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Space tapering call also be extended to planar arraysof other shapes and to three-dilnensional arrays such ascylinders and sp!leres. 111 drawing the annular ring gridwhich was used to locate the elem ents, we were essen-tiall>- making a contour plot of the llumination func -tion over the aperture. This same procedure can he fol-lowed for otherplanararra>.sand llunlination f u n c -tions. Obviousl\-, howe\ler, the construction of the con-tours will not be as str;lightfornmd as for the circular;ma>-vir11 a circular1)- s>'mm etric illunl i~~at ion functi on.

\\Then ;Ippl>.ing th ismethod to three-dimensionalarra!.s, theonlyextrastep equired is toproject heelements i n the reference array onto a plane that is tan-gent to the center of the arr a?.; this will give a "new"

referellce ar ray with which to work. Since the new refer-ence arra?; is planar, the procedure from this point on!vi11 be the same as t h a t described previously.

IS. ~ ( . I ~ ~ o \ ~ - L ~ ; I ~ G ~ ~ I ~ ~ T'I'he author is indebted to many who aided i n the

gathering of data for this repo rt: to \Y. chneider, whoperfomled man!.of the calculations and made severalsignificant contributions; to J . Best and H. Dantzig,who offered mall>-helpful uggestions: to L. Ehudin,wh o performed the graphical design of the nrral-s; andto E. Reed, X. Hart , H. R y n a n , E. Crizer, a n d R . Sel-lcrs, who \\-ere responsible for the construction and test-ing of the tes t array.

A Quasi-"Isotropic" Antenna in the Microwave Spectrum"

Summary-Thegeneralproblem of radiation by apertures onspherical urfaceswas first considered theoretically by Strat tonand Chu in 1941.' More recently, the problem was again consideredby Bailin and Silver,? and Mushi ake and Webster.3Al l numericalandexperimental efforts to datehavebeenre-

stricted to the special case of a very small sphere as measur ed inwavelengths. This paper is in effect an extens ion of the previousresul ts to the case of large sphere s uniformly excited by an equa-torial slot of varying size. The detailed numerical and exp erimentalresults are restricted to the fi s t TM and TE modes.

In general, t s possible oshow that he arge spherical slotantenna can be made unusually "isotropic" with a far-field coveragefactor of over 98 per cent.

If the antenn a is to be characterizedby a n ornnidirec-tional broad-beam radiation pattern, i t is evident thatthesupportingstructuremightbespherical i n shape.hope full^., an ten nas of this kind c o u l d be made quasi-"isotropic" b\. properl~r exciting he spherical shell.

I he generalproblem oi radiation t)!, apertures onspherical surfaces was first considered heoreticall>- bStratton and C h u i n 1941.' >lore recentl?., the prob lem\vas again considered b,. L3ailin and SillTer i n 19.56? andby i'ebster and 1Iushinke i n l s )j i .3

,.


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Julysidered in detail. These modes could be excited by theEoand HI modes5 of a radial waveguide feeding the slot.

I t will be shown that l arge anten nas of this type canbe made unusually sotropic, particularly for he firstTM mode.Forexample,asphere 20 wavelengths i ndiameter can be used to obtain a 3-db coverage factor6of over 98 per cent.

b2a

a = radius of the sphere, = height of the slot,E O elec-tric field intensity in the slot, = 2r /X , h n @ ) ( x ) spheri-calHankel unction, P,,= Legendrepolynomial,and77 = d p , / e o . I t is of importance to note that the sum on can be bound by some AT such that X - k a for ka>>l .For reference , the slot field intensity is given by :

cr = sir1-- 7 and

-Th e spherical slot antenna can be characterized b,- a

perfectlyonducting 2 2in 4 by a slot located in the equatorial plane. The ge-omet ry is illustrated by Fig. l. As noted by Stratton and B. The TElChu,l and Bailin and Silver, the electromagnetic fieldat some point P can be written as the sum of a se t of This mode can be excited by the H1 mode in a radialorthogonal TE and ~ y qodes. The magnitu de of any 1yaVeguide feeding the slot. The far-fie ld components ofmember of the set is determined by the boundary condi- the a n t e n n a are giventionsverhelot. From symmetryonsidera- Utions t is evident hat heoptimum sotropycanbeob- &,(r, 0, t ) = 2 E 1- -i(ui-kr)tained by exciting the first TM an d ?E modes: TMo r

E , = h ( r - a ) ~ o for- r 5 e 5- aiT n-

Iand TE1,since these are independent of the angle 4.

