On a Uniform Geometrical Theory of Diffraction based Complex Source Beam Diffraction by a Curved Wedge with Applications to Reflector Antenna Analysis

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1 On a Unfom Gomtcal Thoy of Dffacton bad Comlx Souc Bam Dffacton by a Cuvd Wdg wth Alcaton to flcto Antnna Analy Dtaton Pntd n Patal Fulfllmnt of th qumnt fo th Dg Docto of Phloohy n th Gaduat School of Th Oho Stat Unvty By Youngchl Km, B.S., M.S. Gaduat Pogam n lctcal and Comut ngnng Th Oho Stat Unvty 009 Dtaton Commtt: Pabhaka H. Pathak, Adv obto oja-tan obt J. Bukhold

2 Coyght by Youngchl Km 009

3 Abtact A unfom gomtcal thoy of dffacton (UTD) oluton dvlod fo dcbng th hgh fquncy (HF) lctomagntc (M) fld uoundng an abtaly cuvd, fct lctcally conductng (PC) wdg, that llumnatd by a ont ouc, n comlx ac, whch gnat a comlx ouc bam (CSB). Th oluton ungly found to b th am a th UTD oluton obtand vouly fo PC cuvd wdg llumnatd by a al ont ouc aft t analytcally contnud fo dalng wth CSB llumnaton; hnc, t dfnd h a th CSB-UTD fo cuvd wdg. Th oluton fo cuvd wdg dvlod fom canoncal HF oluton fo a taght wdg wth lana fac. Fo a al ont ouc, th canoncal UTD oluton obtand va a ml aymtotc HF bad ft od Paul-Clmmow mthod (PCM) of tt dcnt. Howv, a CSB fld conttut a comlx wav, and th ft od PCM tctly not vald fo olvng th canoncal wdg oblm wth CSB llumnaton. Hnc, fo comlx wav, a dffnt, l comact, ft od aymtotc aoxmaton, known a th Van d Wadn mthod (VWM) of tt dcnt, nd to b mloyd. Th ft od VWM lad to a canoncal wdg oluton whch may b

4 vwd a an xtndd UTD (o UTD) wdg oluton fo th CSB llumnaton, dfnd h a th CSB-UTD oluton, bcau t can b xd a CSB-UTD = CSB-UTD. Howv, found to b nglgbl fo th wdg ca; t fo th aon that a ft od PCM bad CSB-UTD man accuat vn though PCM not tctly vald fo th ca. Aft havng tablhd th fact that th CSB-UTD oluton fo th canoncal wdg th am a f th canoncal UTD wdg oluton fo a al ouc xctaton mly and dctly analytcally contnud to dal wth a comlx ouc locaton (o a CSB), th CSB-UTD fo th abtaly cuvd wdg can thu alo b mlaly dvlod by analytcally contnung th coondng UTD oluton fo th am cuvd wdg xctd by a al ont ouc, to now dal wth th xctaton by a ont ouc n comlx ac (o CSB). Th cuvd wdg CSB-UTD oluton mloyd to analyz th adaton fom an offt aabolc flcto llumnatd by a CSB. Th CSB-UTD ult fo th flcto a comad wth a numcal hycal otc (PO) aoach to llutat th accuacy of th CSB-UTD. Th CSB-UTD xtmly fat comad to th numcal PO mthod; alo, th latt bcom ncangly tm conumng wth nca n fquncy, wha th CSB-UTD man ntally ndndnt of fquncy. Th comlx ont of ufac flcton and dg dffacton on th comlx xtnon of th flcto a comutd h ung a latvly fat and ffcnt ocdu, thu makng t vy actcal to tac comlx ay wth almot th am d and ffcncy a tacng al ay.

5 Ddcaton To H gloy, kngdom and ol v

6 Acknowldgmnt I would lk to x my nc gattud to my advo Pofo P. H. Pathak fo h gudanc, hang h valuabl knowldg and xnc, and dcuon thoughout th cou of th ach. Alo, I wh to thank hm fo h hl n th fnal aaton of th manuct. I would lk to thank th oth mmb of th dtaton adng commtt, Pofo obto oja-tan and Pofo obt J. Bukhold, fo th hlful commnt. Scal thank to go to my fom uvo D. Th-Hong L fo h gudanc dung my ft fw ya of tudy and ach at th lctoscnc Laboatoy. Alo, thank go to Andw O bn fo vwng th matal n at of chat 4 and all th Andc. Futhmo, I would alo lk to thank vyon who ha ayd fo m, cally my ant and my wf, th. Wthout th ay and tual uot, I would not hav comltd all th cou of my ducaton at Th Oho Stat Unvty. I alo thank my on, Ian, fo havng bn tayng hay and halthy, v nc h wa bon. Abov all, my dt gattud go to Hm who bought m to th Untd Stat and wokd though m fo H gat lan. v

7 Vta Al 4, 974. Bon Kwangju, South Koa 00...B.S. n Avonc ngnng, Koa Aoac Unvty Kyoungg, South Koa M.S. n lctcal ngnng, Th Oho Stat Unvty, Columbu, Oho 00nt..Gaduat ach Aocat, lctoscnc Laboatoy, Th Oho Stat Unvty, Columbu,Oho. Publcaton Y. Km and T-H. L, Shad Cculaly Symmtc Dual flcto Antnna by Combnng Local Convntonal Dual flcto Sytm, I Tanacton Antnna and Poagaton, vol 57, No, Jan 009. Fld of Study Majo Fld: lctcal and Comut ngnng v

8 Tabl of Contnt Abtact Ddcaton...v Acknowldgmnt..v Vta..v Lt of Fgu...x Chat:. Intoducton..... Canoncal oblm of D CSB dffacton by a taght wdg wth lana PC fac xctd by a comlx ln ouc.9.. Dffacton coffcnt of a CSB dffactd by a PC wdg fo a comlx ln ouc xctaton.... Numcal ult fo a comlx ln ouc xctaton Canoncal oblm of M CSB dffacton by a taght wdg wth lana PC fac xctd by an abtaly ontd comlx ont ouc Comlx ay-fxd coodnat ytm Dyadc dffacton coffcnt of an M CSB dffactd by a PC wdg Numcal ult fo an M comlx ont ouc xctaton ad analy of offt aabolc flcto antnna va CSB-UTD and CSB-UTD mthod 9 v

9 4.. GO bad CSB (CSB-GO) flctd fld by a PC cuvd ufac Dyadc CSB-UTD and CSB-UTD dffacton coffcnt fo a CSB xctd PC cuvd ufac Analytcal aoach to analyz offt aabolc flcto antnna va CSB-GO and CSB-UTD oluton Gomtcal Paamt aocatd CSB-GO flctd fld Gomtcal Paamt aocatd CSB-UTD dffactd fld Numcal ult fo analy of offt aabolc flcto antnna CSB ncdnt on nfnt aabolc flcto CSB ncdnt on fnt offt aabolc flcto Concluon and futu wok. 5 Andc: A. Two unfom hgh fquncy mthod of th tt dcnt...57 A.. Summay of valuaton of an ntgal va two unfom hgh fquncy mthod...57 A.. Summay of valuaton of I ( κ ) va th PCM mthod...6 A.. Summay of valuaton of I ( κ ) va th VWM mthod...67 A.. Mathmatcal dtal of th oluton bad on two mthod...69 I κ aco SB bad on I PCM...73 A... Chck on th contnuty of ( ) A... Chck on th contnuty of ( κ ) I aco SB bad on I VWM...77 B. Dcontnuty n th tanton functon at th hadow bounda...8 C. Comlx dffacton ont on th taght dg...8 D. Comlx flcton ont on an offt aabolc flcto ufac Comlx dffacton ont on an offt aabolc flcto dg...89 F. Comlx ouc bam...9 G. Shadow bounda...03 v

10 Lt of Fgu Fgu Pag. A ln ouc ncdnt on th wdg...0 ± and th comlx ξ lan toology fo a al ln ouc llumnaton of a canoncal wdg. Th tajctoy of ξ ndcatd. Stt dcnt ath, ( π ) along th al ax Comlx ln ouc bam llumnat on th wdg. b th bam aamt and bˆ th dcton of bam ax. Comlx ay and ouc locaton a dawn ymbolcally Stt dcnt ath, ( π ) tajctoy of ξ ndcatd fo ( ) 0 ± and th comlx ξ lan toology. Th Im ξ < Tajctoy of hadow bounda whn a CSB llumnat a half lan na th dg (a) Symbolcal comlx ay gomty at th nc of a PC wdg. (b) Comaon of total CSB fld and dffactd CSB fld obtand va CSB-UTD and CSB-UTD. Th ncdnt and flctd CSB a hadowd at ISB and SB Th tajcto α Im wth ct to th angl of th obvaton fo ISB (to) and SB (bottom) x

