pure.tue.nl · CON'l'L:N'I'S. I 1-1-1 1-1-2 II II-2-1 II-2-2 III Introcluciion ril'Ld (llluLy~;i~; lly ~>ll\li!',111 [l)l\-Jdnl illl('!',Y\ll i()ll of ~;Pl'('tr'UI1l run'I ion~; Uniform-pha~~ecl

Eindhoven University of Technology

MASTER

General plane wave spectrum analysis of the near field of circular aperture antennas

Pruijsen, H.M.

Award date:1977

DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Studenttheses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the documentas presented in the repository. The required complexity or quality of research of student theses may vary by program, and the requiredminimum study period may vary in duration.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 03. Jun. 2018

..

General plane wave spectrum analysisof the near field of circularaperture antennas.

by

H.I"'!. Pruijsen

This study has been performed infulfillment of the requirementsof the degree of Master of ScienceOr.) at the Eindhoven Universityof Technology, Department ofElectrical Engineering

under supervision ofDr. l'1.[.J. Jeuken.

HP/mpET 3 januari 1977ET-1-1977

CON'l'L:N'I'S.

I

1-1-1

1-1-2

II

II-2-1

II-2-2

III

Introcluci ion

ril'Ld (llluLy~;i~; lly ~>ll\li!',111 [l)l\-Jdnl illl('!',Y\ll i()ll

of ~;Pl'('tr'UI1l run' I ion~;

Uniform-pha~~ecl ilpertur'(' d lstr ibu t iorm

Aperture distributions with qUddnltic plklsedi~,lribut~()nr

Field analysif.; with Fourier-Bessel SFTlCS orDini series

Fourier-Bensel G('1'1C8

Dini series

Example 1.

Example 2. (the paraboloid reflector)

Near-field approach by steepest descents method

References

Appendjx A

Appendix B

"

'1

27

68

bi

-1-

INTRODUCTION

Over the years aperture integration techniques have been used

almost exclusively for the calculation of antenna-near fields.

'Illis technique is extremely time-consuminG in terms of computer time,

especially for aperture diameters greater than, say lOA . Recently

plane-wave-spectrum'(P~S)methods have been shown to be quite

advantaGeous as a calculations approach in the case of circular

apertures. We will refer to Rudduck, Wu, Intehar [1] and Brown [2J.rfhese methods appear to provide less restrictive approximations

than, for instance, the Fresnel approximation used by Hansen (3].

'I1le ~VS methods discussed here make use of simplifications entailinG

neglect of the reactive portion of the spectrum, the so-called

inhomogeneous of invisible region.

In the present study a near-field approach is d~velopped In

detail for the special case of the corrugated horn radiator in which

a HvS formulation is obtained by means of the steepest-descents technique

recently presented by Rudduck illld Chen [4] , based on the more general

treatment of basic plillle-wave theory by Clemmow [5] , resulting In"

more efficient and rapid calculations than was possible with previous

methods offered.

Furthermore a general theory for the analysis of the neal"-field

of circular aperture antennas has been presented here, based on the

utilisation of Fourier-Bessel series for aperture distributions which

are zero at the rim of the aperture, or Dini serles for distributions

with a pedestal. 'I1lis theory not only allows almost every practical

aperture distribution to be dealt with, providing the starting point

of a reliable near-field expression both on-axis and distant' from the

axis, but also considerable reduction in computation time for those

aperture distributions which have no spectrum function in closed form.

L-},J-

'111(; electric field Ln tJw dpcrtUl:'l' of d len'iT eorr'W',dlc'cl Wdv('["uidl'

radi.aLor under baldnc('cI hytlrid concJiti()n~; if; I~iven hy ,Jc'ukc'n [H] rmel

Clarricoats, SaJH [9] as..

(1-1-q)

J 0 ie; a 13esr;el func t.i on, rmd :i 01 Lf-: the fi n_;l~ 7,(~r() of Ihi,) rune I ion.

Comhining (j-1-:1) dnd (I-l-Q), we Jim] d ~;ilflplc c]o::l,d rOl'Tll fop [('0)

0-1-5)

Hence

00

(l-I-(i)

in which ci - r .6_z--a 'r-a- (1-1-7)

1ne reactive portion of the spectrum corresponds to )<) ka; thus

jf,J.fk'a'-x2. is real, "resulting in the inhomogeneous plane-wave contrihution.

This inhomogeneous or "invisible" rer;ion of the spectrum yields the

reactive portion of the aperture impedance. Because of the evanescent

decay exp(-II3V~-K2.lz), the inhoffiOf,eneous contribution is ,;~gnificant

only for near-field distances less than one or two wavelengths from

an aperture. Consequently the contribution for x>ka can usually be

neglected in (1-1-6). [1]

-2-

1. FIELD ANALYSIS BY STAAIGHT- FORWARD INTEGRATION OF SPECTRUM

n.JNCTIONS.

".

gl.1Uniform-phased aperture distributions.

L,------­IIIIIIII

I

SA: 2a

___-----'>-., I

~--

To find the radiation field of a

corrugated waveguide radiator

CFig.l/1l we expand thi~ field for

z> 0 in elementary cylindrical

waves such as shown by Goubau

and Schwering [6J . For a

linearly polarised field the

following result is obtained

(1-1-1)

The function fCr) lS the amplitude spectrum functi~n. JOC .. ) lS a

Bessel function.

... (1-1-2)

Assuming a linear-polarised field

in the aperture SA CFig.l 12) we

have

ExCr,cr,o) =Fer) ,r<a= 0 , r >a

Now f CT) may be found by applying

a Fourier-Bessel transform as done

z

(1-1-3)

by Stratton [7J. The result is

a

£(0)= 2~jE, (r,o) JoCor) rd,

We must realise that there is a harmonic time-dependence expC-jwt)

everywhere which is omitted where its role is inessential.

-3-

We have made cl nwncI'icdl illvesLigdtion of 1J1l' r;p(~ctt'UJrl fUllCUOll (1-1-~)).

The l~sults are plotted in Fig. 1/3 ...For Ex(r,z.) let us write

(1-1-8)

We can now calculate the arnplitude and also the equiphase lines by

computing the quantity ep(aj - <PCo). The calculations are carried out. ,

for the case flare angle 6.51, a = 6.25 em, A= 1.998 an (15 01Z).

Some results arc given in Fig.1J4and FigJ;1.) for jJ = 1, 2 and (;, the

amplitude and the equiphase lines, respectively. These results were

obtained by computing (1-1-6) for the values ~ = 0; 0.1; 0.2; .... ; 1.5.

0,10

j0,05"

0,01

- 0,01

Fig. 1/3

11.

!...-I

-5-

90

. ,

180

27

l-------~5-------1;_-__:=~r~--~1~.55360 .5 a-ao

Fig. 1/4

-(j-

0___

-30

z{j=rr-

l-------O:s-------~11-~~~r~----;1.:s5~~ ~5 a-ao

Fig. 1/5

-'1-

a

d

Fir,. 1/[;.

in whichr

t =- anda

If e 1S very small then the equiphase­

plane approximates a parabolic function. r'!

and 1S f = 2L

31.2 Aperture distributions with quadratic phase distributions.

The theory in ~ 1.1can br.\ verified by considering, a small quadratic

pha~;c distribution, a[; it occurf; Ln reali ty.

'lhis phase d_is tribution can be

explained by Fig,. Vo and consists

of a factor to be added

exp( -j cSt1 ) •....... ('1-;-1 ),

For this case

In combination with 0-2-3) no closed form now results, buta

fCo)= 21JOgOl~) 'Jo(~r) exp(- j <I ~~) r clr ( j-?-")

oThus the resultinr; expl'2ssion for the electric field is:

(1-2-5 ).III _j~t2 -y:i'la~-)(~

£x(r,2)=jJ J09011:) Jo~t.) e -ldt] Jo(xet) e xdx....o 0

In the first form of (1-2-5) an accent is added to the first integration

of r because of the fact that r plays a different role in each integration.

It is only in the inner interrral that r is variable. Further,r' r z

t = d ,X =Oa , a = a ' (3 = aIn the second inteeral r1... and;3 are parameters.

-8-

Expression (1-2-5) can also be calculatc:'d, but the nWTll'rical il1vc~;tigclt:ion

is much more time consuming for the computer than the expression (1-1-6).

However, we again make use of (1-1-8) and compute the amplitude and

the equiphase lines for /J = 1, 2 and fj using the dimension faY' the

corrur;ated conical horn givon on par;e ~ and ~ = 1.998 crn as we1] .

Results can be studied on pages IO~l1 ~ 12. l 13,14 and IS • On pages

13, Il.j and 15 we have drawn the amplitudes I Ex (r,z)I for

/3= 1,2 and 6, respectively and on pages lo~11a11d12 the equiphase lines

( <fCtt-) - epeo)) , for the same values of j3.In each figure we have plotted with a dotted line the measured values

of IL)( (r, z) \ or CfCo() - epCO) • He also r;ave with a dashed line in all

figures the calculated quantities of 11.1 Haking a comparison between the

results of S1.1and ~.2 it is interesting to note the effect of a phase

distribution consistinr; of a certain spreading out of the figures

showing the amplitude. The figures with equiphase lines are narrower

for small values of fi and have the spreading out only for rather large

values of f3.

