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JOURNAL OF RESEARCH of the National Bureau of Standards- D. Radio Propagation Vol. 65D. No.2, March- April 1961

Fresnel Region Fields of Circular Aperture Antennas l

Ming-Kuei Hu

(July 5, 1960 ; r evised July 29, 1960)

A different approach to the Fresnel region fi eld approximation i in t roduced. Instead of using t he conventional t run cated power series expansion approximation, thc Ne wto n's iteration formula for square root is used. By using such an app roximation for circular a pertures with tapered illumination of the form (1- 12)", the Fresnel r egion fields can finally be expressed in terms of a new class of function s Woe')', u). The function wg is s hown to be related to th e Lomm el's funcLion s of two variables, and the fun ction Wo is then obtained from the fun ction TVO- l by a simple recurre nce relationsh ip. Fi eld dist ribu t ions for n = O, 1, 2, 3, 4 and at distances}~ D2/A, ~ D2,'A, Y:l D2jA, D2/A, 2D2/A, "" have been com puted and prese nted as sets of curves. General and qu antitative properti es of the fi elds are clearly demonstrated by t hese curves. It i~ a lso shown that the fi eld of any other nonuni form illumination with circular symm etry can be ex pressed in term s of t he fi elds of t he basic i ll umination of the form (1 - 12) ".

I. Introduction

It is weIl known that t here is a close similarity between diffra ction problems in optics find aperture field problems in antenna t heory. In opties, two diffmction problems- the diffract ion from a single edge or a sli t and the diffract ion from a circular aperture- have been well treated. These two problems were first solved by Fresnel and Lommel, respectively, in terms of the now so-called Fresncl illtegnlls and Lommel 's i'unctions of two variables . The corresponding problcms in antenna thcory arc t he Fresnel region fields produccd by rectangular ftnd circu1ftI' i:LIJ I:Jn llrUtL However, in optics, t ile primary illumination is ftssumed to be uniform over the aperture, but in antenlla theory, t he case of nonuniform illumination is of great inLcrest.

For the ca e or unii'orm illumination , the allillyt ical methods developed in optics can also be applied to antenn a problems. For F]'esnell'egion fields o[ rectangula,r apertures, genemlizations Lo include nonunii'orm illuminat ion througll expansions in terms of different bftsic forms are not too diffieulL, polynomial and trigo llometric sum types of illumination were treated by the author [1] .2 On Lil e oLher hand , t il e analysis oi' Fresncll'egion fields of circular aperture antennas with nonun ii'orm illumination is cons iderably more diffi cult . However, with the use of illumination oi' Lhe form (1 - i'2) n, the Fresnel region field was solved by the present author [1] . A short communication about this resul t has been published [2]. The complete analysis and numerical results are presented in this papel' . A refined but more complicated result for the case of uniform illumination was given by Hansen and Bailin [3]. It may be noted that the corresponding far field problems have been t reated by many authors with the use of different basic forms [4, 5, 6, 7], including the form (1- i'2) n l4 , 5] used in this paper.

2 . Fresnel Region Approximation

In the study of field distributions of linearly polarized uniform phase plane aperture anLennas [8] the following formula [1] or a slightly different form given by Silver [8] may be used as a starting point.

jlc J e- j kTo ( Z) E(x'Y' Z)= 47r A F(~,7)) -----:;:;; l+ro clA,

1 Contribution from Department of Electrical Engineering:, Syracuse Unh'crsity, Syracuse 10, N .Y. by Rome Air D evelopmen t Center uncleI' Contract No. AI' 30(602)-928 with Syracuse U ni v.

2 Fi gures in brackets indicate the literature references at the end or thi s paper.

137

(1)

rrhi s work is part of a project spon sored

where

E(x,y, z) = the complex magnitude (amplitude and phase) of the electric fi eld intensity at the field point (x,y,z) ,

F(~,71) = the illumination at the point (~,71,O) in the apperture A. For uniform phase illumination, F(~ , 71) can be taken as a real function of ~ and 71 ,

ro= .y'(x-~)2+(Y-71)2+z2= the distance between the field point (x,y ,z) and the source point (~ , 71,O) , and

k= 27r A

with A= the wavelength.

