High-frequency Incremental Techniques for Scattering and Diffraction. ALBERTO TOCCAFONDI Dept. of Information Engineering University of Siena - Italy

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1 High-frqucy Icrmtal Tchiqus for Scattrig ad Diffractio ALBERTO TOCCAFONDI Dpt. of Iformatio Egirig Uivrsity of Sia - Italy Sia, Fbruary 23, 2005

2 SUMMARY Backgroud Basic trmiology GTD vrsus PTD Basic cocpts of Icrmtal Thoris Ovrwiv of Icrmtal Thoris Currt basd mthods Fild basd mthods Icrmtal Thory of Diffractio (ITD) Localizatio procss Wdg-shapd cofiguratios Cylidrical cofiguratio Doubl dg diffractio

3 HF SCATTERING FROM LARGE OBJECTS A fficit ad accurat dscriptio of scattrig phoma at high frqucy is of itrsts i may applicatios. dsig ad aalysis of atas (.g., rflctors) prdictio of Radar Cross Sctio propagatio i complx viromts (.g., urba propagatio) ata istallatio ad itr-ata couplig (.g., o ships, aircrafts, spac platforms ) Rx

4 HF EDGE DIFFRACTION TECHNIQUES Th most importat thory that has domiatd this scario i past fw dcads is: Gomtrical Thory of Diffractio (GTD) i its Uiform vrsio (UTD, UAT) Othr importat thory is: Physical Thory of Diffractio (PTD*) i its origial vrsio Both UTD ad PTD ar ray-optical thoris: at high frqucy, th scattrig from a complx objct may b obtaid as a suprpositio of rays maatig from isolatd flash poits. * P. Ya Ufimtsv Mthod of dg wavs i th physical thory of diffractio Sovit Radio Publicatio Hous, Eglish Vrsio: Air Forc Forig Tchology Divisio, FTD-HC , Spt (ormal icidc).

5 GEOMETRICAL THEORY OF DIFFRACTION Th xact fild is th sum of a gomtrical optics fild (GO) ad a diffractd fild. Th GO fild accouts for dirct, ad rflctd ray-filds. go diff E= E + E RSB Th spac surroudig th objct is dividd ito rgios sparatd by th rflctio ad shadow boudaris. Th diffractd fild is th cotributio which must b addd to th GO to obtai th xact fild. ISB Th GTD fild provids a asymptotic approximatio of th diffractd fild. E diff E gtd

6 PHYSICAL THEORY OF DIFFRACTION Ufimtsv s basic approach: First, w will ivstigat th rigorous solutio of this problm (caoical). Th w will fid its solutio i th physical optics approach. Th diffrc of ths solutios dtrmis th fild cratd by th o uiform part of th currt (frig wav currt). Th scattrd fild is th sum of a physical optics fild (PO) ad a frig wav fild. Th PO fild ariss from th iducd surfac currts o th po objct. J Th FW fild ariss from th o uiform surfac currts o u th objcts. J i po fw E= E + E + E J po u J

7 PHYSICAL THEORY OF DIFFRACTION Th NU currts must b addd to th PO currts to obtai th total currts, that radiats th scattrd fild. po u J = J + J Th PTD fild provids a asymptotic approximatio of th frig wav fild. E fw E ptd O major advatag of th PO approach is that both th icidt ad th PO filds ar cotiuous across th RSB ad ISB. As a cosquc, also th FW fild ad its asymptotic approximatio (PTD fild) ar cotiuous across th sam SB. J po J fw

8 GTD VERSUS PTD Th PO fild is dividd ito a PO surfac cotributio ad a PO po diffractd fild. E d E = E + E po po po s d Th PO surfac cotributio is th ladig trm of th asymptotic valuatio of th PO surfac radiatio itgral for k. po E s J po E po s = po Es = E i E r po E s = 0 As a cosquc, th sum of th frig wav ad th PO diffractd fild is qual to th diffractd fild. po fw diff Ed + E = E E = E E fw diff po d

9 GTD VERSUS PTD Th PO diffractd fild is asymptotically approximatd by th PO dgdiffractd cotributio po. E d E po d E Th PO dg-diffractd cotributio is th ladig trm of th asymptotic valuatio of th PO radiatio itgral for k. Th diffrc btw th GTD fild ad th PO dg-diffractd fild is th PTD fild. po d E = E E fw diff po d E = E E ptd gtd po d At high frqucy, GO ad PO prdictios may b augmtd by GTD ad PTD fild rspctivly. E go E + gtd E E + E + E i po ptd