11 .8 Dcontnuou tanton functon aco ISB and SB. H, F and F 4 a tanton functon n th cond and fouth tm n dffacton coffcnt, whch a motant to ncdnt and flctd CSB, ctvly Contnuou cotangnt functon aco ISB and SB. H, C and C 4 a cotangnt functon n th cond and fouth tm n dffacton coffcnt, whch a motant to ncdnt and flctd CSB, ctvly Nglgbl addtonal CSB-UTD tm n th cond (to) and fouth (bottom) of CSB-UTD dffacton coffcnt whch a motant to th ncdnt and flctd CSB...4. Tajctoy of ξ na SB a th coondng ol, ξ, aoach ξ S = π 3.4 (b) P ( ξ ) 0 ξ aoach ξ S = π (a) Tajctoy of ξ na ISB a th coondng ol, ξ, aoach ξ S = π 3.4 (b) P ( ξ ) 0 ξ aoach ξ S = π (a) Symbolcal comlx ay gomty at th nc of a PC wdg. (b) Comaon of total CSB fld and dffactd CSB fld obtand va CSB-UTD and CSB-UTD. Th ncdnt and flctd CSB a hadowd at ISB and SB Th tajcto Im α wth ct to th angl of th obvaton fo ISB (to) and SB (bottom) Nglgbl addtonal CSB-UTD tm n th cond (to) and fouth (bottom) of CSB-UTD dffacton coffcnt whch a motant to th ncdnt and flctd CSB (a) Tajctoy of ξ na SB a th coondng ol, ξ, aoach ξ S = π 3.4 (b) P ( ξ ) 0 ξ aoach ξ S = π (a) Tajctoy of ξ na ISB a th coondng ol, ξ, aoach ξ S = π 3.4 (b) P ( ξ ) 0 ξ aoach ξ S = π (a) Symbolcal comlx ay gomty at th nc of a PC wdg. (b) Comaon of total CSB fld and dffactd CSB fld obtand va CSB-UTD and CSB-UTD. Th ncdnt and flctd CSB a hadowd at ISB and SB x

12 .9 Th tajcto Im α wth ct to th angl of th obvaton fo ISB (to) and SB (bottom) Nglgbl addtonal UTD tm n th cond (to) and fouth (bottom) of CSB-UTD dffacton coffcnt whch a motant to th ncdnt and flctd CSB (a) Tajctoy of ξ na SB a th coondng ol, ξ, aoach ξ S = π 3.4 (b) P ( ξ ) 0 ξ aoach ξ S = π (a) Tajctoy of ξ na ISB a th coondng ol, ξ, aoach ξ S = π 3.4 (b) P ( ξ ) 0 ξ aoach ξ S = π Tajctoy of CSB ISB and SB a th obvaton ont mov away fom th dg wth dffnt locaton of a ln ouc n al ac. Th CSB ISB and SB fom away fom ISB ( φ = 50 ) and SB ( φ =50 ) fo a al ouc xctaton of a wdg, ctvly A CSB ncdnt on th wdg wth t ax httng th ufac. Tajctoy of CSB ISB and SB a th obvaton ont mov away fom th dg wth dffnt locaton of a ln ouc n al ac. Th CSB ISB and SB fom away fom ISB ( φ =40 ) and SB ( φ = 0 ) fo a al ouc xctaton of a wdg, ctvly A CSB ncdnt on th wdg wth t ax httng th dg. Tajctoy of CSB ISB and SB a th obvaton ont mov away fom th dg wth dffnt locaton of a ln ouc n al ac. Th CSB ISB and SB a dntcal to ISB ( φ =40 ) and SB ( φ = 0 ) fo a al ouc xctaton of a wdg, ctvly A CSB ncdnt on th wdg wth t ax mng th ufac. Tajctoy of CSB ISB and SB a th obvaton ont mov away fom th dg wth dffnt locaton of a ln ouc n al ac. Th CSB ISB and SB fom away fom ISB ( φ =40 ) and SB ( φ = 0 ) fo a al ouc xctaton of a wdg, ctvly Th dffnc btwn ult fom CSB-UTD and CSB-UTD oluton fo ach ca hown n Fgu.6,.3 and.8, ctvly Comlx ont ouc bam llumnat on th wdg. b th bam aamt and bˆ th dcton of bam ax. Q a unqu comlx ont on th dg fo a x

13 gvn ouc and obvaton ont. Comlx ay and ouc locaton a dawn ymbolcally A ont ouc bam llumnaton of a PC wdg. H, Q a unqu al ont of dffacton on th dg fo a gvn ouc and obvaton ont (a) z-dctd comlx ont ouc llumnaton of a wdg. Comlx ay and ouc locaton a dawn only ymbolcally. (b) Comaon of total and dffactd CSB fld obtand va CSB-UTD and CSB-UTD Th tajcto of Im α wth ct to th angl of obvaton fo ISB (to) and SB (bottom) Nglgbl addtonal CSB-UTD tm n th cond (to) and fouth (bottom) of CSB-UTD dffacton coffcnt whch a motant to th ncdnt and flctd CSB (a) z-dctd comlx ont ouc llumnaton of a wdg. Comlx ay and ouc locaton a dawn only ymbolcally. (b) Comaon of total and dffactd CSB fld obtand va CSB-UTD and CSB-UTD Th tajcto of Im α wth ct to th angl of obvaton fo ISB (to) and SB (bottom) Nglgbl addtonal CSB-UTD tm n th cond (to) and fouth (bottom) of CSB-UTD dffacton coffcnt whch a motant to th ncdnt and flctd CSB (a) z-dctd comlx ont ouc llumnaton of a wdg. Comlx ay and ouc locaton a dawn only ymbolcally. (b) Comaon of total and dffactd CSB fld obtand va CSB-UTD and CSB-UTD Th tajcto of Im α wth ct to th angl of obvaton fo ISB (to) and SB (bottom) Nglgbl addtonal CSB-UTD tm n th cond (to) and fouth (bottom) of CSB-UTD dffacton coffcnt whch a motant to th ncdnt and flctd CSB A CSB ncdnt on a cuvd ufac wth dg. Th ncdnt, flctd and dffactd comlx ay a hown ymbolcally. Th ont of flcton wth th bam ax ( bˆ ) al...94 x

14 4. flcton of a comlx ay at a cuvd ufac hown ymbolcally Cautc dtanc aocatd wth th dffacton by a cuvd dg Plan tangnt to th cuvd ufac at th ont of dffacton Q An offt aabolc flcto llumnatd by an ncdnt CSB catd by a comlx ont ouc at wth an abtay ontaton, ˆ Incdnc and flcton of comlx ay on th aabolc offt flcto antnna. Th comlx ouc coodnat ytm, ( x S, y S, zs ) a dfnd wth ' at t ogn dg cuvatu aamt of an offt aabolc flcto. (a) 3D vw (lft) and th dg cuvatu n x-y lan (ght). Th ogn, O ', locatd at th cnt of flcto atu n x-y lan. (b) dg cuvatu and aamt n th lan contanng th dg Th comlx angl φ and φ a hown ymbolcally n th lan ndcula to th dg tangnt vcto ê at Q. Th dcton of ncdnc and dffacton, and unt ufac nomal at Q a ojctd onto th lan and alo ndcatd ymbolcally Ontaton of a ont ouc n al ac ndcula to th ncdnt CSB ax Incdnt CSB llumnaton fom th focu on an nfnt aabolc ufac wth a bam ax httng Q on th ufac...5 B 4. Comaon of CSB flcton fom an nfnt aabolc ufac. Th ncdnt CSB launchd at th focu Incdnt CSB llumnaton away fom th focu on an nfnt aabolc ufac wth a bam ax httng Q on th ufac...7 B 4.3 Comaon of CSB flcton fom an nfnt aabolc ufac. Th ncdnt CSB launchd away fom th focu Incdnt CSB llumnaton fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th mathmatcal xtnon of th ufac...3 B x