The coordinate system used for writing formula (1) is given in figure 1. Formula (1) holds for any illumination function F(~, 71) and for apertures of any shape. In the present paper, it is assumed that the aperture is circular and the illumination is of circular symmetry.

For a circular aperture of radius a, as given in figure 2, formula (1) can be rewritten as follows:

jk i 2"i a e-jkro (1' ) E (1', e, </» =-4 . F (p,</>' ) -- l +-:;- cose pdpd</>' 7r 0 0 1'0 10

(3)

where

The evaluation of the above integral is a very difficult one without further approximation. In order to facilitate its evaluation, different approximations are generally used. Depending upon the approximation used, the result may be classified as the far field approximation, the Fresnel region approximation, etc. The most widely used approach to such approximations is to expand the distance 1'0, appearing in the exponential , into a power series involving 1', e, and </>. The result obtained, by neglecting all terms of order higher than one or of order higher than two, is generally considered as the far field approximation or the Fresnel region approximation. As far as the remaining 1" S under the integral sign are concerned, much more crude approximations are satisfactory. If the field point in consideration is not too close to the aperture and e is not too large, then the factor 1/1'0 (1 + (1'/1'0) cos 0) can be approximated to good accuracy by using the relation,

- 1+- cos e ~-. 1 (1' ) 2 1'0 1'0 l' (3)

lx, y, z)

lr, 9, ." )

FIGURE 1. Coordinate system for A plane aperture. FIGURE 2. Coordinate system for A circular aperture.

138

A rather differ ent approach to the Fresnel region approximation is used in the present. paper . Instead of using the truncated power series expansion, the well-known Newton's iteration formula for findin g square root of a given number is employed. T his formula has t he following special property. If an approximation BI , of .J A is known to be correct to n slgnificant figures, then the approximation B2 of .fA obtained by using the iteration formula ,

(4)

will b e correct to 2n significant figures. If this formula is applied to the approximation of 1'0 in Fresnel region, we have

, 1 { + (1' sin 8 cos ¢ - p cos ¢' )2+ (1' sin 8 sin ¢ - p sin ¢' )2+ (1' cos 8)2 \' 1 0~2 r r ) .

After simplifying, eq (5) b ecomes

p2 ro~r-p sin 8 cos (¢-¢')+-2 .

r

(5)

(6)

This approximation of ro gives a impler form than that given by Silver [8] and also by Hansen and Bailin [3]. It should be noted that this par ticular form makes the followin g relatively simple analysis possible.

Using the approximations (3) and (6), formula (2) gives:

E ( ) 'k e- JkTf 2". £a 1"( ') ik [ psin o cos (<I>-<I>')- -2P' ] d d 1 1r 8¢ = J - - I p¢ e r pp¢ .

, , 27rT 0 • 0 ' (7)

If it is further assum ed that the aper ture iJlumina,tion F(p,¢I ) is of circular symmetry, i .e., F(p,¢' ) can be written as F (p); then E(r ,8,¢) will be independen t of 8 and can be denoted by E(r,{}). H ence we have

E(r,8) = j k e~:kria F (p) J o(k sin 8p)e -jk (~) pdp . (8)

If a normalized variable

is defined, t hen eq (8) can be written as

wb ere

and

u = ka, sin 8

ka2

'Y=-' r

(9)

(10)

(11 )

(12)

From eqs lll ) and (12) it is clear that U= 'Y implies r sin 8= a; t herefore U= 'Y means that th e field point is at a distance, equal to t he ra,d ius of the aper t ure, from the aperture-axis.

