Diffraction by dielectric wedges: high frequency and time domain solutions

Transcription

1 UNIVERSITÀ DEGLI STUDI DI SALERNO Facolà d Ingegnera Dparmeno d Ingegnera dell Informazone ed Elerca e Maemaca Applcaa Doorao d Rcerca n Ingegnera dell Informazone XIII Cclo Nuova Sere TESI DI DOTTORATO Dffracon by delecrc wedges: hgh frequency and me doman soluons CANDIDATO: MARCELLO FRONGILLO COORDINATORE: PROF. MAURIZIO LONGO TUTOR: PROF. GIOVANNI RICCIO Anno Accademco 14 15

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3 A u professor che ho nconrao nella ma va. Una pare d ognuno d loro è presene n queso lavoro. Soprauo, a mglor professor che ho avuo: mo padre e ma madre.

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5 Conens Chaper 1 Inroducon 1 Chaper Hgh frequency echnques 5.1 Geomercal Theory of Dffracon 5. Geomercal Opcs 8..1 General expresson for he feld 8.. Reflecon from surfaces 11.3 Dffraced Feld Dffracon by perfecly conducng surfaces Slope dffracon Dffracon by fne conducvy surfaces: UTD heursc soluons.4 Physcal Opcs 1.5 Unform Asympoc Physcal Opcs approach: 3 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge Dffracon by delecrc wedges: sae of he ar Dffracon by an arbrary-angled delecrc wedge GO feld model for < ϕ < π/ Equvalen problems Dffraced feld: UAPO soluons for < ϕ < π/ Formulaon of he problem for π/ < ϕ < π-α Numercal resuls Formulaon of he problem for H-polarzed plane waves Parcular cases: rgh- and obuse-angled delecrc wedges Comparsons wh a rgh-angled delecrc wedge Comparsons wh an obuse-angled delecrc wedge 93 Chaper 4 Tme Doman soluons for dffracon by delecrc wedges 98

6 4.1 Tme Doman dffracon by an acue-angled delecrc wedge Formulaon of he problem for < ϕ < π/ Tme Doman UAPO dffracon coeffcens Numercal examples 18 Chaper 5 Conclusons and fuure works Summary Fuure sudes 119 Appendx A Acue angled wedge: equvalen surface currens 1 Bblography 137

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9 Chaper 1 Inroducon The knowledge of he propagaon characerscs of he elecromagnec felds s fundamenal n he analyss and plannng of anennas and opcal sysems. The propagaon predcon models based on ray-racng are he mos employed echnques n modern rado communcaon sysems. The ray-based models allow one o calculae magnude and phase of he receved elecromagnec feld and also he delay of each ray due o he propagaon mechansms. These las are ofen complex and can be generally arbued o he phenomena of reflecon, ransmsson, dffracon and scaerng. As well known, when he dmensons of he sysems are large n erms of he elecromagnec wavelengh, dffracon conrbuons due o maeral dsconnues can be neglgble and mus be accuraely calculaed. By usng he numercal dscrezaon mehods such as he Fne-Dfference Tme-Doman (FDTD) mehod, he Fne Elemen Mehod (FEM), and he Mehod of Momens s possble o oban relable soluons for he scaerng problems. Unforunaely, such echnques have he drawback of he ncreasng wh frequency of he number of unknowns and so he compuaons become neffcen or nracable a he hgh frequences. Furhermore, he physcal undersandng of he propagaon mechansms s dffcul wh hs knd of approach. The asympoc mehods resul o be an effcen alernave for solvng hgh frequency scaerng problems. The Geomercal Theory of Dffracon (GTD) [1] has receved grea aenon n he las decades because of s smplcy and srengh. I s based on he rayorened heory of he Geomercal Opcs (GO) and descrbes he dffracon a he edge of a srucure lke a local phenomenon a he hgh frequences basng on a generalzaon of he Ferma prncple. The major GTD dsadvanage s he presence of feld sngulares a he GO shadow boundares. Therefore, a unform verson of GTD was

10 Chaper 1 Inroducon elaboraed, he Unform Theory of Dffracon (UTD) []. I uses he same GTD rays bu furnshes unform asympoc soluons for he dffraced feld ha are vald also n he ranson regons. As consequence of he local properes of hgh frequency elecromagnec felds, he dffracon phenomenon depends only on he geomerc and elecromagnec feaures of he obsacle n proxmy of he dffracon pon. Accordngly, for many canoncal srucures (half-plane, wedge, ec.) he correspondng dffracon coeffcens, assocang he dffraced feld o he ncden feld, are avalable n leraure. I s mporan o hghlgh ha he rgorous, closed form dffracon coeffcens have been derved only for parcular canoncal confguraons and maeral. In [1], [] he auhors produced he exac dffracons coeffcens for srucures wh perfecly conducng surfaces. In he nex research acves he maeral dsconnues have been modelled by mpedance or ransmssve boundary condons, and he soluons are generally expressed n erms of he Maluzhnes funcon ([6], [14], [19]-[1]). Ths las s almos complcaed and can be easly evaluaed only for parcular geomerc confguraons. Therefore, her applcaon o he new praccal engneerng problems s lmed. Ths work has he purpose o produce approxmae bu relable, closed form Unform Asympoc Physcal Opcs (UAPO) soluons for several canoncal problems of dffracon concernng wedges made of delecrc maerals. UAPO dffracon coeffcens are que accurae and easy o handle snce expressed n erms of he GO response of he srucure and he ranson funcon of he UTD. Furhermore, her knowledge allows one o evaluae he correspondng dffracon coeffcens n he me doman (TD) for he same geomeres. The hess s srucured as follows. In Chaper are presened some of he mos common hgh frequency echnques for analysng wave propagaon n presence of obsacles. In parcular GO, GTD, Physcal Opcs (PO) mehods are dscussed and he UAPO echnque s nroduced. In Chaper 3 he sae of research abou dffracon by delecrc wedges s examned and a hgh-frequency soluon for he problem of plane wave dffracon by an arbrary-angled delecrc wedge s shown. Alhough such a problem has been already ackled n leraure [6]-[3], he soluon proposed here s fully formulaed n he UTD

