the current distribution is uniform in the circumferential direction

and that I t is thus given by

I. = if t J sin kd,

t o (58)

where t is the probe diameter.

In evaluating eq (56), it is convenient to write Z in as a mode sum be- cause G is a mode sum.

ooE n=l ,3,5

E m=0

Z nm’

where Z nm

R nm

+ ^nnr

(59)

R nmis the resistance associated with the nm waveguide mode, and X nm

is the

reactance associated with the nm waveguide mode. The probe excites only TM^

m modes that are transverse magnetic to y. The fields of the TM^ mode have n half cycles in the x direction and m half cycles in the y direction. The

terms for n even do not contribute to Z

n

- n because the probe is located at the

center of the waveguide (x = a/2). For propagating modes, the propagation

constant 3 nmis given by

3 = j / k

2 - ( n n/a

)

2 - (mn/b) 2

, (60) nm

where a and b are the waveguide dimensions indicated in figure 23. For evan- escent modes, the attenuation constant r nm

is given by

r = / (mr/a) 2 + (rWb) 2 - k

2

. (61) nm

Only the propagating modes contribute to the resistive (real) part of Z in , and

Collin treated the in-band case where only the TM^q mode was propagating. If we apply his method of evaluation to an arbitrary nm propagating mode, we obtain the following:

25
( 62 )

nm

2 C

-- -

m

- sin 2

3 Si, 3 nm

* nm

( = -—- si n (23 sl) , nm 3

v nm

nm

n e om where C

= -—- —r

-?——

m k ab sinz kd

[cos kd - cos ( —^ ) ]

rmr

and e = I !’ om (2,

1 , m = 0 m t 0

*

1 -

< i?b

>

The result in eq (62) agrees with Collin's result for the special case, n and m = 0.

= 1

The evanescent modes contribute only to the reactance X^

n

. For sufficiently large values of m, we follow the Poisson summation formula of Collin

and obtain the following:

Z n=l,3,< nm

a C

r

2tt K 0< t/2),

o m

(63)

where k = / (mrr/b) 2 - k

2

, m and K

Q

is a zero-order, modified Bessel function [14]. The result in eq (63)

is valid for m > M where k^ a >> 1. The sum on m from M to 00 converges rapidly because K

Q decays exponentially for large argument [14]. For m < M, we perform the n summation using the same asymptotic technique that Collin used

for m = 0. The technique is based on the following asymptotic behavior

of r for large n:

Jim r

- 1 ~ -i -V2 ( -± )

2 [(

STL )- k 2

l . (64)

n+°° nm nu 2 nn

' L ^ b

> J

However, we must include a term, exp (- r nm

^)» which Collin dropped because it was exponentially small for in-band frequencies. When this is done, the sum

for m < M is given by

00 M-l M-l a C

X * Z An nm A ziT m=0

(1 —7T t

z

n=l ,2,« •

Z

' m=0

00z

n=l ,3,* •

M-l

y

' 1 - exp (- 2 r f

r si) nm

Li m=0

m > r nm

( 65 )

P P

J - Z Z C —

nu J m nn

n m

26
The primes on the summations indicate the omission of the propagating modes

that are computed by eq (62) and the P's on the final sum indicate only the

propagating modes. The terms in the infinite double sum go to zero as n“ J for

large n because of the asymptotic behavior of r nm

~'*' as given by eq (63).

A computer code for

n has been written and the double summation over n and m is performed as indicated in figure 24. The boundaries between the

three regions depend on the parameters a, b, and k, but the computer code

determines the appropriate boundaries automatically. The computer code is

fast because both the n summation in eq (65) and the M summation in eq (63)

converge rapidly. The code has been checked against Collin's curves for X^

n and R.j n

, and agreement has been obtained to graphical accuracy.

We now consider the transmission of power from the coaxial cable to the

waveguide. We assume that the coaxial cable supports only a dominant TEM mode

and that the coaxial cable has a real characteristic impedance R

c

. Then the

voltage reflection coefficient r

y is given by

r

V

( 66 )

The power reflection coefficient is then given by |

v

| 2

, and the transmission

coefficient for the total power supplied to the waveguide T^ is

T

t

4 R. R

l n c Jr. +TT“+ x.