It is more convenien t for t he following discussions to r ewri te eq (10) as:

E(r, 8)= e- j kT ('Yej"'~1') i l F W J o(une i ; (1-,2) !;dr

139

(13)

l

3 . Uniform Aperture Illumination

For the case of uniform illumination, F (S) = l, eq (13) reduces to

E(l', fJ) = e -jkr ('Y/r~1') il Jo(uS) ej i- (1-,2) . retr

In the above equation, the integral may be denoted by W~ ('Y ,u),

(14)

(15)

This integral has an imaginary part as well as a real part. It is closely related to the Lommel's functions of two variables [9]. The Lommel's functions of two variables can be expressed in in tegral forms as:

(16)

UI('Y,u) and U 2 ('Y,u) are of order one and order two, respectively. By integrating by parts and using the recurrence relations of Bessel functions, it can be

proved that UI('Y ,u) and U 2 ('Y,u) may be expanded into the following two series :

(17)

Generalizations of these functions to higher orders are possible, but these generalizations will not be used in the present paper.

In terms of UI ('Y,u) and U2 ('Y,u) , we bave

(18)

Using the above relation for W~ ('Y,u), formula (14) can be written simply as:

(19)

This result is essentially the same as the first term in the series solution of Hansen and Bailin [3] . It can be seen from their figure 6 that the two results are close to each other within the region considered in this paper.

4 . Nonuniform Aperture Illumination of the Form (1 - S-2 )n

If the aperture illumination is assumed to be of the form (1- r2) n, the field expression can be obtained by substituting F W=( 1-r2)" into eq (13) , that gives

(20)

140

Now if we define

(21)

then

(22)

When n = O, eqs (20) and (22) reduce to eqs (14) and (19), respectively. Tn other words eqs (20) and (22) include uniform illuminfl,tion as a special case. For n~O, these give the field distributions for the special tapered illumination (1- t 2)". The integration of the integral (21 ), or the evaluation of the function lV~'('Y,u), can be carried out by noti.ng the following simple property: If differentiation of HT~'('Y ,U) wit.h respect to 'Y is carried out, we have

(23)

Equation (23) gives the reCUl'l'ence relation,

W n+ 1 ( ) -~ ~T;rr n ( ) o 'Y ,U - . 0 ' r 0 'Y,U . .7 'Y

(24)

In terms of lIvg('Y ,u), we have

(25)

or

(26)

It is possible to derive expressions of 1iTl~('Y,u) in terms of Lommel's functions of orders higher than two. But as far as numerical evaluation is concerned, it was found that the simplest way is to use eq (26) and the series expansions (17) for U\('Y,u) and U2('Y,u). The series expansions of Co n/o'Yn)(U1 ('Y,u)h) and (On/O'Yn) (U2 ('Y,u)/'Y), for n = 0,1,2,3 , and 4, are obtained by differentiating the power series of U\('Y,u)/'Y and U2('Y,u)/'Y, term by term, n times with resp ect to 'Y.

5. Results

In the last section, for the apertme illumination of the form (1- t 2) n, the field expression is given by

(27) where

(28)

and u = ka sin o. (29)

E(r,O) in eq (27) can always be rewritten in the following form:

E(r,O) = A ei 27r q, (30)

where A is the amplitude and 27r-q) is the phase angle in radians. Plots of the apertme ilhunination (1-r2)n for n = 0,1,2,3,4, are given in figure 3. The

field distribution curves, both amplitude A and phase 27r¢, are computed for the following cases (see fig. 4) :

141

1.2

"""'Iiiii

~ ~ ( 1_ ~2)~"'" I

1.0

\

~'" 1\."'" ]'-., I

~ 1\'\ ~~2) .8

1\\ I'l l -). (1_ ~2)21\.

\ \ \ (,-t')' ~I\\ 1\ (H2)~ ,\ \ 1\

.6

.4

\ \ \ [1\ 1\ [1\'\ 1\.'\

.2

"" ~ ~ o .2 .4 .6 .8

1\ ~

1.0

r sin 8 ; ~ or u; r

r sin8 = 0 or u= 0

FW"CRE 4. FTesnel region, in which field distributions have been computed.

FIG URE 3. ApeTtuTe ill1i1ninalion distTibutions, uniform and tapeTed.