10 RAY METHODS AND CANONICAL PROBLEMS Both GTD ad PTD ar ray-optical mthods: upo a idtificatio of statioary surfac ad dg poits (ray tracig), th propr ray-fild is associatd to ach ray. Th locality pricipl is sstial: wh a radiatig objct is larg i trms of a wavlgth th scattrig ad diffractio is foud to b a local phomo,i.., it dpds oly o th atur of th boudary surfac ad th icidt fild i th immdiat ighborhood of th flash poits. S P Th actual scattrr is substitutd by local caoical cofiguratios at th poits of rflctio ad diffractio. Q' L z

11 Obsrvatio poit is locatd at a caustic: all poits alog th dg, or o a fiit portio, ar statioary. LIMITATIONS Ray-Optical mthods ar ot applicabl wh isolatd statioary dg poits caot b idtifis. ar ar fild fild caustic caustic caustic caustic at at ifiity ifiity RCS of poligoal flat plat Obsrvatio poits is locatd outsid th cos of diffractio: o dg statioary poits xists.

12 INCREMENTAL THEORIES FOR EDGED BODIES Th origial ida is that a illumiatig fild udrgos a diffractio procss at ach icrmtal lmt of th dg (Youg, Maggi, Sommrfld). Th obsrvd fild pattr (x: diffractd fild) is thus causd by th itractio btw th suprpositio of lmtary cotributios distributd alog th actual dg, ad th GO fild. ( i i E, H ) Th mathmatical formulatio of this ida may b asymptotically xprssd as a li itgral alog th dg cotour. ê Q ' β sˆ' sˆ R = r r z r P diff diff E () r F ( r ) dl l scattrr β r dg cotour l rˆ ˆr x y

13 INCREMENTAL THEORIES FOR EDGED BODIES diff At high frqucy, ach icrmtal fild cotributio F ( r ) is stimatd from a local uiform, ifiit, caoical cofiguratio that mor appropriatly modls th actual discotiuity. diff diff E () r F ( r ) dl l S F diff c ( r ) Q' L z Th two prvailig tchiqus to driv icrmtal cotributios ar th Currt basd ad th Fild basd mthods.

14 CURRENT BASED METHODS Icrmtal filds cotributios ar dduc from th currts of local caoical problms. Icrmtal Lgth Diffractio Cofficits (ILDC): K. M. Mitzr, Northrop Corp., Tch. Rp., Apr R. A Shor ad A. D Yaghjia, IEEE AP, Ja A. D. Yaghjia... Equivalt Edg Currts (EEC): A. Michali, IEEE AP, Mar A. Michali, IEEE AP, July,1986. Equivalt Edg Wav (EEW): D. I. Bouturi ad P Ya. Ufimtsv, Sov. Phys. Acoust. Jul.-Aug P. Ya. Ufimtsv, Elctromagtics, Apr.-Ju 1991.

15 Icrmtal filds cotributios ar dduc from th fild of local caoical problms. Icrmtal Diffractd Fild: FIELD BASED METHODS A. Rubiowicz, Acta Phisica Poloica, Equivalt Currt Mthod (ECM): R. F. Millar, Proc. IEE, Mar C. E. Rya ad L. Ptrs, Jr., IEEE AP, May E.F. Kott ad T. B. A. Sior, IEEE AP, Spt Icrmtal Thory of Diffractio (ITD): R. Tibrio, S. Maci, IEEE AP, May R. Tibrio, S. Maci, ad A. Toccafodi IEEE AP,

16 EQUIVALENT CURRENT METHOD diff Th icrmtal fild Fc ( r) attributd to th dg lmt at r may b thought as gratd by travlig-wav lctric I ad magtic M li currt, locatd at r ad dirctd alog th dg tagt ê. F diff c ê I M Q r sˆ' sˆ jk R jk ( r ) = [ sˆ ( sˆ ˆ) ζ I ( Q') + ( sˆ ˆ) M ( Q')] 4π R R = r r ' Th costat propagatio factor is k cos β. It assumd that th valu of I ad M dpd liarly o th icidt lctric ad magtic fild.

17 EQUIVALENT CURRENT METHOD Each travlig-wav currt radiats a coical wav jπ /4 I kζ ' ˆ E ( P) I( Q')siβ 8π k jk R jπ /4 M k ' ˆ H ( P) M( Q')siβ ζ 8πk R jk R R ê I M Q r sˆ' sˆ R = r r If R λ ad R ρ, th compots of th GTD dg-diffractd fild at P by a local caoical cofiguratio at Q, ca b approximatd as: ˆ E P ˆ E Q D d i ' ' ( ) [ ( ')] s(,, ) ˆ H P ˆ H Q D φ φ β φ φ β d i ' ' ( ) [ ( ')] h(,, ) jk R R jk R R

