bEEE Antennas and Propagatlon Society Newsletter. April 1986

Edltor'r comments (cont'd. from page 2 )

short courses in his section. If you have an item tha t should be included here, please send it directly t o Ray. Ed Millerls co lumn takes you to t h e "electromagnetic movies" this issue. Actually, he has me a little worried. I'm afraid someone may find out why so many of us think numerical e lec t romagnet ics is such fun. If you are trying to plan a program for a c h a p t e r or sect ion meet ing, be sure to read the art icle on AP-S Dis t inguished Lec turers who are available to you. The abstracts of their talks are given.

The December and February issues reached the

problem with the December issue. We usually have membership late. I'm sorry. I had anticipated a

trouble because of the holiday mail. However, a similar problem occurred with the February issue. Kurt Synder and his crew in New York have been doing their usual fine job of getting the material to IEEE printing and mailing on schedule, but it appears that the balance of the process is now experiencing delays. He and I will be watching this quite closely, and we will try to get back to the timely delivery we worked so hard to achieve.

F ina l ly , a personal note. I demonstrated my capability for breathing and walking simultaneously by fall ing and tearing a tendon in my ankle whi le crossing the parking lot at the National Radio Science Meeting in Boulder in January. If you must do this, I recommend the University of Colorado as the bes t place: I was back at a Commission business meeting, complete with X-rays, crutches, and ice, exactly one hour after I fell. More importantly, do it where you have the caring friends in URSI and the U of C who were present. To all of you who made a difficult and frustrating situation quite tolerable, thank you very, very much.

I hope April finds everyone well, and that I will see many of you in Philadelphia in June.

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Notice to Potential Advertisers The AP-S Newsletter publishes paid advertisements on short courses, job.opportunities, e tc . , which a r e of interest t o the AP-S membership. The price schedule for ads in the Newsletter is:

eighth page $85.00 quarter page $135.00 half pege $225.00 full page $335.00

T h e e d i t o r reserves the r i g h t t o reject a d v e r t i s e - ments based on c o n s i d e r a t i o n s of s u b j e c t matter OK a v a i l a b l e s p a c e . P l e a s e contact W. Ross S t o n e , IRT C o r p o r a t i o n , 1446 Vista C l a r i d a d , La Jo l la , CA 92037, w i t h q u e s t i o n s r e g a r d i n g t h e p l a c e m e n t of Newsletter a d v e r t i s e m e n t s .

Introducing Robert S. Elliott

Feature Article Author

R o b e r t 5. El l io t t has been a P ro fes so r o f lectrical Engineering a t UCLA since 1957. He holds n AB in English Literature and a BS in Electrical ngmeering from Columbia University ('42, '43), a n MS nd PhD in Electrical Engineering from the University

he University of Cal i fornia at Santa Barbara ('71). f Illinois ('47, '521, and an MA in Economics f rom

H i s p r i o r e x p e r i e n c e i n c l u d e s p e r i o d s a t t h e .pplied Physics Laboratory of John Hopkins and at the lughes Research Labora tor ies , where he headed the

een on the faculty of the University of Illinois and n t e n n a r e s e a r c h a c t i v i t i e s . Dr. El l io t t has ' a l so

'as a founder of Rantec Corporation, serving as its .rst Vice President and Technical Director. He is

listinguished Lecturer for the IEEE and was Chairman r e s e n t l y s e r v i n g a s e c o n d t w o - y e a r s t i n t as a

f the Coordinating Committee for the 1981 IEEE AP-S lternational Symposium, held in Los Angeles. He urrent ly serves as a consultant to Hughes, Canoga ark. In addition to being a Fellow of the IEEE, Dr.

>e New York Academy of Sciences. He has received lliott is also a member of Sigma Xi, Tau Beta Pi, and

umerious teaching prizes and is the recipient of two est Paper awards by the IEEE.