(i) Field distributions along the circular arcs:

where D= 2a. The corresponding values for "I are:

It should be noted that the case r=ro or "1 = 0 is the conventional far field result. For the convenience of comparison , the field distribution curves are normalized in the

following way: Let

(31)

th('n the normalized E(r, e) for a fixed r is given by

E(r,e) _ A j27r(¢-¢o)= A j27r¢n.

E(r,O) - Ao e ne (32)

The normalized ampli tude and phase distributions along the arcs are given III figure 5 to figure 14. (ii) Field distributions along the aperture-axis:

In this and the following case, the distributions are not normalized but are plotted with the factor e- jkr excluded. The two distribution curves, amplitude and phase, in the present case are useful in correlating the normalized distributions given in case (i). (iii) Field distributions along a line p arallel to the aperture-axis and at a distance equal to the radius of the aper ture:

142

1.0

\ rin (0:) k02 r~ -,

.8

.£ ..

\ I CD 0

\ 2 2 Tr/4 , 3 I Tr/2

." .6 ~

~ g ." .. N

.4 C E 5 z

\ 4 112 Tr

5 1/ 3 3Tr/2

\'" ~ ........ 6 1/4 2 ..

~ "-~ ~

. 2 \~4 "'" \ \3

~

~ ~ # -...........

~ V ~ V ~ o 2 4 6 8 10 12

u : k a sin 8

FIGURE 5. N onnalized arnplitude distTibution curves with apeTtuTe iLlurnination = (1- 1"2)0, (constant).

1.6

( 0 2 k 0 2 r in T) r;-,

I CD 0 f" t;:= 2 2 .. 14 'I; ~

1. 4

3 I ,,/2 /.V 4 112 ..

'I 5 1/ 3 3 .. 12

1. 2

6 114 2 .. II J

/ v::: W c 1. 0

~

II; ~ V

Vi '1/

~ N

.c . 8

.j'

~ .. J

~

. 6 0 .c 0-

." .. 1;-; ~~ V

N

C E a .4 z

~

6 \ 4 3~ W;II . 2

M ~ ~ fh o 2 4 6 8 10 12

u: ~ a sin 8

FIGURE 6. Nonna lized phase dist,·ibution curves w·ith apeTture il/urnination = (1- 1"2)0, (constant).

1.0

. 8

. 6

.. ~ ~ . 4 o

i: .!:! C E 5 z .2

o

, \ \ ~\ ~i\ l\\ 1\\ ~ , ,"" ~

1\"" 4 ............

"'--~ ~~

~

2 4 6

u ~ k Q sin 9

r in (~) kr! y= r

I CD 0

2 2 ".,04

3 I "./2

4 1/2 ... 5 1/3 3"./2

6 1/4 2,..

f'..... ~ ~ t---.......;

8 10 12

Fre UR;;; 7. N onnalized amplitude disll·ibution curves w·ith ape,·tuTe illumination = (1- \2).

1.6

'in (f) ka2 r~-,

I CD 0 1.4

2 2 ,,/4

3 I ,,/2

1.2 4 112 " 5 113 3.-/2

1.0 6 114 2,. IJ

~ ~ V ;; • N .8 ~ ..

~/ ~ /~

;;

f .6 /~ '/

~ 1: .. i .4

z

/ ~ '" Ira ~ ",

.2

6\ 5 Y ~ ~ ~ ~2 ~I

o 4 6 10 12

u ;t; kolin8

FIGURE 8. N ormalized phase distribution curves with apeTture illumination = (1- 1"2) .

143

1.0

~~ r in (~) k 0 2 y ' ,

\ I 00 0

2 2 ,,/4 . 8

1.0

.8

""'" 2 k02

I\. r in (~) y', , I 00 0

~ t--I

'\ 3 I ,,/ 2 1\ 2 2 .,,/4

J . 6

~

" " <i E o .4

" .. N

"0

~ z .2

o

,

2

~ ~ ,~ ! l

I~ ~ 4 3

/ \(~ ~ ~ lU::

4 6

u::ko sine

4 112 ."