---- --------....

........"­ ,,, ,,,

\\

\\

\\ ,,

\ ,\ ,

\\

\,\\,,

\\ \

\. \\\ \

\ \

\. \\ \

\. \\ ,

\ \\ \.,

\ \'. \

\. "\ \

\ \

\\\\\

....

..

1'1~q;r,qt 10'1 ~~, I.I~

\V1tfgrll.! io'" b= 0

rnE'c1~lJrfa

90

o180

27

1.5360L--- .L-- .L-- L...-

ora-­a

Fig. 1/7

-10-

..

.... ........"

......" , ~' "- "-

......,..,. ""-...'.......... "-',

............ ",....,..

\\.\. '"'\ \

\, \

'''\. '\... '\ ,

... ,\. "'. \

\. \" ,\~ ',

\.. ,,\ \~ ,,,,,

\.. \\ \

\ ,\ ,

... ,\ ,\\\

-::==--==..=...- - -.-:-------

90

o180

27

Il1tE'§rabon S'" 1.12­

1\·,-tE'gr81JoVl S= 0

ll1l'il"weol

1.51.5360"-- --L ....&- .L...-_

o

Fig. 1/8

-11-

90

o180

27

inteBI'f\1 iOI1 l~"--I.t~

H'1t<28>clttoV1 £-=0

VI'\!' aS U\'ed

l-------~~-------~1~-_:~~r~--11~.5i360 .5 (1:'_ ao

Fig. 1/9

-12-

\.\ .,\.

\\ ..

\\"

.

&'" 1.12

..

I

intqvaboll

l"-tq~r8"hoY1

measured..

~..~...•,

.....•,...,"','\\

....\' ..~

\'\\

\:',\\.\\\\

\\".\ \'\" \~\ .

\\

\\

\\

\\\

\

\\\\\

\\'\

\\\

\\\\ \\\ \

\ ,\ ':

\ \.\ \

\ \\i,

\\\\\

\ \'. \\ \

... \\. \, \

-10

-30

-20

1·510.5

\

\

-4JI.. ...L- ...L- ...\ __

orQ:'=­a

Fig. 1/10

-13-

-30

\,\.

\,'.

\\

\\

\

\ '.

\.

,

"

0'---':::0--_~'~'"

.....:::..:', ........... .

...... ""...... '.

........ ,

" "'"........ ""

"- ""." "-." ...."" .....,....

..'\.. ....'\. "'"

"" .....\ .....

\ "'"

\ .....\

\\

\\

\\

\\

\\

\\

\\

\\

\\

\\

\\

\\

\\

\

-10

-20

Inte <gration

lI.t"l?raiidn

mea,uyed

1·510.5-40

1L- ......... --L ...-!-.==o

or(l's::-a

Fig. 1/11

O't---__

-10

-20

•

~II

1iJ--30

inte.~,...borl S~ 1.12

intE''6tll{lon &=0

mel'S l.<Tect

1·510.5

-40,L.- .L-- 1...- ---'1...-_

or(l'-=­a

Fig. 1/12

(2-1-1 )

-1S-

II. FIELD ANALYSIS WITH FOURIER-BESSEL SERIES OR DINI-SERIES.

~ 2.1. Fourier-Bessel series.

In the precedinf, section we learned that an aperture distribution

of the type defined in (1-1-4) gives rise to a spectrum function in

closed form. This observation is also valid when we replace the first

zero of JOin 0-1-4) by the mth zero j Om of the Bessel functions.

1~is makes it possible, to analyse the near-field of a general aperture

distribution f( r) with f( a) = 0; r = a denotes the rim of the aperture.

In this case an expansion of f(r) in a Fourier-Bessel series is possible

[10J . Considering of r to be an arbitrary function, but summable in the

sense of [11] of the real variable r we find00

fer) :; mS am Jo(jDmr)

Where j01' j02' .....denote the positive zeros of J O (z) arranged in

ascending order of magnitude. In the consideration which follows we prefer

the form

( 2-1-2)

(2-1-3)

1:= ra

The coefficients a in trlis expansion are determined by multiplying bothmsides of (2-1-2) by tJO(jOmt) and integrating between the limits 0 and 1.

It follows that

~n = -+ J-t. ttf 1Jo(jamt) dt.:J1 Yom) 0

For the spectrum function one readily finds (again with t = ria)

1

fey) = 2'ffa';;', a,fJogomt) JJOa-l:) hit.o

(2-1-4)

-1G-

Here we used C1-1-5) and assumed that changing the sequence of the

L andoj1Sign was permissible.

Substitution of (2-1-1+ l .i n 0-1-1) .r;ives tlw fi n,11 r('~;ull 1 ni I~; rnOGt

(2-1-5 )

Ex(r,z.) = 2£ [{~ .j:flt) Jo(iolll-t)c1tl.{j~(X)JQ(l<£) ;j~Vk'a~d~ 11. . (2-1-G)WId J100rrY 0 ':J J 0 J0l\11_)«1 ~

1 .

In this form X ="0 a, t = ria in theo! integration; for the integrationl ria is not a variable but a parameter.o

It is of particular interest to see that a double integral as shown in

(1-2-5) is replaced by a series of products of single integrals. For quite

a number of functions fCt) this can then shorten the computation time

if we are looking for numerical results.

Obviously there is nO need to use the theory formulated in C2-1-6) where

fCr) can be expressed in a closed form, so we apply this theory to the

aperture distribution already mentioned in the preceding section

= 0 1 r >a.

-17-

t11en

00 [ 1

fCr) = 2 b {~im .jJ, (jo;t) 1>-9 cos ( u') t: dt:]QllI) 0

-j)~~:\ jlJo (JoJ)']o(jo\,>\-t). 5in(~.t'l) tdt}r~~~iJjl 0"") Jl U (')-1-'7)o •••••••

. ,SubstitutinG this in (2-1-5) we find with

....... <7-1-8)

....... (2-1-9)

In this form ~ 0 r a; 13 0 l ; "0 ~ in Thej:tegral; t 0 ~. in thef~\tegral.~ 0 0

In the integra~j we again replace OQ by ka as we did in the previous

s(:ci:ion (and for the same reason). It is now possible to make a numerical

investigation of (2-1-9).

-18-

00 sThe numerical investigations resulted In replacing the series L. by ~i

while for m = 6, m = 7 etc. the teJ1TIS are s~lller than 0.1% of ~~e 5thterm. With [11] it is easy to prove that! is convergent, we however

..' ll'~1.do not know how fast this convergence is. So the rip;ht nwnber of tcnns

of the series is an approxbnation based on trial.

Results are shown in Fig. 2/1, 2/2, 2/3 (equiphase lines for ~ = 1, 2 dnd G)

and Fig. 2/4,2/5,2/6 (amplitude forf3= 1,2,6). Each figure gives the

values calculated by means of repeated integration (dashed lines), the

values measured (dotte~ lines) and the valuc~; calculated with tlll' Fourielo­

Bessel series (solid lines).

The correspondence between each set of three graphs is striking for small

values ofj3, the area we are most interested in. Moreover the computertime

using Fourier-Bessel series is for instance for the cases (3 = 0.5, ;3 = 1

1/ th f . d . .6 0 the tlIDe that ouble lntegratlon takes.

-19-

......: ....... , .........."' ..,"' "" '." '-'"" ., \, '\

,".. ,\

"",\'\

\\.,

~

I

90

\

\\\\'\\\\\\ '\,\

\ ....

\ \",\., \,\ \

\ \,\ \

\

\ \,\ \\ \.,

\\..

Thune.r- Bessel method.1,,1~8rat10Vl rne{{,c,.,[

mell su.reol va\u€.

180

27

1.51.53601-- --L.. --'L...-. ......~

ora-­a

Fig. 2/1

-20-

..... .............".,:-.,...... "-

"':-',;, () --1- -2",,, a

". "'\. "", ".... ,

.......,,,,," >\"~""

\'" "\ .... \

... \\

\\

\\\

\~

~

~~

\\

\

"~\\\\\\\\ \\ \,

--------.-,. -~

90

180

27

Tau."" - 'Bess" I O',H, ol·1

l~.(.%.aj,,,~ l'\'I,.ltlOol

I~easurea\ value

1.51.5360L- L- L.- ---JL-

ora-­a

-21-

90

,

f3=i- -6

180

27

fOune~.J)esse\ \'II.t~oal

\,,~e8.al1o.., ~eJl\ocl

i"\E'ssurrt! \Ia\ue

1.51.5360L-- -'- -JL-- ..&-.._

ora-­a

Fig. 2/3

-22-

O't---_~

1'(lUYler - :Res'el \MHwe)

1~.Jf? 'iir84,Ol<\ ",d~ool

I>lea'>u,ed vlll~e

-10

\\

"\\.

....

\.\I

\i\-4010- ---1. ..£- _

o 0.5 1 1·5

-30

-20

r(}'c-aFig. 2/4

-23-

,

....................

......... ....

......' ....." ......