11 3 framework and hen, s a user-frendly worhwhle alernave for he consdered dffracon problem. I s obaned by a decomposon of he consdered scaerng problem no wo sub-problems, respecvely exernal and nernal o he wedge regon. For each of hem, s consdered a PO approxmaon of he elecrc and magnec equvalen surface currens n he radaon negral and a unform asympoc evaluaon of hs las s performed. The found UAPO soluon has been mplemened n a compuer code, and he smulaons show ha compensaes he GO feld dsconnues a he shadow boundares. Moreover, s accuracy s weghed by he good agreemen acheved wh numercal resuls obaned va he commercal ool Comsol Mulphyscs based on FEM and va a FDTD compuer code. Chaper 4 s devoed o he sudy of he dffracon of plane waves by an arbrary-angled delecrc wedge n he TD framework. Dfferenly from he Frequency Doman (FD) framework, n he TD one no closed form soluons are avalable n leraure for hs knd of problem, excep hose concernng rgh- and obuse-angled lossless wedges [58], [59]. The nverse Laplace ransform s appled o he FD- UAPO soluons for he dffracon coeffcens found n Chaper 3 by akng advanage of her UTD-lke naure. Then, he ransen dffraced feld s evaluaed va a convoluon negral. The resuls provded by smulaons for hese soluons have verfed her goodness. The TD-UAPO formulaon has he same advanages and lmaons of he FD-UAPO ones, arsng from he use of a PO approxmaon. Anyway, a he wrng me of hs hess he TD- UAPO formulaon s he only one analycal soluons o TD scaerng problems nvolvng penerable wedges. Alhough unavodably approxmae, he FD-UAPO soluons are smple, effcen and que accurae for solvng several dffracon problems nvolvng penerable obsacles. Furhermore, he correspondng TD-UAPO soluons are relable and he only avalable n closed form for solvng he same problems n he TD framework. Accordngly, hey can be surely useful from an engneerng vewpon when no rgorous and effcen soluons for he dffraced feld are avalable.

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13 Chaper Hgh frequency echnques The nen of hs chaper s o dscuss some of he mos common hgh frequency echnques for analysng he propagaon of elecromagnec waves n presence of obsacles. As well-known, when an objec s nsered n he pah of a propagang elecromagnec wave modfes he feld dsrbuon n he surroundng space. Then, he oal feld a a gven observaon pon s gven by he superposon of he ncden feld and he scaered feld. The evaluaon of he scaered feld can be performed va dfferen analycal echnques. In parcular, Geomercal Opcs (GO), Geomercal Theory of Dffracon (GTD) and Physcal Opcs (PO) are llusraed n he followng, because hey consue he bases for he UAPO mehod shown a las..1 Geomercal Theory of Dffracon When he dmensons of a radang elemen or a scaerng objec are large n erms of wavelengh, s possble o use he hgh frequency asympoc echnques o solve n a que smple way many problems ha would oherwse be mahemacally nracable. In he pas years GTD, defned by Keller [1] and laer developed by Kouyoumjan and Pahak [], has receved grea aenon. I s an exenson of GO whch s an approxmae and well-known hgh frequency echnque based on he heory of propagaon of elecromagnec waves along rays. Accordng o GO, only drec, refleced and refraced rays exs; as a consequence he feld exhbs sharp ransons n correspondence of he shadow boundares and s null n he space regons no drecly llumnaed by he ncden feld. In he framework of GTD, a new ype of rays, namely he dffraced rays, s nroduced o overcome he GO naccuraces.

14 6 Chaper Hgh frequency echnques The GTD s based on he prncple of localy and on a generalzaon of he Ferma s prncple. The frs one saes ha, a he hgh frequences, reflecon, refracon and dffracon are local phenomena dependng only on he geomerc and elecromagnec properes of an objec n proxmy of he reflecon, refracon and dffracon pon, respecvely. Therefore, many dffracon coeffcens relang he dffraced feld o he ncden feld have been deermned for several canoncal problems. For example, he smples confguraons are he wedges of nfne dmenson, made of perfecly conducng maeral and llumnaed by unform plane waves. Is dffracon coeffcens have been deermned by rgorous soluons of Maxwell s equaons. As consequence, he basc mehod of GTD o ackle and solve a scaerng problem s o decompose he orgnal geomery n smpler geomerc confguraons, whose dffracon coeffcens are known. In fac, s possble o oban he fnal soluon as he superposon of he conrbuons relevan o each canoncal problem. The generalzaon of he Ferma s prncple allows one he ray racng and he deermnaon of he dffracon pons. As wellknown, saes ha n a homogeneous medum a ray moves along he shores pah from he source o he observaon pon. Such a prncple can be also generalzed o dffraced rays. The applcaon of GTD o solve scaerng problems s smple n prncple bu some problems may arse. A frs, he number of rays o consder grows rapdly wh he geomercal complexy of he problem. Moreover, even dffracon coeffcens relaed o smple srucures are sll unknown. Fnally, s dffcul o deermnae he lower frequency lm beyond whch he resuls obaned by GTD can be consdered accurae. Accordng o GTD, when an elecromagnec wave a hgh frequency s ncden on a runcaed surface, generaes a refleced wave, an edge dffraced wave and a surface wave propagang along he srucure. Ths las, can also be exced n he shadow regon of a curved surface. Such a phenomenon s represened n Fg..1, where a plane perpendcular o he edge a he dffracon pon D s shown. A ray mpngng on he edge a he pon D produces edge dffraced rays (ed) and surface rays (sr). Wh reference o convex geomeres, he surface ray produce rays dffraced from he surface a any pon Q

15 .1 Geomercal Theory of Dffracon 7 along s pah. As a resul, ES s he borderlne beween he rays dffraced from he edge and he rays dffraced from he surface and s angen o he surface n D. Moreover, SB bounds he zone llumnaed by ncden feld and RB ha llumnaed by refleced rays. The srucure n Fg..1 s non penerable oherwse also refraced rays and dffraced rays n he nner regon mus be consdered. When boh he surfaces are llumnaed by he ncden feld, here are no shadow boundares bu wo reflecon boundares arse. Snce he behavour of he opc rays s dfferen n he wo regons delmed by a boundary (ES, SB, RB), a ranson regon wheren he feld exhbs an abrup varaon s found near each boundary. Fgure.1 Incden, refleced and dffraced rays. Shadow boundares. I s assumed n he followng ha he feld sources and he observaon pon are far enough from boh he surface and he boundary ES n order o neglec he conrbuons due o he surface rays. Consequenly, he oal elecrc feld can be expressed as: wheren space whch s null n he shadow regon, r d E E E E (.1) E denoes he ncden elecrc feld, namely he feld free d E s he dffraced feld. r E s he refleced feld and

16 8 Chaper Hgh frequency echnques. Geomercal Opcs..1 General expresson for he feld In lne wh he GO, rays can be nerpreed as flux lnes of he power densy and he varaon of he feld nensy along a ray can be deermned by applyng he prncple of conservaon of he power flux n a ube of rays. Le s suppose ha a pon source emanaes soropcally waves and denoe wh L and L L wavefrons a he me and, respecvely (see Fg..). Fgure. Prmary and secondary wavefrons of a radaed wave. I s possble o consder he ube of rays beween he cross-seconal areas d a P and d a P hrough whch he power flux s consan. Therefore, he followng deny holds: wheren Sd Sd (.)

