analysis of rect horn ant via uniform asymptotic theory

7/27/2019 Analysis of Rect Horn Ant via Uniform Asymptotic Theory

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AND PROPAGATION, V O L . AP-30, NO. 2, MARCH 1982Analysis of Rectangular Horn Antennas via Uniform Asymptotic Theory

RONALD C . ME N E N D E Z , MEh lBER , LEE , A N D SH U N G - W U LEE, FELLOW, IEEE

Abstract-Approximate analyticxpressionsorhear-fieldpattern ndmodal eflection coeffkient ofhehree-ntenna ave een eveloped. Based upon a

modelf thewo-dimensionalorn-waveguideapproachoffersgreatergenerality than previously

nd accuracy. Theasymptotic heory CAT),which semployed hroughout,

accuracy in the solution. It is demonstrated that, in most cases, theof the horn is reducible to the pattern of a simple slit

an absorbing creen lluminated by anarrayofpatterned ineThemeasured patternsofseveralhorns are convincingly

ered by the prediction of the analytical model. The same model iso calculatehemodaleflectionoefficient of the horn-

and, again, comparison with measured results is

I . I N T R O D U C T I O NY R A M I D A L andsectoralhornantennashavebeenana-lyzed by a variety of app roach es [ l ] -[ 61 wh ich use rayMost a re ybr id pproacheswhi c h oup l e ay

iding a comp le te ray p ic turet he ho rn a n t e nna . The goa ls o f h i s pa pe r a re to d e m o n -e ha t suc h a p i c t u re ma y be fo rmul a t e d , a nd t o ob t a i ndom inant far- field pa t te rn and the mod a l re f lec t ion coef-

of th e horn-waveguide anten na s t ruc ture . Fur the rm ore ,a comple te ray p ic ture will be show n to be more genera lThis paper is organized as fol lows. After a concise summarythe re levant d i f f rac t ion theory in Sec t ion 11, a s t a t e me n t o fproblem i s presented n Sec t ion 111. I t is fo llowed by th eesults of th e far-fie ld adiat ionpattern Sectioncompari sonsof hepredicted patterns wi thme a s -

IV-D). A discussion of the meri t s of th i sh e la tive to previouswo rk is given in Section IV-E.heappl ica t ion of th i smult iple magem o d el t o o b t a i n h eeflect ionoeffic ients resentedn ec t ion V-A,owed by a compari son of e xper imen ta l resul t s and predic -in Sec t ion V-B.T h r o u g h o u th isw o r k ,he n i fo rm sympt o t i ch e o r y) is e mpl oye d t o de sc r ibe hediffract ionprocesses n-The nota t ion and te rminology will be those employ ed[ 8 ] andma nyof he esul t spresentedheredrawheavi ly

Severa l onvent ions used throughou this apero l l ow.( - i w t ) and i s suppressed. 2 ) The d i f -problem i s lways educed to wod i m e n s i o n s n oManuscript eceivedMarch18 ,1980; revised Ju ne 2 5 , 1981 .Th i s\vas supported by he National Science Foundation under GrantR . C. Menendttz is with Bell Labora tories , Whippany Road, Whip-S. W . Lee is with t h e Elec t romagnet ics Labora tory . Department of

NG 77-20820 .NJ 07981 .

niversity of Ill inois, Urbana, IL 61801 .

24 1

y var ia t io n) . 3) Both he t ransverse magne t ic (TM)case (non-ze ro f ie ld components H y , E,, E,) an d the transverse electric(TE) case (E, , , H ,, H z ) are rea ted s imul taneou s ly , wi th hehe l p o f t he t wo symbo l s a nd r such tha tfo rh l : u = H y and T = + If o rE : u =E, , and r = - l .

11. S U MM A R Y O F G T D A N D U A TF or onve n i e nc eof e fe rence nd nt roduc t ionofno t a -

t ions ,hee levant or t ions fhe eomet r ica lhe o ry fd i f f r ac t i on (GTD) a nd t he UAT a re summa r i z e d i n t h i s s e c t ion[71 - [141 .

A . Geometr ica l Theory ofDif frac t ion [9]As show n in F ig . 1 , a wedg e i s i l lumina ted by the inc identfield u' f rom a l ine source a t 7 = Ti. The prob l e m is t o d e t e r -mine the asym ptot ic so lu t ion of the to ta l f ie ld u r m u p t o t h eo r d e ro f k - ' I 2 inc luded re la tive t o u'). According to t h eG T D , u t is given byG T D : ur(?) = ug(?) + ud(?) + O(k-l), k + m. (1)

Here ug i s the geom et r ica l opt ics f ie ld , comp osed of he in-cident field, u z and the reflected fie ld u r . Corresponding to t h efac t tha t u z s prod uced by the source a t ?iF i g . l ) , u r ma y beident i fied wi th the f ie ld radia ted f rom an image source a t T r .The presence of th e wedge casts a shadow with respect to uiand one w i th respec t ' to ur in the geom et r ica l opt ics sense . Toexpress this , fact mathematical ly, le t us i n t roduc e t wo sha dowindica tors E' and 8 , uch tha t '

+ 1, if i i s in he geom et r ica l shadow region&zr(r) = of u2rr

-1, if ? is in the geo metrical l i t region of u'".(2)

The nma y ewrit ten s~gc)O ( - E ~ ) ~ ~ ( ? ) + e(-Er)ur(?j (3 )

whe re 8 is a uni t -s tep func t ion: B ( x ) = 1 if x > 0 an d e ( x ) = 0if x < 0. The se c ond t e rm ud in (1) is the (Keller ' s) diffractedfield, given b yud(?) =g(kr){X,,,(Jr')u'(;= 01 + X , ~ ( W ) U ' ( ; = 0)). (4)

He re g ( k r ) is a cyl indrical w ave factor,

0018-926X/82/0300-0241$00.75 0 1 9 8 2 IEEE


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2 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 2 , MARCH 1982

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POINT T

1 . Point source at f i illuminating wedge of exterior angle wn.OBi and OBr are the shadow boundaries.

is the diffraction coeff icient of a wedg e of exterior angle( = [ F ( @)- (ti)] u ' ( 7 )+ [F(E') - )I u ' (? ) .