, '"'.. ," ,

", ,". ,

,.'. "-"-

"-"­'.~"',

""'~',\

'\.~'"-y,

"\'\\\,

\\\\"\\\

\:\

\\\\

\\\\\\\\

\ \\ \\. \

'., \\ \\ \

\ \

-10

-20

\

-30

Foun", - :Brssel YlI f 4kl'lol

1Y\+q,;ratl"" l\o11"4\'00lY\1fa~\I'(,(~ V~Il\C

r(le-a1·510.5

-40L..- ---&. .....L- -'--_

o

Fig. 2/5

-24-

{3 =t- 6

,

..........",.:::-.:::::-:.-::::::...-....~ ... .............

.......:::.~::-.~ ..............................

.................,....... "

" .........:-- ...." ....

" ..........' ........ "­

''', "....~,.... ,.... ,....0..... "' ......,

"''''....~~..~

" .,'.....,.

',~,

o --__._._

-10

-20

-30

Tou'ter- :B~.,~eJ ~~~~od

,,,~e8' ahOV1 meLl-,nd

I~easureol value

r(l'-=­a1·510.5

-40'-- --&. -'- "'-_

o

Fig. 2/6

-25-

S2. 2. Dini-series.

111e Fourier-Bessel. series is not always the most cali tdlJl (' way t-o

analyse the near-field of an aperture field distributioll HI') wi 111

f(a) ~ O. For functions HI') where no phase-> distribution is involvl~d,

it is easier to decompose f(r) into a constant part (uniform field

distribution) and a part which is zero at the rim. 'l'hl~n 1lH' F'O\1l'iC'I'­

Bessel method can s~ilt be used, for the constant part r;ives u

spectrum function in closed form and there the comnon integration method

of chapter 1. ~ 1.1 is preferred. When, however, a phase distribution

is involved the best way to examine a near-field problem is probably to

use a Dini-series [12J. We used this method, though we realised that it

might not be the best method for both examples offered in this section,

but we wished to introduce and make practicable this approach to the

near-field problem. So now this method will be introduced in greater

detail.

The most general expansion containing Bessel-functions lS

denote the pJsitive zero (in ascending order of

l(t) = i tr¥' JoCAmt)II'I={

where A A .- ....l' ~,

magnitude) of the function

<2-2-1 )

<2-2-2 )

(2-2-3)

and when Y,>:- ~ and H is any given constant, was investigated by Dini

[13]. The nDde of determination of the numbers Am subjects Ht) to what

is known as a "mixed boundary condition", namely that

HD + Hf(t)

would vanish at t = 1 as used by Fourier in the problem of the

propagation of heat in a circular cylinder when heat is radiated frorrl

this cylinder [14]. We will follow this ,met-hex'! w_Lthout being sure wether

this is the appropriate manner in this case of electromagnetic radiation.

The best way to detennine H and Am in this situation requires a sepe..rate

and cautious consideration.

-26-

However having found a real constant value for H and chosen an order

in which we want to develope f(t) as a series of Bessel functions we

have yet to distineuisQ betwe~n three situations [121:

(2-2-1)

(2-2-4 )

(2-2-5)

B<:!:)has to beI 0

1130ft.) = 2 (V+1) +.j rV+1. £(i')dt'

oinserted on account of the zero at

(2-2-6)

the origine of (2-2-2).

(2-2-5)

jj (j:)"" 2. Ao'l.Il),ot) f:f&')I0~')dto Q.o1.+/)I)J2.(AoJ->'~I;l\) .~ ..... (1-2-7)

fJot.) has now to be inserted an account of th~ purely imaginary zeros

±:jf'oof (2-2-:2).

This theory has been applied to several fairly simple aperture.distributions (parabolic in form) which had spectrum functions in closed

form and could easily be checked by the method of integration introduced

in chapter I. ~ 1.1. The result was a rather good agreement between both

methods which encouraged us enough to apply the theory to some more

relevant examples.

(i-i-B)

-27-

Lxample 1.

The theory mentioned in this p:1rawaph can be applied to an aperture

distribution of the form~' ·~t1

E",(r,o) .--.::;. £(i:) =)1- JO~DI{)})

It is clear that f(a) # 0 here, moreover this aperture has an

amplitude and phase d.iL;IT.ilJutioll wh.ich is thl' r.'(~Vcr'L;(' of t!loD(' til

expression 0-2-3). The main purpose of applyiIll', the developed nlrthod

to this particular 'exJmple is to examine the devergence of phase aru

amplitude. tbre explicitely, we are looking for a value of z which Eives

ooth optimal uniform phase and/or optimal uniform amplitude distribution

and we are curious to learn whether this prublem can be solved using the

method of application of a Dini-series and whether there will be any

gain in computation time when we compare this method with that of

repeated inter,ration.

Let us follow the solution of the problem in detail.

Applying (2-2-3) on (2-2-7) is possible when we split (2-2-8) up into

and

Now

fr(t) ={1 - JO(jOlt)} cos St'

[f;(t) + Hr frLt.)] =0t=t

(2-2-9)

(1-2-10 )

II = 3.3923 .. (2-2-3)r

Further,

and then

fill) ~ {1 - JO(JOlt)} sin &-1:'­

[f{(t) + Hi f[ (tjt~O

<2-2-11 )

gives H.:- 2.3312 .. (2-2-3)1

Hence we meet two types of Dini-series for solution of thi~ problem

when we develope in Bessel-functions of order zero:

fAt) =f ~~ Jo(A~t)m::l

( 2-2-12)

bh'l-

1

2 0/ t £/i:) Jo~~ t)dt

"J:(A~) + J~2(~:) (2-2-13)

-28-

<2-2-1lt )

C2-2-1G)

4 ' 1

= 2:.I.o(0,!l__ jt :R({) Io(>'ot:)dtI o

2( Ao) - .r'o1,xo) 0

1

2 it f&) Jo(A~t)dt= ----_.._Q-----_ ....-._._-----_. .... ------ - -_._-

~o~(~) + J~\I\~),10 is the Bessel function of the first kind with imaginary ar8Uffient.

For the series g, and b. we need to know the values of a. nwnber of

coefficients ~ .,A~ and Ao • These we determine with C2-2-2). It 1S

not difficult to prove that between every two zeros of JOCz) there is

one zero of JOCz) and vise versa and that all zeros of the function

(2-2-17)

are real and single-valued when V>-1 and H is real. [IS]i

-1 Fig. 2/7

The first ten-zeros of GOCz) for f.(t) and f (t) are determined with1 r

the help of a very simple computer program. AD is determined in the same

way from

(1-7-18 )

The last expression comes from (2-2-17) when z = i~o and Ao is real.

-29-

Table I.

>.; = 1. 842 -I-~~ = ;~-;~~------ -r------~o =;-.-;05-

~~ = 4.522 . A~' = 6.688

A; = 7.45 5 A~ = 9.446

~~ = 10.491 \~ = 13.150

~ = 13.571 A~ = 16.330

~ = 16.673 A, = 19.498

A~ = 19.787. A~ = 22.658

~ = 22.908 A~ = 25. 814

~ = 26.034 A~ = 28.967

~o= 29.163 A~ = 32.117

Now it is possible to determine the spectrum functions fr(l) and fiCO)

of f (t) and f. ( t ) .r 1

1

FrC O) = 2'ITaljf~).Jo(Oat)dto

See [16J for the derivation-of this last expression.

If we write J3 ~ bO·IO(\t) we likewise obtain:

With (2-1-5) and (2-1-8) it is now possible to write down the complete

expression for Re {Ex(r,z)} and Irn{Ex(r,Z)} .

-30-

-31-

For numerical investigation of (2-2-21) we made usc of 1h08(, d impm, i.onG

for a (non-('xisting) llorn WP had dlr'(~ady u[;c'd in chant (~r I and i 11 ~ 7. 1

of this chapter. ThE? c~prcGsion (7-7-21) Wil~'; evaludll'd for'd Idr',!'/, rnun!ll'l'

of values of (3. For j3 = B. 77 th(~ ('(juiplldGC 1inc w,u; fOUl1d t u hI' LI <1 I. Ll':; I •

The area to be searched WiU:; found by the fo1 towinp r'C'dr;c)J]:i 11)',:

It took the length of the horn to convert'/' frem the uniforJTl ph IG('. !ort~

diBLribution La the dj.~;tribut:ion eJe

uo .iL pn-)!Jdbly would ttlkl' "boul1.s~2

the length of that horn also to diverge from e to c1 next uniform. I

phase distribution. As can be seen in Fig. 2/8 and 2/9 this "theory"

lS shown to be highly.probable for this casco

An LnlJX)rtant conclusion i~; now thIt, prov i lkd we hWI' d tcchn i qt\(' to. jcSP

perform the phase distr'ibution e ill Ull~ apl'rLuI'l' of " cOl'TUgdt l'd h01'11

we are capable of making a unifonn phase' plane at any distance from 1hir;

horn; this distance being dependent on ~ = ka" . A further conclus ion lS:2L

that the area of flattest amplitude docs not coincide with the vdlue for

/-J with the roost uniform phase dbtr:ibution, so prohably in th0 funct ion

(2-2-8) (j does not have the right value. (fig. 2/10 (mel 2/11.)

Only a few points of the space in front of the apprtun' were checked hy

the method of repeated integration. Good ar;recment,wcls found oetween

both methods.