17 . Geomercal Opcs 9 E S (.3) s he power densy and he nrnsc mpedance of he medum. Eq. (.) can be rewren as: or E d E d (.4) E d E (.5) d I can be useful o express d and d n erms of he curvaure rad of he wavefrons. To hs end, le us consder he general case of an asgmac ube of rays shown n Fg..3. Fgure.3 Asgmac ube of rays. The surfaces d and d have prncpal rad of curvaure 1, and 1 s, s, respecvely, where s denoes he dsance along

18 1 Chaper Hgh frequency echnques he ray pah from P o P. Snce boh d and d subend he same sold angle, resuls: d d s s 1 1 (.6) or equvalenly E E 1 s s 1 (.7) I s evden n Fg..3 ha rays focus (cross) a dfferen pons. As a maer of fac, he ube of rays degeneraes no a lne and he feld nensy approaches nfny when s 1 or s. The locus of pons where hs occurs s called causc. I mus be sressed ha felds have always a fne value and, as a consequence, neher GO or GTD can be used o predc he feld srengh a causc pons. As he phase varaon along a ray s gven by ks, he elecrc feld along a ray can be wren as follows: E E A s e (.8) () jks wheren E denoes he elecrc feld a a reference pon P ( s ), k s he propagaon consan and As () s he spaal aenuaon facor, also known as spreadng facor. Ths las reduces o he followng expressons n he specal case of plane wave ( 1, ), cylndrcal wave ( 1, ) and sphercal wave ( 1, ): 1 plane-wave As ( ) / s cylndrcal-wave / s sphercal-wave (.9)

19 . Geomercal Opcs 11 I s worh o noe ha when a causc s crossed, s changes he sgn and hs corresponds o a phase varaon of π/. Eq. (.8) allows evaluang he feld a a gven pon P n erms of s value a a reference pon P. I has been derved by usng he prncple of conservaon of energy n a ube of rays. Alhough s a vald hgh frequency approxmaon for lgh waves, could be no accurae for elecromagnec waves of lower frequences... Reflecon from surfaces GO laws such as Snell s law can be used o deermne he feld refleced from surfaces. Le us consder a unform plane wave mpngng on a planar surface S assumed perfecly conducng. Ths case s llusraed n Fg..4, where s ˆ and s ˆr denoe he un vecors n he ncdence and reflecon drecons, respecvely. The oal feld a he reflecon pon Q on S mus mee he boundary condon: ˆn beng he un vecor normal o S. r r n ˆ E E (.1) Fgure.4 Reflecon from a planar surface. I s opporune o express he ncden and refleced felds E and n erms of her componens parallel and perpendcular o he ncdence plane (ha s he plane defned by s ˆ and ˆn ): r E

20 1 Chaper Hgh frequency echnques E Ee ˆ Eeˆ (.11) r r r r r E Ee ˆ Ee ˆ (.1) Equaon (.1) can be rewren n a compac form: r E E R r E E (.13) 1 where R s he dyadc reflecon coeffcen for a perfecly 1 conducng surface. Wh reference o Fg..4, ê s a un vecor normal o he ncdence plane, and eˆ, eˆr are un vecors parallel o he ncdence plane defned by: eˆ eˆ sˆ (.14) r r eˆ eˆ sˆ (.15) In agreemen wh he prncple of localy, he feld refleced from a surface does no change when hs las s no more planar, beng always possble o consder he local plane angen a Q r. Obvously, he spreadng facor changes o accoun for he varaon of he refleced wavefron. Therefore, s possble o express he refleced feld as follows: wheren r jks r E () s R E ( Q ) A() s e (.16) r r 1 As () (.17) r r 1 s s

21 . Geomercal Opcs 13 and s s he dsance beween he reflecon pon Q r and he r r observaon pon P. In addon, 1 and are he prncpal curvaure rad of he refleced wavefron n Q. They are relaed o he prncpal curvaure rad of he ncden wavefron 1. and o hose relevan o he surface S ( a 1 and a ). I can be shown ha 1 r and r are gven by []: r r 1 f r 1 1 f1 (.18) (.19) wheren he parameers f 1 and f n he parcular bu more neresng case of sphercal wave ncdence smplfy o: 1 1 sn sn f1, cos a1 a 1 sn sn 4 cos a a a a 1 1 (.) In eq. (.), s he ncdence angle, 1 and are he angles formed by s ˆ and he drecons assocaed o he prncpal rad of curvaure of S. In he far feld approxmaon, eq. (.16) reduces o: jks r r r e E () s R E ( Qr ) 1 (.1) s I s mmedae o verfy ha n he case of plane wave ncdence resuls:

22 14 Chaper Hgh frequency echnques r r 1 aa 1 (.) Accordngly, GO fals predcng he refleced feld when one or boh he curvaure rad of S are nfne..3 Dffraced Feld In agreemen wh eq. (.8), he dffraced feld can be expressed as: ' E () s E ( O') e d d jks s ' s (.3) d E ( O ') beng he dffraced feld a a reference pon O '. I s convenen o choose he dffracon pon Q d as reference pon. Le us consder he case of dffracon from an edge whch s, obvously, a causc for he dffraced rays. Accordngly, he lm d lm E ( O' ) ' ' (.4) exss and s no zero. Moreover, he dffraced feld mus be proporonal o he ncden feld n Q d and hus s possble o wre: d lm E ( O' ) ' DE ( Qd ) ' (.5) wheren D s usually referred o as dyadc dffracon coeffcen. As a consequence, he feld dffraced from he pon Q d on he edge s gven by: E () s DE ( Qd ) e s d jks (.6)

23 .3 Dffraced Feld 15 beng he dsance beween he causc a he edge and he second causc for he dffraced rays. In [] has been demonsraed ha: 1 1 nˆe sˆ sˆ (.7) e a sn ' where e s he curvaure radus of he ncden wavefron n Q d, aken n he plane conanng he ncden ray and he un vecor ê angen o he edge n Q d (see Fg..5). Moreover, a s he curvaure radus of he edge and n ˆe s he un vecor normal o he edge dreced away from he cener of curvaure. When he dsance s posve, he dffraced ray encouners no causc along s pah. Indeed, such a dsance s negave only when he second causc s locaed on he same sde of he observaon pon. I s opporune o remark ha he dffraced feld expressed by eq. (.6) s undefned a he causc pons and crossng such pons mples a phase varaon of π/. Fgure.5 Dffracon from a curved edge. In he case of verex dffracon, dffraced rays are orgnaed from a pon causc. Accordngly, s possble o wre: d lm E ( O' ) ' DE ( Qv ) (.8) '