(9 ) is defined by,-in14 -

F(.$)=- i t2 d t , for real t .6For a la rge argument ,F(E) has an asympto t ic expansion

~ . ' ( t )e(-- [) + : ( t ) + NE- ), I t I+ 03here

Th e d e to u r p a r ame te r l l ( l r )n (9) measu res the phase differ-ence be twee n the inc iden t ( ref lec ted) f ie ld and the d i f f rac tedfield, as defined in [ 1 3 , eq . (5.5)] . For the specia l case whenthe source is a l ine source (Fig . l ) , ti>'' ssume s imple forms,namely ,p(;) = 2 , ' Iv'k(a + b - "') I ( 1 3 )

where & e r ) i s the d i rec t d is tance f rom ?@') t o 7, while (a -I-b ) is that via the edge.hus, is th ehasei f ferencebetween ui n d ud a t ?. The to ta l f ie ld ut given in (8) is validfo r a l u', i n c lu d in g p o in t s o n th e sh ad o w b o u n d a r i e s o r n ea rthe edge . In fac t , u t in (8) satisf ies the edge condition at ?= 0.

The fo l lowing observat ions can be m ade when com par ingthe GTD so lu t ion (1) and the UAT so lu t ion8).1) If I i f " I + 00, ( 1 1 ) may b e u sed n (9) and uG + $.Thus , th e UAT so lu t ion reduces to tha t of t h e GTD.2) If I [ 1 1 r I + 0, the GT D fa i ls because u t in (1 ) becomesinfiniteanddiscontinuous. ncontrast , heU A Tsolu-tion in (8) remains valid.

In other words, the GTD agrees asymptotically to t h e o r d e r ofk - l 2 with the UAT as long as I tJ>"are large enough so t h a tthe asympto t ic expansion (1 1) may be used in (9) . I t is usefulto in t r o d u ce two t r an s i t i o n r eg io n sT' and T'. When the obser -vation point 7 falls inside p>',he GTD solution differs, signif-ican t ly rom heU A Tsolution. When u' isoutside p s r , h eGTD olu t ion greeswi th heUAT o lu t ionwithin pre-scribed tolerance. The transit ion regions are defined byTi": I $i*r(Q I < M ( 1 4 )

where M isasuitably argenumber . See [7] and [ 8 ] f o r amore com prehensive discussion.) -

111. STATEMENT OF THE PR OB LEMConsiderectangularwaveguidendyramidalornoriented such that the z-axis is in the ax ia l d i rec t ion , and the

normals o he waveguide walls a re o r ien ted o ie a long hex or y axes, Fig. 2 . We approximate th is fu l l th reed imensionh o r n wav eg u id e s tr u c tu r e b y wo wo d imen s io n a l s tr u c tu r e swhichare heprojections of theh o r nwav eg u id eo n to h ex - z or y - z plane . The pat tern of the hree-d imensionalhorn in the x - z plane is approximated by that of the two-d imensional s t ruc ture pro jec ted on to the x - z plane. Thus, apyramidal o rnspproximated by twowo-dimensionalh o r n s (of possibly different flare angles), while a sectoral hornis modeledbya wo-d imensionalhorn noneplaneanda(9 ) parallel-plate waveguide in t h e o th e r . In addi t ion to projectingthe physica l s t ruc ture of the horn on to two p lan es , the inci -dentmodemust tselfbe eso lved on to hese wo p r in c ip a lplanes. In effect , this corresponds to ignoring he f ield varia-t ion in the y-direction when considering the two-dimensional(10) horn whic h is the pro jec t ion n he x - z p lane . The approx-imatio n of the f ields of three-dimen sional structures by thoseof two-dim ensional structures is common in ray analyses, buti t does imi t he analysis here to the two pr in c ip le p lanes of

Th e r e f o re , n wh a t f o l lo ws , we an a ly ze a two d imen s io n a lhorn-waveguidestructurewith he ntention of apply ing he' resu lts to th ree-d imensional s t ruc tures . Our goal here i s to p re-(1 2 ) sen t a generalcohesive aypictureof his tructureas well

(1 1) th eh o r n .

as to learn the shor tcomings of such an approach .


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MENENDEZ AND L E E : ANALYSIS OF KECTANGULAR HORN ANTENNAS 243

Fig. 2. Three-dimensional ectangular hor n and i t s two two-dimen-sional horn projections.

Fig. 3 . Geometry of two-dimensional horn a n te n n a . (a ) Horn-wave-guide geometry. (b ) Cascaded aper tures of horn-waveguide s t ruc-t u r e .We assum e ingle rbitrarilypolarizedguidedmode spropaga t ingwi thin hewaveguideand s nc identupon hehorn . The geom et ry of he genera l wo-dimensiona lproblemis sketched nFig.3(a).Thecompletehorn-waveguide struc-

ture is specified by three parameters: guide half-width-a , flarelength-L,and lare angle-@. Th eguide i s compo sed of tw oparal le luns taggeredperfect lyconduc t inghalf-planes C1 andCz , separated by awidth 2a. The single propagatingmodeinc ident upon the aper ture f rom z = -O O i s of the formTM: H, .(x , z) = 2 co s & (X + a ) ) ex p ($,a)

n = 0, I I 2, ... ( 1 5)

T E : E I s n ( x , ) = 2i sin (l,(x +a ) ) e x p [ifl,.)n = 1 : 2 , 3, ... ( 1 6 )

where n i s the mod e number , 4, is the transverse wavenumber( E , l ,= n7r/2a), and p,, is he ongitudinalwavenumber p,, =+ ( k - - E 1 z 2 ) 1 / 2 . T h e TM expression ( 1 5) a lsoc on t a i ns heTE M m o d e ( n = 0). The inc ident f ie ld may be readi ly decom-pose d i n t o i t s t wo p l a ne wa ve c omp one n t s a nd , wi t h t he c on -ve n t i ons on u and 7 descr ibed previous ly , both pola r iza t ionsof the incident fie ld may be wri t ten as

ui = u+i + u- i ( 1 7 4whe re

u+ = e x p i (x+ a ) +& z)and

u- = 7 e x p ( - t n ( X + a ) + &z). (1c )Clearly, u+ i (u - i ) represents a p lane wave propaga t ing a t angle@,(-@,) with respec t to the -axis ,

@,l = sin- (n7r/2kn). (18 )Not ice ha t u p i epresen ts he nc iden t f ie ld for he wedgeW1 whi le orresponds to he f ie ld e f lec ted rom W1 . orth esecondwe dge W, he ro l e s of u+ and u - i are eversed(Fig. 3(b)).As show n nFig .3(b) , hehorn -wa ve gu i des t ruc turec on-s i st s of two aper tur es , A an d A 2 , in se r ies. Aper ture A con-sistsof thega pbe t we e n W , and W 2 while aperture A2 isformed offlareedges El a nd E 2 . This pa i r of aper tures m-media te ly sugges ts a number of ques t ions regarding the ana l -ysis of such a structure by ray techniques.