XFig. 1-Spherical slot antenna.

whereB , = 2s 9os [ - cos e] sin Ode

ir12-L-i

2aand where El is the maximum valu e of the elect ric fieldintensit): n heslot.For eference, heslot field in -tensity is given by:

A . T h e TMoM o d e E , = & (r - a)E l CO S n-- OS e(aa )This mode could be excited by he EOmode n he

radialwaveguide eeding heslot.The far-field corn-pone nts of the ante nna are give ny:1,2

Ee(r, 0, t ) = 2Eo- e i ( w f - k r )a C. d X e a s w e of I s o t r o p y7 TWO ethods are commonly used to measureheela-x 212 + 1 -4 d Y , , tivesotropy of anntenna:

. p z n ( n + 1) [ K a h , , y K a ) ] B (1) I ) Gain e lat ive to Isotropic, G.M: Thissimplydirectcomparison of the maximum value of the gainfunction G ( 0 ,4 ) , o unity.2) H a t f P o w e r or 3-db B e a m w i d t h : Either method is

quite useful in t he evalua tion of an antenna with onewhere of simplemnidirectionalharacteris-tics like the dipole. During the course of this investiga-

~ si n BdB tion,hough, i t becamevidenthateither of th eabove two methods was useful a s a measure of isotropyof the spherical slot antenna. I n view of this difficulty

6 The notion of coverage factor will be defined in this paper. possiblemethodandusedwithconsiderablesuccess.

1t

H + ( r , 9 , t ) =- e ( T , 8, t )

,4 = 2J d P nsp--a de

Co., Inc. , Xew York, N. Y . 1931.5 N. Marcuvitz, Waveguide Handbook,3,rdhxv-Hill Book the notion of coverage factor was introduced as a third


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3 ) Cozlcrage Factor Rel nti ze o sotr opi c: 'I'he notionof coverage factor is of interest since some antennas areusuallydesigned as integral parts of an over-allcom-munication system. The transmission characteristics ofsuch a s\-stem remost onvenientl>- alculatedbJ-assuming an isotropic antenna at bot h ends of the cir-cuit. I n view of this, it is conveni ent to define a iunc-tion, expressible a s a simple number, which could serveas a measure of that pnr t of tlze t ot al a n t e m a r a d i a t i o njield surface mhichexreeds a partic -ulnr IeaeI re lative t oisotropic. The par ticu lar level will i n general be deter-mined by the over-all s~ - st e~ noise figure, etc.

The coverage factor C.F., i n per cent relative to iso-tropic, can be defined as

The urface of integration S is detenninetlbyheparticular level of interest.

S such that K G ( 8 , 4 ) 2 G Owhere G(0, 4) is the usual gain function relative to iso-tropic and K is the polarization loss factor defined byS~helkunoff.~ For the caseof a linearly polarized s>'s-tem, it is evident that K = . I t is also evident t h a t atruly isotropic antenna, G(0, 4) = 1, u-ould yield a C.1:.of 100 per cent for G o5 1.I n order to gain some insight into the role of C.F. itis convenient to consider a fen. examples. As a reference,let the comparison be made for a G Oof -3-db relativet o isotropic.

a ) I tz jinitesimal electr ic dipole: \4,'e have,$32G ( 8 , 4 ) =- in?0, G X = 1.76 db.

I t is evident that G ( 0 , r$)> 3 db or $ for 35.25'50- 144.75". Consequently:

1 4 4 . 7 5 'C.F. = - 2aoo s sin 0d847T 35 . 2 6 'or C.F. = 81.5 per cent. This antenna couldlso be char-acterized adequately by the econd method, asan omni-directional antenna with a 3-db beamwidth of 90'.b ) Ha(f-wa.oe d ipo le : IlTe have,8

G.tf = 2 . 1 1 dbI t is evident hat G ( 0 , 4 ) >_ -3 db or + for 39.5"-depends o n its order and he magnitude ofK a . Some comparisons of the exact solution to the geo-nletricalopticalareavailable i n the iterature for hecase of slots on cylinder^.^

I n view of the speed of modern digital computers, therange of kn i n question,

the accuracy required, and the fact that over 100 pat-terns were actually calculated, i t became evident thatthe original harmonic series could be used. I t is of in-terest to note that the time of calculation for each pat-tern with incrementsof f" in 0 was abou t 40 seconds.

The necessaryprogramandcomputation were per-formedunder hedirection of A'Iiss RI . Gray of BellLaboratories..-I. T X oM o d e (EB,H6j

T h i s mode is oi particular interest since t can beusedto generate an unusually "isotropic" linearly polarizedfar field. As noted in Section11, the field in the slot couldbe excited by an E O ode in a radial waveguide. A par-ticular aunchingstructure isdiscussed i n connectionwith the experimental results in Section 11,'.