24 16 Chaper Hgh frequency echnques and he feld dffraced from he verex can be evaluaed as: jks d e E () s DE ( Qv ) (.9) s.3.1 Dffracon by perfecly conducng surfaces The canoncal problem of plane wave dffracon by a perfecly conducng edge wh planar surfaces has been reaed by many auhors bu he mos popular resuls have been presened by Keller [1]. The relaed dffracon coeffcens are non-unform, namely no vald a he GO shadow boundares. Ths lmaon has been overcome hanks o he soluon provded by Kouyoumjan [], [3]. Le us consder a radang source a a pon S n presence of a perfecly conducng wedge and observe he feld a a pon P. In accordance wh he Ferma s prncple, he dffracon pons can be deermned by mnmzng he dsance SQd P. Ths leads o he law of dffracon: sˆ eˆsˆ eˆ (.3) where ê s he un vecor dreced along he edge, s ˆ and ŝ are he un vecors n he ncdence and dffracon drecons, respecvely. As consequence of eq. (.3), when a ray mpnges on he edge a oblque ncdence formng an angle ' wh respec o ê, he dffraced rays le on a concal surface whose sem-aperure angle s sll '. The ncden ray drecon s usually descrbed by a couple of angles ', ' and he poson of he dffraced ray on he cone s fxed by he angle (Fg..6). As suggesed by Kouyoumjan and Pahak, s convenen o defne he ncdence plane (ha conanng he ncden ray and he un vecor ê ) and he dffracon plane (ha conanng he dffraced ray and he un vecor ê ). A reference sysem n he ncdence plane s nroduced and fxed by he un vecors ˆ', ˆ ', s ˆ. In a smlar way, a reference sysem fxed by ˆ, ˆ, ŝ s nroduced for he

25 .3 Dffraced Feld 17 dffraced ray. The un vecors ˆ' and ˆ are perpendcular o he ncdence and dffracon planes, respecvely. The un vecors ˆ ' and ˆ are parallel o he ncdence and dffracon planes and defned by: ˆ ' sˆ ˆ ' (.31) ŝ ˆ (.3) Fgure.6 Reflecon and dffracon by a wedge. When adopng hese ray-fxed reference sysems for descrbng he ncden and dffraced felds, resuls: d d E Ds E ' jks e d D d E h E ' s( s) (.33) wheren, sh D are he scalar dffracon coeffcens for he Drchle (sof) and Neumann (hard) boundary condons, respecvely.

26 18 Chaper Hgh frequency echnques The unform dffracon coeffcens for a perfecly conducng wedge [] are repored n he followng for reader s convenence: where: j/4 e Dsh, (, ', ') n ksn ' ' co FkLa ' n ' co FkLa ' n ' co FkLa ' n ' co FkLa ' (.34) n jx j F( x) j xe e d (.35) x s he UTD ranson funcon, a and a are defned by: a nn x cos (.36) N beng he neares negers sasfyng he followng relaon: n N x (.37) The dsance parameer L n (.34) can be deermned by mposng he connuy of he oal feld a he shadow boundares [4], hus obanng:

27 .3 Dffraced Feld 19 where 1, e s 1 sn 1s s s ' L e (.38) are he prncpal rad of curvaure of he ncden wavefron and e s he curvaure radus of he ncden wavefron aken n he ncdence plane. Dffracon coeffcens for he half-plane can be easly found by seng n = n eq. (.34). I can be easly verfed ha n correspondence of a shadow boundary one of he coangen funcons becomes sngular whereas s produc wh he correspondng ranson funcon s fne. Grazng ncdence mus be consdered separaely. In hs case, Ds and he expresson of D h n eq. (.34) mus be mulpled by he facor 1/. When he wedge surfaces are curved (see Fg..1), s possble o consder n accordance wh he prncple of localy a wedge wh planar surfaces angen o he faces of he curved wedge. Moreover, he dffracon coeffcens (.34) can be sll appled bu he parameer L mus be properly modfed o accoun for he curvaure of he refleced wavefron []..3. Slope dffracon In addon o he edge dffracon conrbuon (.33), s ofen necessary o consder a hgher order erm whch s proporonal o he normal dervave of he ncden feld a Q d. Ths conrbuon may be no neglgble when he amplude of he ncden feld a Q d s small. Such ype of dffracon s usually referred o as slope dffracon. By smlar argumens, s possble o consder hgh order slope dffracon erms. When only he frs slope dffracon erm s consdered, he oal dffraced feld reads as: 1 D U Q Ud UQd A() s e jk ' n sh, d jks (.39)

28 Chaper Hgh frequency echnques where U d denoes a componen of he dffraced feld, Dsh, / ' are he slope dffracon coeffcens and U / n s he normal dervave of he ncden feld n Q d. Compac expressons for he slope dffracon coeffcens relevan o a perfecly conducng wedge can be found n [3]..3.3 Dffracon by fne conducvy surfaces: UTD heursc soluons In he pas years, he UTD has been successfully appled o solve a large varey of elecromagnec wave neracon problems wh perfecly conducng surfaces, such as he analyss of radaon characerscs of smple and complex anenna sysems [5]. However, s exenson o fne conducvy srucures s n general a non-rval problem and sll subjec of connung research. Ths neres s jusfed snce n many applcaon felds, such as rado propagaon, Radar Cross Secon (RCS) predcon, analyss of novel anenna sysems, dffracon from no perfecly conducng objecs s very frequen and he use of dffracon coeffcens (.34) could lead o naccurae resuls. As a consequence, s essenal o have relable soluons for he feld dffraced by no perfecly conducng srucures n order o make accurae feld predcons. I s opporune o noe ha rgorous and exac dffracon coeffcens have been developed for dfferen surface mpedance srucures wh he Maluzhnes heory [6]. These are raher cumbersome and, due o her complexy, ofen no easy o use n roune applcaons such as propagaon predcon ools. Thus, he dffculy arsng n usng such soluons forces smplfcaons o be made and o use approxmae echnques. Heursc soluons have been proposed n he leraure wh reference o varous scaerng problems. They are no based on a rgorous soluon of Maxwell s equaons bu on a suable modfcaon o he dffracon coeffcens (.34) wh he am o compensae he geomercal opcs feld dsconnues a he shadow boundares. Moreover, hey are easy o handle and compuaonally smple. The problem of plane wave scaerng by a hn lossless delecrc slab has been reaed n boh he normal and oblque ncdence case [7]. The derved dffracon coeffcens are an

29 .4 Physcal Opcs 1 exenson of dffracon coeffcens for he perfecly conducng halfplane. Followng he same dea n [7], wo-dmensonal dffracon coeffcens have been proposed by Luebbers for he fne conducvy wedge [8] and are currenly used n many ray racng ools. Based on hs soluon, UTD slope dffracon coeffcens have been laer developed by he same auhor [9]. As dscussed by Luebbers, he accurae use of hese dffracon coeffcens s resrced o applcaons nvolvng wedges wh large neror angles, and o he observaon n proxmy of shadow boundares. Alernave heursc UTD dffracon coeffcens for no perfecly conducng wedges have been proposed n he recen years o mprove he Luebbers soluon accuracy whn a wder angular regon [1]-[1]..4 Physcal Opcs The feld scaered n he far zone by a perfecly conducng objec llumnaed by an elecromagnec wave s gven by: s S ˆˆ S, E jk I RR J G r r' ds (.4) where J s denoes he curren dsrbuon nduced on he surface S of he objec, k and are he wavenumber and he mpedance of free jk space, r r ' Grr (, ') e 4 r r' s he Green funcon and r, r ' denoe he poson vecors a he observaon and negraon pons respecvely, ˆR s he un vecor from he radang elemen a r ' o he observaon pon and I s he (3 3) deny marx. Physcal Opcs approxmaes he curren dsrbuon J s by usng geomercal opcs and provdes accurae resuls only when he objec s far enough from he sources so ha he ncden feld can be descrbed n erms of wavefrons and rays. Accordng o hs approxmaon, he surface curren s null n he shadow regon. Whereas, a any pon P on he objec surface n he llumnaed regon can be deermned by assumng ha he ncden elecrc feld s