18 EQUIVALENT CURRENT METHOD Comparig th abov xprssios outsid th shadow boudary trasitio rgio: M sˆ' jπ /4 i 8 π [ ˆ E ( Q')] ' ' I( Q') = D (,, ) ' s φ φ β sˆ R = r r ζ k si β ê Q jπ /4 i 8 [ ˆ r ζ π H ( Q')] ' ' I M( Q') = D (,, ) ' h φ φ β k si β Th currt ar quivalt sic it dpds o th agls of icidc ad obsrvatio at th poit of diffractio. Ulik GTD, th filds radiatd by EC rmai valid v withi th caustic rgios of dg-diffractd rays. Ths currts ar dfid oly for obsrvatio poits lyig o th Kllr ' co of half-agl. β A huristic xtsio to iclud poits outsid th Kllr co hav b suggstd (Kott ad Sior). ˆ I ( Q') (,,, ) jπ /4 i 8 π [ E ( Q')] ' ' = D ' s φ φ β β ζ k si βsi β ˆ M ( Q') D (,,, ) jπ /4 i ζ 8 π [ H ( Q')] ' ' = ' h φ φ β β k si βsi β

19 EQUIVALENT EDGE CURRENT Th diffractd fild by a local caoical cofiguratio (pc) locally tagt at Q is (half-pla): y r jk R diff ˆ ˆ Ec ( P) jkζ R [ R J( rs)] ds x 4 π R R= r r S c s S c total iducd lctric currt dsity Th surfac radiatio itgral is approximatd i far fild as: Q ê r s ( ) Jr s z y Q ê û x r ( ) Jr s E z diff c icrmtal strip ( ) ( ) ( ) jkr jk ( rˆˆ u ) u P jkζ uˆ ˆ rˆ rˆ s du dz 4π r J r 0 Th icrmtal diffractd fild associatd to Q is th d-poit cotributio to th fild gratd by th surfac currt dsity o a ifiit xtdd icrmtal strip, startig at that dg poit.

20 EQUIVALENT EDGE CURRENT It is foud that th quivalt dg currt ar ˆ ˆ ˆ ˆ ˆ u ( ') ( ˆˆ) 2 ( ) ( ) jk r u u IQ = r r Jrs du rˆ ˆ 0 ˆ ˆ ˆ ˆ u ( ') ( ˆˆ) 2 ( ) jk r u u M Q = ζ r Jrs du rˆ ˆ 0 Wh th total currt is applid, th abov xprssios provid a asymptotic approximatio to th diffractd fild. Wh th PO surfac currt dsity is applid, th abov xprssios provid a asymptotic approximatio to th PO dgdiffractd fild. Wh th FW (o uiform) surfac currt dsity is applid, th abov xprssios provid a asymptotic approximatio to th PTD fild. gtd I I ( Q') = L{ Jr ( )} gtd M M ( Q') = L { Jr ( )} po I po I ( Q') = L{ J ( r )} d po M po M ( Q') = L { J ( r )} d ptd I u I ( Q') = L{ J ( r )} ptd M u M ( Q') = L { J ( r )} s s s s s s

21 EQUIVALENT EDGE CURRENT Origially th strip was chos ormal to th dg Michali (1984) drivd a st of EEC for th wdg diffractd fild. 1 1 I = E ( Q') ˆ D ( ') ˆ E + Q D jkζ H jk i ζ M M = H ( Q') ˆ DH jk i I i I H β ' Q uˆ = xˆ z Th crosspolar compot vaish at statioary poit β = β'. Valid for arbitrary dirctio of obsrvatio. Th icrmtal cotributios ar sigular o a half-co with tip at th dg, axis alog th icrmtal strip ad itrior tip agl θ = β '. p

22 INCREMENTAL LEGTH DIFFRACTION COEFFICIENTS Mitzr (1984) first cosidrd th frig wav surfac currt dsity alog th strips ormal to th dg. H providd xprssios i trms of Icrmtal Lght Diffractio Cofficits (ILDC). β ' uˆ = xˆ F ptd ptd i jkr F D ' D β ββ βφ ' E '( Q') β ( Q') = ptd = i F 0 D φ φφ ' Eφ' ( Q') 4π r Th ILDC s ar sigular o a half-co with tip at th dg, axis alog th icrmtal strip ad itrior tip agl θ = β '. It is foud (Kott, 1985) that if this cofficits is cast i th form of EEC, th two mthods ar quivalts (thy diffr by trms du to PO). p Q z

23 REDUCTION OF SINGULARITIES I gral, for arbitrary strip oritatio, th sigularitis li o a halfco alog th strip with itrior agl: [ ˆ ˆ ˆ] θ p = 2arccos (cos β' x+ si β' z) u To rduc th umbr of sigular dirctios, Michali (1986) ad Ufimtsv (1991) cosidrd th lmtary strip i th dirctio of th grazig diffractd ray. x β ' (cos β' xˆ+ si β' zˆ ) = uˆ θ = 0 p Q β ' β ' z Th rsultig icrmtal filds ar sigular oly alog th strip dirctio, wh th icidt fild is grazig.