2

‘ in c in

(67)

We can also define the transmission coefficient for the nm mode T

pm as the

ratio of the power transmitted to the nm mode to the incident power in the

coaxial cable. By using the relationship between base current I t and R nm

[26], T

pm

is found to be

4 R R nm c nm (R. + R ) 2 + X. 2

* in c in( 68 )

From eqs (67) and (68), it is easy to see that the total power transmitted to

the waveguide is equal to the sum of the power transmitted to the individual

propagating modes:

T

t

E

n

E T nm

m

(69)

27
In the receiving antenna problem, we are interested in the reciprocal problem

where a propagating waveguide mode transmits power to the coaxial cable. By

reciprocity, it can be shown that the power transmission coefficient T nm

as given by eq (68) also applies to this case.

A computer code was written to compute T nm from eq (68), and it was applied to an S-band coax-to-waveguide adapter. The adapter parameters were chosen to match those used in the reflector antenna which was studied experimentally by Cown, et al. [5,27]: a = 7.112 cm, b = 3.302 cm, d = 1.9 cm, and

i = 2.4 cm. The radius of the probe was not given, but we assumed a value of

t/2 = 3.5 mm in order to yield a small reactance at 3 GHz. The assumed value

of t is not critical because the results are only weakly dependent on the

probe radius. The transmission coefficients as a function of frequency are shown in figures 25 and 26. Mote that the transmission coefficient for the

dominant TM^q mode is nearly unity from about 2.5 to 5.0 GHz. Above 5 GHz,

higher order modes begin to appear, and the results are very frequency

sensitive. Some related calculations have been done by Cown and Ryan [27],

but they did not compute the reactance of the probe which is required to com- pute transmission coefficients as indicated by eq (68). At 10 GHz, there are five propagating modes (TM^q, TM^, TM^q, TM^, and TM^) which are coupled to

the adapter. In addition, there are propagating TE^

m (transverse electric)

and propagating TM^

m modes with n even [27], but these modes do not couple to

the adapter. All of the transmission coefficients have a null at 7.9 GHz

because the sinusoidal current assumption in eq (57) yields an infinite value

of Z

n

- n at kd = tt.

In the earlier comparison between theory and experiment for on-axis gain

in figure 14, the experimental values were well below the theoretical results. The main problem with the comparison was that the theory included only

the antenna response while the experiment included the antenna plus the

adapter response [27], No attempt was made to separate the antenna and

adapter responses because the out-of-band response of the adapter was not

known. Even though we now have theoretical results for the adapter response,

it is not clear how to combine the results with the reflector antenna theory

at out-of-band frequencies where higher order modes are propagating in the

waveguide. If we take the total transmission coefficient Tj. as given by eqs

(67) or (69) and multiply by the effective aperture of the antenna, then a

28
corrected frequency response is obtained as shown in figure 27. The correction can be made in other ways, but no precise correction can be made because

the antenna theory gives only the total waveguide power, not the modal content. However, the adapter correction does improve the agreement significantly, particularly below 7.5 GHz. The null in the adapter response at 7.9 GHz

is not apparent in the experimental data, but no measurements were made between 7.5 and 8.0 GHz. Above 8 GHz, the agreement is not as good, but this is

probably because the adapter theory is less accurate when the probe is longer

than a half wavelength.

It might be possible to improve the adapter theory for higher frequencies

(kd > it) by using a more accurate distribution for the probe current [26], but

it is doubtful that the detailed structure above 8 GHz in figure 27 could be

obtained. The thin wire and small coaxial gap assumptions are not really

valid above 8 GHz, and also the constructive and destructive interference of

the higher order waveguide modes is probably responsible for the rapid frequency variations in figure 27. The main point of figure 27 is that the

theory matches the experimental results fairly well in the region where the

adapter theory is valid (below 7.5 GHz), and we can expect the reflector

antenna theory to remain valid for higher frequencies because it is a high

frequency theory. It might be desirable to study the isolated adapter re- sponse experimentally, but it might be difficult to perform a meaningful ex- periment because of the higher order modes.