1 ) What will be the role of the pr imary inc ident f ie ld ( themod a l rays w i th in the guide) n the f la re region (be tween41 a n d A 2 ) n d n h e a r f ie ld ( b e l J o n d A 2 ) ? 1 9 a )

2) Can the in te rac tions be tween the throa t wedges and theflare edges be correct ly accounted for in l ight of the dif-ficul ty in cascading U A T solut ions?1b)

3) What will be t he role of the diffra cted sour ces near thethroa t in thea rieldbe yond A 2 ) ? (1c )

Theabovequest ionsar eanswered n he ol lowingsections.In answering those ques t ions , we d raw upon two conc lus ionsderived in [8 I . We now summarize the two conc lus ions .F i rs t ,anarbi t ra ryaper ture l lumina tedbyaplane wavegeneratesa ar ieldwhichcontains n o d i rec t geomet r ica l)ray on tribution. notherword s, he perture ar ie ld iscomple te ly descr ibed by he Ke l le r d i ff rac ted rays emana t ingfrom the edges . Fur the rmo re , for an aper ture which i s of theorder of a typica l open-ended waveguide ( l ike aper ture A , inFig. 2 , 2 k a < IO) , th eun i fo rme xpre s s i ons o r hepa t t e rn[ 171and he ar-fie ld Keller pat ternconvergewithina ewguide widths of the aper tu re . S ince aper ture A is i l luminatedby two planewaves, u+ and u- I , the d i rec t cont r ibut ion ofthese plane waves in the far fie ld is absent . Except very near


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IEEE TRANSACTIONS ONat, the f ield in the f lare region is,wellymodeled by th e

u+(u-) inc iden t uponW 2 (W I ). Since hese ields re ay ields, hey re

A 2 u n d e r h e o n -of the UAT.Another conclusion der ived in [SI conce rns the use of theheransmissionhrough an absorbing(a screen which produces no ref lec ted f ie ld for an n-t ray f ield, i .e ., ref lectioncoefficient 7 = 0). Referr ingFig. 4, consider a l ine source having a radiation pattern ( 0 )7 behind two absorb ing screens. (The z igzag l inesFig. 4 are meant t o represen tabsorbingscreens.)Let the7 = ( Y , 6 ) b e ex -ed b y, as Y + =,

U ( Y , 6 )g ( k r ) P ( 6 ; ? ) + O ( k - 3 / 2 ) . ( 2 0 )[81 leads to a f ina l so lu t ion for

e far-field patte rn P(e ,? ) given by

* e x p [ i k r cos (e + e ) ] p ( e )

S represen ts the s li t ha l f -wid th ; he de tour parametersre= sin*1,2/2)> (22)

, r l ,2, and Ql2 a r e sh o wn in Fig . 4. The incident f ield isof the am e ormas (20). We use he xpressionin (21) la ter .I V . FAR - FI ELD PATTER N

. Ana ly t ic a l Resul t sTh e r aypicture of thed if f rac t ion processesoccurring ne horn waveguidestructuremaybeshown to be ormallycal to ha t o f much implerproblem: slit in n

creen l luminatedby n rrayofpatterned ineP ( 0 ) of the horn is repre-y a uperposi t ion of te rmso f h e o r m o f211,9

~ ( 0 ) p ( e ; + o(k - ( 2 3 )n= 1

ere, in m ost cases the sl it s the ape rture A 2 ( th e sl i t half-S = a + L s in@ ), and the se t o f line sources loca ted a t, ~ represen ts hed if f rac ted ources twed g es W l an d W 2E l W1 and2 W 2 . Thesemultiply eflected aysmay be convenientlyof images [31 . Conceptual ly , the

bouncing to and f ro in the la re eg ionmay be equiva-described as the result of a set of image sources in free. The interaction of these image sources with the secondA 2 provid es the far-f ield pa ttern.

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ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 2 , MARCH 1982I

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Fig. 4 . Singlepatternedine ource at r illuminating a slit in anabsorbingscreen.Hornmay be treatedassuch a slit lluminatedby several such sources.Consid er the horn show n in Fig. 5 . Fo r th e ang le 0, larelength L , and gu ide wid th a chosen here , there are ef fec t ive lyfou r sources: the tw o wedges Wl ,2 and the i r images W3,4. Ifwe egard the aysd i f f rac ted rom W1 ,2 asarising from apat terned ine source a t W 1 2 , i t s no t d i f f icu l t to see tha tsource W3 is imply the imageof W 2 in theupper la rewhile W4 is the imageof W 1 in the ower la re . The se t o fsources lies on a circle of radius b(b= a/s in 9) cen te r ed ab o u tth e p o in t a t wh ich th e u p p e r an d lo wer f l a r e su r f aces wo u ld