Not all properties of the C'..xamplp- were gone into in detail. This example

is intended only as an illustration of the utility of the Dini-series

method.

Some observations can be made about the nwnerical procedure ..1. It was foumto be sufficient to take into account only the first five

terms of each series. The value of each next tenn was smaller' than 0.1%th

of the 5 term. ka

2. Integrations foowere again replaced by J for the same :r;>eason as haso 0

already been mentioned.

3. 'The accuracy of the numerical standard integrations procedure "INTT~GML"

in "Beathp-"language (dcomputer language used at this university) could

probably improved by cautious study of the size of the crron~ in l~i.lch

integration in dependence of a number of parameter~-,.

4. Co~paring tl1e Dini-series with the repeated integration method the

computation time is shortened with the factor 2.

+18

+90

-9

-32-

8.0/\, \, \, \

I \, \I \I \

7.0-... I III " ,

I '" I: "', , Ii ." \I ,.... \

j I '.I , '~,! I"i Ij ,i ,I I

I I, I

f :I :! Ii Ij Ii ,: ,i II ,

j ,i I

II

Il---------:lj--------:1;----:~ir~----:11:i.55-180 .5 a-ao

Fig. 2/8

-33-

+18

zf)-a-

..

+90

8.77

-9

oo

.......,\............ 0"'. 10,., \. I

\.... I' \,,\ /\ \,...-/ 19.37

• , I, I\ I, I

\ I' .... '"

l----------:5-------~1;_-_:~~r~--11:5.5-180 .5 a-ao

Fig. 2/9

-34-

-10

-20

-30

............

.... "'~ ..............

10.0 ..._....-

• 9·3;' .----­8·H--

..\

\

1·5r(}'-=­a

10.5-40'- --' --.L ...I--_

o

Fig. 2/10

-35-

I,,I,

I,It

I,,I,

t,,,IIII,I

......- ....~ ......~

"""...

/iI

/.

,

i

\"

\.

,,",

\ ,\\\

\\\\,

\\\\\\\,

\\,\I\I,III I, , .

\. ~ /\ Ii\ ~I ~

i f,I ni, i'

: 1\; i I

i ,! 1

'\ i :I '1

\/ II\II\\,

\\\ I

\ "oJ

S.1T --R - Z _ 8.00 ---~ ...fJ -1f 1.00 ......-- .

;,.,. ........ '"'I ...........

\'"" \

... "\\

-10

-30

-20

l-------(1.5-------~11-~~~r~--11i:!.5:i~ M a.~o

Fig. 2/11

-3b-

/"<--,-~I------,

If' '~

rig. 2/12.

Examp1(' 2.

The parabolo:id r'Cf1('C' tOY'.

/\ scC'onlexdmple we want to ('vdludh' by 1lI('dm; uf d [)ini :;('1'1(':; I:; Illdl~ , '

of a put'abolo'i.d r'eflcc tor' wit h l' i ('ell 1<.11' C!'O: ;r;-:,('c t i Ull. f\:; ill llm i 11<1 lUI'

or feed has been made use of a corn.lf~a t cd horn wi th ], I r'p'<' f 1ilf'(' (1 np, 1(1 •

'1110 ckctrie field hd[~ 1>('<'n 5tulli/'ll by ,Lm:;('Il, d('llkl'll dllll 1.lm}Jr'<'<'li1r;<'

l17J . \.VlWI1 wc.' tellq ~ Ill<' (11)< '1't llY '(' :;f\

of such a horn as part of cl sphere

with radius 1" (Fig." 2/12) the far

field of this horn can be deduced

1n a simplified form as

E8 = constant . F( e, 80 , kr' ). co<, <p

Er = constant . F( e , eo, kr' ). sin epThen also \Eel:l + )E Cf[2 is independent of <p.I-Ience we conclude that this form has a

I 81,

~,-- ~i~_."--__Il::-

. £ IFig. 2/13

T-D/l

of the reflector can be calculated by

evaluation of the ap~ture field, .

expression in a Dini series. As a

first approximation to the situation

outlined above we will make the following

linearly polarised field which has a power spectnlm independent of r.It can be proved that the field in the aperture of the reflector is now

also linearly polarised. This is done in appendix:B , where we also

prove that the linearly polarised plane wave propagates with positive

z-direction (Fig. 2/13). \I}hen there

1S a way to determine the aperture

field of the reflector the near field

assUIllptions:

1. TIle reflector 1S located 1n the Fraunhofer reglon of th~ feed.

2. The field in the plane z = 0 is the same as if the radiation of the

field was reflected at a perfectly conducting reflector according to

geometrical optics.

3. TIle radiation field of the total antenna is the field that W<:.Juld

result from a field distribution 1n the plane z = 0, determined

according to assUIllptions 1 and 2.

-:37-

The first llnPJrtant step we make to de terminetl1c c1p(~rjure field of the

reflector is measuring the gain function of th(" f('eel. '[lLi i, ]S dC)T1C' d L

a frequency of f = 15 GH~ for tbe

feed sketched in Fie. 2/1'1. '111(' n:.>l,U It

of measuring th.ls gain fune t ion is

to be seen in Fig. 2/15. Fla.re •an~lego

For the reflector D = 50 ern and flD = 0.4 (Fif,. 2/13). Making usC' of the

assumptions 1 and 2 on page 36 it is PJssible now to determine the aperture

field of the reflector if we are able to approximate tr1e measured values

by a mathematical form. We call the measured p,ain function F<e); this is

a function with e as varidble. Now with Fig. 2/13 we find for' ()

paraboloid

r'p = 1.L1+ CDS e

e = ~r}..§ .r1 + easEl

= £coste W,2)

(2-2-23)

(2-2-21+)

Using only geometries it is to be seen that in a plane perpendicular

on the z-axis the phase is constant. Thus there is a uniform phase

distribution in the aperture of the reflector [18]

E = lEI. e- jk .2£ .

The amplitude 1S attenuated due to the scape of the reflector, so now

E = f(e) =relabve-.1..- r1(8)._

cos'(t}'2)(2-2-75)

By trial we find that a very good approximation of the measured function

F(e) is (Fig. 2/15)

2( .) .'sin 122 -8 - Sin 122.2

(2-2-2(:; )

( 2-2-27)

fITa)

0.2

0.1

-38-

\

~\

\\\

\\

\\

measured 'value

apprOl('ll1atICYl

36"12° 4BO

--- ....- ···~e

Fig. 2/15

6Cl

-39-

With (2-2-22) and (2-2-23) We' find

.. ~6L~ - 25Psin e = ~-~-~- =

L +~2 -b,-;~;S't.1l

~r'

~ 80 t,case = £ = --~._--

Ii

~2 bLl + 2~P+~f~

U-i-7fn

( ?-?-i'1)

Taking ~ = ~t we find thLlt the rim of the r'eflcctor hd~; the value t = 1.

How

Using <2-2-3 )

(2-2-2)

( 2-2-18)

it appears that H = -3.3232.

So the Dini serles we have to use is of the type 3 described on page 26 .

Then ~vith Z ']:(2) + HJoCz) =: 0

and Z..Ilz) + HIo(L) = 0

the following values for \tl and AD in d. Dini series (2-2-8) arE' found.

Table II.

AI = 3.081

;\1= 9.562

>'3 = 9.851+

\= 13.078

\ = 16.271

\= 19.448

\= 22.615

\= 25.77G

\= 28.933

A= 32.08710

"\= 3.872

___--1- ---------

-40-

With (2-2-20) we can directly Wl'ite down the expressions for f(O):

. ISo with (2-1-5) and (2-1-8)

-L~l-

111e conclusions which can be deduced from Fig. 2/16, 2/17, 2/18 and

Fig. 2/19, 2/20, 2/21, 2/22, 2/23, showing the numerical results of

this e~amination, are that the results are not so very good. But f,ood.. '

resul ts were not to be expected due to the (,[fpet of LlH' h01"11 and tlw

struts in front of the reflector, called "hlocld ng". L:~;pt'c idlJy cClInpdri nl'.

measured and calculated amplitude values is difficult, for we do not

know the exact attenuation at thE; axlS as effect of the "blocking".

As for the numerical approach the same remarks are valid as mentioned

on page 31. Looking bac~ at the general remdrks at the beginning of 32.2.

we must note, that for this special case probably the application of a

Fourier-Bessel series, after decomposir~ the function (2-2-30) as

described formerly, would have been easier. 111e reason we applied the

Dini series was that the computer program made for the case described ln

example 1 could very simply be converted into one suitable for this case.

-42-

+18

+90

. ,

o

O'r-----.....__

-9

:Di,.,1 -series

l----------:!~------~1;_~~::~r~'--~~1l:.55-180 .5 a- a. 0

Fig. 2/16

-43-

+18

...

+90

o

-9

,,/

II

II

II

II

I,,.. '

:Dini - serir S

\'l\PllsUHd vslLlt

r....I \

I \: \I \

/ II ,

I \

I I

" II ,

IIIII,ItII,II

I~I II I

I I,I II ,, II II II II II II I: I, II IIII

IIIIIIII,IIII

l---------:~-------~1~-~~~r~---1.1.55-180 .5 a-ao

Fig. 2/17

-44-

+1,8

f3-t -6

..