The numerical analysis was restricted to spheres offrom 1 to 20 wavelengths i n diameter in increments ofone wavelength. Slot heights of three-quarter, one-half,an d infinitesimal size were considered for each diameter

9 L. L. Bailin and R. J. Spellmire, "Convergent representationsfor the radiation fields from slots, in large circular cylinders," I R ETRANS.N AKTENNAS ASD PROPIGITIOK,ol. AP-5, pp. 374-382;October, 1957. In particular, see Fig. 3 o n p. 377.


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other than the first two. Th e height of the slot was re-stricted to less than one wavelength by multimode con-siderations in any practical launching structure.

In general, the points of interest are the changes ofantennagain, adiatingpatternandcoverage actorwithchanges n hediameter of thesphere,and heheight of the slot.1) d n t e n n u Gain: All an tenna gai ns were calculated

relative o sotropic. The gain n he equatorial plane(8= 90) for the case of a 6 (infinitesimal) slot has beenplotted as a function of the dia meter d of the sphere inFig. 2. Somearlieresults ueo lushiake ndb:ebster3 for small spheres has been included for refer-ence. I t is evident that the ga irin the eq uatorial p lanefor this size slot is asymptotic to about - 2 db as thediameter of the phere ncreases o 20 wavelengths.The reason for the negative value will be apparent froman inspection of the radiation patterns to follow.

The change i n gain in the equatorial plane with in-creasing slot size b is also of interest. From elementa r>-aper ture theor!. one would expect the gain i n this direc-tion o ncreasewith ncreasingslotsize. I n order oillus trate this change, some of the resul ts of the exac tcomputationhavebeenplotted as Fig. 3. Results or

D 2

FIG.2 - G a i n at 8=90 as a function of the diameter of the sphere.TMomode, 6 slot.&-Bugnolo0-Mushiake & bebster3

Fig. 3-Gain in the equatorial plane asa function of theaperture height and in wavelengths.

other diameters are available n request. The results foth e TE1 modeare ncluded or uture eferenceanddirect comparison.2 ) Rudiation P a t t e r m : A411radiationpatterns werecalculated with incrementsof t o n 8 and ind b relative tthe m agnit ude sq uared in the equatorial plane. Some ofthe results have been plotted as Figs. 4 an d 5. The gainin the equatorial plane relative to isotropic is includeothat the various patterns can be compared with eachother.

An inspection of these results shows hat the gain asfunction of 8, G(8),exceeds som e arbitrary level su ch as-3 d b relatire to isotropic over a very large part of th efar field. Also, the relati ve isotro pyof the antenna increaseswith ncreasingdiameteranddecreaseswithincreasing slot height.

The special case of a 6 (infinitesimal) slot isof particu-lar interest. From Figs. 4 and 5 i t is evident that thisan ten na is quite nearly isotropic provided the sqrstemis not adversely affected by the oscillatory nature of t hefield intensity.Keedless o s a y , somecommunicationsystems can be so designed.3) Coaeruge Factor: The coveragefactor npercent

was efined by (3). I n ordero omparehe T Mmode to the dipole antenna, t is convenient to calculathe C.F. for a Go of -3 d b relntive to isotropic. The re-sults are summarized in Table for the principal or firsequatoriallobe.Two tendencies are immediately evident. For spheresof diameter less than 6 wavelengths the coverage factorincreaseswith ncreasingslotsize; orspheres of di-ameter greater than 6 wavelengths the opposite is true.Also, the argestcoverage actorcanbeobtainedbyusing a very small slot on a large sphere.4 ) Conzments: From nspection of the radiat ion pat-terns i t is evident hat he arge coverage factors areobtained at the expense of oscillations of the pa tt er n.The depthof the minima can be quite large as compawith he magnitude of the maxima. However, t is ofimportance onote hat hedepth of theminima isnever large relative to isotropic. This is the characteristicthat results in a large coverage factor. The usefulnessof this ntenna heref oredepends xplicitlyon heability of the over-all system o handle hese fluctua-tions. One possible example of such a system is an ac-tive satellite communication network.B. TE1 X o d e (E+,He)

This mode is of interest since it can be used t o gen-erate a far field in space and time quadrature with theTA.10 mode. As noted in Section 11, the field in the slotcould be excited by th e H1mode in a radial waveguide.A particular launching structure isdiscussed in connec-tion with the experimental results ir , Section IV .

The numerical analysis was restricted to spheres offrom 2 to 20 wavelengths in diameter in increments of


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382 IRE TRANSACTIONS O N AIYTENNAS AND PROPAGATION J d Y

Fig. 6-Gain in the equatorial plane as a function of thediameter of the sphere. T EI mode, X/2 slot.

Fig. 8-Theoretical radiation patterns, TEI mode.

for diameters of from 4 o 20 wavelengths. Hence86 .6 forb = X/2

%,c.F. 7 9 . 9 fo r b = 3x/4for 4X I I OX.