5 1/ 3 3,,/ 2

6 114 2"

2

~ r-=:::: t;::-t:::-

8 10 12

J .6

f Ci

~ .4

'0 ~ N

~ !; z .2

~~ 3 I .,,/ 2

4 112 ."

1,\ 5 1/ 3 3.,,/2

'\ t\ 6 114 2 "

I~ "-'\~ ~5

4 3

'\ ~ k 2

--

~ ~ ~ ~ o 2 4 6 10 12

FIGURE 9. N ormalized amplitude distribution curves with aperture illumination = (1- \2)2 .

U :: k a sin 8

FIGURE 11. Normalized amplitude distribution curves with aperture illumination = (1- \2)3.

1.2 ( 0 2

y= \02 r in ):"")

c 1.0 c

~ c

4-.8 • N

~ .. .J .6

~ " .4 . N

I z

. 2

I '" 0 I 2 2 ,,/4

/' ~ 3 I v!2

L ~ 4 1/2 " V/; V 5 1/3 371"/2

6 1/4 2 " It tJ' ~

~ ~ I~ V

~'fI' \ 5 .\. ~f7

~ ~ ~ ~2 1'1 o 4 10 12

u=k osinB

" .. N

~

I. 2

1.0

. 8

.6

.4

z .2

o

r in (~)

I 00

2 2

3 I

4 112

5 113

6 114

k02 y' ,

0

,,/4

L ,,/ 2

11 / "

IA 3 ,,/ 2

2." ~~ J'

~ ?7 !t~

~ 1\ 5 ~ ~ r

-::::::;:; ~ k K< I I

2 ~I I 4 6 8 10 12

u =k a sin8

F IGURE 10. Normalized phase distribution curves with aperture FIGURE 12. Normalized phase distriblltion curves with aperture ilillmination= (1- \2)2. illumination = (1- \2)'.

Distribution curves for case (ii) and cases (iii) are given in figure 15 to figure 18. From these fi eld distribution curves, the following properties are observed.

(1) For the normalized field distribution curves along circular arcs: (a) For each aperture illumination of the form (1-t2)n, the main beam width is essentially a constant in the range of r from 7H2D2j}..) up to infinity. (Note: For a fixed direction, the parameter u is a constant. ) (b) There is no perceptible difference in the normalized amplitude distribu tion curves at 2D2j}.. (conventional far field limit) and at infinity for the same aperture illumination, except near those directions where the infinity case has zero values or nulls. On the other hand, there are differences between the phase distribution curves; these curves change from a stepped form at infinity to a smoo th continuous form at 2D2j}.. .

144

c

'"

0

8

4

2

o

.......

" ' \. ,

2

r in (f ) ka2 Y =,

1.0

r in (DA2) 2

Y ' ~ r

I Q) 0 :g I aa 0

2 2 ... /4

0 .8 'C

~ 2 2 .,,/4

c

\ 3 I 11'/ 2

4 1/2 ." , 5 1/3 3,,"12

-&. ~ .6

N

.c ' j

3 I ... 12

4 112 ."

5 1/3 311"/ 2

~ 6 1/4 2 ... .J. .4

6 1/4 2.-

~ ~ .. . 0 .c c.

~ ~ 6 5

~ ~ 4111 ]

.2 0 E l; z

6~, ~ ~ ~ 8::: t:::= --k:::::: ~ ~

4 6 8 10 12 o 2 4 6

u= kas in8 u z ka sin8

~ / V ~

~~ ~ VJ K-'~2 I

8 10 12

! FIGU RE; 13. ~: Normali zed amplitude distrib1ttion curves with FIGU Rl, 14. Normalized phase distribution curves with aperture aperture i llwnination = (l - r2) '. i llumination = (l - r2)'.