15 4.5 Comaon of offt flcto antnna adaton, wth CSB llumnaton fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th mathmatcal xtnon of th flcto ufac n th ymmty lan Dtmnaton of hadow and lt gon whn two hadow bounda xt. (a) d t tajctoy of n th Im (b) flctd, ( ) dffactd ( ) and total CSB ( ) α vcnty of two SB Incdnt CSB llumnaton fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac fa away fom th dg B 4.8 Comaon of offt flcto antnna adaton, wth CSB llumnaton fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual flcto ufac fa away fom th dg n th ymmty lan Incdnt CSB llumnaton away fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th mathmatcal xtnon of th ufac outd th B ymmty lan Comaon of offt flcto antnna adaton, wth CSB llumnaton away fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th mathmatcal xtnon of th ufac outd th ymmty lan Incdnt CSB llumnaton away fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac fa away fom th dg outd th B ymmty lan Comaon of offt flcto antnna adaton, wth CSB llumnaton away fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual ufac fa away fom th dg outd th ymmty lan Incdnt CSB llumnaton fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac na th dg n th ymmty B lan Comaon of offt flcto antnna adaton, wth CSB llumnaton fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual ufac na th dg n th ymmty lan...43 xv

16 4.5 Incdnt CSB llumnaton away fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac na th dg outd th ymmty B lan Comaon of offt flcto antnna adaton, wth CSB llumnaton away fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual ufac na th dg outd th ymmt y lan Incdnt CSB llumnaton away fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac na th dg outd th ymmty B lan Comaon of offt flcto antnna adaton, wth CSB llumnaton away fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual ufac na th dg outd th ymmty lan Incdnt CSB llumnaton away fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac na th dg outd th ymmty B lan Comaon of offt flcto antnna adaton, wth CSB llumnaton away fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual ufac na th dg outd th ymmty lan Incdnt CSB llumnaton away fom th focu on a fnt aabolc ufac wth a bam ax httng Q on th actual ufac na th dg n th ymmty B lan Comaon of offt flcto antnna adaton, wth CSB llumnaton away fom th focu, va th numcal PO, CSB-UTD, CSB-UTD. Th ncdnt CSB ax ht th actual ufac na th dg nd th ymmty lan...5 A. Banch cut fo a ud n dfnng F( ± κa )...65 A. Two banch of F. Th o banch th on coondng to hadd gon...66 C. ay gomty to fnd a comlx dffacton ont on a taght wdg...83 F. Th al at of th comlx dlacmnt contant on hod (hafont jk of th CSB functon ), whl th magnay at contant on th xv

17 jk hybolod (quamltud ufac of ). It notd that and a nomalzd wth b n th Fgu...94 F. A hcal wav adaton and a CSB adaton...99 F.3 Coodnat of a comlx ouc...0 G. Th tt dcnt ath () and th tajctoy of ol n th comlx ξ lan G. Th tt dcnt ath () and th tajctoy of ol n th comlx α lan...04 xv

18 Chat :Intoducton A unfom gomtcal thoy of dffacton (UTD) oluton dvlod fo dcbng th hgh fquncy (HF) lctomagntc (M) fld uoundng an abtaly cuvd, fct lctcally conductng (PC) wdg, that llumnatd by a ont ouc, n comlx ac, whch gnat a comlx ouc bam (CSB). Th oluton ungly found to b th am a th UTD oluton obtand vouly fo PC cuvd wdg llumnatd by a al ont ouc aft that analytcally contnud fo a CSB llumnaton; hnc, t dfnd h a th CSB-UTD fo cuvd wdg. Th oluton fo cuvd wdg dvlod fom two and th dmnonal canoncal HF oluton fo a taght wdg wth lana fac llumnatd by ln and ont ouc, ctvly. Fo a al ln o ont ouc, th canoncal UTD oluton obtand va a ml aymtotc HF bad ft od Paul-Clmmow mthod (PCM) of tt dcnt. Howv, a CSB fld conttut a comlx wav, and th ft od PCM tctly not vald fo olvng th canoncal wdg oblm wth CSB llumnaton. Hnc, fo comlx wav, a dffnt, l comact, ft od aymtotc aoxmaton, known a th Van d Wadn mthod (VWM) of tt dcnt, nd to b mloyd fo olvng th canoncal oblm. Th ft od VWM lad to a canoncal wdg

19 oluton whch may b vwd a an xtndd UTD (o UTD) wdg oluton fo th CSB llumnaton, dfnd h a th CSB-UTD oluton, bcau t can b xd a CSB-UTD = CSB-UTD. Howv, found to b nglgbl fo th wdg ca; t fo th aon that a ft od PCM bad CSB-UTD man accuat vn though PCM not tctly unfomly vald fo th ca. H unfomly vald f to th aymtotc oluton to th ft od ovd a contnuou total fld aco th o calld ncdnt and flctd bam hadow bounda a wll b laboatd futh n th chat. Aft havng tablhd th fact that th CSB-UTD oluton fo th canoncal wdg th am a f th canoncal UTD wdg oluton fo a al ouc xctaton mly and dctly analytcally contnud to dal wth a comlx ouc locaton (o a CSB), th CSB-UTD fo th abtaly cuvd wdg can thu alo b mlaly dvlod by analytcally contnung th coondng UTD oluton fo th am cuvd wdg xctd by a al ont ouc, to now dal wth th xctaton by a ont ouc n comlx ac (o CSB). Th cuvd wdg CSB-UTD oluton mloyd to analyz th adaton fom an offt aabolc flcto llumnatd by a CSB. Th CSB- UTD ult fo th flcto a comad wth a numcal hycal otc (PO) aoach to llutat th accuacy of th CSB-UTD. Th CSB-UTD xtmly fat comad to th numcal PO mthod; alo, th latt bcom ncangly tm conumng wth nca n fquncy, wha th CSB-UTD man ntally ndndnt of fquncy. Th comlx ont of ufac flcton and dg dffacton on th comlx xtnon of th flcto a comutd h ung a latvly fat and ffcnt ocdu, thu makng t vy actcal to tac comlx ay wth almot th

20 am d and ffcncy a tacng al ay. Th abov commnt a ntally xandd, n mo dtal, n th manng at of th chat, whch alo nclud a ummay of th latonh of th nt wok to that otd vouly by oth. Th CSB-UTD oluton fo th cuvd wdg dvlod h utlzd fo th ad and accuat analy of lctcally lag offt aabolc flcto antnna llumnatd by a CSB. On not that th total fld comutd va convntonal UTD fo al ay (a wll a by th unfom aymtotc thoy of dffacton (UAT) [4]) fal n gon of ay cautc, whl th total fld comutd va th CSB-UTD man vald th nc th cautc a now uhd nto comlx ac fo th CSB xctaton ca. Fo xaml, a convntonal UTD (o UAT) ay analy of a tycal focu-fd offt aabolc flcto antnna fal n th dcton of th man bam n th fa zon bcau of th ovla of th cautc of al (a ood to comlx) flctd and dffactd ay, ctvly, along th flcton hadow bounday [5]. A a ult, th hycal otc (PO) ufac ntgaton mthod oftn mloyd to comut th fld na th man bam and ft fw dlob to comlmnt th UTD (o UAT). Howv, nc th PO mthod bad on th ntgaton ov cunt nducd on th flcto ufac [6] and th ntgaton n gnal ha to b fomd numcally, t cla that a numcal PO valuaton comutatonally vy nffcnt, cally fo th analy of lctcally lag flcto antnna. In contat, th nt CSB-UTD oluton xtmly fat, nc t xd n clod fom and do not qu any numcal ntgaton. Alo, nc th comlx ay ath of a CSB a ndndnt of th fquncy, th comutaton vy ffcnt comad to th fquncy dndnt numcal PO mthod of analy. 3