5. Conclusions

The response of reflector antennas to out-of-band (above-band) frequencies has been studied in a two-step analysis. In Section 2, the electric and

magnetic fields and the Poynting vector in the focal region of a paraboloidal

reflector have been determined by a physical optics integration. The large

f/D approximation is found to be sufficiently accurate for most realistic

antennas (f/D > 0.3), and numerical integration is therefore not required.

The second step of the analysis is an integration of the Poynting vector over

the aperture of the feed horn to obtain the received power. As shown in

29
Section 3, this integration can also be done by an analytical approximation in

order to eliminate the need for numerical integration. The results yield a generalized effective aperture (the total received power divided by the incident power density) for the antenna, and a fairly simple expression gives both

the frequency response and the receiving pattern.

The theoretical results have been compared with published measured re- sults for a symmetrical paraboloid with f/D = 0.32 over a frequency range from

3 GHz (in-band) to 10 GHz. The theoretical pattern shape matches the envelope

of the measured pattern, but the measured gain falls well below the theoretical result. When the transmission characteristics of the coax-to-waveguide

adapter are taken into account, the agreement between theory and experiment is

much better. Two important results of the theory are that the beamwidth at

out-of-band frequencies is approximately the same as the in-band beamwidth and

that there are no high sidelobes. These results are in agreement with the

published measured data [27].

In Section 4, a number of extensions to the basic physical optics model

are discussed. The level of the distant sidelobes is examined using GTD and

is found to be in approximate agreement with physical optics. The transient

fields in the focal region are examined for pulse excitation, and the pulse

stretching is in agreement with simple geometrical arguments. The effect of

reflector roughness is examined, and the reduction in gain or effective aperture can be calculated if the surface tolerance is known. Extensions of the

theory for a symmetrical reflector to the cases of offset or dual reflectors

is accomplished by using equivalent focal length and diameter. Also, the previous theory for the response of a coax-to-waveguide adapter is extended from

the in-band case [26] to the out-of-band case. This theory might be useful

for other microwave antennas using the same type of adapter in the feed system.

Other extensions and improvements to the model, such as a more precise

treatment of the feed horn, are certainly possible, but the physical optics

model presented here is probably adequate for typical out-of-band applications. The analysis of other common microwave antenna types, such as phased

arrays, is recommended as having highest priority.

30
The Defense Nuclear Agency (DNA) supported the research described in this

report. The author is grateful to the technical staff of the National Bureau

of Standards for numerous technical discussions during the course of this

work. Many thanks also go to Captain David Stone of DNA for his continuous

interest and support.

6. References

[1] Lind, W.R. Out of band performance of antennas. Moore School of

Electrical Engineering Report No. 65-18; 1965.

[2] Levinson, D.S. Transmission-line measurement techniques. Rome Air

Development Center Technical Report No. RADC-TR-65-187; 1965.

[3] Cown, B.J.; Cain, F.L.; Duffy, E.G. Statistical Drediction model for EMC

analysis of out-of-band phased array antennas. IEEE Trans. Electromag.

Compat., EMC- 1 8 : 163-170; 1976.

[4] Wells, T.C.; Ryan, E.C. Out-of-band reflector antenna model. Georgia

Institute of Technology, Engineering Experiment Station, Contract No.

DAAG29-80-C-0083; 1981.

[5] Cown, B.J.; Ryan, E.C.; Weaver, E.E. Antenna pattern measurements to

characterize the out-of-band behavior of reflector antennas. Georgia

Institute of Technology, Engineering Experiment Station, Report ECAC-TR83-003; 1983.

[6] Kennaugh, E.M.; Ott, R.H. Fields in the focal region of a parabolic

receiving antenna. IEEE Trans. Ant. Propagat., A P - 1 2 : 376-377; 1964.

[7] Minnett, H.C.; Thomas, B.M. A method of synthesizing radiation patterns

with axial symmetry. IEEE Trans. Ant. Propagat. AP-14: 654-656; 1966.

31
[8] Minnett, H.C.; Thomas, B.M. Fields in the image space of symmetrical

focusing reflectors. Proc. IEE, 115: 1419-1430; 1968.

[9] Thomas, B.M. Matching focal region fields with hybrid modes. IEEE

Trans. Ant. Propagat. AP-18: 404-405; 1970.