in tersec t . The sources are separa ted f rom hei r ne ighbors byangle 20. The set of im age source s is alwa ys f inite in n u m b e rbut can be large if 0 s small.Aside from ach sources individual ontr ibu tion in t h ed i rec tion of the observat ion po in t , o ther fac tors are a lso im-p o r t an t .Th e ay s r o me a c h m a g eso u r ce o l lo w a u n iq u ep a th o h eo b se r v a t io np o in t .T h ep a th en g thd i f f e r en cesare crucial . The r icochetted paths from flare to f lare are pre-cisely acco unted orby he elativeposi t ions of the mag esources . naddi t ion , ac torsaccount ing or henumber ofbounces, he po lar iza t ion , and he d i f ference in t h e n c id en texci ta t ions of W l and W 2 must en c lud ed to ma in ta incor rec tphase e la t ionsamong he mage ource ie lds .Th ecomple te set of mages adiating in free spaceautomat ica l lysa tisfies the oundary ondi t ion longh elare surfaces.Rela t ive to the kO-ord er inc iden t mod e in the waveguide, thefield produced by the wedge sources is of k- /*-ord er . How-ever , s ince none of the kO-ord er f ie ld penet ra tes aper ture A Ito the ar ie ld , hek- /* -order ie ld of the imagesourceswill be the dom inant te rm in the far f ie ld of the th roa t aper -tu re .The ie ldofan mag esourcemust , n u rn ,penet ra teaper ture A , ; the geometr ica l op t ics contr ibu t ion of the t rans-mit ted f ie ld s o f p r imary mpor tance to the far f ie ld of thehorn . The cascading of the throat-diffracted f ields with aper-tu re A , ra ises the impor tan t quest ion of wheth er the th roat -diffracted f ields may be represented as an optical f ield (or sumof optical f ields) in order that the UAT may be applied at 42 .If the f lare angle 0 and f lare length L are bo th la rge enoughthat edges El and E 2 arecompletelyoutside he ransit ion


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D L E E : ANALYSIS OF R E C T A N G U L A RO R NN T E N N A S 245

5. Modeling of mul t iple images as point sources i l luminating a s l i tin anabsorbingscreen.Source W 1 adiates hrough sli t form ed ofedges E l an d E,, (a) an d (b). Source W4 (image of K 1 in lowerflare) radiates through sl i t formed of edge E 2 and wedge W 2 . (Simi-larly for W 2 an d W 3 . )withhewaveguidemode ,here i s n o

mul t i p l ea re of he proper form to cascadewithedgesHowever, hor ns do not n general meet hiscri terion.E l ,, re bur ied n he rans i t ion regions a t tached t oW1 ,2 . Fortun a te ly , the guide width i s typica l ly qui tel compared to he lar e e n g t h L , andherans i t ion

ons over lap to suc h an ex tent ne a rEl ,2 tha t the d i f f rac tedare a good approximat ion to the uni form solut ion . Tha tl thoughedge El may be so n e a r o B I Z small detour)di f f rac tedraycont r ibut ionfrom W l o E , isexces-E l is simultaneously near B z r and the d i f f rac ted cont r i -W2. ill largely com pensa te for he s ingula rbe-B 1 . ( In the inf in i te ly fa r f ie ld , the compensa t ionPerfectcompensa t ionfor hesingulari ty is n o tcted in the near field. As a result , he edge diffractedrising fro m E l , 2 , due o l lumina t ion by the Ke lle rW l 2 , may be som ewhat larger than would be derivedrigorous cascading of UA T solut ions. Since the diffract ionth e image source fie ld by E l ,2 is of k- -order (relative to

e ko guide mode) , the e ffec t of the e rrors men t ioned aboveminimal n th e mainbeamwhere hek-12-orderof t he t h roa t sou rc e s i s domi na n t . Howe ve r , fo r 0 S 9,

ed rays are dominant . Discrepancy in this region is notsing. Ou r goal s t o obtain he dominan t fie ld behaviori n t he u ll pa t t e rn , i s k - l I2 - o rde r .Large Flare Horns (@>30)Returning to Fig. 5 , we cons id er the in te rac t ion of sourceand i ts image W4 with aper ture A 2 . It is c lear that sourc el adiatesdirect ly hrough he perture ormedof dgesE 2 , Fig. 5(a ) . Other d i f f rac ted rays n te rsec t he oppos i te

the upper f la re again bu t pass o ut the aper ture wi th-

out ur the r n te rac t ion .Obvious ly , hedi rec t n te rac t ionofthe ays i f f rac ted rom W 1 wi t h pe r t u re A2 (F i g . ( a ) )can be re in te rpre ted as source W 1 in f ree space i l lumina t ingth e s l it shown in Fig. 5(b). The rays reflected from the low erf la re (F ig . 5(c )) may be cons idered to a r i se f rom image sourceW 4 which i s t ransmi t t ing through an aper ture be tween W 2 ndE, (F ig .5(d)) . n hecomple te aypic ture ,sources W 1 n dW 2 adia te hrough aper ture E l E 2 whilesources W3 a nd W4radia te hrough aper tures E2W2 and El W 1 , respectively. Nosingle source aperture nteract ion sat isfies he bound ary con-dit io ns alon g the flare . However, the aggregate does satisfy th eboundary ondi t ion . For example , ons ide rheayromW 1 o E2 shown in F ig . 5(b) . S ince the sc reens an absorber,on l y he nc i de n tpor t ion of theU A Tsolution isnecessaryt o obta in the d i ff rac ted f ie ld of edge E,. However, th e diffrac-t ion of the ray from W4 o E2 (Fig. 5(d ) ) by he a bso rb i ngscreen wil l combinewi t h hea bove osa t i s fy hebounda rycondi t iona long ine W 2 E 2 . (The nte rac t ionof W1 with W 2does not involve an absorbing screen.) The ray diffracted fromW 2 a s no subse que n t i ma ge a nd , a s suc h , t he c ompl e t e UATwedge iffractionxpressionmust e sedor this finalin te rac t ion . This fac t is depic ted by the nonabsorbing w edgeinserted in Fig. 5(d).C. Narr ow Flare Horns ( Q


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246 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, V O L . AP-30, NO. 2 , MARCH 1982

w. ./31

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31

- - -- - -- -& . ...