21 In addton, th nt mthod dvlod n th dtaton ovd a hycal nght nto th vaou adaton/cattng wav contbuton wthn th famwok of UTD. It alo notd that th PO mthod fo flcto antnna u th aoxmat gomtcal otc (GO) cunt n th xact ntgal ntaton fo th cattd fld; howv, th GO cunt dcontnuou at a hadow bounday and not coct na an dg [7,8], but th PO tll a aonabl aoxmaton whn th adatng/cattng ufac wll llumnatd and lowly vayng wth ct to th lctcal wavlngth. Nvthl, th PO mthod xctd to b naccuat n gon wh th GO cunt do not oduc th domnat cattng [5]. A a conqunc, whn a convntonal M ont ouc n al ac llumnat a aabolc flcto, th numcal PO mthod not a accuat a th convntonal UTD fo al ay cally fo dctng th fld n th dlob gon away fom th man bam. On th oth hand, th PO mthod dct th man bam and ft fw dlob accuatly whl th ay mthod (UTD and UAT) fal n th man bam gon fo th aon mntond al. Th naccuacy of th PO mthod n th gon away fom th man bam can b movd by addng an addtonal dg dffacton cocton tm bad on th hycal thoy of dffacton (PTD) [9]. Howv, whn a CSB, a ood to a al ouc, llumnat an offt aabolc flcto na th dg, th PO mthod may not b uffcntly accuat n th gon of man bam fo th followng aon. Whn th ax of an ncdnt CSB m th flcto and ht ntad th mathmatcal xtnon of th aabolc flcto at t dg, thn th flctd CSB not domnant, bcau th dffacton of th ncdnt CSB can now bcom a motant a th flctd CSB fld 4

22 contbuton. In addton, nc th ncdnt CSB dcayng adly away fom t ax, th aoxmat ufac cunt nducd by th ncdnt CSB a concntatd na th dg wh th nducd cunt bad on th PO aoxmaton a ncoct [7,8]. Thfo, whn an ncdnt CSB ax dctd outd th actual flcto ufac na th dg, th numcal PO can lo t accuacy n th gon of th dlob and obly vn th man bam, fo th ty of CSB. Whn an ncdnt CSB ax ht th actual flcto ufac na th dg, th flctd CSB much mo domnant than th dffactd CSB, and fo th ca th PO mthod lo only nglgbl amount of accuacy (< 0. db) n th man bam gon comad to th ca whn th ncdnt CSB ax m th actual flcto ufac. Th PO mthod of cou cov t accuacy n th gon of man bam whn th ncdnt CSB ax ht th actual ufac fa away fom th dg. Conquntly, th nt CSB-UTD mthod dvlod n th dtaton ovcom th lmtaton of UTD (and UAT) fo al ay, and alo of th numcal PO. Th nt CSB-UTD oluton dvlod n th dtaton can alo b gnalzd to tat an atgmatc Gauan bam (GB) llumnaton of abtay wdg. Alo, th CSB-UTD oluton can b ald to analyz flcto antnna llumnatd by an abtay M fld fom a fd o a ubflcto, aft xandng th M fld llumnatng th flcto nto a t of CSB. Howv both th toc a not tatd n th nt wok. Fo an ncdnt CSB whch tk a flctng ufac uffcntly fa fom any ont on th dg, th bam dffactd by th dg bcom nglgbl n comaon to flcton ffct bcau th ncdnt CSB dcay adly away fom t ax. Howv, 5

23 whn th ncdnt CSB tongly llumnat th dg and whn th ncdnt CSB ax ntct th flcto ufac, thn th bam dffactd by th dg not nglgbl, and th only oton of ncdnt CSB that tk th flctng ufac gt flctd. Th ath of th CSB whch flctd fom th ufac found va gomtcal otc (GO) aft analytcally contnung th locaton of ouc nto comlx ac n th wok; hnc, th flctd contbuton wll hncfoth b fd to a th CSB-GO fld. To obtan th flctd bam fld va CSB-GO, t ncay to fnd th comlx ont of flcton fo a gvn obv that obvouly locatd n al ac. Snc th ncdnt CSB comod of comlx ay that oagat n comlx ac fom th ouc oton n comlx ac to th obv n al ac, on ha to tac th comlx ay n comlx ac to fnd th comlx ont of flcton on th comlx xtnon of hycal flctng ufac. Thu, th tatonay ha condton mloyd and th comlx ont of flcton fo a gvn al obv achd numcally. Only fo a wdg wth lana fac can th comlx ont of flcton b found n clod fom mly va mag thoy ud n conjuncton wth a comlx ont ouc locaton. Th ont of dffacton on th dg of a cattng objct alo bcom comlx fo a CSB llumnaton. Th Kll law of dg dffacton [,3] mloyd to fnd a comlx dffacton ont, fo a gvn al obv, whch alo ha to b numcally achd. It notd that fo a two dmnonal (D) wdg ca wth a CSB llumnaton (du to a comlx ln ouc), th ont of dffacton bcom al and concd wth th dg, a can b hown analytcally aft mly calzng th oblm of fndng a dffacton ont on a 3D taght dg to th D ca. Th comutatonal ffot utlzd n th wok 6

24 fo achng both comlx ont of flcton and dffacton nglgbl and th mthod ud h fo localty uch comlx ont aocatd wth comlx ay mak t almot a fat a tacng convntonal ay n al ac. It notd that all oth M fld aamt aocatd wth comlx ont of flcton and dffacton gnally bcom comlx n comutng th flctd and dffactd fld va CSB-GO and CSB-UTD, ctvly. Th dtal of dvlomnt of flctd and dffactd fld contbuton to th fld ang fom a CSB llumnaton of a cuvd ufac, and th aocatd comlx ay aamt (ncludng ont of flcton and dffacton) a dcbd lat n th dtaton. Whn an ncdnt CSB tk a flctng ufac wth an dg o t mathmatcal xtnon na th dg, only th oton of comlx ay aocatd wth th ncdnt CSB that a catud by th ufac a tanfomd nto th flctd CSB. Thu, th ncdnt and flctd CSB bcom dcontnuou at th ctv hadow bounda whch a taght ln n comlx ac. Unlk th adaton of M fld xctd by a al ont ouc, th CSB oagat though comlx ac untl t av at th obvaton n al ac. Thfo, th locaton of GO lk ncdnt and flcton hadow bounda, aocatd wth th ncdnt and flctd at of th CSB, namly th ISB and SB, ctvly, gnally do not aa to b at th am locaton a fo th ca of th M fld xctd by a al ont ouc. Futhmo, th CSB-GO latd ISB and SB can not b found aly vn fo a latvly ml oblm, uch a fo th ca of a D CSB xctaton of a half lan. Hnc, th xlct xon to locat th ISB and SB a dvlod n th wok fo th CSB xctaton of a gnal 7

25 cuvd wdg, and th condton dfnng th xtnc of th lt and hadow gon fo th ncdnt and flctd CSB a alo tablhd fo th ca o that a comlt CSB- UTD mthodology bcom avalabl fo th u. It motant to not that only fo th canoncal taght wdg wth lana fac and fo a wdg wth cuvd fac n D, th flctd CSB alo a CSB ognatng fom a comlx ont ouc locatd at th mag ont. On th oth hand, an ncdnt CSB that flctd fom a wdg wth cuvd fac n 3D, th flctd bam bcom atgmatc vn n comlx ac. Th nt CSB-UTD oluton ft dvlod va a unfom aymtotc valuaton of an ntgal ntaton fo th M fld uoundng a canoncal PC wdg xctd by a D and 3D CSB. Two dffnt but wll tablhd unfom aymtotc tchnqu, bad on th mthod of tt dcnt, a mloyd, fo aon gvn blow, fo valuatng th canoncal wdg fld ntgal at HF. Th unfom aymtotc tchnqu a tycally fd to a th Paul-Clmmow mthod (PCM) [3, 9-] and th Van d Wadn mthod [-5], ctvly. Th tchnqu hav bn comad mathmatcally n th at [,6-0]. Both tchnqu a utlzd to valuat th ntgal that nt th fld n th canoncal oblm of th dffacton of an ncdnt D and 3D CSB by a taght wdg wth PC lana fac. Thn, th oluton of th D and 3D canoncal oblm whch a xd wthn th UTD fomat a gnalzd ung aoat ocdu to dal wth th ca of CSB llumnaton of an abtaly cuvd PC wdg. Th latt dctly alcabl to th analy of aabolc offt flcto xctd wth a CSB, and alo to gnal (not 8