[10] Landry, M.; Chasse, Y. Measurement of electromagnetic intensity in focal

region of wide-angle paraboloid reflector. IEEE Trans. Ant. Propagat.,

AP-19: 539-43; 1971.

[11] Valentino, A.R.; Toulios, P.P. Fields in the focal region of offset

parabolic antennas. IEEE Trans. Ant. Propagat., AP-24: 859-865; 1976.

[12] Bern, D.J. Electric-field distribution in the focal region of an offset

paraboloid. Proc. IEE, 116: 679-684; 1969.

[13] Kraus, J.D. Antennas. New York: McGraw Hill; 1950.

[14] Abramowitz, M.; Stegun, I. A., ed. Handbook of mathematical functions.

Nat. Bur. Stand. (U.S.) Handb. AMS 55; 1964.

[15] Stubenrauch, C.F.; Yaghjian, A.D. Determination of mutual coupling

between co-sited microwave antennas and calculation of near-zone electric

field. Nat. Bur. Stand. (U.S.) Interagency Report 80-1630; 1981.

[16] James, G.L. Geometrical theory of diffraction for electromagnetic

waves. Stevenage: Peter Peregrinus; 1976.

[17] CCIR, International Radio Consultative Committee. Documents of the XIVth

Plenary Assembly. Geneva; 1978.

[18] Roberts, E.C.; Kaufman, H. Table of Laplace transforms. Philadelphia:

W.B. Saunders Co.; 1966, p. 297.

[19] Papoulis, A. The Fourier integral and its applications. New York:

McGraw-Hill; 1962, p. 274.

32
[20] Morse, P.M.; Feshbach, H. Methods of theoretical physics. New York:

McGraw-Hill; 1953, Sec. 11.3.

[21] Ruze, J. Antenna tolerance theory - a review. Proc. IEEE, 54: 633-640;

1966.

[22] Rahmat-Sami i , Y. An efficient computational method for characterizing

the effects of random surface errors on the average power pattern of

reflectors. IEEE Trans. Ant. Propagat., AP -31: 92-98, 1983.

[23] Tripp, V.K. A new approach to the analysis of random errors in aperture

antennas. IEEE Trans. Ant. Propagat., AP-32: 857-863; 1984.

[24] Hannan, P.W. Microwave antennas derived from the Cassegrain telescope.

IEEE Trans. Ant. Propagat., AP-9: 140-153, 1961.

[25] Doan, D.L.; Vu, T.B. Focal region fields of dual reflector antennas.

IEEE Trans. Ant. Propagat., AP-25: 392-396; 1977.

[26] Collin, R.E. Field theory of guided waves. New York: McGraw-Hill; 1960,

Sec. 7.1.

[27] Cown, B.J.; Ryan, C.E. Near-field theory and techniques for wideband

radiating systems at in-band and out-of-band frequencies. Georgia

Institute of Technology, Engineering Experiment Station, Contract No.

DAAG29-78-C-0029; 1982.

[28] Wheelan, A.D. Tables of summable series and integrals involving Bessel

functions. San Francisco: Holden-Day; 1968, p. 8.

33
Appendix A Evaluation of the <t>i Integration

The first step in the evaluation of eq (6) is to write the phase term in

the following form:

n

s + R = Mo ~ M

1 cos ^l" (Al)

where M

Q

, M^, and ij> are independent of 4>]_. From eqs (7) and (8), we find that

the three quantities are given by

M = A +

o o

5

M = / (A + B )

2

+ B

2

1 CCS (A2)

and = sin~^ »

where A

Q

= (f - rj) cose

s

.

A

c = rj. sinO^ sin0$

.

B

c = - r

2 sin0j cos^^

and = - r

2 sin0^ sin<t> 2

.

The next step is to express the amplitude terms in eq (6) in terms of a Fourier series in <)>].. When this is done, it is found that the series truncates at n = 2, and dE_ and d]l_ can be written:

dE = - (Mo

+ M

1 cos4,)

.