Feature Articles Solicited for Newsletter Arlon T. Adarm Department of Eieclrlcal and Computer Engineering 111 Link Hail Syracuse Unlvenity syracuro. NY 13210 (315) 423-4397

The e d i t o r i a l S t a f f of the AP-S N e , ~ ~ b t & z Y e h c o n t i n u e s t o a c t i v e l y so l i c i t f e a t u r e art icles which d e s c r i b e e n g i n e e r i n g activit ies t a k i n g p l a c e i n i n d u s t r y , g o v e r n m e n t , a n d u n i v e r s i t i e s . Emphasis is b e i n g p l a c e d on p r o v i d i n g the r e a d e r with a g e n e r a l u n d e r s t a n d i n g of the technical p r o b l e m s b e i n g a d d r e s s e d b y v a r i o u s engineering o r g a n i z a t i o n s as w e l l as t h e i r c a p a b i l i t i e s t o cope with these p r o b l e m s . I f y o u o r a n y o n e else i n y o u r o r g a n i z a t i o n is i n t e r e s t e d i n s u b m i t t i n g m ar t ic le , w e encourage you t o c o n t a c t Arlon T. Adams t o d i s c u s s the a p p r o p r i a t e n e s s o f the t o p i c . He may b e r e a c h e d a t the above addres s .

1

4

IEEE Antennas and Propagation Society Newsletter, April 1986

Array Pattern Synthesis

R. S. Elliott University of California

Los Angel-

Part II: Planar Arrays

(Part I of this article appeared in t h e October 1985 issue of t h e AP-S Newsletter.)

I f the mnth element of a planar array of identi- cal, similarly-oriented radiators has its representa- t i v e p o i n t a t (xm,ym, 0) the a r ray fac tor is given bY

wi th @ measured f m t h e p s i t i v e 2 axis and $ mea- sured in the XY plane fran the posit ive X ax is toward the posit ive Y axis.

A useful reduction of (10) yields the collapsed dis t r ibu t ion . If one introduces the planar rotation

When the special condition B = Q is impased, (11) reduces to

with e; measured from the x ' ax i s i n the x ' Z plane. In words, (12) implies that i f a l l elements in the planar array are projected onto the X ' ax is and given their or ig ina l exc i ta t ions Im , the pattern of the resul t ing l inear array, in the X' Z. plane, is the same a s t h e p a t t e r n o f t h e a c t u a l p l a n a r a r r a y i n t h a t p l ane (i-e., the Q plane). This result is t rue for sum, difference, and shaped patterns, and whether or not the elements are regularly spaced. mis concept of a collapsed distribution underlies several impor- tant synthesis procedures. Notably, i f t h e elements are l a id ou t in a rectangular or t r iangular g r id (see Fig. 61, regardless of the shape of the boundary the collapsed dis t r ibu t ions on the X and Y axes are those of equispaced linear arrays, for which a l l the proce- dures discussed in Part I are applicable.

Sum Patterns - If the array elements are arranged i n a rectangular grid and if the array boundary is rec- tangular, the assemblage can be viewed as a family of linear arrays placed side-by-side. If fur ther &/&,,o = I on/I oo , the d is t r ibu t ion is sa id to be separable, s ince a l l rows (columns) are excited alike except for leve l . In th i s case the array factor (10) assumes the form

wherein I, = I,o/I,o, I, = Ion/Ioo, COS^^ = s i n e cos@, a n d cosay = s i n @ s i n g ; dx and dy are the element spaci!?gL I n the two direct ions.

The array factor (13) is seen to be the product of the patterns of an equispaced linear array parallel to the X ax i s and an equispaced linear array parallel to the Y axis. Thus a l l the synthesis procedures dis- cussed in Par t I can be brought to bear on the design of such planar arrays. In particular, if 1, and I n are given synntstrical amplitude, equiphase distribu- tions, the conical main beams of the two linear arrays i n ( 1 3 ) lie along the 2 axis and their product is a p e n c i l beam. Further, since the nulls which inter- sperse the lobes of the pattern produced by the X - d i - rected l inear array are cones about the X axis, where- as the nul ls which intersperse the lobes of the pat- t e r n produced by the Y-directed array are cones about the Y axis, the null contours for (13 1, plo t ted on a la rge hemisphere i n 2 '0, appear in projection as por- trayed in Figure 7a. The damin of the main pencil beam is s h cross-hatched. Regions A and B represent respectively the product of the main beam of the X - d i - rected (Y-directed) l inear array with a side lobe of t he Y-directed (X-directed) l inear array. Region C represents the product of conical side lobes.