30 Chaper Hgh frequency echnques refleced n he same way as would be from he nfne plane angen o he surface a P [13]. When he surface S s perfecly conducng, he surface curren dsrbuon can be evaluaed accordng o: ˆ ˆ r ˆ J nh n H H n H (.41) S where ˆn s he un vecor normal o he surface a he ncdence pon and H s he ncden magnec feld. As a maer of fac, when a plane wave mpnges on a perfecly conducng surface, resuls for boh polarzaons (see Fgs..7(a) and.7(b)): r nˆh nˆ H (.4) (a) (b) Fgure.7 Reflecon from a perfecly conducng surface. Parallel polarzaon (a). Perpendcular polarzaon (b). PO can be also employed o evaluae he feld scaered from fne conducvy objecs. In such a case, boh elecrc J s and magnec J ms surface currens mus be aken no accoun n he radaon negral and so eq. (.4) becomes: ˆˆ ˆ s ms, s E jk I RR J J R G r r' ds S (.43) The accuracy of he resuls aanable wh he PO mehod depends on he degree of approxmaon for he surface currens and on he observaon drecon. As a maer of fac, when he conrbuons due

31 .5 Unform Asympoc Physcal Opcs approach 3 o he pars of he objec no drecly llumnaed by he ncden feld are no neglgble, some lmaons n he feld predcon are expeced..5 Unform Asympoc Physcal Opcs approach In he recen years, Unform Asympoc Physcal Opcs soluons have been developed o solve varous dffracon problems [14]-[18]. The basc dea o oban such soluons s he use of a PO approxmaon of he equvalen surface currens nduced by an ncden feld on a srucure. As well-known, he scaered feld s expressed by means of he radaon negral (eqns. (.4) and (.43)) and ncludes boh he geomercal opcs and he dffracon conrbuons. The applcaon of he seepes descen mehod and a unform asympoc evaluaon of he radaon negral allow dervng he dffracon coeffcens n closed form. These las are expressed n erms of he UTD ranson funcon and perfecly compensae he GO feld dsconnues. I s opporune o pon ou ha hese soluons are nevably approxmae snce he surface currens n he radaon negral are no exac bu based on a PO approxmaon. In spe of hs hey are que accurae when compared wh more rgorous soluons and are smple, easy o handle and o mplemen. For hese reason, hey are poenally useful for praccal applcaons when no exac analycal soluons are avalable or when hese las canno be evaluaed n an effcen way. Alhough heursc and effcen soluons have been proposed n he leraure (see Subsec..3.3) o solve dffracon problems havng no exac soluons, hey are no based on a rgorous analycal procedure and, herefore, should be used wh consderable aenon. The key pons showng how o consruc a UAPO soluon are revewed n he followng wh reference o he problem of plane wave dffracon by a penerable half-plane surrounded by free-space [18]. Le us consder he oblque ncdence of an arbrarly polarzed plane wave over a penerable half-plane surrounded by free-space (see Fg..8). The ncdence drecon s fxed by he angles ' and '. In parcular, ' s a measure of he ncdence drecon skewness wh

32 4 Chaper Hgh frequency echnques respec o he edge ( ' / corresponds o he normal ncdence). The observaon drecon s specfed by he angles and. I s convenen o nroduce he ray-fxed coordnae sysems sˆ, ˆ ', ˆ ' and s ˆ, ˆ, ˆ for he source and he observaon pons, respecvely (see Fg..8) and o assume he observaon pon P on he Keller s dffracon cone,.e. '. b^ s^ ^ f z y x b' ^ s ' ^ ^ f' b' Fgure.8 Dffracon by he half-plane edge. A any observaon pon P, he oal elecrc feld E s gven by he s superposon of he ncden feld E and he scaered feld E. By applyng he equvalence heorem, he surface currens nduced by an ncden plane wave on he surface S of he half-plane can be s nerpreed as sources of E and herefore, n he far-feld approxmaon, he scaered feld can be expressed by means of he followng radaon negral (.44) s ˆˆ PO PO E jk ˆ ( I RR) J s J ms RG( r, r ') ds S

33 .5 Unform Asympoc Physcal Opcs approach 5 n whch he observaon and source pons are denoed by r xxˆ y yˆzzˆ zzˆ and r' x' xˆ z' zˆ ' z' zˆ. By usng a PO approxmaon for he equvalen elecrc and magnec surface currens on S and expressng he felds n erms of her parallel and perpendcular componens, resuls: PO Js * jk ( x'sn 'cos ' z 'cos ') Jse Jms PO * jk ( 'sn 'cos ' z 'cos ') Jmse (.45) (.46) and he hree-dmensonal Green funcon can be wren as: jk rr ' e Grr (, ') 4 rr' jk ' zz ' e 4 ' ' z z (.47) To evaluae he edge dffracon confned o he Keller cone for whch ', s possble o approxmae ˆR by he un vecor ŝ n he dffracon drecon [19] Rˆ sˆsn 'cos xˆsn 'sn yˆ cos ' zˆ (.48) Accordngly, resuls: e s jk ˆˆ * * E ( I ss) Js Jmssˆ 4 jk ' zz ' jkx'sn 'cos ' z 'cos ' e ' zz' * * ( I ss ˆˆ) J s J mssˆ I s dz' dx' (.49) The expressons of he PO surface currens n erms of he ncden elecrc feld are here obaned by assumng such currens as

34 6 Chaper Hgh frequency echnques equvalen sources orgnaed by he dsconnues of he angenal GO feld componens across he layer,.e., ˆ ˆ S S r jk x'sn 'cos ' z 'cos ' s PO r J y H H y H H H yˆ H H H e (.5) PO ˆ r ms ˆ S S r jk x'sn 'cos ' z 'cos ' E E E yˆ e J E E y E E E y (.51) As well-known, s convenen o work n he sandard plane of ncdence and o consder he GO feld componens parallel ( ) and perpendcular ( ) o. Therefore, resuls: r cos ˆ r J ˆ s E E E e E E E ms r cos ˆ r ˆ (.5) J EE E e E E E (.53) wheren E and E are he ncden elecrc feld componens (a he orgn) parallel and perpendcular o he ordnary plane of ncdence, s he ncdence angle n such a plane, eˆ sˆ' yˆ sˆ' yˆ s he un vecor normal o he ordnary plane of ncdence, s ˆ' beng he un vecor of he ncdence drecon, and ˆ yˆ eˆ. The refleced and ransmed feld componens can be expressed n erms of he ncden feld componens by means of he reflecon marx R and ransmsson marx T, and her expressons can be found n [18]. As well-known, n he hgh-frequency approxmaon, he PO negral (.44) exended o S can be reduced asympocally o a sum of ray feld conrbuons from (solaed) neror saonary phase