I \

[C xn cos n +l

+ S xns1n n +l>

u-x

+ \

[ V°S "*1 + S

yn

s1n n *l ] n=0 n=0 J j hj

2 2 sin0, + Zn

tC

zn cos "*1 + S zn s1n n *1> -

u z> 1 + cose.

d0 > n=0 1

34
and (A3)

E. f

dH _• _i -jk (M + M, cos^)

" J > c U 1 •

1

[D xncos n *l

+ F xns1n n *l ] -x

+ K

C V°S "*1 + F yn

s1n n *l ] un=0 n=0 J J

2 sine. + s.

[D

zn C0S n *l + F zn sin n4,l

] a 2

l l + cos

' e, de ’ d<f l n=U I

The Fourier coefficients are obtained from the terms inside the brackets in eq

(6) and found to be:

sin 2

e, C ~ = a (cosQ-i + V? t—

r—). S n = 0, xO y

v 1 2

1 + cos0^ ; * xO *

C

xl

0

sine.

> S

xl

= - a

z 1 + cose^ *

C x2

" 1/2

si n

2

e

1

yo

y 1 + cose^

si n

2 0

,

a (cose, + V? i—T —

)

x^ 1 c1 + cose

^

J

si

2 e^

^x2 ^2

a

x 1 + cose, * 1

S y0

= °»

sine

1 'yl z 1 + cos 0

1 sin 2

0. >2

1 1 ^ a

x 1 + cose^

C

z0

= 0

- a sine.,

y 1

C 9 = 0

z2

and

S , = 0, yi

a sin 2

e, s * 1 yz z 1 + cose^ *

S z0

°»

’ S zl

= a

x sin0l’

*z2 S_o = 0,

(A4

)

35
D xQ 2 (1 + cos0

l) » F xO

xl

cosQ-^ sin©!

D xl

a

z 1 + COS© * F

1

D

a sin 2

©., x 1 x2 21+ cos0

1

Dy0

' ~2 (1 + cos6

i)

D , = 0 yi

a sin 2 ©. n = _y. L_

y2 2 1 + cos 0

1

si n

2

0

1 D z0

a

z 1 + cos0

D zl

= - a x sin9

l

D z2

=°

, F x2

, F

, F

yO

yi

, F y2

, F yO

, F

, F

yi

y2

o,

o.

sin 2 ©!

1 + cos©! 9

0

sin©! cos©!

a

z 1 + cos©! *

a (cos©! -1)

0,

- ay

S i n 6

!

»

0.

From Fourier-Bessel expansions [2.7], we can derive the following useful

integral

I | cos n 4

> . -jk M, cos( <*», 1 1 sin nct>

; ^^d<t>! = 2 tt j n

{ cos ni

f } 1 J (k M,), sin m|i J n

v V * (A5)

where J n is the nth order Bessel function. By using eqs (A5) in (A3), the

integrations can be carried out to yield the desired expressions for E_ and H_

in eg (13).

36
Appendix B Open-ended Waveguide

Consider the two-dimensional, open-ended waveguide of width w in figure

28. A plane wave is incident in the xz plane, and the incident electric field

is y polarized:

E , = E e yi o jk (z cose + x sin9)

(Bl)

where 0 is the incidence angle measured from the z axis,

the incident Poynting vector S

z is given by

The y component of

E I 2 o|

2n

cos 0

(B2)

From eg (23), the approximate received power per unit length P is given by

P = P n COS0,

r o

*

( B3)

where P Q

= w S Q

and S 2n

‘

If we make the Kirchhoff approximation in the waveguide aperture (z = 0,

x| < w/2), then the electric field is given by

'y U_

= E e jkx sin9

( B4

)

z=0

A general waveguide mode expansion for the electric field E^ is

r

_ v r (2 n-1) ttx _ J

' k xe

z E y

- I lc n cos e 2n ttx jk xo

Z

! + s

n S1nw

e f (B5)

. . . o r (2 n "^) 77 i 2 where k = / k 2 - xe w

and k = / k 2 -

( )

. xo w

The cutoff wavelength A

c is determined by setting k xe = 0 for the even (in x)

modes and by setting k xo = 0 for the odd (in x) modes:

37
Even: = 0 xe

, 2w

c 2 n-1 »

X o

II o X = - . c n

( B6)