+90

,'"I '1 \

1 \"'---_..... ,1 \

\\\\\

~~

\ ,,,,\\ I\ J

\ I

" /, I, I, ,"-,

1)ll1i· 5e",~~

V\1~esur.J vi\lue

-9

-1.80L--------=-------~1~------l-11~.5io ~ ra-­a

Fig. 2/18

·-45-

..

])jni - series

-10

-4oL------IO'~55-------~1--~a~c~~~--1j5o

-30

-20

Fig. 2/19

-46-

-10

. ,

-20

-30

l-------Il5--------11-~~~r~--11w.55-40 0-5 a~ao

Fig. 2/20

-47-

-20

IIC'l

1i:r--30

____ mr8sured value

-4U'- ----I .L-- ........_

o 0.5

Fig. 2/21

1r(l'-=­a

1-5

-10

. I

-20

II0/

1iJ--30

-48-

l-------~ii------~11-~:~r~--11~.55-40 ().5 ll' c ao

Fig. 2/22

-49-

-10

-20

-30

____ measured value

1·510.5-40·L.- ---" .....&... """---_

o r­et&:­aFig. 2/23

-50-

III. NEAR-FIELD APPROACH BY STEEPEST-DESCENTS HErnOD.

We ar,ain consider the pla!l('-wave-spectnlln notation for' thE' !1('df' field

of t)'I(' corrueated cortj C~1 Ilorn ~

with

and

j"" ·z~:l.

E)((r,~) = i~ f(y) JoCOr) e-J adro a

£(0) = 2Hf [/F,O) JoC1rr)r.drI 0

Ex (l:-,o) ::: J o( JOI~) , r< a

(1-1-1)

( 1-1-:3)

Now f (0) can be written in closed form so that a stcl'pesi..-clcscents

approach is dir<.~ctly ilpplicdblc'.

= 2'1li . jOI J 1 (1 01) J O(T8..)Jo~ -o'l.i J ·v

J 0 ( ..... ) and J 1( ..... ) are Bessel functions of the first kind and j 01 is

the first zero of J O' 1118 complete formulation for the x-polcTI'isE'd

electric field by substitution of l a = X but not %::: ct becomes

00 _ jf3A~- )0;"

C)(Q:-,z.) = JOI J1(jol) r"J~Q5)~olx·5).e xd..x 0-1-6)1) JOI->1;o

(3-1-1)

Jk1a1._~:l. ==. kz. :::. ka sin «x ka cos oc.

For application of the steepest-dE~scentsmethod we fin;t have to

transform expression (1-1-6) to a contour integral in the complex

-plane using [If]

-51-

Fig. 3/1 illustrates the aper1:ure

and its near-field region and fir.;.

3/2 shows the path Co in the c~mplQx..«.--plane with

fI

2.a.

"Bo-",

/clrcularaperture

I

r

Fig. 3/1.

, Co

Fig. 3/2.

TI1e application of the modified steepest-descents approach [4] , [5]

requires the possibility to deform the path Co to a proper steepest­

descent path. If we call the exp~ession

Pcco:, (() = JoC kacosoc.) Jo(krcosx) . sJnt£k~~St£2......... (3-1-3)JOI - cos ex:..

we note that Co ln formulation (3-1-2) cannot be directly deformed

to a steepest-descent path. We shall explain this statement below.

The spectrum P(cos~) can be seperated using the Bessel-function

identity

(3-1-L~ )

-52-

'rhus the field can be pxprer;sed uS

(3-1-£;)

(3-1-7)

(3-1-8)

111e convergence properties of (3-1-G), (3-1-7), (3-1-8), (3-1-~1) can be

detennincu by cormid(~ratj()n of the df--;ymptut ic fonn for Ln'g(' dr!,.um('llt~;

corresponding to each integranu when Icos (X \-- 00

'TIIUS with 6\ . r;Hv(z) ~V#z. ~P{j(z-~})rr-~'TT)r -[1- C.1(~)+O(~.}-··l·. (3-1-10)

-Ti <arg, z<~".

(3-1-11)-'-~<ar8' z<'fI'" •

fonnulations are obtained in which the sin8ularity at ()l. = ~ n~sultinr;

from the product of two Hankel functions is cancelled by cos ~ in the

denumerator but in which the' sin8~larity for(j01 - kacos IX) = 0 is

not cancelled now by JO(kacos oc) in the denwnerator of e..xprcssion (3-1-2)

It is easy to verify that ar8% luaains within the permitted area and,

in fact, on path Co

-53-

We find, introducing kO = 11' ~.Jra

..-jk (-r-a)co!>oc -jk-z. sr\'\£x

e

-jk (r-a)c05~ -jkz <;;inO(e

. .. 0-1-1))

... (3-1-1:1)

Fig. 3/3 shows the deformation of the

path Co to C~ as a result ~f the

singularity at Cf = arccos ,~ the

contour modification can be chosen

to pass below of above. the role. The

resulting sum (3-1-5) is quite the

... (3-1-1[1)

... (3-1-15)

same.c'o

Fig. 3/3.

Usin['; R1 = Vf:f. _a)1 ... Z1

R2 = Vr.t+a)1 + Z2

........ (3-1-16)

6t = arc tan 2:-r-a

8,2. = arct.a.n ...Lf+a

-54-

:expressions (3-1-12), (3-1-13), (3-1-1'+), (3-1-15) can be written as

· .. (3-1-17)

• .. (J-1-Hl)

· .. <:3-1-1 Cj)

.•• (J-1-2())

The convergence of each tenn 1S detennined by the C'xponentidl Lw t or

in its asymptotic form; thus consider the convergence of

-jkR cos (e-«.) -k'R sin(D<.r-e)sinh O(.t -fRo C05(O(r':S) cosh OCt

e = e . e

'Ihe exponential factor converf,E~s for regions where

kR sin (OCr -e) sinh .xl> 0

(3-1-21)

· .. (3-1-)7)

descent point O:s = e lies on a

3/l.t

the convergence

A.

-55-

Thus EX1 has the steepest-descent point 0(.101 - '11'-81

EX2 has the steepest-descent PJi.nt "'-Sl = 91

Ex3 has the steepest-descent PJint IX. 53 = rr - 8tI

EX4 has the steepest-descent point 0( 54"" e1

In the following observations we must never Im;e sir,ht of the fact

that the near-field under consideration if'; so near tht~ dpcrtun~ thert

always 82< <p = arccos Ij~l (see Fig. 3/1). This condition can be. K.a

expressed as o/Irl+z1>~ which clearly shows the interdependance of

z, k and a in this situ~tion. A number of possibilities still have

to be distinguished for 81 and ~-~ leading to different results

in accordance with the fOTIns in which E are expressed.xThese situations are

~ 81 < '!T/2 and ej <0/ or 61>epIT-e1>~

2 81 >% and 1T- 61<0/ or 61 '/<p'fr-~<%

For this study the situations 81 =; , 81 =0/ and rr-~= ep are not of great

interest because the electric field at the outside of the aperture

will have the appearance of a smooth curve, so we can easily drop three

PJints without fear of loosing infonnation.

For each of the situations mentioned at 1 and 2 we will now make a

sketch to illustrate the defoITnation of Co to the steepest-descent path.

j"'t

A-1. EX1 has the steepest-descent

PJint (W. s = 'IT - 82.

IT-e~ >~C~ closed with C1 to contour.

IC1 can be called C1 + Cr where

Cr is only that part of C1close round ep

..

A- 2. LX2

has the st(~ClX~f;t -dl'SCl~lll

jx)int tt I = 61

A- 2-a 81 <epS \.l 'cpcs t descent J)<11 h (..'ld·

C~) closed with C7 + S1d to

contour.

A-2-h 8J >epStccpest-dpscent path Sl

, . bCo closed wlth C2 + Sih

contour.

A-3. EX3

has the steepest-d~:;scent

point 0:: s '" 'IT- 61 , IT- ~ >~C~ closed with C1 to contour.

C1 can be called.C~ + Cr where

Cr is only that part of' C1close round <p.

A-4. EX4 has the steepest descent

point ~~ = el

81 <<pStpepest descent pelth ~~')

I

C' .o closed wlth C2 + S2

contour.

, 1./ //,'

-" I, './/~/.{ /~ ~/ '

/;/'// t;,/,.

/ '//}"-Q// /'////1////

/ ri 1',. 3/7. '

B-1. [xl har; the stPCJK'Llt d('[icl.:nt J)()ild ~~='Jr-e2

1r-e, >~

C() clor;l>c1 witt1 CJ

tll nmlllUl'.

C1 can be' cdlh~d C~ + Cr

whl 'l'l , Cl' I:; lllllytlldt 1>,11'[ of C:1 l'llJ::l' I'olmcl

( St"e ri~'). :3/!»).

. ,B-2. E

X2has the s!:eepc::,t dc:.;cpnt pOlint- (X.s"",91

el>~C~ closed with C1 to contour.

C1 can be called C~ + Cr, wh~~rc Cr

1 i:; only Uld L pen'! of C1

cit J::e round

(See Fig. 3/5).

B-3. r:x3

hew the' stCl'!X'!;t dl'F;C('rlL puint· oc\ = 1l-e1

B- 3-a ". -81 <epC~ closed with C2 + S1a to contour. (See Fig. 3/6).