35 .5 Unform Asympoc Physcal Opcs approach 7 pons on S and an edge dffraced feld conrbuon. To hs end, s necessary o perform he evaluaon of he followng negral: jk I s 4 jk ' zz ' jkx'sn 'cos ' jkz 'cos ' e e e dz' dx' ' zz' (.54) As repored n [19], by makng he subsuon z' z ' snh and usng one of he negral represenaons of he zeroh order () Hankel funcon of he second knd H resuls: e jkz ' cos ' e jk ' zz ' ' zz' dz' jk z cos ' () j e H k 'sn ' (.55) The nvolved Hankel funcon can be now wren wh he useful negral represenaon () 1 H k 'sn ' e d C jk 'sn 'cos (.56) where C s he negraon pah n he complex -plane (see Fg..9). The angle s beween he llumnaed face and he vecor ', and he sgn (+) apples f y ( y ). If, accordng o

36 8 Chaper Hgh frequency echnques he geomery shown n Fg..1, 'sn sn and 'cos cos x ', so obanng 1 jk sn ' cos 'sn ' () jkx'sn ' cos C H k e e d (.57) Im[a] C p Re[a] Fgure.9 Inegraon pah n he complex α-plane. and hen: C C I s k e 4 jkz cos ' jksn ' cos jkx'sn ' coscos ' e e d dx' 4 k e jkz cos ' jksn 'cos jkx'sn ' coscos ' e e dx' d (.58)

37 .5 Unform Asympoc Physcal Opcs approach 9 By applyng he Sommerfeld-Maluzhnes nverson formula, resuls: jk x 1 e dx' (.59) jk sn ' cos cos ' 'sn ' cos cos ' y P r r - r' f q x z r' Fgure.1 Inegraon pah n he complex -plane. so ha Is e 1 e sn ' j cos cos ' C jkz cos ' jksn ' cos d (.6) where he sgn (+) apples n he range ( ). Such an negral can be evaluaed by usng he Seepes Descen Mehod [19]. To hs end, he negraon pah C s closed wh he Seepes Descen Pah (SDP) passng hrough he pernen saddle pon s as shown n Fg..11. Accordng o he Cauchy resdue heorem, he conrbuon relaed o he negraon along C (dsored for he presence of sngulares n he negrand) s equvalen o he sum of he negral along he SDP and he resdue conrbuons Re s p assocaed wh all hose poles ha are nsde he closed pah C+SDP,.e.,

38 3 Chaper Hgh frequency echnques 1 f I s g e d j C 1 f Re s p g e d j SDP Re s p I (.61) n whch 1 f I g e d j SDP f e s ffs g e d j (.6) SDP g jk e z cos ' 1 A sn ' cos cos ' cos cos ' (.63) k f jsn ' cos. Noe ha s ypcally large, p ' and s ( s ) f ( ). Moreover, by pung ' j" and mposng ha Im f Im f s and Re f Re f s, he consdered SDP s descrbed by: and 1 1 s s gd ' sgn( '')cos ( '') (.64) cosh '' where gd " s he Gudermann funcon. By usng now he change of varable f f s, eq. (.6) can be wren as

39 .5 Unform Asympoc Physcal Opcs approach 31 I G e d (.65) wheren 1 f s d G g e (.66) j d Im[a] SDP C ap Re[a] as p When Fgure.11 SDP n he complex -plane. p s approachng s, he funcon G canno be expanded n a Taylor seres. To overcome hs drawback s convenen o regularse he negrand n (.65) usng he Mulplcave Mehod. I requres nroducng he regularsed funcon wh G p G p (.67) sn ' 1 cos ' p f s f p j

40 3 Chaper Hgh frequency echnques jsn 'cos ' j (.68) and s a measure of he dsance beween p and s. Accordngly, Snce p e I Gp d p (.69) G s analyc near, can be expanded n a Taylor seres. By reanng only he frs erm (.e., he snce 1, resuls: n whch G I F j 1 -order erm) p p (.7) p Gp 1 f s d G g s e j d p jsn ' 1 Ae j 4 e j cos cos ' sn ' (.71) and j F j j e e d (.7) s he UTD ranson funcon []. By subsung (.68) and (.71) n (.7), he explc form of he asympoc evaluaon of I s:

41 .5 Unform Asympoc Physcal Opcs approach 33 I j 4 jksn ' z cos ' e e k sn ' sn ' cos cos ' ' F k sn 'cos j 4 jks e e 1 k s sn ' cos cos ' ' F kssn 'cos (.73) where he denes s sn ' and z scos ' are used on he dffracon cone. The above analyc resul conrbues o he UAPO dffraced feld o be added o he GO feld and s referred o as a unform asympoc soluon because I s well-behaved when p s. In he GTD framework, he marx formulaon for he scaered feld can be wren as s ' E E s E M I s E E ' s (.74) As a consequence he UAPO edge dffracon conrbuon d E s: d jk ' ' s E E E d e E M I D d E E ' E ' s (.75) where he UAPO soluon for he dffracon marx D D D ' ' D D ' '

42 34 Chaper Hgh frequency echnques M11 M /4 1 j 1 e M 1 M k sn ' (cos cos ' ) ' F ks sn ' cos (.76) where he sgn ( ) apples f ( ). The explc expresson of he coeffcens M j (, j = 1, ) can be found n [18]. Accordngly, he UAPO soluons have he same ease of handlng of hose derved n he UTD framework and, n addon, hey have he nheren advanage of provdng he dffraced feld from he knowledge of he GO response of he srucure. In oher words, s suffcen o make explc he reflecon and ransmsson coeffcens relaed o he consdered srucure for obanng he UAPO dffracon coeffcens.