One can see f r m t h i s t h a t t h e a r r a y f a c t o r is a sum pattern, consisting of a single main pencil beam surrounded by a two-dimensional family of mund-type s i d e lobes. The s ide lobe heights in the two princi- pa l p lanes are those of the X-directed or Y-directed linear array, b u t the other side lobes (such as C ) are greatly reduced, being the product of two conical s ide lobes. This fact betrays a serious defect of separa- ble dis t r ibut ions. I f one designs the a r ray fac tor to g i v e desired s i d e l& heights in the two pr inc ipa l p lanes , one m u s t accept reduced side lobes elsewhere with a conamitant beambroadening and loss in d i r ec t - ivi ty . This defect , p lus the fact that separable dis- t r ibu t ions a re s t r ic t ly on ly appl icable to rectangular g r ids and rectangular boundaries, limits their use.

Despite this, much can Se learned about planar a r r a y s by studying the separable d is t r ibu t ion case 1101 and the array designer can p r o f i t from delving i n t o t h i s case exhaustively since its principal fea- tures are shared by the non-separable distributions. Beam placement via the introduction of uniform pro- gressive phases in the two l inea r distributions is

RECTANGULAR GRID

TRIANGULAR GRID CIRCULAR

GRID

Figure 6. Three Common Arrangements of the Elements i n a Planar Array

5

IEEE Antennas and Propagation Society Newsletter, April 1986

I

X I Figure 7. Projections in X Y Plane o f Null Contours on Far-Field Hemisphere. a) Separable Distribution.

b ) Q Symmetric Non-Separable Distr ibut ion.

r e a d i l y u n d e r s t d , and the bearn-pointing formulas and scan-limit equation have simple f o m . The nature of the main beam ( tha t its 3 dB contour is an ellipse) , the def in i t ion of a rea l beamwidth, and the concept of principal cuts throught the main beam are features t h a t can be demonstrated convincingly. A simple for- mula fo r d i r ec t iv i ty emerges which can be used a s a ya rds t i ck for the mre e f f i c i e n t non-separable dis- tr ibutions.

Given tha t one wants to learn about planar arrays by studying the separable distribution case, but only seldom wants to use such designs in practice, what type of sum pattern caused by a planar array would be

value of e a l l Q di rec t ions are equally j-rtant with more desirable? As an exanple, suppose fo r a given

respect to side lobe reduction. Then one desires an a r r a y f a c t o r t h a t l m k s the same i n all Q -cuts. I n

gested in Figure lb. o t h e r words, one wanering-type side lobes, as sug-

Taylor 1111 was the f i r s t to address this p m b lem. He assumed a linearly-polarized continuous dis- t r ibu t ion in a planar aperture with a circular bound- ary and required a pattern with Q-symnetry. The pat- tern w a s to cons is t of a pencil beam pointing along the Z axis, surrounded by a family of ring side lobes o f a quasi-common s p e c i f i e d h e i g h t . H i s a n a l y s i s proved to be the circular analog of h i s earlier linear aperture developnent. Taylor's 9-symnetric pattern and the corresponding tsymnetric aperture distribu- t ion are given by

n-1 n n

n = l 1 1 1

I n (14) and (151, Jo and J 1 a r e Bessel functions, u = (Za/X)sin 8, w i t h a the radius o f the c i rcular aper-

* Y

t u r e , i s a root defined by J1(Tyln) = 0 with n = 0,1,2,.y?and p i s t he r ad ia l coordinate i n the a r- ture . The roots u n are given by u g = yf ; [A2+(n-+P'I/ [A2-+(n-+) ' j with -20 log cosh T A the desired side lobe level. As in the {Inear aperture case, Taylor 's side lobe level droops sl ightly; n i s a selectable in- teger which marks the boundary between the inner , drooping side lobe region, and the outer region where the side lobes decay a s r 3 / 2 . A typical Taylor c i r - cular pat tern and its corresponding aperture distribu- t ion are shown i n Figure 8.

Taylor c i rcular aper ture dis t r ibut ions can be sampled to y ie ld the exc i ta t ions of planar arrays. For arrays with circular boundaries, this is of ten done for rectangular, tr iangular, and c i rcu lar g r ids . Pattern degradation is inevitable. It increases as the planar array size diminishes. There is more de- gradation if the elements are arranged in rectangular or t r iangular gr ids than i f they are laid ou t i n a c i r c u l a r g r i d . Indeed, i n the latter case, pa t te rn degradation is virtually eliminated if the elements are less than 0.75 Xapart 1121.