13- 3-b 'Tf - 81>'fC~ closed with C2 + ~;1b to contour'. (Sec Fir,. 3/7).

G-4. EX4

ahs the steepest der:;cent point IXs =e~

e2 <epC·.''· ( • /'0 closed wl.th C2 + ;';7 to contour. S('l' rig. 3 ~l).

Table III.

..r>a or Sl <1Ji~ r<a or 81) 1Ji,2

Ext Cr+C~ C +C'r 1

---~---SI<<P 81>ep

Cr+C~EXl S1a+Ci,

~b+C1..

n-e·)Cf 1f-fll <<r

EX3 Cr +c; S1\tC1 S1lL~

Ex~ S2+Cl Sl+C.2

A B C D

Fig. 3/10 illustrates the n~r,lOn,~ cOllcerned vIi th each of the combinations

of contours in table ill expressed as A, B, C or D.

Clrc.ularaptrl:un:

D

z:>

'If-a-'" If--­.-----._---._--

Fig. 3/10

He can verify that for all deformed contours argz remlim; wi thin the

limj t s for which the largf' arr,umcn1 ('xpall,; j ems of th(' 11.1 "k('1 fund i.on,;

are vdlic1.

-59-

1. When we consider situation D (Fig. 3/10) we will be ablE' to demonstrate

the mathematical technique used to achieve a fairly simple fonn for E .x

We now apply (3-1-10) and (3-1-11) to the deformed paths:

E:2 = -K'a' j~ JI~bLJ (\ i~)(.ka t:lS rx.)

J.C1+Cr

E:,~ -k"'j~JIY'iH:\ka<o, "'-)SIQ.f-C~

E:',= -k'.'j~.Jl:plj I{\ka,",~)

S:z+C

It is readily shown that the sum of the contributions to C1 of (3-1-23)

and (3-1-24) is cancelled by the contributions to C2 of (3-1-25) and

(3-1-26) for the reason that, using (3-1-4) to combine pairs (3-1-23),

(3-1-24) and (3-1-25), 3-1-26) to give one formulation each, we obtain:

+ j dt<J on (1

c1~ =- j J \Xl' oVl (~

(3-1-27)

-60-

Cornbininr, the rcnuinder of (3-1-23) and (J-1-2lj) we filldlly OlJl,lill:

..1-·k :dn IX_'~,\~ ,~_( ~y~ d(:( (,

i,~ - k'<'l' (('1\1r.<

- j;.bi'HX<,i~\ il<,_( C1'>.t5 dIX.. ('-- -T~~ ,k';l' '(1~'{'(

(3-1-;>fj)

The first part of (3-1-28) is the residue contribution in

cos D( =~ = cos ~ found applying the Cauchy residue theorem

with

sin D< (OS·D<

jOI ~ - k'a~ (os'O(.=: ~il'l 0(. cos rx..-----

k'a' (cor::Cf- COS'IX)

we find

e :1.'iTj

::: 111 j

:= 'IT';2.

- JIVkta'-jJe ......... (3-1-30)

The last simplification in (3-1-30) comes from

Wrons~ian 'W JJv(z) , ~(Z.)} =: J\I+-l(Z)'y;(z) -J~)'(v+I~)=2 (3-1-31)I 'liz

--61-

A complete and partly apprDximated ('xprcssion for (3-1-28) Cdn be deduced

by again using the asymptotic apprDxirrutions for thc lIimkcl functions with

larr;e arVlffiento

Hcre

jOI ku J,gol) j _.____ _~in 0< _4. (cos I.f -(O\o()( cos cf' H ':'$ K)

~ia

- JO }r k -JeJo,} 1 ..__:>il1 D<.. _

~ 1 «OS~)-(O\o<)((()~(r+(O'iI'{)

52-(with appendix )

_jkR, '"\Qr-0,-0<)nlrx e

jk'R~ (C'$ QJ~-1'\)

(~iX. e ° 0 o. (J-~1-37)

Hence

000. (3-1-3'1)

000 ° (3-1-35)

. k J ' -pTiL, -JkRt . - e - -_j )01 0 ,go,JVIT. e . e Sin 82. 1[-0;(k'R~co\~~~1. ° o(3-1-3G).2. Sj"(e.'l.;Cf)~oS~+COSe2) R °/j

If kR> 1 a further simplification can be achieved by using the large-

argument fonn of the Fresnel integral (appendix Ab )

. O. ° .(3-1-37)

This glves (k - 2 )0- --'ilkfi

--62-

or

SII'1S,(cos ~+ Z;S60CCos <f-=--(O~ 81)

(3-1-38)

This last approximation, however is not quite satisfactory near the axis

of the aperture (r = 0 j because of the fact that the argument of the. -

Fresnel integral can be too small for application of the large argument

approximation.

-63-

If we make use of the procedure in determining E~ we soon come to

expressions for E~ , E: and E~ .

(3-1-39)

(3-1-l~0)

Hence

and

"2. We proceed with E~ . From table I and Fit~. 3/10 we sec that the only

difference with the former situation is an E,xtra residue for 0( = cp •This becomes evident when we write Sib = Sia + Cr , allowing E~ to be

deduced by the expressions (3-1-23), (3-1-2l~), (3-1-2~)) cllld (3-1-26)

and adding IRes (Ex~)l to them.L y=<p II ('\ J \0

Now . rl}l (z) = vC?) + j ,(z)

H(2) ] \/v (2) == _ y (z) - j r'('2)

{"Res (E~ l = - {'Res (E~3 )}JC(~lf ()( =t.p

The only residue remaining is that of the function E~l at the :[X)int ()( = cp .Thus all we have to do is replace the first part of (3-1-38) by the value

appropriate to the residue of Ex~ in IX = ep.\Je find

-64-

3. Determining E~ 1S found to be slightly more complicated, for othel'

paths have to be considered. Now

Now

,- k'a' ]'; J,gi H:\ka cos"')

C\+Cr

E,~ = -k'a' Jy~,gfIC(b'OS<X1~+~b

E~= -k'a' jOI ]g01) H~)(kacoSLX)'1 I

(;1 +-Cr

:ll~ll / J J(~)EX4 = -k.alj~~gJ' 110 (kaco'iC<)

C1 i 52. .In this case we again see from (3-1-27) that the contributions OVer C; in

(3-1-42) and (3-1-43) together cancel those over C~in (3-1-44) and (3-1-45).

M .. f . f . IS d.Boreover, the contr1but1ons rom res1dues or tx: r1n Ell, an EX3 together

glve zero (see (3-1-39) and (3-1-49)).. .. f c. . . f'l1ms Just as 1n the Sltuat10n or E x there J.S only onC' contrl.but Ion . rom

,.:Ba residue, that of EXl •

\Je find! ] - j ~~k'a'-jo~

E)<~ = \ "Joyol~J - jY.ogOl~J i .ea.2 -

(lli-Tf-<I\ -j(05 _. ;2.)_

-65-

'"4. E" is finally found by omitting the contribution fr'oT!l the rer;lJuc ln

LX:!. •

We thus directly write

A' -t% -jk'R~E, = - J" k~::J ,Go,) Vii e e -<in(" ,;i'F)R((~-;~:. ,,;ie,{[4>1<ii,

On sUbstituting e,=ry'll. in (3-1-/+1) and (3-1-46) exactly the same result

is obtained. This should not come as a surprise for e, == ~ is not a

singularity, so that in Fig. 3/10 region B and C need not be seperated

by the point 8, = '!Th. • At this particular point Lh!:y an~ equal.

5. For the on-axls field I' = 0) we have to combine the two intq"l'dls

(3-1-25) and (3-1-26) and set I' = 0 in them.

This gives

The steepest descents paths Sla and S2 merge for I' = 0, so we find

Here

JjTliZj( - jk'"R w sC9-t><.)e SinO( coso<. dO<.. e

~ cos 0(., (COS~(D -Co'5~iX)S \: 1 ••••••• (3-1-49)

(3-1-50)

-66-

Then

- 'kRJ ~IVl e C05 ee :r (~ • )\j (O~e Cos ep-COS e

(3-J -51)

or USlng (3-1-50),

. J (.) ka7z-J' .JDI 1,1 01 '.:.J J ~'t)':l k'l 4

tJl" - a

(3-1-52)

Conclusions:

The formulations found in 1, 2, 3, 4 and 5 for this specific type of field

are very efficient when considered in terms of computation time. They are

also reliable when we will determine the amplitudes of the fields.

Applying this method for determination of equiphase lines however gives

no trustworthy answers. See Fig. 3/11 and Fig. 3/12 where numerical results

of (3-1-38) concerning region D are shown for amplitude and phase respective­

ly. In these figures a tomParison is made with the results of chapter I ~ 1, ,

where the results were obtained with the integration method ..Probably the approximation used to sjmplify the Fresnel integrals were the

reason for this., Therefore a further study from a pure nwnerical point of

view is necessary. Herein the exact integrals have to be calculated.

-67-

___ sh.e<>est descel1tsmelhod

--- - - ,nt~9ra.t,on method

o

-1

-2

-3

-I.(

~ -5

it~ 1-6-..;::.