43

44 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge The problem of dffracon by a delecrc wedge has grea relevance for praccal applcaons. The accurae pah loss predcon n rado wave propagaon envronmens requres a correc characerzaon of dffracon conrbuons arsng from he presence of edges and corners. However, he research acvy focused manly on mpenerable srucures a hgh frequency and rgorous soluons have been repored n he leraure for perfecly conducng wedges or mpedance wedges. Sarng from hese soluons, here have been some aemps o exend her valdy o he penerable srucures. Ths mehods resul o be complcaed and manageable jus for smple confguraons, so has been necessary o buld up new approaches for hs knd of problems. The lack of an exac and, a he same me, compuaonally effcen soluon whch can be employed n ray racng ools forces smplfcaons o be made and o apply approxmae soluons. The am of hs chaper s o provde a UAPO soluon for he feld dffraced by he edge of a lossless arbrary-angled delecrc wedge n he case of normal ncdence. The soluon s derved by means of he decomposon of he consdered scaerng problem no wo subproblems relevan o exernal and nernal regons of he wedge. For each of hem, proper equvalen currens, whch can be nerpreed as sources for he scaered felds, are deermned by akng no accoun he penerable naure of he srucure. Then, unform asympoc evaluaons of he radaon negrals allow dervng closed form expressons for he dffraced feld. As demonsraed by numercal examples, he here developed UAPO soluon compensaes he GO feld dsconnues n he exeror and neror regons. Furhermore,

45 3.1 Dffracon by delecrc wedges: sae of he ar 37 s accuracy s confrmed by he good agreemen aaned wh resuls provded by numercal mehods. 3.1 Dffracon by delecrc wedges: sae of he ar The problem of plane wave dffracon by wedges has receved grea aenon due o he mporance of s soluons n rado propagaon plannng, analyss and desgn of radang srucures and wavegude heory. The frs sudes have concerned wedges wh perfecly conducng surfaces [], [6] or mpedances faces [14], [], [1]. Heursc soluons have also been proposed n [8], [9] for solvng dffracon problems n he case of delecrc srucures. Alhough effcen, hey are no based on a rgorous analycal procedure and, herefore, should be used wh consderable aenon. Rawlns [] presened an approxmae soluon for he feld produced when an elecromagnec plane wave s dffraced by an arbrary-angled delecrc wedge, whose refracve ndex s near uny. I was obaned from an applcaon of he Konorovch-Lebedev ransform and a formal Neumann-ype expanson. The resuls were n agreemen wh hose derved by he same auhor wh reference o a rgh-angled wedge [3]. The dffraced feld of an E-polarzed plane wave by a delecrc wedge was formulaed n erms of negral equaons by Bernsen n [4]. These were ransformed no Fredholm negral equaons and solved by erave mehods for lmed values of he delecrc consan. Joo e al. [5] proposed an asympoc soluon for he feld dffraced by a rgh-angled delecrc wedge based on a correcon of PO approxmaon o he edge dffracon. The correcon n he far-feld zone was calculaed by solvng a dual seres equaon agreeable o smple numercal evaluaon. The exenson of he approach o arbrary-angled wedges was addressed n wo companon papers [6] and [7]. Burge e al. [8], sarng from a PO verson of he GTD, provded an edge coeffcen for he exernal and nernal dffracon by an arbrary-angled delecrc wedges. Rouvere e al. n [9] mproved he Luebbers heursc soluon [8] n he UTD framework [] by addng wo new erms ha compensae he dsconnuy creaed by he feld ransmed hrough he srucure. Anoher heursc soluon for he dffracon coeffcen

46 38 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge of a penerable wedge has been proposed n [3] by Bernard e al.. I s obaned by a modfcaon of he exac UTD dffracon coeffcen relaed o he meallc wedge. The FDTD mehod was appled n [31] and [3], provdng numercal resuls for he dffracon coeffcen of rgh-angled wedges made of perfec elecrcal conducor (PEC), lossless delecrc and lossy delecrc maeral. Anoher neresng aemp o solve he problem of he dffracon by a rgh-angled delecrc wedge was proposed by Radlow [33]. He used muldmensonal Wener-Hopf equaons o model he problem, bu he facorzaon of hese equaons needs funcon-heorec echnques employng wo complex varables ha are cumbersome o handle. Baes [34] nroduced erave formulas o oban a se of basc wave-funcons useful o represen he sources suaed on he plane of symmery of an arbrary-angled delecrc wedge. Seo and Ra [35] proposed a remarkable mehodology o evaluae he scaerng from a lossy delecrc wedge: refleced and refraced GO felds are obaned by nhomogeneous plane wave racng n he lossy medum, whle PO approxmaon n he edge dffraced felds are accuraely correced by addng he mulpole lne sources a he edge of he wedge o sasfy he exncon heorem. The unknown expanson coeffcens used a he p of he wedge o make dsappearng he exncon negral are evaluaed numercally. Salem e al. [36] exended Rawlns approach [] o a more general form of excaon (lne source) of a delecrc wedge. They expressed he Konorovch- Lebedev ransform funcons as a Neumann seres and represened he scaered feld as a Bessel funcon seres, so exendng soluons o he case of real valued wavenumbers and arbrarly posoned source and observer. Danele and Lombard [37] analyzed he problem of dffracon of an ncden plane wave by an soropc penerable wedge n he general case of skew ncdence. They used generalzed Wener- Hopf equaons, and he soluon s obaned usng analycal and numercal-analycal approaches ha reduce he Wener-Hopf facorzaon o Fredholm negral equaons of second knd. Vaslev e al. [38], [39] developed a numercal approach based on he mehod of negral equaons o solve he problem. They represened he unknown surface currens on he delecrc wedge as a sum of unform and non-unform componens (usng he Ufmsev s mehod), makng

47 3. Dffracon by an arbrary-angled delecrc wedge 39 possble he dervaon of he numercal soluon. Budaev [4] developed a hybrd mehodology o solve he problem of dffracon by a delecrc wedge n an exac sense, whch combnes analycal and numercal echnques. UAPO soluons were proposed n explc closed form n [41] and [4] for evaluang he feld dffraced by rgh- and obuse-angled lossless delecrc wedges n he nner and ouer regons. The UAPObased approach has been also appled o acue-angled wedges n [43] consderng a specfc range of ncdence drecons for boh cases of E- and H-polarzed plane wave. Then, n [44] he analyss has been exended o all possble cases of ncdence drecon provdng generalzed UAPO soluons whch are vald for he dffracon by a wedge wh arbrary aperure. 3. Dffracon by an arbrary-angled delecrc wedge In hs secon he UAPO soluons for he dffraced feld orgnaed by an arbrary-angled lossless delecrc wedge are deermned for an ncden plane wave. The geomery used as reference n he followng s relevan o a wedge havng an acue nernal apex angle. Ths choce s jusfed by he complexy of he propagaon mechansms (mulple nernal and exernal rays) whch nclude hose concernng rgh- and obuse-angled wedges [41], [4], as wll be demonsraed a he end of hs chaper. Le us consder he problem of he dffracon of an E-polarzed plane-wave by he edge of an acue-angled delecrc wedge (see Fg. 3.1). Is nernal apex angle s and s maeral s lossless, nonmagnec ( r 1), wh relave permvy r and propagaon consan kd kd r. The wedge surfaces are denoed by S and S n. A Caresan reference sysem x, yz, s nroduced wh he y-axs normal o he face S and he z-axs dreced along he edge. The ncdence drecon s assumed perpendcular o he edge (see Fg. 3.1) and defned by he angle ' ( ' corresponds o he grazng ncdence wh respec o S ).