(c) The "side lobe level" in crea es a Lhe d istanee l' is reduced , but th o ampliLude dis tribution curvo as a whole becomes "smoo ther." It is also true that tho pha e distr ibution curve beromes smoother too as r is reduced. (d ) When the distribu Lion curves of differen t aperture ill uminat ions ar e compared, it is seon that tho main lobe beamwidth increases as n is increased. At Lile same Lime , within a fixed m easuro of the angle fJ , the number of side lobes is reduced. (0) T he side lobe level is definitely red ueod as n is ineren,sed . T he ratio or Lh e normalized maximum amplitudes of tho firs t s ide lobes wi th n = 4 and n = O is approximately 1 to 10. (r) As n increasos, the difl'erence in tll e normaJizecl ampliLud e distribution curves at differen t T beeomes smaller .

(2) For thr,fiold clistribuLion curves along the aper ture-axis : (a ) Along Lho aperture-axis, tho ampli tUde distributions vary closely as 1/ 1' in tho range T= %(2D2/}..) to 1' = infinity for all cases s tudied. As for Lhe particular case n = 4, this is true evon when l ' is reduced Lo %(2D2/}.. ). (b ) Along the aperture-axis, all tho p base distribuLion , excluding the oxpon onLial factor e-1kT, aro approx'imately straigh t lines, i .e., the phase varies linearly with r espoct Lo 1/1'.

(3 ) For the field distribution curves along a line parallcl to the aperture-axis and at a distance equal to the radius of the aperture; (a ) In the range, 1'= 2D2/ }.. to 1'= infini ty, bo th the ampli tude and ph ase distribu tions h ave approximately the same values as those obtained along Lho aperture-axis. (b ) At l' = }~ (2D2 /}..), the ratio 0 f the corresponding ampli tudes for th 0 prosen t case and case (ii ) is roughly one to four for all cases computed.

6 . Generalization to Nonuniform Aperture Illuminations of Other Forms

In tho last section, only a special type of nonuniform ill umination of the form (l -nn was given . It is possible to general izo tbe analysis to include nonuniform illuminations of othor forms either through expansions by t ho Schmidt orthogonalization proeess [5] or by the following simple algebraic procedure. For any circularly symmetric aperture illumination , i t is always possible to approximate it by an even polynomial of the variable t. Such poly-

145

.. ~

~

t E «

2.0

1.8

1.6

1.4

I., :

1.0

. 8

6

. 4

2

o

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

""'I'-

f- -r "'-

I I

"'---t----. r

..... ~ ....... ~

r------1"---I'--- )'--..

r---r--- -----K r- t--

r---1'--- n=3

r--- n=4 r---

2 .. 3 .. /2

-Hf)

I'\. '\ n=O

\ l'i

\

~

~ ~ "" "- ~ 1'-

......... I"'-- b. ~

i'--- f'--. ~ \ r----

-------

I'---

------

'-..... ~ 1\

-------::--1'--...... I---- I", \

....... r.::: ~ ~ \ ~ ~ .. ,,/2 .. /4 0

k 0 2 --- y= ---r-

i(~) ~~) ,

2(~) Q)

Distance r_

.3

. 2

-&-~ . 1

N

.c

'i

~ 0

t:::::== I--

r-

..------~ 21T .c a.

1

I

I n=4

!==--I-;;;"~ I--~2

kI V

---./

V

l1

i : 1

V -:::- ;:..--

!--!--t:-~ V ...-

I---- .....--

V .-;;;;

V

,",4 o (

l(~) 2 A

co

Distance r ----r- I FIGURE 16. Phase disl1'ibulion C1l1'ves (exc luding thejactor e -i k,)

along the aperture axis with aperture ilht1nination = (1- 1;2) ". ,

c o

.6

.4

.2

o

...-/

-2 ..

l-- V ...... "-.. ,/

I'---r--: ~ -~

3.,,/2 .. k 0 2

~ Y:---r-

J.(.!t) , 2

t(-'tl 3 ,

Distance r ~

I-- '0 I---- I

'\ n=1

1\ --.... n=2

---- '"" \ :::---

---~ :::::::: ~ 1\ f::::: ~ ,,/2 ,,/ 4 0

' 2 ' 2

(~ ) 2(~) 00

FIGURE 15. Amplitude dislTibulion CUTves along the aperture axis with aperture illumination = (1- 1;2)" .