The unknown coefficients are determined by matching the electric field at y

=

0 in eqs (B4) and (B5)

E ejkx s1ne = l [c cos

n

-

L 1) ” + s n ui n ui -* n=l

(B7

)

By using the orthogonal ity relationships for the cosine and sine functions in

eg (B7), we obtain the following expressions for c

n and s n

:

c = E n n o

r kw . „ . (2 n-1) tt sin [ sine + A —

]

^ r kw . ~

, (2 n-1) tt [ —

2

" sine + —]

and

sin [-^ sine - -y-—

]

sine -

(B8)

s_ =

r kw sine , ni^

sin [ 2 + ~-i

r kw sine nn-.

L 2 2J

. r kw sine nir-, sin [ 2 2

-*

r kw sine nn -. L 2

" 2

J

The sin ( )/( ) in eq (B8) have peaks where the transverse wave numbers of the even and odd waveguide modes, (2 n-ljn/w and 2mr/w, match the transverse wave number of the incident field, k sine.

In order to compute the power in the waveguide modes, we need an expression for the magnetic field H:

H

= — V x u. E...

GJU -y y

From eq (B9) the magnetic field components H

x and H

z are

1

3 E

1 d L H = -r^ and H

= -? x jwp 3 z z

1 ‘-"‘

z

3E

jwp 3x*

(B9)

(BIO)

The power per unit length transmitted by the waveguide Pg is given by

w/2

P = / Re (E H *) dx.

9 -w/2 y z (Bll)

38
By substituting eqs (B5) and (BIO) into (Bll) and carrying out the x integration, we obtain

P 9

where P ne

= 4F7 l c

rj

2 Re (k xe>

and P no

= 4R l s nl

2 Re (k xo>-

00z n=l

+ P no

), (B12)

For modes above cutoff, k xe

and k XQ are real. For modes below cutoff, k xe

and

k x0 are purely imaginary. Thus, only the propagating modes carry power, and

the summation in eq (B12) can be truncated.

A check on the accuracy of the Kirchhoff approximation for the aperture

field in eq (B4) can be made by comparing the waveguide power P

g in eq (B12)

with the received power approximation P p in eq (B3). In general, the evaluation of eq (B12) must be done numerically for specific parameters. However,

the limit for large kw can be evaluated analytically for on-axis incidence,

0=0. In this case, the mode coefficients are given by

=

4 (-D

n-1

c

n (2 n-1) tt E

( and s„ = 0. (B13)

For large kw, k xe « k, and P

n is given by

P =

9

4w E 0

2

TT"- n

Z (2 n-1)'

2

. n=l

The infinite sum has a known result [28]:

l (2 n-1)

n=l

-2

8

*

Substituting eq (B15) into eq (B14), we have

w|E

P =

9 2n

W S

o

* P r*

(B14)

(B15)

(816)

Thus Kirchhoff approximation yields a consistent result for the received power

and the waveguide power when kw is very large.

39
In figures 29-31, we show results for the received power and the waveguide power as a function of 0 for various values of w/x. In all cases, the

various powers are normalized to the incident power P Qat 0=0. As w/X is increased from 1.5 to 10, the agreement between P r and Pg improves for all incidence angles. In all cases, the powers in the first even and odd modes are shown for comparison. As w/x is increased, the mode patterns show more rapid

oscillation, and the number of propagating modes increases. For w/X = 1.5,

there are only 2 propagating modes, but for w/X = 10, there are 19 propagating

modes.

40
P2 (X 2 ,Y 2 ,

Figure 1. Geometry for a plane wave incident on a symmetrical parabolic

ref 1 ector.

41
1.0

0.8

0.6

0.4

0.2

0.0

- 0.2

k P2

Figure 2. Normalized Poynting vector for on-axis incidence.

42
Figure 3. Normalized Poynting vector for off-axis incidence.

43
Figure 4. Geometry for a circular feed horn.

44
Figure 5. Normalized Poynting vector and received power for on-axis

incidence.

45
Figure 6. Approximate integration surface (dashed) for off-axis incidence.

The geometrical optics point is located inside the circle.

46
Y;

X 2

f sin e<

Figure 7. Approximate integration surface (dashed) for off-axis incidence.