Taylor c i rcular aper ture dis t r ibut ions can also be sampled to obtain the exci ta t ion of a planar array wi th an e l l i p t i ca l boundary. One merely employs the one-way stretch 5 ' = 5 , rl' = (a/b)q to go from ( 5 , n ) m r d j n a t e s i n the plane of the actual array to ( 5: n ' ) coordinates i n the plane o f a v i r tua l a r ray ; v i r tua l a r ray has a c i r c u l a r boundary and , i t s yn! t%l element has the radial ccwrdinate hn= [(E,,!, )'+(n,!,,,) q z. me Taylor pattern is modified to have a %eawtdth i n the YZ plane (a/b) times as great as the beamwidth i n the XZ plane. The ring side lobes have a n e l l i p t i c a l contour. The remarks already made about pattern de- gradation when the boundary is circular apply equally well when the boundary is an e l l ipse.

The use of sampled Taylor c i rcular dis t r ibut ions is l i m i t e d to a p p l i c a t i o n s i n w h i c h t h e s i d e l o b e l e v e l is the same i n a l l Q -cuts, with the side lobes a t a quasi-ccnrrmn height. It has proved possible to generalize Taylor's analysis so as to produce "roller- coaster" s ide lobes (d i f f e ren t l eve l s i n d i f f e ren t Q - cu t s ) or a @-independent but otherwise a rb i t r a ry side lobe topography [13]. IAe latter can be applied

6

IEEE Antennas and Propagation Society Newsletter, April 1986

f u = (za/hl sin 8 I

F igu re 8. A T y p i c a l T a y l o r C i r c u l a r Sum P a t t e r n and the Cor respond ing Aper tu re D is t r ibu t ion . SLL = - 25dB, R = 5.

to circular grid arrays with negligible pattern degra- dation [141.

A technique that works well with rectangular or triangular grids and an arbitrary boundary uses the concept of collapsed distributions. If we return to equation (11) and assume that x = mix, ymn = nd then for $ = g o and a=~=rr/2, we gel" Y '

M(n)

m= 1 The c o l l a p s e d d i s t r i b u t i o n s I,,,,, = an and

Imn m = b can be so chosen t o g i v e e x a c t l y t h e

n= d e s i r e d p a t t e r n s i n t h e XZ and YZ planes. This avoids, i n t h e p r i n c i p a l p l a n e s , t h e p a t t e r n d e g r a d a - t i o n i n t r o d u c e d when sampl ing con t inuous p lanar d is - t r i b u t i o n s . It also pe rm i t s t he sum p a t t e r n t o have a d i f f e r e n t s i d e l o b e t o p o g r a p h y i n t h e t w o p r i n c i p a l planes.

With Imn and I,,,,, known, the remain ing p rob- m n

lem i s t o dbduce how 't'o spread ou t these co l lapsed d i s t r i b u t i o n s , t h a t i s , how t o d e t e r m i n e t h e i n d i v i - dual Imn values. There i s no unique answer but a use fu l way t o proceed i s a s f o l l o w s : L e t a s t a r t i n g d i s t r i b u t i o n I,,,, be determined from

This can be recognized as the average of spreading out each collapsed distribution so that a row (column) is uniformLy excited. The result is usually a distribu- tion that is not too far fmn what will ultimately be achieved. ?he correspmding starting pattern is given bY

A n i n t e r a t i v e t e c h n i q u e d e s c r i b e d i n t h e l i t e r a - ture [ I 51 can be used t o improve on (19). A l te rna - t ive ly , the conjugate gradient method can be employed. An example i s shown i n F i g u r e 9. The p a t t e r n s a r e f o r a 20 by 20 r e c t a n g u l a r - g r i d a r r a y w i t h c o r n e r s l o p p e d - o f f t o f i t a c i r c u l a r b o u n d a r y . T a y l o r - l i k e d r o o p i n g

s ide l obes were spec i f i ed , w i th a -21 dB s i d e lobe l e v e l i n X Z plane and a -30 dB s ide l obe level i n t h e YZ plane.