F I GURE 17. Amplitude distl'ibution curves along a line parallel to the aperture axis and at a distan ce a jrom the aperture axis with apeTt1l1'e illumination = (1- 1;2) ".

. 5 I

'" '" ......... n=O

r--------

~ "'" n::: I

~ 1"\ . 4

~ I~ n=2 I'" 1\ ..............

~ n=~ ~ "" \ l'---

.. c 0 'i:i .3 0 L

C

n= .. I---- 1\ V -- -

-&-~ .2 N

.c

'i

~ .. .1 0

~ Q.

o 2... 3 .. /2 .". .. 12 ."./4 0

k 02 -r=-r-

2 2 ' 2 1(..Q...) (~ ) 24) 00 2 A

, 2 2

Distance r-FIGU RE 18. Phase distribution CUTves (excluding the jactoT e-ik ,)

along a line parallel to and at a distance a jTom the aperture axis with aperture illumination = (1- 1;2) n .

146

nomials are 01 the follow ing form :

(33)

It is clear that such a polynomial can always be transformed i Illo the fo ll owi ng form:

(34)

For a given set of a's, the corresponding b's call be uu iquely determi ued by solving a system of l in ear simul taneous algebraic equations. In particular, for the case of n= 4, the coeffi cients b's obtained in terms of the (L'S are as follows:

b4= a4

b3=-a3-4a4

b2= a2+ 3a3+ 6a4

b1= - al - 2a2-3a3-4a4

bo= ao+ al + a2+a3+ a4'

vYlic ll bo, bl , • •• , b4 a re cl ete rmined as above, Lhe field expression is l hen g iven b~'

E(r, B) =e- jkT (yejrr;')J'l t b,,(l - .\2)" Jo (u.\)i~ (1-\2) .Id.\ o ,,= 0

III terms of 11'0'(A,u), (36) gives

Equation (3 7) ca n lhcn be eval uated by ll s ing t he results already obLained in t hi s paper.

7 . References

(:37)

[1] i\l. I\: . Hu, tltud.v of ncar-zone field s of large apcr t urc antennas, Syracuse Univ. Hesearch Inst. , Syracllse, l\elV ·York, final rept., pt. 2 (Apr. 1957).

[2] i\l. K. Hu, F resnel region ficld distribution of circula r a perture antellnas, IRE Trall H. 011 Antennas and Propagation, AP- 8, No.3, pp. 344- 346 (i\la.\· 1960) .

[3] H a nsen a nd Bailin , A ne\1' method of ncar fie ld a llalysis, IRE Trans. 011 An tennas ancl Propagation AP- 7, special suppl. , pp. S458- S",,67 (Dec. 1960) .

14] R. C. Spenccr, Paraboloid d iffract ion patterns from thc standpoint of pln's ical optics, i\lIT, Radiation Lab . RL Hcpt. 1'- 7 (Oct. 19-12) .

[5] C. T .. \ _.Jo l1ok, S.vnthcsis of apcr t urc a ntc nn as, U niv. , Ill., E ng. Exptl. Sla., T cch. Rcpt. 1 (1954) . [6] A. I shim aru a nd G.Hcld, Analysis and sy nthcsis of rad iation pattcrns from ci rcular apcrturc8, Ca n. J .

Phys. 38, pp. 78- 99 (J a n. 1960). [7] T . T . Taylor, Dcsign of circular a pcrturcs for narrolV bcamwidth and low sid clobcs, IRE Tra ns . on Antcnna

and Propagation 8, No.1, pp . 17- 22 (J an . 1960). [8] S. Silver , Mi crm\'ave a ntenna thcory and dcs ign (M cGraw-H ili Book Co., Inc., New York , N.Y., Radi ation

Lab. Ser 12, pp. 170- 173, 1949) . [9] G. N. Watson, A t reatisc on the thco ry of Bcsscl fu nctions (Cambridgc Un iv . Prcss, second cd., pp. 537- 540,

1952) .

(Paper 65D2- 111 )

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