26 ncaly aabolc) flcto gomt. In th nt wok, th CSB-UTD ald only to aabolc flcto to llutat t accuacy and d. Th PCM and VWM a tycally utlzd fo aymtotcally valuatng a cla of comlx contou ntgal, by th mthod of tt dcnt, whn thy contan ntgand chaactzd by a ft od addl ont and a naby ml ol. A n fom [3,6], uch tt dcnt mthod can alo b utlzd fo tatng hgh od addl ont, hgh od ol, and multl ol, tc.; howv, th tuaton a not condd n th dtaton. It known that th ft od aymtotc oluton bad on th PCM tctly not vald fo tatng any of th canoncal oblm tanng to comlx wav [4,7,8], wha th ft od VWM vald vn fo th ca. Howv, t found h that vn whn t ald to dal wth comlx wav aocatd wth CSB ty wdg llumnaton, th ft od PCM wok ungly accuatly fo analyzng th oblm of CSB dffacton by a wdg; an xlanaton fo th unxctd ult ovdd n th wok. Th latt outcom of th PCM bad oluton fo th CSB wdg dffacton oblm ha an motant mlcaton on th oblty of xtndng th unfom gomtcal thoy of dffacton (UTD) oluton fo a PC wdg xctd by a ouc otond n al ac [,3,], to dctly tat th ca of CSB wdg xctaton (tanng to a ouc otond n comlx ac) mly va analytc contnuaton of th avalabl UTD oluton fo th al ouc. It obvd, n gnal, that th PCM mo convnnt fo numcal comutaton than th VWM, cally fo tatng oblm nvolvng al (a ood to comlx) wav; th u bfly dcud n Andx A, and a lghtly mo convnnt fom 9

27 of th VWM alo ntd h. Th latt xd a th um of th PCM bad ult togth wth a cocton tm; th aangd VWM fd to n th wok a th xtndd PCM (o PCM). Th PCM (whch ovd an xlct cocton to PCM) thu vald fo both al and comlx wav, nc t mly aangmnt of th VWM whch mhaz th dffnc btwn th ft od PCM and VWM. Snc th convntonal UTD fo al ay ognally bad on th aymtotc valuaton of a tnnt wav ntgal va PCM [,3], th oluton found va PCM (=VWM), whch alo vald fo comlx wav, fd to a th xtndd UTD (UTD) oluton n th dtaton. In th PCM and VWM, ctvly, th ognal ath of ntgaton n th comlx lan dfomd to th tt dcnt ath () on whch th ha of th ntgand tay contant o a to allow an ffcnt clod fom valuaton of th wav ntgal at hgh fqunc fo whch th ha of th ntgand othw hghly ocllatoy. A ol of th ntgand can b cod n th contou dfomaton to th ; hnc, t contbuton mut b ncludd va th Cauchy du thom []. Oth banch cut contbuton, f thy xt, mut alo b ncludd whn thy a cod n th contou dfomaton; howv, uch banch cut contbuton a not catud fo th canoncal oblm of ntt whn thy a fomulatd n tm of ctal ntgal n ola (o angula ctal doman) fom fo th total fld. Tycally, n th ctal ntgal fomulaton of th oluton to th canoncal oblm of ntt, th mov latv to th addl ont (o vc va) a th obvaton ont mov n ac; hnc, th a om obvaton dcton (o act angl) fo whch th lvant ol 0

28 catud n th contou dfomaton but not o fo oth angl, thu dfnng a hadow gon wh th ol wav contbuton abnt lavng only th contbuton. In atcula, lt u nt th total cala wav fld nt n th canoncal oblm of CSB dffacton by a PC wdg, whch a addd n th dtaton. Alo lt u PW and u nt th catud ol wav contbuton (o th du of th ntgand) and th wav contbuton, ctvly. Thn, u = u H, (-) PW u wh th Havd t functon, H, ha a valu of unty whn th ol catud, and zo whn th ol not catud, ctvly, n th dfomaton of th ognal ntgaton contou to th. vn though u PW H a dcontnuou functon, u not; hnc, u mut contan a comnatng dcontnuty to k th total fld, u, contnuou vywh n th ac uoundng th adatng o dffactng tuctu. Whn u aoxmatd aymtotcally, thn on obtan an ntally clod fom aymtotc xanon n nv factonal ow of fquncy fo th tm [4,6,8,]. To facltat a ytmatc dvlomnt of th aymtotc, th ntgal n th ognal comlx ctal angula lan, dnotd h a th ξ lan, tanfomd va a tandad mang to th al ax n th nw tanfomd comlx doman, dnotd h by α, wh now th addl ont alway man fxd at th ogn ( α =0) of th comlx α lan [4]. Th ognal ctal ntgal thn ducd to a gnc ntgal along th nfnt al ax n th α doman; uch an ntgal can b

29 ducd ytmatcally va th PCM o th VWM to av at th dd unfom aymtotc xanon [9,-4,6-8]. H, unfom man that th ultng aymtotc uod to ovd a total hgh fquncy oluton that man unfomly vald aco th ol wav hadow bounday (SB); th latt SB dlnat th uoundng ac nto a hadow and lt gon coondng to wh th upw tm xt and wh t do not (accodng to whn H o 0 n ()). In th α doman, th alway man fxd along th al α ax (wth th addl ont at th ogn ( α =0) ndcatd abov; a uch, only th ol can mov aco th n th α lan a th obv chang oton. Th locaton of th SB can thu b dfnd a that obvaton (o act) dcton at whch th ol jut co th. In th gad, t notd that f only th ft od ult n th aymtotc valuaton tand n th PCM, thn th aymtotc ult fo u to th od contnuou only f th ol co th though th addl ont at th ogn ( α =0) ; othw, th PCM bad ult not tctly unfom (.., do not guaant contnuty) aco th SB. On th oth hand, t can b hown that th ft od, VWM alway man vald aco th SB,.. t ovd a contnuou total fld u aco th SB, vn f th ol co th away fom th addl ont. Fo all th canoncal oblm bng condd h nvolvng comlx wav, th ol alway co th away fom th addl ont. Conquntly, only th VWM bad aoach wll man tctly unfom fo comlx wav bng tatd h. Howv, a mntond vouly, t hown n th wok, that th PCM wok ungly accuatly fo th oblm of CSB dffacton by a ) a

30 wdg. Of cou, t ha bn vouly dmontatd that th comlt PCM bad aymtotc xanon can b tanfomd nto th comlt VWM bad aymtotc xanon [,6,8]; thu, t ndd obl that contnuty of th aymtotc tmat fo th total u can b tod n th PCM whn th ol co th away fom th addl ont, by ncluon of hgh od tm n th PCM. On th oth hand, uch addton of hgh od tm to th ft od PCM mak t fa mo comlcatd to u than f on mloy only th ft od tm of th VWM. It n that n ca wh th ft od PCM bad oluton vald, t gnally ml fo alcaton than th ft od VWM bad oluton. Aft th valuaton of th ntgal contbuton, u n (-) va PCM and PCM (=VWM) who oluton a dcbd n dtal n Andx A, th PCM (o UTD) oluton contan th cocton tm fo th PCM (=UTD) oluton and t ymbolcally xd a wh PCM u and PCM u u PCM = u PCM, (-) a oluton of u found va PCM and PCM, ctvly. A xland bfo, a cocton to PCM u wthout whch PCM u alon can not man tctly vald fo comlx wav wh th ol gnally co th away fom th addl ont. If th ol co th though th addl ont a t gnally do fo oblm nvolvng al wav, only thn do = 0 at th SB o that PCM PCM u = u n tho ca. In th comlx wav oblm nvolvng CSB dffacton by a wdg, 3

31 found to b ungly nglgbl a mntond al, and o fo th cal ca PCM PCM UTD UTD u u ( u ) u a dcud n Andx A. Many ach n th at hav xlotd th CSB conct o t Gauan bam (GB) fld aoxmaton wthn th aaxal gon [3][38]. A vaty of oblm hav bn tatd by thm fo bam oagaton n comlx nvonmnt ncludng GB tanmon though adom by Gao and Fln [3], GB tanmon though lan and cuvd dlctc lay by Macl and Fln [4,5], and though nhomognou and anotoc mda by Hyman [6]. Moov, a owful combnaton of th CSB mthod and th aymtotc ay tho ha bn ald fo th analy of bam flcton fom th 3-D aabolc flcto ufac by Halman and Fln [7] wh an nfnt aabolc ufac ud and hnc th dffacton abnt n that wok. Du to th dlacmnt of th ouc locaton nto comlx ac n [7], ach ont of flcton tacd n comlx ac by ung th tatonay ha condton and th comlx flcton ont a found numcally. Gn, Bton and Fln [8] ntoducd th conct of hadow bounda fo D CSB llumnatng a half lan, and ovdd clod fom oluton n two cal ca aft ntoducng ctan aoxmaton; on whn th fld ont locatd n fa zon of th dg and th oth whn th bam wat locatd fa away fom th dg. In th lat ca, th dlacmnt of a ouc n al ac nto comlx ac do not hft th hadow bounda gnfcantly, bcau th bam fld may b n a lan wav at th dg manatd by a ouc n al ac fa away fom th dg. Sudan and Jull n [9, 30, 3] hav utlzd th combnaton of th CSB mthod and an analytc contnuaton of 4