48 4 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge Fgure 3.1 Geomery of he dffracon problem. The observaon pon s denoed by P(, ). The srucure dvdes he space n wo regons: he exeror regon and he neror regon GO feld model for < ϕ < π/ Closed form expressons for he GO response when ' are derved n hs secon. Only he face S s drecly llumnaed by he ncden feld. A ransmed ray exss n he nner regon and, n absence of oal nernal reflecon on S n, s ransmed hrough hs las. The refleced ray ravels oward S and gves rse o followng reflecon and ransmsson mechansms nvolvng S and S n. When he value of he nernal angle of ncdence makes o occur he nverson he rays begn o go far from he apex. Accordngly, wo seres of nernal and exernal rays exs: he frs conans rays (black arrows n Fg. 3.) ravellng owards he apex and he second comprses hose (red arrows n Fg. 3.) gong far from he apex, as llusraed n Fg. 3.. The exernal rays exs unl he nernal oal reflecon doesn occur. The number of pre- and pos-nverson rays s deermned by he values of ', and r. In order o beer undersand he ray propagaon nsde and ousde he wedge, he reader can refer o Fg If ' s he exernal ncdence angle, he wave

49 3. Dffracon by an arbrary-angled delecrc wedge 41 peneraes no he wedge hrough S wh he ransmsson angle L sn sn and propagaes oward S n. 1 r ' S Fgure 3. Inernal reflecon and exernal ransmsson. The ray mpnges on S n wh he ncdence angle 1 and, for he r Snell s laws, s refleced ( 1 1 ) and ransmed hrough S n wh he angle 1, as shown n Fg Accordng o he consdered values of ' and r, he oal nernal reflecon can occur and no ransmed wave hrough S n exss. In such a case, an evanescen wave propagang along S n and aenuang n he drecon normal o arses. The condon ensurng he absence of oal reflecon a P 1 s: c 1 arcsn 1 r (3.1) c where s he correspondng crcal angle. By denong wh ˆ s cos ', sn ', he un vecor n he ncdence drecon, resuls: jksˆ r jk cos ˆ ' E Ee z Ee zˆ (3.) S n

50 4 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge wheren E s he amplude of he ncden feld a he orgn. The feld E r refleced by S can be deermned as: r jk dr E R E P e (3.3) where R s he reflecon coeffcen a P and dr s he dsance ' S Fgure 3.3 Rays ransmssons/reflecons hrough he wedge from P o P along he refleced ray: dr cos ' x cos' (3.4) n whch x r s he coordnae of P. As a resul, eq. (3.3) reads as: As regards he feld y x sˆ1 1 E 1 sˆr E r 1 1 r r jk cos' E REe z (3.5) ˆ E L ransmed hrough S, resuls: P L jk d r E x r P sˆr1 E L P 1 1 ˆ L s S n E T E P e (3.6) ˆ s E

51 3. Dffracon by an arbrary-angled delecrc wedge 43 T beng he ransmsson coeffcen a P and d s he dsance from P o P along he ransmed ray: As a consequence, resuls: L sn L sn L d x r (3.7) cos sn sn L L jk sn d L jkx r ' r jkd ˆ E T E e e z (3.8) TEe zˆ The feld E r 1 ransmed hrough S and hen refleced a P 1 ( 1, 1 ) on S n wh 1 s gven by: r1 1 L jkddr E R E P1 e 1 (3.9) wheren R 1 s he reflecon coeffcen a P 1 and dr 1 s he dsance from P 1 o P along he ray refleced from S n : Now, s smple o verfy ha 1 1 sn 1 sn 1 dr (3.1) 1 L (3.11) and so eq. (3.9) reads as r1 jk sn d 1 1 ˆ E RT E e z (3.1) If 1 c, he feld E 1 ransmed hrough S n can be evaluaed as:

52 44 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge 1 1 jk 1 d E TE P e 1 (3.13) wheren T 1 s he ransmsson coeffcen a P 1 and d 1 s he dsance from P 1 o P along he ransmed ray: d1 1sn1 sn 1 (3.14) 1 where 1 sn r sn1 s he ransmsson angle a P 1. Therefore, eq. (3.13) reads as: The feld r 1 jksn 1 1 ˆ E TT E e z (3.15) E refleced a, P on S wh s gven by: r r jk dr E RE P e 1 d (3.16) wheren R s he reflecon coeffcen a P and dr s he dsance from P o P along he refleced ray: sn sn dr (3.17) where. I can be easly verfed ha: L r jk sn d 1 ˆ E R RT E e z (3.18) If c, he feld E ransmed hrough S can be evaluaed as:

53 3. Dffracon by an arbrary-angled delecrc wedge 45 r jk d E T E P e (3.19) 1 wheren T s he ransmsson coeffcen a P and d s he dsance from P o P along he ransmed ray: 1 Snce sn r sn sn sn d (3.) eq. (3.19) reads as: jksn 1 ˆ E TRTEe z (3.1) The ray ransmed hrough S ravels oward he apex undergong N In L reflecons/ransmssons ( In denoes he neger par of he argumen) before movng away from. Accordngly, r n n L n for n 1,,..., N and he oal nernal reflecon c doesn happen f n. Wh reference o he frs se of rays propagang owards he apex, he feld conrbuons for n 3,..., N can be evaluaed by reerang he prevous approach. The expressons of he correspondng nernal ( E ) and exernal ( E ex L ) GO felds are: L n N n L r E n L E E n1 sn N n sn jkd L jkd n ET e Rp e n1 p1 neven

54 46 Chaper 3 Hgh frequency dffracon by an arbrary-angled delecrc wedge and N n jk sn d n Rp e zˆ (3.) n1 p1 nodd N 1 ex r n L n1 jkcos( ') zw ˆ (, ') jkcos( ') E E E E Ee REe zw ˆ (, ') N1 n1 jksn p n ET R p Te n n1 p1 nodd 3 W pn, Uc n zˆ N 1 n1 jksnqn ET Rp Te n W, qnuc nzˆ n1 p1 neven (3.3) where N N N 1 f N odd f N even (3.4) he funcon W () s defned as:

55 3. Dffracon by an arbrary-angled delecrc wedge 47 W, 1 U () s he un sep funcon: 1 f 1 (3.5) elsewhere U c c n n 1 elsewhere (3.6) and he parameers p and q have been defned as: 1 f nn p (3.7) 1 elsewhere 1 f nn1 q (3.8) 1 elsewhere The expresson (3.4) allows one o keep n accoun boh he possble cases of nverson so ha (3.) and (3.3) are generally vald, he un sep funcon W () s useful o keep no accoun he shadow boundares and he parameers p and q have been nroduced o have smlar expressons for he wo ses of neracon pons, as wll be shown n he followng. A crucal neracon pon s he N 1 -h one whch corresponds o he nernal ncdence angle N 1 N1 L. The nverson occurs a hs pon and he ray ravels far from he apex, as llusraed n Fg The use of he parameers p (eq. 3.7) and q (eq. 3.8) allows one o consder he frs conrbuon of hs se as arsng from S. Then, s opporune o descrbe hs seres of neracons sarng from hs ray. The angle R N r 1 N 1 plays now he role of L for evaluang he feld conrbuons. The number of neracon pons of he ray propagang far from he apex s