-8

-g

-10

-11

,

~\\\\\\\\\\\\\,

\\\\\~

Fir,;. 3/11

rif,. 3/17st. desc. m.int. melhod

...........

....... -----_/o ,\

\

\\,,

\,\\\,

\\\\\

\\\\\\

-Il;'--_-:':-_~::__-~-~----'---_:__----.......:...\ ~~0.1 0,1 0 3 0,4 0 ~ 0 6 0, 08

~ O{:La

_J~O

-10

-68-

REfERENCES.

[1] R.C. Rudduck, D.C.r. Wu, Cl[ld M.l~. Intihar':.."N(~ar-fi(~ld !\nalysiL:; hy the Pl(Ulc-Wdv(~ ~;p('ctrum I\ppl'O<1ch" ,

IEEE Transactions on I\ntpnnas dnd propdr,dL Lon, flkU,<'h 1973,

pages 231 - 234.

))J Alexander C. Brown, Ir.: "t1avenumber - bandwidth - limi tied

near-field", Rad"io'Scicnce, Volwne 11, nwnbcr 7, pdgCS ~B3 - ~(Ll,

jUly 1976.

[3] R.C. Hansen and L.L. I3a.ilen: "A new method of near-field

analyses", IRE Trans .Antennas Propagat. Vol AP-7 pages S458 - S467,

december 1959.

[4] R. C. Rudduck, Chin-Long 1. Chen: "New plane wave spectrum

formulations for the near-fields of circular and strip apertures".

IEEE Transactions on antennas and propagation, Vol. AP-24, no. 4,

jUly 1976.

[5] P. C. Clerrnnow: "The Plane wave Spectrwn, representation of

electromagnetic fields", Oxford England, Pergamon Press, 1966, pp.ltJ - 58.

[61 Goubau, G. and SChwering, f.: "On the guided propagation of, .

electromagnetic wavebearris", IRE Trans. Antennas and Propag., AP-9,

pp.248 - 256, 1961.

L7J Stratton, LA.: "Electromagnetic Theory", Hc. Graw Hill, N.Y., 1941,

360 - 371.

[8J Jeuken, M.E.J.: "frequency - independence and symmetry properties of

corrugated conical hornantennas with small flare angles", Ph.D. Thesis,

1970, Eindhoven University of Technology, Netherlands.

[9] Clarricoats, P.J.B. and Saha, P.K.: "Propagation and radiation

behaviour of corrugated feeds", part 1 and part 2, Proc IEEE, 1971,

118, 1167 - 1186.

-69-

G.N. Watson: "A treatise on the theory of Bessel functions", Cambridge

University Press (second edition 1966), pp.576 - 596.

[11] •• AIG.!L Watson, llnd. p. ~)~J1.

G.N. Watson, ibid. pp. 596 , 5~l7.

G.N. Watson, ibid. p. 571.

G.N. Watson, ibid. pp.577, 606.

Prof. Dr. J. Boersma: "Toe8cpe1.ste /\naly<;c I, sylla!lu,,;to the college' cuun;l~

of Prof. Dr. J. Boersma, pp 75, 76.

or G.N. Watson, ibid. pp. 480, 481, 482.

~6J. Prof. Dr. J. Boersma, ibid. pp. 61, 62.

or G.N. Watson, ibid. pp. 123 - 137.

r7J Jansen, J.K.M., Jeuken M.E.J"., Lambrechtse, C.W. : "The scalar feed",

report Department of Electrical Engineering, Te~hnological University,

Eindhoven, Netherlands, 1972.

[18J S. Silver: "Hicrowave Anterma theory and design", Me. Graw Hill, N.Y.,

1949.

[19] A.T.W. Titulaer: "Cross-polarisation properties of paraboloidal

reflector pntermas with elliptical cross-section", report graduate

work ET-l0-1973, augustus 1973, Eindhoven Unive.rsity of Technology.

-al-

(A-4)

(1\-2)

(A-3)is real. With

a. I\E5SUffiiJng the :J:'(~P~;~~~~\(~J~)npf th<' (-' L(~('tric fi('ld t () b{'

= 1\050<..) e . . . . . . . (1\-1)c . .

the method of steepest descents proceeds by flrst dlstorting the original

path of integration C into a new path everywhere along which

jkR{1- c.os(e-o<.)} .......

0<. = lX.r-+ jlX..i •••••••. ,(A-2) can be expressed as

-jkRcoS(lX-8) = - jk'R cos(D(r-8) cos'r1 ()(i - k'R siv1(DCr-e) sinh 0(1

(A-5)

It is therefore necessary for the convergence of the integral that

sin(El-lX.y)sinh()(.{ be negative when O(c_±oo ; and consequently the extremes

of any path of integration obtained by distorting C must lie in the

shaded sectors of Fig. All which are specified bye < o<'r < e+'IT '.J~eY1 lXt >D

-TT+e < O(r < e w~en 0<:,«0

To come now to the specific possibility that a path can be found such

that everywhere on it jk"R \I - cos (e-t>()} is real, it need only be noted

that the requirement, evident from (A-4) is simply

cos(D<r-e) C.OShlX( = 1

By comparing the graphs of cos x and sech x it 1S readily seen that the

path specified by (A-5) passes through 0< "" 8 at an angle 1J4to the axis,

and has IXr-=tl+ "IT/~ and O(r" e-'l}'Q as asymptotes;

it is shown diagrammatically 18 Fig. A/2 and 1S henceforth designated as

S(8) •

I

C:

Fig. All Fig. A/2

its IIlc"1XlmUm vdluc, zero, at

-a2-

Moreover, (A- 5) is equivalent to

siV\ (!x, - 8) :: hnhcxI

and when (A-5) and (A-~) are Qveyed (A-4)

-jkR (OS (0<.-8) =: - jkR - kR ~inhlX': tanh o<,~

Hence on S( ) the real part of (A-7) has

«(\.-6)

states

(A-7)

0<. := 8, there 0<.( '" 0, and decreases monotonically to - 00 away from 0<. '" e

on eather side. It is therefor'e possible, after distorting the path to

S(61), to change the variable of intep;ration from ()( to'l where

-jkR ros( tX.-e) =' - jlR - k'R<C . . . . . . . (A-8)

and runs through real values from-«>to DO. Evidently (A-8) is equivalentto 't";: I.{i . ejTT;/j. sini1(oc.-e)l ••••••• (A-g)

.t'TTA Jso that ott: *e-J 4.V1-~j'[' doc. ••••••• (A-10)

and the explicit transformation of the integral 1SpO

j -j k'R (OS ~ -0<) •r;; V(4 -JkiJl' - k'R'C~"P(cos 0<) e doc. = \/2 e e (cosO() e d'l

S(fI) -<>0 vi 1- 1J '[~ \ .. .. .. . (A-l1)

where on the rir,ht hand side P( COS{)() is understocxl to be interpreted

as the corresponding function of or •

See) is called the path of steepest descents, implying that path along

which the real part of -jk'R LOS(e -IX.) decreases most rapidly

as ()( pl~ceeds away from the saddle point; the quantity increases most

rapidly along a path ort~gonal to S(e) at 0< "" e , and rerTBins constant

(zero) along the real axis. The merit of the steepest descent path is

basically that, when kR.J>1, only that part of the path in the vicinity

of the saddle point contributes significantly to the integral. In this

respect its importance is akin to that of the stationary phase path.

It has, however, an added advantage in securing the transformation (A-11),

since this form leads directly to the complete asymptotic expansion in

descending powers of k'R , as will be shown furtheron.

It turns out that in a nwnber of problems the spectrum function P(COSIX.)

contains a simple pole, and that this partiCUlar singularity plays a

major role 1n determining the nature of the field.

(A-11)

then

(A-:18)

(A-19 )

-a3-

b. Ib~_~~~pl~_It§§D§1_iD!§gr§1~~~9_~!§_~E2E2~~!~~~§.

If''''' ·Con[~l d(~r the function F(d) os eJ8 eJ'r d,.a-

d is concievcd as tdK.i.ng <w1>itnlry cUlllplcx vdlli('~;, ,Ifill till' td\'l(ll\ ('XP\jF?)

has been included in the definition of Lhe fUllction ill en'de't' to S()(~Ut\C

boundedness as \iX\""",OO when arg a lies in the n.lll[';e -11 1'0 111;:l, which covel'[';

the cases of physical inter'est considered subsequently. Consider also- ji\j<l-j'f'the related function f o@)= e e d<r ..••••• (A-13)

JIT;' 'a'f(a) +~(a) = VTY,. e-'It) lJ • • • • • • • (A-111)

. j iVi 'a~"Rg) + t(-a) ~ fir t: Ie]

When a is real FOCa) is C'xpn~ssibl(' Ul terms of the fh;'[3ncl :inh'f'}Yl1f,.