32 UTD fo cala ln and ont ouc. Howv, n [9], th SB a found aoxmatly only fo th ca wh th dtanc fom th dg to th obv much fath than th dtanc fom th dg to th bam wat. In addton, thy dd not tat th ca wh th bam ax clo to th dg,.. whn th bam tongly llumnat th dg. In [30], thy dmontatd th ca whn th dg on th bam ax, n whch th dg ht only by a al ay (axal ay). In atcula, thy dd not comut a combnaton of th flctd and dffactd bam fld to how th comlt total fld and th contnuty of th total fld uoundng a -D aabolc flcto n [3], but howd only th dffactd fld. Thy mloyd atu ntgal (AI) mthod fo calculatng th man bam and ft fw dlob ntad of calculatng CSB flctd and dffactd fld n th fowad gon. Anatau and Pathak [3], and Zogb and Pathak [33], dvlod an aoxmat aoach fo analyzng th flcton and dffacton of an ncdnt GB by a fnt cuvd flcto; t bad on an aymtotc valuaton of th aoxmat PO ntgal fo th ufac cunt nducd by th ncdnt GB on th D and th 3D flcto gomt, ctvly. Th aoach ovd clod fom oluton fo both flctd and dffactd fld, a wll a fo th comlx flcton and dffacton ont. A a ult, th oluton vy fat to analyz lctcally lag flcto antnna a comad to numcal valuaton of PO and atu ntgal (AI) fomulaton. Th ult n [3] vald only fo analy of -D flcto n th fa zon. Th oluton n [33] ovd 3-D dg dffacton fom flcto, but jut a th wok n [3], th oluton of [33] alo contan ctan aoxmaton n th PO ntgand; futhmo th ultng xon fo both th 5

33 flctd and dffactd fld a mo comlcatd than th UTD xon dvlod h. Chou and Pathak [34] dvlod a mo comact and gnal 3-D oluton to ovcom th dfcncy n [33] and ovdd a mo comact oluton. Th oluton n [34] alo alcabl to mo gnal 3-D cuvd ufac than tho whch can b tatd va [33]. Th comlx ont of flcton and dffacton n [34] a found n clod fom oluton bad on a local aabold aoxmaton at th flcton ont thby avodng comlx ay tacng. Howv, th accuacy of th oluton n [34] wthn th am lvl a th numcal PO oluton, bcau th GB analy dvlod fom an aymtotc clod fom valuaton of th PO ntgal tlf. Hyman and Ianconu [35] mloyd th CSB mthod n th tm doman to analyz th dffacton of a uld bam by a wdg. Thy ovdd uful hycal nght nto th bhavo of th cattd wavackt, but th oluton lmtd to a taght dg wth lana fac. Plo and Sll [36] dvlod a numcal aoach though th alcaton of th aabolc quaton to obtan th oluton of GB dffacton fom a PC wdg. A ln ntgal ntaton of PO dvlod by Matn t al [37] to fnd th cattd fld fom a fctly conductng qua lat. Th ncmntal thoy of dffacton (ITD) ha bn ald to comut th dffactd fld by an dgd objct by a CSB n th wok of Polm t al [38]. Th ITD tchnqu avod fndng comlx ont of dffacton, but ntad qu th valuaton of a ln ntgal on th dg; nvthl t ha hown t ffctvn to comut th fld adatd and cattd fom fctly conductng qua lat and dk llumnatd by CSB. Howv, th wok n [36-38] tctd to an dg n tuctu wth lana fac. Thfo, th 6

34 nt CSB-UTD oluton, dvlod h n clod fom to dcb th wav flctd and dffactd by an abtay 3D cuvd PC wdg llumnatd by a CSB ovcom th lmtaton of afomntond contbuton and ovd a mo gnal ult obtand ytmatcally fom an aoat xtnon of aymtotc oluton to aoat canoncal oblm. Th fomat of th dtaton a follow. In Chat, th canoncal oblm of a CSB dffacton by a PC taght wdg wth lana fac llumnatd by a comlx ln ouc tatd. Th cala D dffacton coffcnt nt wthn th CSB-UTD and CSB-UTD oluton that a dvlod n chat a xlctly dntfd. In Chat 3, th dyadc dffacton coffcnt of CSB-UTD and CSB-UTD a obtand fo th canoncal oblm of a 3D M CSB dffacton by a PC taght wdg wth lana fac. In both Chat and 3, numcal ult a ntd to how th valdty of two unfom oluton namly th CSB-UTD bad on th PCM and th CSB- UTD bad on th VWM, ctvly, and ov numcally th cocton tm n th dffacton coffcnt of CSB-UTD nglgbl o that CSB-UTD bacally and ungly duc to CSB-UTD. Th latt jutf obtanng th CSB-UTD oluton fom a dct analytc contnuaton of th coondng UTD wdg oluton whch wa obtand vouly va a PCM bad valuaton of aoat canoncal wdg dffacton oblm [,3]. Wthout th abov jutfcaton obtand analytcally, on could not hav bn u that CSB-UTD would ovd contnuou total fld aco ISB and SB a tctly guaantd by CSB-UTD, but not by CSB-UTD. Th xlct xon to dtmn th ISB and SB a alo dcud n both analytcally and 7

35 numcally. Alo, th cta to dtmn th lt and hadow gon fo th ncdnt and flctd CSB a hown fo th gnal ca of CSB llumnaton of an abtay cuvd wdg. In Chat 4, th oluton fo th oblm of a fully 3D cuvd PC wdg xctd by a CSB dvlod by gnalzng th canoncal oluton obtand n chat 3 ung analytcally contnuaton agumnt. Th gnal CSB-UTD oluton thn ald to analyz a 3D aabolc flcto llumnatd by a CSB, and numcal ult obtand va both th CSB-UTD and CSB-UTD a ntd fo th ca and comad wth th numcal PO mthod to dmontat th accuacy and comutatonal ffcncy of th clod fom CSB-UTD oluton. It notd that th aymtotc aoxmaton ud n chat and 3 fo olvng th canoncal taght wdg oblm, and hnc thn xtnon ud n chat 4 to dal wth an abtaly cuvd wdg, a all vald a long a th obvaton ont not xtmly clo to th dg. Th latt ntal to guaant that κ, th aymtotc lag aamt ud n chat and 3 nv too mall to tan th aymtotc HF CSB-UTD oluton fo all ca tatd n th wok. Som concluon and toc fo futu wok a ntd n Chat 5. An j t ω tm dndnc aumd and ud n th dvlomnt whch follow; h t dnot tm and ω dnot th angula fquncy of th wav. 8

36 Chat : Canoncal oblm of D CSB dffacton by a taght wdg wth lana PC fac xctd by a comlx ln ouc In th chat, t of ntt to dvlo th CSB-UTD and CSB-UTD dffacton coffcnt fo a canoncal D oblm of a CSB dffactd by a taght wdg wth PC lana fac xctd by a ẑ - dctd comlx ln ouc. Th taght dg locatd along th z ax. Alo, th wdg uoundd by f ac (wth wavnumb k). Th ctal ntgal ntaton of th Gn functon fo ẑ - dctd lctc o magntc ln ouc xctaton gvn n [] ud, aft analytcally contnung th locaton of th ouc nto comlx ac to tat th comlx ln ouc xctaton oblm. Th dffacton coffcnt a found by aymtotcally valuatng th ntgal va two mthod of tt dcnt, namly, th PCM whch lad to th wdg oluton CSB-UTD and PCM (= aangd VWM) whch lad to th CSB-UTD wdg oluton; Th PCM and PCM (= VWM) mthod a vwd n Andx A. 9