(dFig. 6. Modeling of small flare angle horn withmany image sources.Sources W , and W4 rad ia te th rough slit E 1 E 2 in absorbing screen,(a) a n d (b). Source W 5 radiates through the narrow aspect slitshow n in (c). (Sources 2 , 3 , and 6 ar eanalogous to 1 , 4, and 5 ,respectively.)cons ider ing he s impl ic i ty of he mode l . The predic ted curvefo l l owsheme a su re desu lt very closelyver th e range0 < 9 < Q. F o r I9 >@,he predic t ion shi f t s upw ard -2 d Bf rom hemeasured esul t ,butc on ti nue s t o fo l l ow he obes t ruc ture . A possible explana t ionfor h i sbehavio r is that nthe region 0 > @ .the d i rec t rays f rom the four image sourcesn o longer exis t and only he d i ff rac ted rays f rom E l and E 2are significant . As discussed earlier, waves incident up on edgesE l and E2 f rom the throa t a re incorr ec t individua lly a s E l , 2arewi th in he rans i t ion egionsof he hroa tdiffract ion.Howe ve r , t o hee x t e n t h a t E l are in the ar ie ld of th et h roa t ( r S a ) , the singulari t ies should izearly cance l . The re -sul t ing rays diffracted from E l ,2 are typica l ly too la r ge , w hichwou l d ac c oun t fo r he +2 dB shi f t . This hypothes i s i s borneou t by the nex t case in which Q a n d a are unchanged but Li s inc reased. T he shi f t obse rved s smal le r a s expec ted .

Case 2: Fo r th e s eco nd case, the relevant parameters areQ = 17.5' ~ = f l

ka = 3 R = 0.k l , = 145 .85 ('5)

The obe s t ruc tu re of the pa t te rn i s now even more nvolved(compare F ig . 7 and F ig . 8) . Again th e agree me nt in the mainbeam region (I9 < 9) i s exce l lent , but a sh i f t (smalle r than thefirst case) is again observed fo r 0 >9. Some disagreement wi ththe lobe s t ruc ture occurs near 0 = 60". No explana t ion for th i sbehavior is available,butotheranalyses [ 1 1 , [SI have alsofailed to recover this po rt ion of t he pa t t e rn .

s8 , - ALCULATEDMEASURED(-5 B I !- . . _.m0 9

W

250 50.0 780 1w.oe ( d 4

Fig. 7 . For hornandexcitation of (24) compositepattern of horn(in dB) is compared with m easured values fo r 0 < 90". (F or clar i tymeasured values are shifted down by 5 dB.)

C A L W U T E DMEASURED( - 5 dB )

S G 50.0 750 loooe ( d e g )Fig. 8. For horn an d excitation of (25) composite pattern of horn (indB) is compared w i t h measured values for 0 < 90" . (For clar itymeasured values are shi f ted do\vn by 5 dB.)Case 3: The th i rd case (F ig . 9 ) i s qui te d i f fe rent from t h e

p re c e d i ng t wo . The ho rn a nd mode pa ra me t e r s a re@ = 0 . 6 " . r = + l

kQ = 3 n = 0.k L = 18.94 (26)

For such a na rrow horn , the re a re s ix image sources . S ince kLis so. small , the ape rtu re effect ively emoves the geomet r ica lopt ics cont r ibut ion of the throa t sources in the fa r f ie ld . Onem i g h te xpe c t hepa t t e rn os h o wan overal l shift ince heedge diffracted rays are dominant and E l , 2 are typical ly largeby a co ns tant ac tor . Reca l l heshi ft in Fig. 7 fo r 0 > 6.)However,since the c ons t a n t fa c t o ra pp l i e s o a l l angles, theshi f t i s not observed in the dB plot . Accord ingly , the pred ic tedresults ol low hemeasuredvalues nd ecover thepa t t e rnlobe structur e very well. (It is l ikely tha t the abso lute value ofth epredicted result is large by some cons tant value.) A pos-sible way to circum vent this difficul ty is t o pe r fo rm a r i go rouscascadingof the uni form hroa t f ie lds wi th he aper ture A 2using the appro ach of Lee and Boersma [ 1 6 1 .

Case 4: The p re v i ous fou r e xa mpl e s ha ve l l involved a TE Mgui de mode . The a pp roa c h t a ke n i n [ 1 1 , d o m i n a n t h o r n m o d eanalysis, is suff ic ient t o desc r ibe the f ie ld in the throa t regionfor the first three cases in the region I9 < @ (pa r t icula r ly t rue


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L E E : A N A L Y S I S OF R E C T A N G U L A R H O R N A N T E N N A S 247

25.0 50.0 1 8 08 deg)

I

9 . For horn and exci ta t ion of ( 2 6 ) compos i t e pa t t e r n o f hor n (indB) is comparedwithmeasuredvalues or 0 < 90 . (For clari tymeasured values are shiftedby 5 dB.)3) . For larger $ horns , the throa t pa t te rn can devia te8 < Q. Higher orderalsoproduce hroa tpat ternswhichareconsiderably

T h e h o r n in case 4 is both large Q and the inc ident mode i se fi rst TM mode. T he param eters employed are l isted below.Q = 2 5 O ~ = + 1

ka = 10 n = + I .

mo de angle Q1 , fo r t h i scase is roughly 9.Reca ll tha t the TM, mode i s ant i symmetr ic in x . As a re -the far fie ld of the thr oa t is zero in the direct ion 8 = 0.e overal l pat tern (Fig. 10) is c learly anisotropic in the region< Q a nd , in fact , fol lo ws he far-field pat tern of the fou rRecall that he Kellerpat terns of W l 2 are ingular a t= Q1 but hat hes e singulari tiescancel n the farf ie ld toce a finite result 181. For th i s cance l la t ion o occur ne f a r fie ld of the sl i t , the geometrical optics factors of the

direct contributions of sources W1 and W 2 ,be dentical orsources W l nd W2. or a TEM (TMo)(G o = 0), the sources an d the singularities of the fieldssymmetr ic wi th respec t to t he z-axis. The modi fying func-(28) are utom atical ly dentical t B = 0 ( f rom hehorn) . For t he TEM mode , he re fo re , healwayscancel n he ar ield of the lit.Suchon does not occur fo r h igher order mod e exc i ta t ion .hemodeangle Qn # 0 fo r n # 0. As aresult , hepair ofies which must cancel in the dire ctio n Q, = 8 is n osymmetr icwith espect t o the z-axis. Themodi fying( 2 8 ) i s not , n genera l , he same for source W , and

W 2 in thedi rec t ion 8 = I f,however , we choose tomi ts of (28) when Itl 2 z I>M and should the transi t ionof edges E l ,2 not overlap over some range of angles,modi fying ac tors (28) areuni formly1over ha tofangles. If theshadowboundarysingularities of W 1nd W 2 ieentirelywithin hislit egion, hesingularities