Difference Patterns - Until recently, difference pat- terns have been the handmaidens of sum patterns in the following sense: Radar antenna arrays that produce both sum and difference patterns typically have their elements arranged with quadrantal symnetry. Wen all four quadrants are excited in phase, a sum pattern re- sults, and it is this pattern for which the proper element excitation is chosen. To get a difference pattern, two quadrants are then fed out of phase with respect to the other two. One takes what one gets as far as the resulting excitation and difference pattern are concerned, and what one gets is far f m optimm. With the advent of new feeding schemes, this handmaid- en status is no loplger necessary, and interest in the synthesis of difference patterns has revived.

The seminal work on th i s p rob lem was done by Bay- l i s s [161 . Us ing Tay lo r ' s ana lys i s f o r t he sum pa t - t e r n as a g u i d e , B a y l i s s f o r m u l a t e d t h e d i f f e r e n c e pa t te rn express ion

shifted root that Bayliss was Abl!"to Connect tO In (21), p n is a root of J'(a ) = 0 and Un is a

and the side lobe level using fitted polynanials [167! The corresponding aperture distribution is given as a truncated series of Bessel functions [161. A typical Bayliss pattern and the continuous aperture distribu- tion which produces it are shown in Figure 10.

When applied to planar arrays, Bayliss' technique poses the same limitations as Taylor's. There is a pattern degradation due to sampling and the side lobe toposraphy is quasi-unifonn. The contour should be circular or elliptical; the degradation is least for a circular or elliptical grid.

If the grid is rectangular or triangular, reg& less of the shape of the contour, the ooncept of a collapsed distribution suggests a useful design tech- nique. The function f(u) in (20) can be chosen as the difference pattern of an equispaced linear array. The side lobe topography can be arbitrary. One needs to

7

IEEE Antennas and Propagation Society Newsletter, April 1986

THETA THETA

THETA THETA

Figure 11. Difference Patterns Due t o a Spread-Out Dis t r ibu t ion for a Planar Array. SLL = - 30dB. 20 x 20 Rectangular Grid, Circular Boundary.

spread out the collapsed distribution that produces f ( u ) so that D(u,Q) is achieved. The iterative proce- dure mentioned in connection with inproving on equa- tion (19) can be followed in this case. An example of this is found in Figure 11. Here again the patterns are for a 20 by 20 rectangulargrid array with corners lopped off to fit a circular boundary. Bayliss-like drooping side lobes were specified with a -30 dB side lobe level relative to the twin peaks in the $I =Oo cut.

Shaped Beam Patterns - The technique of spreading out collapsed distributions can also be applied to the de- sign of planar arrays which produce shaped beam pat- terns. A practical example is suggested by Figure 12. One can picture a nose cone radQRe behind which re- sides the planar array of a ground-mapping radar. The

vertical pattern is to be csc%-cos aover a specified range and the horizontal pattern is to be a pencil beam with same desired side lobe level. Once these principal plane patterns are specified, Orchard's pro- cedure 191 can be used to determine the two collapsed distributions. If the pattern is to change snoothly through successive Q cuts, enough information is known to solve for the spread-out distribution.

An added advantage in the shaped beam case is that there are 2M collapsed distributions that will give the same csc 2 e .cos e pattern. One is able to choose the distribution that offers the least diffi- culty in overming the effects of mutual coupling. Figure 13 shows the synthesized pattern for a 26 ele- ment array. The pat tern is csc28.cos e over a 40" range with a 2 2 dB ripple. Eleven of the twenty-five roots need to be off the Schelkunoff unit circle in

= ( 2 m sin e 0 I 9 2

P = zp'a

Figure 10. A Typical Bayliss Difference Pattern and the Corresponding Aperture Distribution. SLL = - 30dB, ii = 4.

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IEEE Antennas and Propagation Society Newsletter, April 1986

0 0

-10 -10

-20 -20

I m

-30 -30

-40 -40

0" 90" -9W -60" -3(p 0" 90 -50

THETA THETA

0 -

4=w -10

c

THETA THETA

Figure 9. Sum Pat terns Due t o a Spread-Out Distribution for a Planar Array. SLL = - 2 1 dB in XZ Plane, - 30 dB in YZ Plane. 20 x 20 Rectangular Grid, Circular Boundary.

Figure 1 2 . Ai rcraf t With Planar Array Behind a Nose Cone. Array Produces a Shaped Beam Pat te rn for Ground- Mapping .

order to synthesize this pat tern so there are 2048 ac- ceptable dis t r ibut ions. The cQoputer can be asked to l o o k a t a l l these dis t r ibut ions and select the one fo r which Imax /I ,,,in is a minimum, since that serves to ease the mutual coupling problem. I n t h i s case the computer selected the distribution shawn i n Table I. f ie amplitude variation is only 3:l although an alrnost mnplete phase reversal occurs near one end of the array.