37 ( ) ρ,φ ρ y φ ' φ ρ ' ( ρ', φ' ) x ( n)π Fgu.. A ln ouc ncdnt on th wdg. 0

38 .. Dffacton coffcnt of a CSB dffactd by a PC wdg fo a comlx ln ouc xctaton Ft, cond a ẑ -dctd unfom ln ouc locatd at ρ ' = ( ρ', φ' ) n al ac whch oduc a cylndcal cala wav that llumnat a PC nfnt wdg, coondng to a D oblm. Th total lctc fld, = z ˆ, and th magntc fld H = zh ˆ z, du to an lctc and a magntc ln ouc, ctvly, at ρ' can b wttn u a, n tm of a two dmnonal cala Gn functon, ( ρ '), h ρ z H z ( ρ, ρ' ) = jωµ I u ( ) ρ ρ' ( ρ, ρ' ) = jωε M u ( ρ ρ' ) z h (.) wh th al obvaton locaton at ρ = ( ρ, φ). H, ω th angula fquncy of th wav. Alo, µ and ε a th mablty and mttvty of th otoc, homognou mdum uoundng th wdg, ctvly. In (.), I and M a th tngth of th lctc and magntc ln ouc, ctvly. Th ubct and h on th wdg Gn functon nt th ca fo whch th ln ouc lctc and magntc, ctvly. Th ctal ntgal ntaton of th Gn u fo z-dctd lctc o magntc ln ouc xctaton gvn n functon ( ρ '), h ρ [] fo th al ouc ca (.. fo a al ouc locaton) a ( ρ ρ' ) = u( ρ, ρ'; β ) u( ρ, ρ'; β ), (.) u, h wh β = φ φ', (.3)

39 u ( ρ, ρ'; β ) 8π jn L L' dξ jk π ( ρ ρ' ρρ' coξ ) ξ β cot n jk ( ρ ρ ' ρρ ' coξ ) (.4) wh k al and lag. Th qua oot quantt n (.4) ult fom an aymtotc lag agumnt aoxmaton of Hankl functon of cond knd, od zo, and agumnt gvn by th qua oot quantty. Th ntgal ath L L' hown n Fgu. th wll known Sommfld contou. Thu, th ntgaton ov L L' atf Sommfld adaton condton and conttut a contou on whch th ntgal convg [3]. Th xonntal n (.4) aoxmatd a by t two tm bnomal xanon a ρρ ' ( ) ( ) ( ) ( co ) ' jk ξ jk ρ ρ ρρ 'coξ jk ρ ρ ' ρ ρ ' (.5) ρρ' wh ( coξ ) ( ) ρ ρ' aumd to b mall; th aoxmaton wll b jutfd lat. Thfo, ncooatng (.5) nto (.4) yld wh u κf ( ξ ) ( ρ, ρ'; β ) g( ξ, β ) L L' dξ, (.6) g ( ξ, β ) = 8π jn jk π ( ρ ρ' ρρ' coξ ) ξ β cot n jk ( ρ ρ ' ), (.7) ρρ' f ξ = j (.8) ( ) ( ) ( co ξ ) ρ ρ',

40 κ = k. (.9) Th addl ont of f ( ξ ) a found va f '( ) = 0 Th ( ξ,β ) ξ. It follow that fom (.8) ξ ξ = ξ = mπ, wh m = 0,,, 3,. (.0) ± g n (.7) alo ha banch ont ngulat at ξ = ξb whch a oot of ρ ρ' ρρ' coξ = 0, namly, ρ ρ' ξb = l π ± j coh, l = 0, ±, ±, ± 3, ± 4, (.) ρρ' and ξ b a not n th vcnty of th addl ont at ξ = ± mπ. In od to facltat th Cauchy thom [] n valuatng th ntgal n (.6), on ha to dtmn to fom a clod ath wth C = L L'. Th a obtand by th condton, Im f ( ξ ) = Im f ( ξ ) and ( ξ ) < 0 f a Im ξ. Thfo, th a dfnd wth ξ = u jv, a fo ( ξ ) < 0 ξ = ±π cou coh v =, (.) f a Im ξ. Thu only that a though th addl ont nd to b mloyd a hown n Fgu.. Th ( ξ, β ) ty ngulat at ± = 0, ±, ±, ± 3, ± 4, ± ξ = ξ = β nn π, g alo ha ml ol N (.3) 3

41 Im( ξ ) ξb ξb ξb ( π ) L ( π ) ( ) β ξ 3π π ξ = π π ξ = π π π 3π (ξ L' ξ b ξ b ξ b Fgu... Stt dcnt ath, ( π ) ouc llumnaton of a canoncal wdg. Th tajctoy of ± and th comlx ξ lan toology fo a al ln ξ ndcatd along th al ax. 4

42 wh ± N dtmn th ol whch clot to on of addl ont. Th uct '± ' on ξ and N n (.3) tan to th addl ont ξ = ± π ntg that mot naly atf th followng quaton. S. ± N dfnd a th N ± ± π β π n ( β ) =, (.4) Th nto wdg angl dfnd a ( n)π n Fgu.. Thfo, th addl ± ont at ξ = ± π, and only th ol at ξ = β nn π that l wthn th clod ath C ( π ) ( π ) ±, ctvly, contbut to th ntgal. It notd that fo uffcntly lag k, th majo contbuton to th ntgal valuatd ov ( ± π ) occu only fom th mmdat vcnty of th addl ont ( ξ = ± π ) ( co ). Thu, ξ bcom vy mall a ξ ξ = ± π, whch th uffcnt condton fo th aoxmaton of (.5). Thfo, th ntgal n (.6) dfomd nto va Cauchy Thom [] a follow wh u u ( ρ, ρ'; β ) u ( )( ξ, β ) u ( ξ, β ) H ( ξ ) π u ± κf ( ξ ) ( ± π )( ξ β ) = g( ξ, β ) ( ± π ) PW ( π )( ξ β ) upw ( ξ, β ) H ( ξ ),, (.5), dξ, (.6) ± ± ( ξ, β ) πj ( ξ, β ) u, (.7) PW = 5

43 ( ) ρ,φ ρ y φ ρ ' φ ' b = bbˆ φ b ( ρ ', φ ') x ( n)π Fgu.3. A comlx ln ouc bam ncdnt on th wdg. b th bam aamt and bˆ th dcton of bam ax. Comlx ay and ouc locaton a dawn only ymbolcally. 6

44 wh ( ξ, β ) ± du of ntgand n (.4) fo coondng ol and th addl ont. Th du contbuton n (.5) automatcally tctd by th ± Havd t functon ( ) H ξ to b aocatd wth only tho ol whch l wthn th clod ath of ntgaton fomd by C = L L' and ( ± π ). Th locaton of th ln ouc n al ac nxt analytcally contnud nto comlx ac va ' = ρ' j b ρ at ( ρ ', φ '). Th bam ax ontd along bˆ = coφ xˆ nφ yˆ wh th lan ndcula to th dg only condd. A b b tld ndcat a comlx quantty. Th comlx ay and comlx aamt, ( ρ ', φ ') tc. a llutatd only ymbolcally n Fgu.3. Th gnaton of a CSB and th comlx quantty of ( ', φ ') ρ a vwd n Andx F. Th total lctc and th magntc fld aocatd wth a CSB du to an lctc and a magntc ln ouc, ctvly, at Gn functon, ( ρ ') ρ ' can b wttn n tm of an aoxmat two dmnonal cala u a,, h ρ H z ( ) = ( ρ, ρ jωµ I u ρ ρ' ) z ( ) ( = ρ, ρ jωε M u ρ ρ' ) wh th al obvaton locaton at ρ = ( ρ, φ) of th Gn functon ( ρ '), h ρ h (.8). Th ctal ntgal ntaton u fo z-dctd comlx lctc o magntc ln ouc xctaton can b found fom th ult n (.5)(.7) by analytcally contnung th locaton of a ln ouc n al ac nto th comlx ac. Thu, 7