08(deg)

Fig. 10. F orhornandexci ta t ion of (27) compos i t epat ternofhornexhibits a significant off-axis peak. In cont ras t , pat tern of horn de-scribed in (30) is a l so plot ted above and exhibi t s an on-axis peakand low sidelobe levels. See text for d iscussion.againcancel n he ar ield of thesl i t .T hecondi t ionsde-scribedabove orwhich he ingularities of ahigherordermode are manageable may be expressed as

(Q- sin- (M/&KC>) >9, PI + 0. (29)F or hehornandexci ta t ion l isted n ( 2 7 ) , the hadowboundarydiscont inui t ie sdo ie n he l i t egion sa t i s fy(29 ) ) , and hesingularities do c omb i ne o y i el da ini te ar

f ie ld for the horn a t = $1 .Case 5: F o rc ompl e t e ne s s , he horn n he p re c e d i ngex -ample sevaluatedagain or the first-orderTEmodeexcita-t ion . The horn and mode paramete rs a re

Q = 2 5 O ~ = - 1ka = 10 n = f l .k L = 1 5 0 (30)

The singular behavior of W 1 2 a t B = Q1 emains in the indivi-dual pat terns, but since the horn and mode described in (30)satisfy (29) , t he t o t a l ho rn pa t t e rn p rove s t o be un i fo rm (F i g .10). Again a strong geom etrical optics contribution from hethroa t sources i s present in the main beam. The ho rn , exc i tedby the istT Em o d e ,p roduc e s hee x t re me l ysmoo t h a r -fie ldpa t tern how n.Themain eatures of thepa t t e rn - t heext remely m oothmain eam lendingnto he raduallyincreasing lobestructureatwideangles-are nmarkedcon-tra st o he TM patterns bservedn th e first our ases.Nonetheless, he TE patterns shown n [2] exhibit he sameoveralleatures. A possible xplanation of the extremelysmooth pa t te rn ( in compari son to t h a t for th e TM case)ex -hibi ted by the TE case wo uld be the absence of significant dif-fracted fields from E , 2 . Since the boun dary condi t ion a longthe flare surfaces requires zero fie ld, the nulls of the com positethroa t a t te rn o in t o E , , 2 . Accordingly ,he iffractedfields ( k - -order) are negligible, a n d essential ly the geometri-cal beam from the throa t domina tes in the forw ard d i rec t ion .E. Discussion of Alte rnate Approaches

The comprehensive analysis of the horn radiator providedby Yu et ~ l .1 ] did not employ the mult iple image model of


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248 IEEE T R A N S A C T I O N S O N A N T E N N A SN DR O P A G A T I O N , VOL. AP-30,O. 2 , MARCH 1982t he f i e l d s in t he f l a re r e g i on o f t he ho rn . In s t e a d , t he y c hoset o c onc e n t ra t e upon t he E-p l ane o f na r row se c t o ra l ho rns fo rwh ich the fie lds in the flare region can be well-represented byt h e d o m i n a n t m o d eof a corner reflector.T hedominantmode na lys i s epresents he ie ld n heflare region as arising from a single sour ce si tuated at the apexof a comer re f lec tor . The source of approach [ 1 ] is an so-t ropic radia tor which l lumina tes aper ture A , to p r o l i d e t h efar-fie ld p at ter n. T he val idi ty of this approximation rests uponthecollect io n of "throat ources"being ufficiently closelyclustered to merge in to a single l ine source. This merg ing willa lsodependupon hewaveguidemodeand t spolarizat ion.Th e degree t o which this condit ion is sat isfied in several prac-tical cases is illustrated in [ 11 . The domi na n t mode a pp roa c hcan recover he pa t te rn of appropr ia te horns and exc i ta t ionsqu ite convincingly. How ever, n he case of wide flare horn s,h igher order m ode exc i ta t ions , or the o the r pola r iza t ion , thedomi na n t mode a pp rox i ma t ion ca n bre a k down .At he cos t of re lat ively i t tle ncrease n complexity, hemul t ip le mage approach can ikewise eva lua te he pa t te rn ofthese narrow flare horns, and offers ncreased general i ty. Forexample , he pa t te rns n F igs . 7-9 requi re e i the r four or s ixsources .Tha t s , ( 2 1 ) mus tb eeva lua tede i the r ouro rs ixt imes . In re turnfor h i sext racomplexi ty , h i sapproach re -moves theassumpt ion ha t hefie lds n he flarecanbeap-p rox i ma t e d by t he domi na n t ho rn mode . The mul t i p le i ma gemode loffersgreater lexibilitywi th espect to he nc id entm o d e ( n , 7). The image ource osi t ions nd eomet r ica lrelat ionsi.e., athengths)emain nchanged.Onlyheindividual p at ter ns of the set of image sources need change t oaccommodate the new mod e rays inc ident upon W 1 2 . ( S o m eres t r ic t ions on the inc ident mode must be appl ied when in te r-ac t ion wi th the aper ture A 2 i s in t rod uced. ) Unl ike the domi-nant mode ana lys i s , the mul t ip le image mode l depends s t ronglyupon a l l threehornparamete rs : a, L , 9. Inpart icular, hen u m b e r o fmages an ecome ui tearge or smalllareangles. Acco rdingly, this mod el is bet ter sui ted for large flareho rns .