The geometrical optics point is located outside the circle.

47
Figure 8. Normalized effective aperture for off-axis incidence.

48
Figure 9. Comparison of analytic and numerical results for the effective

aperture.

49
Relative

Power

(DB)

Figure 10. Comparison of theory and experiment [3.3] for the in-band

frequency of 3 GHz.

50
Relative

Power

(DB)

Figure 11. Comparison of theory and experiment [5] for an out-of-band

frequency of 6 GHz.

51
Relative

Power

(DB)

Figure 12. Comparison of theory and experiment [5] for an out-of-band

frequency of 8 GHz.

52
Relative

Power

(DB)

Figure 13. Comparison of theory and experiment [5] for an out-of-band

frequency of 10 GHz.

53
Gain

(dB)

Figure 14. Comparison of theory and experiment [27] for gain as a function of

f requency.

54
Figure 15. Geometry for diffraction from the edge of the reflector.

55
Figure 16. Geometry for direct illumination of the feed horn.

56
0

-10 -

-20 -

D-1.22 m

P - 3.39 cm

f/D = 0.32

freq. = 6 GHz

Phys. Opt. Direct Ilium.

1 20

Figure 17. Various contributions to the wide angle receiving pattern.

57
T/b

Figure 18. Normalized impulse response.

58
Roughness

Loss

(dB)

0

Figure 19. Roughness loss as a function of normalized frequency.

59
Figure 20. Geometry for an offset paraboloid.

60
Figure 21. Geometry for

(dashed) has a focal

a Cassegrain

length f'.

antenna. The equivalent paraboloid

61
Figure 22. Geometry for a Gregorian antenna. The equivalent paraboloid

(dashed) has a focal length f'.

62
>«z

Figure 23. Probe type of coax-to-wavegu ide adapter.

63
Figure 24. Computational scheme for evaluating Z 1n . The nm modes are grouped

into three regions: propagating modes, evanescent modes for m < M, and

evanescent modes for m > M.

64
Figure 25. Transmission coefficients for an S-band coax-to-wavegu ide

adapter. The cutoff frequencies for the dominant ( 0

) and first higherorder (TM^) modes are indicated by arrows.

65
0

Freq. (GHz)

Figure 26. Transmission coefficients for the higher order modes of an S-band

coax-to-waveguide adapter. The cutoff frequencies for the higher order

waveguide modes are indicated by arrows.

66
(ap) wee

67

Figure

27.

Comparison

of

theory

(with

adapter

correction)

and

experiment

[27]

for

antenna

gain

as

a

function

of

frequency.
- yi v

X

Figure 28. Plane wave incident on an open-ended, parallel-plate waveguide.

68
Relative

Power

(dB)

0

Figure 29. Total received power and the contributions from the n=l even and

odd modes. Only two modes are propagating.

69
Relative

Power

(dB)

Figure 30. Total received power and the contributions from the n=l even and

odd modes. Five modes are propagating.

70
e

Figure 31. Total received power and the contributions from the n=l even and

odd modes. Nineteen modes are propagating.

71
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 85-3021

2. Performing Organ. Report No. 3. Publ ication Date

April 1985

4. TITLE AND SUBTITLE

Out-of-Band Response of Reflector Antennas

5. AUTHOR(S)

David A. Hill

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Defense Nuclear Agency

Washington, DC 20305

10.

SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The response of reflector antennas to out-of-band frequencies has been analyzed

using physical optics. A simple approximate expression has been obtained

for the effective aperture, and this expression yields both the receiving pattern

and the frequency dependence of the on-axis gain. The theory has been compared

with published out-of-band measurements, and the pattern agreement is good, but

the measured gain falls below the theory. This discrepancy is caused by mismatch

loss in the coax-to-waveguide adapter, and this mismatch loss has been analyzed

theoretical ly. The basic physical optics model for symmetrical reflectors has been extended to include offset and dual reflectors, reflector surface rouqhness and transient excitation.

12. KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolon s) coax-to-waveguide adapter; effective aperture; focal region; paraboloid, physical optics; Poynting vector; out-of-band response; receiving pattern; reflector antenna.

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