A -20 dB side lobe level was desired in the other principal plane. The spread-out distribution gave in- termediate Q-cuts which show the f i l led- in pat tern giving way to a typical sum pattern. Several of these $-cuts are displayed in Figure 14.

This pattern synthesis w a s undertaken f o r a sp - c i f ic p ro jec t . The ult imate resul t was the planar ar- ray shown on the cover of the October 1985 issue of the Newsletter. A prototype 6 by 26 ar ray w a s b u i l t and tested a t Rantec. Since the shaped pattern w a s to be in the E-plane, mutual coupling was par t icu lar ly severe, but the experimental results were favorable [171 and the f inal array met al l spec i f ica t ions .

Conclusions - Pattern synthesis for equispaced l inear and planar arrays has been a f e r t i l e f i e l d of inquiry for the past half-century. The advent of the modern computer has made possible the uti l ization of sophis- ticated techniques which will generate, at reasonable cos t , sum and difference pat terns with arbi t rary s ide

1 3 5 O 157.5c 1

Figure 13. A Shaped Beam Pat tern for a 26 Element Linear Array. CSC28 .COS0 over

a 40" Sector. d = 0.71.

lobes. Shaped beam pat terns can a l s o be synthesized with a specif ied r ipple and with a choice of excita- t ions which a s s i s t s i n overcoming mutual coupling. The concept of spreading out collapsed distributions, so

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IEEE Antennas and Propagation Society Newsletter, April 198.6

-30

-40

0

-10

-20 dB

-30

-40

-50 00

-10 - @ = 900

-20 dB

-30

4 0

-50 00 30" 60" 909

dB -30

4 0

-50 00 900

-1 0

-20 dB

-30

-40

-50 30° e 60°

-10 I I

@ = 60°

F igu re 14. F i v e $ - C u t s f o r t h e Spread-Out D i s t r i b u t i o n t h a t Gave a Shaped Beam i n t h e X2 Plane and a Penc i l Beam with 20 dB Sidelobes i n t h e YZ Plane. 26 x 26 Rectangular Gr id , C i rcu lar Boundary.

t h a t d e s i r e d p r i n c i p a l p l a n e p a t t e r n s a r e a c h i e v e d t o - g e t h e r w i t h a smooth t r a n s i t i o n t h r o u g h o u t i n t e r m e d i - a te $ -cu ts , shows g r e a t p r o m i s e f o r a p p l i c a t i o n t o p l a n a r a r r a y s i n s i t u a t i o n s where o the r t echn iques a re impotent.

Acknowledgments - The author wishes t o exp ress h i s t h a n k s t o Y. U. K i m who generated a l l t h e s p r e a d - o u t d i s t r i b u t i o n s a n d t h e r e s u l t i n g p a t t e r n s . The suppor t o f Hughes and the generous counsel o f v a r i o u s a n t e n n a engineers i n t h e Hughes research group i s a l s o warmly acknowledged. Rantec i s t o be thanked fo r sponsor ing t h e shaped beam p l a n a r a r r a y s t u d y .

TABLE I: CURRENT DISTRIBUTION FOR 26 ELEMENT ARRAY TO G I V E CSC28 .COSBPATTERN * Phase

14 1 ) .834 I 128.39'

15 .914 131.08"

16 .895 121.45'

17 .865 117.09"

18 .566 108.06'

19 .589 52.49"

-

-

25 1.218 -45.15"

26 1.000 0"

References

10. See Reference 3, pp. 197-213. 11. T.T. Taylor, '&sign of Circular Apertures f o r

Narrow Beanwidth and Low Side Lobs", IRE Trans., v0lAF'-8 (19601, 17-22.

12. See Reference 3, pp. 230-231. 13. loc. cit., pp. 218-225. 14. loc. cit., pp. 233-237. 15. loc. cit., pp. 243-249. 16. See Reference 6 and Reference 3, pp. 250-255. 17. J.J. Erlinger and J.R. Orlow, Waveguide Slot

Array with csc2 e cos e Pattern", Antenna Applica- t ions symposium, University of Illinois, Septem- ber 1984.

10