V. R E F L E C T I O N C O E F F I C I E N TThe modal fie ld propagating within he guide undergoes are f lec t ion upon encounte r ing aper ture A l . This throa t re flec -t ion is a unctio n of the horn-waveguide unctionand nde-pe nde n t of L. The f ie ld t ransm i t ted in to the f la re region i s inturn re f lec ted a t aper ture A , . This reflected fie ld couples t oth eeflectedwaveguidemodes .hesew oontributions

provide thedomi na n tbehavior of the o ta l e f lec ted ie ld .I t is possible to describe hese nteract ionsus ing he am emultiple-imagemodel of the horn-w aveguide unction as wasemp loyed n he far-fie ld pat tern alculat ion.Comple tede-tails are presented in [ 71.A . Formulat ion

In thecalcula t ion of the eflect ion a t he hroa ta nd hecouplingof he ie lds eflected rom he lare dge to heguide, we make use of he ray-to-mode conversion factor pu tforward by Lee [ 141 for parallel-plate waveguides. Th e ampli-t ude CT",'' of the re f lec t~ed mode m excited by a unit amp li-t ude r a y (mode n ) nc ident upon one edge of the gu ide is given

byCTn2J= [ X,J9k ' ) + 7x , v ( \ k y ) ]

[f(-n + @ n ) f ( ~ @m)l[ 4 ~ , a co s @,I - (3 1)

9' 8 = -(n -@,,, t n) f o r w = 2. ( 32 )He re, he first bracket is the famil iar Keller diffract ion coef-f ic ient l inking the inc ident mode ray and the re f lec ted mo deray.The econd racke t onta inshemodi f ica t ion ohediffract ionoeffic ientdescribedarl ier)risingromheproximi tyof hesecondhalf-plane whichmakesup heguide .The final bracket is termed the ray-to-m ode conversion factor.Within this factor, the term E,= takes the value 2 f o r m = 0 ,and1,otherwise .The ac tor ka represents hegu i deha l f -width nd Qr n is themod e angle ssoc ia tedwi thmode m.Stric t lypeaking,heay- to-modeonvers ionactorw asderived fo r the pa ra lle l-p late guide as was the mo di f ic a t ion tothe iffract ion oeffic ient . orhe aral le l-platetructure(w = 2 ) , ( 3 1 ) i s dent ical to he exac t so lu t ion obta ine d bythe Wiener-Hopf techn ique.T hegeneral izat ion of (31 ) o he h roa t e f l e c t ion o f ahorn i s obta ined by the subs t i tu t ion X + x , Lee has shown[ 141 ha t fo r the flanged waveguide (LC = 1.5) the above sub-s t i tu t ion agrees to order k- ' l2 with he results obtained byothers. In addit ion, Jul l [61 has obta ined the fo l lowing throa treflect ion coeffic ient r r y matching the dominant guide andhorn mode s a t t he t h roa t

H , (* )(kb)-jH,(Z)(kb)H l ( 2 ) ( k h )+ jHO(*)(kh)rT = ( 3 3 )

where b = a /s in @.Since he coordina te surfaces over whicht he ma t c h i ng oc c u rs on l y c o i nc i de fo r Q, = 0 , (3 3) i s val id fornarrow la reho rns .Fur the rmo re , 33) s e s t r ic ted o TEM-to-TEM re f lect ion . For n = m = 0 and in the l imi t q5+ O(w+l ) , (31) and (33) provide the ame ef lec t ion oeff ic ient(rT= CToo) .Accord ingly, we a ccep t the generalized form of(31) as the e f lec t ionc oup l i ng rom nc i de n tm o d e n t o r e-f lec ted m ode m for arbi trary flare angle @.In a dd i t i on o he h roa t e f l e c t i on , he e t u rn rom hef la ree dge sente rs n to he o ta l e f lec ted ie ld .T hemul t i -bounce ray fie lds originat ing at wedges W1 , are representedas arising from the same series of ima ge sources as employ edin the far-field patterncalculat ion.Here , he i rs tbracket isthe familiar Keller diffract ion coeffic ient l inking the incidentmo de ray and he eflectedmode ay .The econdbracketc on t a i ns hemod i f i c a t i on o hed i f f r a c t i onc oe f f i c i e n t de -scribed earl ier) arising from the proxim ity of the seco nd half-p lane . S ince the horn i s symmetr ic , i t i s suff ic ient t o cons iderthe re turn produced by jus t one edge as the magni tudes of there turn of the two edges a re ident ica l . Each of the N sourceslaunches a ray to he edge n ques t io n . The f ie ldarriving atth ee dge romsource p fo ra n nc i de n tm o d e n i s denotedZ P " . The f ie lddiffractsas if from he edge of an absorbingscreen. T he edge-diffracted ieldmay ollowan y of N pa t hsback to ntersectwedges W1 , 2 andc oup le t o he e f l e c t e dmodes. Each of these N pa ths corresp ond s to the s tra ight -l ine


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A N D LEE: ANALYSIS OF R E C T A N G U LA R H O R N A N T E N N A S 2490 5 I I I

- ALCULATED0 0 MEaSUREDk 020-

wuz2 0 1 5 -w

I I I7 8 9 10 I

X ( C dFig. 11. Magni tude of reflection coefficient of horn described in ( 36)versus free-space wavelength of excitat ion.f rom t he e dge t o one o f t he N image sources . The cou-of the f ie ld f rom source p ba c k t o sou rc e q is given bye Keller diffraction coeffic ient a t th e ed ge a nd s denoted byThe ray propagat ing toward source q will cou ple to th e

d mode a c c o rd i ng o (31 ) wi t h an e f f i c ie nc y de no t e dC P . In al l , there are N. t e rms c on t r ibu t i ng t o t he r e t u rn

hr N( 3 4 )

,= I p = lz t o m o d emgiven by

ern, n = cTm, + cFn l , ( 3 5)The above resul t i s for the twodim ensiona l horn . To incor-the threedim ensiona l na ture of the phys ica l horn , theof the mod e in the second principal plane (at rightes to he f i rs t ) mu st be inc luded.T heprojec t ionof herays n h i s p lane i susedas the nc ident f ie ld and he

For asec tora lhorn , hemode l ingc ompose d of a wo-dimensiona lhornandaparal le lplateThecomplete eflected fieldof the hree-dimen sionall horn is givenby an expression of t h e f o r m ( 3 5 ) a n d

umerical resultsThe preceding analys i s has been mplemented for a rangef re que nc y romne a rc u t o f f o heonse t of the econdmode in the o l lowing ec tora lho rn -wa ve gu i de

q5 = 16.5, 2a = 3.404cmL = 36.16m , 2c = 7.214 c m .3 6)

e first hreevariables havebeen definedearl ier; c i s heto t he p l a ne o f a and. The i nc i de n t mode i s TEl o . The p ro j e c t i ons o f t h i s mode

to th e tw o principal planes of he physical horn prov ide aM inc ident f ie ld for he wodim ensiona l horn and a TE,de for the paral le l-plate guide.