\Jhen arg a is - y,/lT ,~ j11j, 'a1 (;co '[2.

r(lalel 'i) = e- ~eJ j e- d'( (A-IG)

and when arg a is + I/l1'IT e- .~ \al \912(laI1

1;;'Ual eJIV~ == - e.J

Zj e- oj e't d't . . . . . . . (A-17)

Anorrther form of the complex Fresnel integrals 1S provided by

00

I = 'bJ3-X<f

i

d't"-= 't~+Jb1

(x lS real and b is arbitrary complex)

It is now pqssible to deduce00 1-

b f eXT cl'l = ± 2VT T (± blJ'X)..-ooJ ;r2.+jb1

where x is real and positive and where the upper sign holds for

-""1T<arsb<~, the lower for T1(4 <arg6 <~'IT •

Further a convergent e~pansion in ascending powers of a for FoCa) can be

by repeated integrati~n by P9I'ts,.startirig from ~:~Td~or by using the exponention~l series under the i~lt:cf'Pd.l sir;n, curl then to

integrate term by term. Results are respectively

"" nt@) == a .2- (:2.ja~)n=o 1.2.3, (2"*1)

r ja:l. -f! ~ja1)nI~ La) = e ,a L '---'--

1'1=0 n\ Gn+i)they give useful approximations when lal «1.

(A-20)

CA-21 )

An asymptotic cxpanSlOTl of r<'::l) 1ll dC~;cclldi.nr; pow('r'~3 of d Cell br'

obtained by rcpPtlted i.nL('I',nlt iOIl lly J,lrt:; :;ldr'l ing from j'~-J~'2d'll~csul L:~ are then a..

and

(/\-2') )

lca) ,... \ffl~ e-j'IV" cjal

+ ~Jd. [1 ~ (- iTa') + 0j~~? ]/\ppr'OxiJlldtioIlS dre liuw'w;dul fur \al),'>l.

m,l '/<1\ s"a <Si1l\ ,It..... /4 ' .. . . . . . • (/\-23)

(A-26)

(A-21+)

0(- e (/\-2'1) dppc~ars

• . . .. .. (/\-25)

c. Reduction to Fresnel integral of steepest descent~; (expression.-------------------------------------------------------------Consider the particular plane wave representdtion

j -Jk'Rcos$-o<)~ec(~rO<7'-?) e do<.

~(e)

In which the only sineularl ties of the spectrum fW1Ct.ioll in Lhe complex

o<-plane are simple poles at rx = 0<.0 ± (2Yl-i)TI , and the' path of integration

is presumed already distorted from C to S(8), with proper allowctnce

understCXld for any pole cdptured in the process.

By changing the variable of integration from ()( to

j -Jk 'R cosO<.as SW) ~ec (0$1.2 - 1X1~ +e/~) e JtI(

or by now reversing the sign of 0(, as

.I -jk'"RC05(l(5(8) ~ec(% .j. CXi2 - %) e dlX

'The addition of (A-25) and (A-2(.), and devision by two, then puts (A-24)

in the form

When we~

nll. -J'k'RCOSlX2 coS(lX o- e) CO'S 'r!}. . e dO<.

.:z $) cos D( +coS(p<o-B)

now make the change 'of variable\ r,:;"' - j'ITILj'( == \}:t e sif'\ %

(A-27 )

(A-28)

\oJe find

j -jk'RcoS(6-O<). -j'f114 _jk1?F[ ,r:i""D ]sec(~_lX~)e. dcx==+{1-re E' ,:hl2.kR,eos(8/;l.-o(°121 ,"(

~) ....... (/\- ... .1)

with the upper sign for (8-iX.) between S(-1T) and S(91), and the lower

slgn otherwise remembering ofcourse that the expression has period 4'IT In

e -0<0 •

The result (A-29) is exact, and is important as a "canonical" formula., \oJheo. 1.'0

that descrlbes preclsely what happens,. tor a glven valut' of l'\ 1'\ much

greater than unity, the asymptotic approximations fail because of thc~

close approach of one of the poles of P( cos lX) to the saddle point.

-a5-

(A-32)

(A-JCl)

Consider

J-0 - Jk'R c.o~ (6 -c<.) d.r (c051X) e c<.

5/)

and suppose now that P(,,~os e ) hiS <l simple poLc' dt 0(.-1r Uk!! /lBY

approach arbitrariyl close to the saddle point e. Suppose also that no

other singularities need special consideration, in particular that there

are none near (X. -TI • Then for k'R ~1 a unifonn asymptotic approximation,

that is, an approxirrB tion which remains valid for all values of e-0(0'

can be obtained by ~i)ing

P(cosc<.) = 1i~oso<) + p sec(~_;()(o) (A-31)

where p is independent of 0(, and is so chosen that thl~ fW1ction P1 has no

pole at 0(0-'iT. In fact,

P = lim cos(cX-~·)1\cosO()ol~ ()(.-'IT :I.

and the resolution (A-31) simply splits off the pole in a way suggested

by the canonical result (A-29). By hyPOthesis, P1 contains no singularities

requiring special treatment, and the unifonn asymptotic approxirncl.tion

to (A-30) is therefore

.~ -jk'R .~ 'k'R.{iii- eJ ~ 1irg,se) _e__ :+ ~ffr.p. e-J ZJe-J T[±~1.k'R c.os(S-cJ (A-33)

~ q.with upper or lower sign as in (A-29).

A slightly simpler version of this result, which we used III chapter III,

can be obtained by factoring out the pole writing

P~s \)() = sec(~) 'E ~os 0<) • • • • • • • (A-34)

It is then argued tha~ since P2(cos Of..) has no singularities in the

vicinity of the saddle point, it may be removed from under the integral

sign with rx. equated to e . This step can indeed be justified quite

straightforwardly by going over to the « integral as in (A-l1) and

developing as a Taylor series in ascending powers of er what remains of

the relevant part of the integrand after the pole has been factored out.

The resulting approxin1ation to (A-30) is

+f'IT e-JlT/1.j P,2,(sos6) e-jk'R T [ ±V;2.k'R c.oS~-;'OI-o)] ••..•.. (A-35)

For k'R.}l, expressions (A-33) and (A-35) are in close agreement as can

be prooved.

-b1-

Appendix B.

Sup:[X.)se a linearly :[X.)lari-sl'.d fidd~proJuccd by d fe>lxl dL z = O,wLlh

<r~-l )

( n-2)

,(B-3 )

Then the electric field can be expressed for an arbitrary point r HI

free space as [ ]

with

Et) = _ b .e-jkrjqCl:,[P) e-jkr'.Qt- dSlifT r

. S

(8-4)

We can write (B-5) as

Q~,r')::: Eo.gr)(~zX~r-gr)(·(~)(9.~.,:. (B-6)

or

Qt,t')::: Eo .9rl{Q~ +~,"XQ)(J (B-7)

Then with §1 =' at- sin 8 -"In c.r + ae (O~esll'l<t' + g'f <-ostp

a~ = gr ~itl e cos cr + go CClSe c0 5Cf - .fl<f sin cr... (B-8)

we find for Q at last

Fig. B/1

r

17,

So (B-9)

and

~(r,r') = EoQr x{aecoses\V\(f+Q.CfLO",,<f +§,,~(OstJ(OsL.f+~aS\ViCf}... (B-9)

can be expressed as

Q(r,r') = Eo~ +COS0){§\f SIl1'f - aecosY']

. -jkr .\ If jbi~lh"'<r~' +k )il\tl )I\'(f >(ECt) =-j ~~··T QHose)(a'fsl"''f-8.~lose.pJe L~~'d'f'

S(B-10 )

-b2-

With assumption 2 of page and Fig. 2/13 we change (B-10) in

(B-:1.2 ;,

... I

_jkrp " iffjbnacoS(fl!:I.ksiVl;l51~<Pir(r,) = -J' kEo,~ G+C.OSS)(j siI1Cf-ae cos<p\ e Ol!'di'=1 ~fT t'? « I) '8""

• • • • • • • \. -..1.."':",

S

We are only interested in the vectorial part of this last formulatio~

We call I;i the wave falling on the reflector and I:r the :-'eflected 'dave

D is the unit vector.pefPendicular on the surface of the reflector

pointing in positive z-direction.

Now tJ x (E. r + E() = 0

D· E; r =!J. Erand

(B-14 )

Then the direction in which the waves are reflected from a parabolic

surface is found to be

~r = ar - :2. (~r .n) n.

when these waves are produced by a feed as sketched, in Fig. 2/13.

So with (B-8) and (B-13)

(B-15)

(B-1S~

(B-16 )

(B-17 :,

(B-18)

further from (B-11) and (B-12) we can find the reflected field.

Formulations (B-12) gl'!e Er = (ll .Ei ) n -0 xE..<) X.D

Using now (B-8) and (B-13) we' find'

U.~t :: SinelrcoS~ .•.••••

(n.El)t:l :, -sin/:% cosep CO.5ep Q.r +.sin%Si\'\GI2COSr ~e

n )< E-: :: -CDS %cos ep sr - cos 8/2 sin <PQ/J - sin %sin Cf Qr

(p k ~;))I!!:: ( - cos %, cos %, sil'llf' - si" 812 .5irl Elh sin <r) 9: r+ cos fJh coseh. cOS~? ae + cos %s1Yle/~ (Osr Sir

(B-17), (B-18), (B-19) and (B-20) in (B-16) give at lastEr :: -,Sly CDScr sll') e - 99 CO.) Cf cos e + ~'t' sin 'f

or

(B-21 :

So we proved that the field in the aperture of the reflector is lineaI'ly

polarised.