In effect, the size of the sphere does not affect the sys-tem capabilities of the a nten na from this poin t f view.4) Commetzts: As noted previously, he nfinitesimal

dipole antenna has coverage factor of 81.15 per cent. I tfollows that theTE1mode compares favorably. This e-sult may at first appear trivial; however, it should benoted hatapplicationsexist so that he TE1 modespherical slot antenna can be constructedas an integralpart of thecompletesystem. A magneticdipolean-tenn a would requirea supporting structure. This and thproximity of the emainder of thesystemwouldde-crease the C.F. of the dipole to a figure well below 81.5per cent. The active satellite communication networkis one possible example of such a system.

IV . SOME XPERIMENTAL EXAMPLESThe experimental models of the TMo and TE, mode

anten nas were designed to operate in theX band (8-12kMc). Two wooden spherical hemispheres 24 inches indiameter were rigidly attached to the platesf the radialwaveguide. Th e structur e was cover ed with thi n sheetsof copperfoil.Over-all oleranceson hehemisphereswere less than 1/10 inch. Access to the interio r of thehemispheres was providedby two removable caps about4 inches in diameter located a t th e poles. This locationwas desirable to minimize the physical variations in the$ direction and to locate the capsn a region of minimumsurface current. A photograph of the model is shown inFig. 10.

ThesourceproblemwassolvedbyusinganEsakidiode resonant-slot oscillator supplied by W. M. Sharp-less of Bell Laboratories. Th e small source and its powesupplywere asilypositioned ompletelywithin hesphere.

Fig. 9-Theoretical radiati on patterns, TEl mode. Fig. 10-A 24inch spherical slot antenna.


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383A . TJlo X o d e (EO,H ,)1 ) Small Slot, b = 0.3 Inch: For this case the oscillator

was coupled directly to the center of the radial wave-guide by a probe from a 50-ohm coaxial ine. One ex-ample of the far-field rad iati on ntensitypatternha sbeen reproduced as Fig. 11 for a frequency of 8.65 kMc .This corresponds to a spherical diameter of 17.6 wave-lengths and a slot height of 0.22 waveleng th. The pat-tern in the quatorial lanewas a circle towithin 0.5 db. Fig. ll--Esperi~nent;lladiationatternlator was coupled to the launching structure through a50-ohm coaxial line. T he launching structure consistedof a 50-ohm biconical . line which i n turn fed the radialwaveguide a t a point where its impedance was 0 ohms.This resulted i n a very broad-band lo w \,'S\VR match,betweenhe oaxialine ndhe adialwaveguide.One example of the far-field radiat ion intensity patternhas been reproduced as Fig. 1 2 for a frequencl: of 9.75khlc.hisorrespondsophericaliameter of about D 10 20 3020 wavelengthsnd a sloteight of ab ou t 0.7 wave- Fig. 12-Experimental radiationatternlength.B . TE1X o d e ( E , , H s )

2 ) L a r g e Slot, b=0.843 I n c h : Forhisasehe oscil- TlIo mode, d = li.6X, h = 0 . 2 X

T310mode. d = ZOX , h =O . i X .

For this case the oscillator was coupled to the radialwaveguide by a transition section fromTElo rectangularto TEol circular mode in a ?-inch pipe. T he purit). of thecircular electric mode was insured by a section of heli-cally oaded pipe. The ransition section was suppli edby J . J-oung of Bell Laboratories . One example of thefar-field adiationntensitypatternhasbeen epro-duced as Fig. 13 for a frequency of 10 k l l c . This cor-responds oa phericaldiameter of ab ou t 20 wave-lengths and a slot height of abo ut 0.7 wavelength.

I n general, i t is possible to conclude that the experi-mental esultsagreewith he heoreticalwithin helimits of accuracy of the experiments.

V. CONCLFSIONSThus, i t is possible to conclude that this antenna is a

useful device for radiation i n the microwave spectrum.The TkIo mode antenna can be designed o 1-ield a

considerable increase in coverage factor over th at of aninfinitesimal dipole. I t is also evident that this antennacould be designed for extremely large bandwidths.

Th e TE l mode antenna has a coverage factor hat

Fig. 13-Esperime1ltal radia tion patternI E , mode, d=20X , b = O . i i ..

compares favorably with thatof a small magnetic dipolewithout the severe bandn-idth limitations usual1~- asso-ciated with small antennas.

V I . . ~ C I i N O \ \ - I . I ~ D G ~ ~ - I E N TNeedless to say, a numerical and experimental proj-

ect of this magnitude requires co~~siderable assistance.I n particular, heauthor would ike to han k A\. B.Crawford of Bell Laboratories for his man). interestingdiscussions. The necessar?- programming and computa-tionswereperformed by hliss 31. Gra>-, Airs. C . L.Heattie and 3Iiss P. Ilnmilton oi I3ell Laboratories.

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