In F ig . 11 , the abso lute va lue of the re f lec t ion coeff ic ienti sp l o t t e dversus he reespace wave lengthof the nc identf ie ld . The re f lec t ion vanes f rom roug hly 20-5 pe rcent and theoverall discrepancybe tween hecalculatedandobserved e-flect ion is 2-3 percent (da ta f rom Jul l [ 6 ). In pa r t icula r , theoveral l pat tern of reflect ion peaks and val leys is bro ug ht outqu ite well. We observe th at he fi t is bet te ra t he h i gh fre -quency end of the graph as expec ted . In addi t ion , the ca lcu-la ted resul t s a re typica l ly too la rge as was the case in the fa r-f ie ld pa t te rn . In the la t te r case , the d i sc repan cy was asc ribedto the Ke l le rdiffracted ays rom he hroatwedges .Thesesamedifficult ies houldexplain he l ightoverest imation inthe ca lcula ted resul t .Theagreementobtainedby hemult iple-imagemode loft h e h o r n in the far-fie ld pat tern predict ion and the reflect ioncoeffic ient calculat ion speaks well for th e val idity of the raypic ture of the horn . The disc repanc ies encounte redn t h e U A Tanalysisapparently temnot us t rom hemode lbut alsofrom the difficul ty cascading the fie lds of the f i rs t aper ture Awith the second aper ture A 2 . How e ve r , f rom t he unde r s t a nd -in g of the single aperture [ 8 ] , i ts associated transi t ion regionsand dominant f ie ld behavior, i t i s possib le to produ ce mean ing-fu l r e su l t s fo r t he s e r i e s o f a pe r t u re s i n t he ho rn de sp i t e t heinherent cascading difficul ty.

REFERENCESJ . S . Yu : R .C .Rudduck, nd L . Peters , J r . , Comprehensiveanalysis for E-plane of horn antennas by edge diffraction theory,I E E E TrUJlS. Antennas Pr op ag at. , vol. AP-1 4, pp. 138-149, 1966.C . A. IMentzer, Analysis nddesign of high-beam fficiencyaperture ntennas.Ph.D. issertation.TheOhioStateUniv.,Columbus,1974.Y . Ohba. On the radiation pattern of a cnrner reflector finite inwidth, IEEE Trans.AntennasPropaga t . , vol.AP-12,pp. 127-132,1963.IM. A . K . Hamid,Near-fieldransmissionetweenornantennas ,Dept .E lec .Eng. ,Univ.Toronto,Toronto,Canada,Res . Rep. no. 43, 1966.P . M. Russo, R. C. Rudduck. and L. Peters, J r . , A method forcomputing -planeatternf horn antenna s,E EE Trans.Antennas Propagar. , vol. AP-1 3, pp. 219-244, 3965.E. V . Jull. Reflection from the aperture of a long -plane sectoralhor n , IEEE Trans . Antennas Propa go t . , vol . A P -20 . pp . 6 2 4 8 ,1972.R.C .Menendez nd S. W . Lee. Uniform symptotic theoryapplied to apertu re diffraction , Dept. Elec. Eng., Lniv. Illinois,Urbana, Tech. Rep. EM76-9, Aug. 1976.


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IEEE T R A N S A C T I O N S O N A N T E N N A S A N D P R O P A G A T I O N , VOL. AP-30, NO. 2 , MA R C H 1982

R. C . Menendez and S . W . Lee, On the role of the geom etricalopticield in apertureiffraction, IEEE Trans.ntennasP ro p a g a t . , vol . AP-25. no. 5 , pp. 688-695 , Sept . 1977.J . B.Keller.Geom etrical heory of diffraction. J. O p r . SOC.Atner. . vol. 5 2 , pp. 116-130, 1962.R. M. Lewis and J . Boersm a. Uniform asymp totic theory of edgediffraction, J . Math. P tzys . , vol. 10, pp. 2291-2305, 1969.D. S. Ahluwal ia ,R . M . Lewis,nd J . Boersma,Uniformasymptotic theory of diffraction by a p lane screen, S f A M J . Appl .M a t h . , vol. 16, pp. 783-807, 196 8.J . Boersma and P . H . hl. Kersten, Uniform asymptotic theory ofelectromagnetic diffraction by a plane screen, Dept. Math.. Tech.UniversityofEindhoven,Netherlands, inDutch)Tech.Rep..1967.S. W. Lee and G. A. Deschamps. A uniform asymptotic theory ofelectromagneticdiffraction by a urvedwedge. l E E E Trans.Antennas Propaga t . , vo l . AP-24, pp. 25-34, 1976.S . W . Lee, Ray theory of diffraction by open-ended waveguides.I . Field in waveguides, J . M a th . P h y s . , vol. I I , pp. 2830-2850.1970.Applications, J . M a t h . P h p s . , vol . 13, pp. 656-664, 1972.S. W .Leeand J . Boersma,Ray-opticalanalysis of fieldsonshadow boundaries of two parallel plates,J. M a th . P h y s . , vol. 16,1 7 4 6 1 7 6 4 , 1 97 5.R . C . Menendez and S . W . Lee,Near ield of th eopen-endedparallel plate waveguide, Wave Morion , pp. 239-243, Jan. 1979.

- Ray heory of diffraction by open-ended waveguides. 11.

Ronald C . Menendez (S75-M77) received theB.S.degree nphysics romWashington Uni-versi ty,S t.Louis ,MO , in 1971,and he M.S .and Ph.D. degrees in electrical engineering fromth eUniv ersity of Illinois,Urbana, in 1973an d1976, respectively.Following earlong ostdoctoralppoint-ment with he Coordinated Science Laboratory,research nterestshave ncludedmagnetic levi-tation of high-speedgroundvehicles, ray tech-niques, lectrostat ic ffects urroun ding EHV power ines, ightning-induced t rans ient mi t igat ion, and inves t igat ions into hu man physiologicalef fects of electrical st imuli .Dr . Menendez i s a member of Tau Beta P i , S igma Xi , and Phi KappaPhi .

%9 6 . heoinedel lelephoneaboratories.is

Shung-Wu Lee (S63-M664M73-F81),orhotographndbiographyleaseeeag e17fhe May 1980ssue of thisTRANSACTIONS.

analysis of rect horn ant via uniform asymptotic theory

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