45 u, h ( ) ( ', '; ) (, ρ ρ = u ρ ρ β u ρ ρ'; β ), (.9) wh β = φ φ ', (.0) u u ( ρ, ρ'; β ) u ( )( ξ, β ) u ( ξ, β ) H ( ξ ) PW π u ± κf ( ξ ) ( ± π )( ξ β ) = g( ξ, β ) ( ± π ) PW ( π )( ξ β ) upw ( ξ, β ) H ( ξ ),, (.), dξ, (.) ± ± ( ξ, β ) πj ( ξ, β ) u, (.3) = g ( ξ, β ) = 8π jn jk π ( ρ ρ' ρρ ' coξ ) ξ β cot n jk ' ( ρ ρ ), (.4) ρρ ' f ξ = j ξ (.5) ( ) ( ) ( ρ ρ' co ), κ = k. (.6) It notd that fo th comlx ouc locaton, th addl ont man th am a th al ln ouc xctaton ca, ξ = ± π, and th a fo th comlx ouc ca alo unchangd a n (.). Th nd ont of th ath S L L' at nfnty a wthn th hadd gon to ovd convgnc of th ctal ntgal and to atfy th adaton condton; howv, L L' not allowd to co any ol o banch ont n th oc of analytc contnuaton fom a al to comlx ouc locaton. Th ol a now comlx n contat to al ol fo th al ouc ca; thy a tll gvn by th condton, 8

46 Im( ξ ) ξb ξb ξb ( π ) L ( π ) ξ ( β ) ξ ( β ) 3π π ξ = π π O ξ = π π π 3π ( ξ ) L' ξ b ξ b ξ b Fgu..4. Stt dcnt ath, ( π ) ξ ndcatd fo ( ξ ) 0. Im < ± and th comlx ξ lan toology. Th tajctoy of 9

47 wh ± ± ξ = ξ = β nn π, N = 0, ±, ±, ± 3, ± 4, (.7) ± N dtmn th ol whch clot to on of addl ont. Smlaly, th uct '± ' on ξ and N n (.7) tan to th addl ont ξ S = ± π. Fo a comlx ouc xctaton, quaton. ± N dfnd a th ntg that mot naly atf th followng N ± ± π ( ) ( β ) β =, π n (.8) Fgu.4 how th and th tajctoy of ol. ± Snc ( ξ, β ) th du of ntgand n (.4) fo coondng ol and th addl ont. Th du contbuton n (.) automatcally tctd by th ± Havd t functon ( ) H ξ to b aocatd wth only tho ol whch l wthn th clod ath of ntgaton fomd by C = L L' and ( ± π ). Th du contbuton nt th ncdnt and flctd CSB contbuton ang fom th comlx ln ouc llumnaton of th wdg. Th du ± ( ξ β ) found to b j j jk ρ ρ ' ρρ 'co ξ, = ± ( ), (.9) πj 4 πk ρ ρ' ρρ ' coξ Thfo, ncooatng (.9) nto (.) yld u, h { π PW ( ρ ρ' ) u ( )( ξ, β ) u ( ξ, β ) H ( ξ ) u ( π )( ξ, β ) upw ( ξ, β ) H ( ξ ) } 30

48 { u u ( π )( ξ, β ) upw ( ξ, β ) H ( ξ ) ( π )( ξ β ) upw ( ξ, β ) H ( ξ )},. (.30) Th ft and th cond du tm ( u PW n (.30)) contbut to th dct CSB lctc/magntc fld fo th n-fac and o-fac llumnaton, ctvly, wha th thd and th foth du tm contbut to th flctd CSB lctc/magntc fld fo th n-fac and o-fac llumnaton, ctvly. Th total bam fld a comod of a uoton of th ncdnt CSB, th flctd CSB and th bam dffactd fld, ctvly a n (.30), n atcula, th addl ont contbuton n (.30) coond to th bam dffactd fld. Thfo, ach u n (.30) fom ξ S = ± π nd to b valuatd and uod to fnd th bam dffactd by th wdg and ubquntly, th total fld n (.8). Nxt, th UTD coffcnt of bam dffactd by a PC wdg xctd by a z- dctd comlx ln ouc a found va two mthod of tt dcnt, namly, PCM whch lad to a CSB-UTD oluton and PCM (=aangd VWM) whch lad to a CSB-UTD oluton. Fom Andx A, on can valuat th Gn functon aymtotcally n clod fom fo th bam dffactd fld n (.) a u u UTD, h UTD, h UTD ( ρ ρ ' ) d jkρ h, u UTD ( ρ ρ ' ) ρ d, h (.3) wh jkρ ' j j u =, (.3) 4 πk ρ' 3

49 d { d ( β, ξ ( β ) d ( β, ξ ( )} UTD, h UTD UTD = β { d ( β, ξ ( β ) ( β, ξ ( β )} UTD d, (.33) { d ( β, ξ ( β ) d ( β, ξ ( )} UTD, h UTD UTD d = β, π UTD { d ( β, ξ ( β ) ( β, ξ ( β )} UTD d (.34) [ ] j 4 ± ± π ± β [ β, ξ ( β )] cot ± d UTD = ± ( β ) π F ka, (.35) n k n d ± UTD π UTD [ ] j 4 ± π ± β [ β, ξ ( β )] cot ± = F ± ka ( β ) n πk n ± ρ ρ ' δ n ± ( ξ ( β ) co( ξ ( β ) ) π β cot n [ ± ± ka ( )] ( F ), β ± (.36) ( ), ( ) ' ± ± ρρ ξ co β a β = (.37) ρ ρ' ± ± ( ξ ( β ) ρ ρ' ρρ ± 'coξ ( β ) δ =. (.38) ( ) wh ( ± F ± ka β ) th convntonal UTD tanton functon fo dg whch not gnalzd to comlx ac a dcbd n Andx A. It notd that u nt th ncdnt CSB at th dg fo lag ρ ' j = 0 ρ. Th 4 () k wh u H ( k ') UTD d, h and UTD h d, a th CSB wdg dffacton coffcnt found va CSB-UTD and CSB-UTD, ctvly, n th ca of a comlx ln ouc xctaton. A dcud n Andx A, 3

50 ach tm of th dffacton coffcnt va CSB-UTD n (.34) contan an xta tm n addton to th coondng tm of th dffacton coffcnt obtand fom CSB-UTD n (.33). It hown analytcally n Andx A that th addtonal tm ngnfcant bcau th ft backt of cond tm n (.36) bcom nglgbl whn th obvaton at and na th SB and bcau th cond backt bcom nglgbl whn th obvaton away fom th SB, whch alo vfd numcally n th nxt cton. Thfo, th oluton va CSB-UTD and CSB-UTD a naly dntcal; th fact wll alo b vfd wth cfc numcal xaml n th nxt cton... Numcal ult fo a comlx ln ouc xctaton Bad on th oluton found n th vou cton, th CSB dffactd by a PC wdg xctd by a comlx ln ouc locatd at ρ ' = ( ρ', φ ' ) comutd. Th total CSB fld at th nc of a wdg obtand by th uoton of ncdnt, flctd and dffactd fld. Th flctd CSB fld aa to b manat fom th mag of th actual comlx ouc locatd at ' ( ', φ ρ = ρ ' ). Th ncdnt ( ) flctd ( ) z z and th CSB fld a du to th ognal z-dctd comlx lctc ln ouc and t mag, ctvly. Th fld dnotd a CSB-GO fld a gvn by th coondng ol contbuton and can b wttn a, z jk ρ ρ ' = 0 j 4 j πk ρ ρ ', ( ρ ) ± C,, (.39) 33

51 wh th ngatv gn fo z du to th olazaton chang val ang fom flcton off th lana PC wdg fac bng llumnatd. Alo, C0 = jωµ I n (.39) a comlx contant nt n (.8). A dcud n Andx A, th SB fo, z can b dtmnd a th locaton of obvaton coondng to whn th ncdnt/flctd CSB ol co th aoat. In oth wod, th SB a found to b th φ coodnat of th obvaton ont whn Im α = 0 n th comlx α lan of Andx A. Th α to fnd ncdnt (ISB) and flcton (SB) hadow bounda a gvn, ctvly, a α, ( β ) = ρρ ' ξ co ρ ρ' ( β ) j 4 π, (.40) wh th banch cfd a dcud n Andx A. It notd that th comutatonal ffot to calculat ISB and SB nglgbl. Th lt and hadow gon fo th ncdnt and flctd CSB hould b dtmnd o to th comutaton of th total fld. A cta of choong th lt and hadow gon dcud at th nd of th chat. Fgu.5 llutat th tajctoy of ISB and SB fo a comlx ln ouc at a fxd comlx locaton llumnatng a PC half lan, a th adal dtanc of obv chang. A llutatd n Fgu.5, th ISB and SB a not a taght ln a n th ca of a ln ouc n al ac xctaton. Th llutaton n Fgu.5 th only ca wh th dtanc to th ouc and th obvaton fom th dg nth fa no clo. 34