24 PTD-EEC Michali drivd a st EEC by asymptotic valuatio of th frig wav currt radiatio itgral for th caoical wdg solutio. β ' x (Half-Pla) Q β ' β ' z µ = si βsi β'cosφ + cos β(cos β' cos β) si 2 β ' It xhibits a o-rciprocal bhavior w.r.t. th aspcts of icidc ad obsrvatio.

25 PTD-EEW Ufimtsv (1991) drivd a st PTD icrmtal diffractio cofficits basd o th cocpt of Elmtary Edg Wav. EEWs ar th wavs scattrd by th viciity of a ifiitsimal lmt of a curvd dg. Thy ar calculatd as th sphrical dg wavs radiatd by th ouiform currts flowig alog th lmtary strips High-frqucy asymptotics ar providd i th form: jkr 1 1 F ( P) = D ( Q', rˆ) ; F = rˆ F 2π r ζ ptd ptd ptd E H E β ' x i i D ( ', ˆ) ( ') ˆ ( ', ˆ) ( ') ˆ ( ', ˆ Q r = E Q M Q r + ζ H Q N Q r) Q β ' β ' It xhibits a o-rciprocal bhavior w.r.t. th aspcts of icidc ad obsrvatio. z

26 OTHER CONTRIBUTIONS Shor ad Yaghjia (1989) dvlopd a simpl substitutio mthod for obtaiig ILDC s dirctly from th 2-D scattrd far filds of PEC scattrr. It allows to obtai th ILDCs without ay itgratio, or diffrtiatio or spcific kowldg of th currts. Ado (1991) itroducd a st of GTD quivalt dg currts i ordr to compltly limiats th fals sigularitis, i which th oritatio of th strip dpds o th obsrvatio aspcts. ILDC s for aprtur (Michali 1995), strips ad slits (Yaghjia 1997) as wll as for cylidrical scattrr (Yaghjia 2001) hav also b proposd.

27 INCREMENTAL THEORY OF DIFFRACTION Explicit ITD formulatios hav b obtaid for: Pc local wdgs. Pc local thi circular cylidrs. Doubl local dgs Huristic ITD formulatios hav b obtaid for: Local dgs i impdac plaar surfacs. Local dgs i thi coatd ad dilctric pals.

28 Diffractd fild E diff ITD LOCALIZATION PROCESS diff ( P) = F ( Q') dl L S S th icrmtal fild ca b dducd from a appropriat local caoical problm Q' L Fid a covit rprstatio for th diffractd fild of th caoical problm. Q' L z E diff c + d ( P) = Fc ( z'') dz'' Assum that at high-frqucy F diff d ( Q') = F ( z'') c z'' = 0

29 A USEFUL REPRESENTATION For a objct, with a uiform cylidrical caoical cofiguratio alog th z-axis, by spctral sythsis: Rciprocity suggsts 3D 1 2 D '' z '' ( ') '' (, ') (, ', ) jk z z = ρρ ρ 2π G r r G k dk z S 3 D 1 '' '' z '' ( ') '' (, ') (, ) (, ') jk z z G r r = L U kρ ρ U kρ ρ dk z 2π Q' z It is foud that G 2D may b rprstd as th rsult of th applicatio of a liar oprator L[].

30 INCREMENTAL FIELD CONTRIBUTION 3 D 1 '' '' z '' ( ') '' (, ') (, ) (, ') jk z z G r r = L U kρ ρ U kρ ρ dk z 2π By Fourir aalysis, this spctral itgral rprstatio is itrprtd as th spatial covolutio product of two fuctios:, ' = ( '', ) ( ' '', ') '' 3D G r r L u z z ρ u z z ρ dz ( ) This abov spatial itgral rprstatio allows to dirctly dfi th local icrmtal fild cotributio F ( z'') = u( z, ρ) u( z', ρ') = F U( k, ρ) F U( k, ρ') d ' c z'' = 0 ρ ρ Ivrs Fourir trasform

31 WEDGE-SHAPED CONFIGURATIONS This "FT-Covolutio" procss is dirctly applicabl to th xact solutio for th wdg 3D 1 2 D '' z '' ( ') '' (, ') (, ', ) jk z z = ρρ ρ 2π G r r G k dk z Pla wav spctral rprstatio 2 D '' 1 ρ α' φ' '' ρ S β' Q' '' ρ β ρcos α jk ρ'cos α' G ( ρρ, '; k ) G ( αα, '; φφ, ') dαdα' 2π = ITD diffractd fild cotributio c α α' c 2 d k dfc ( Q ') = G ( αα, '; φφ, ') u( z, ρ) u( z ', ρ') dαdα' 2 2( j2 π ) c α c jk 1 ' cos ' ' jkρ ρ α jkz z ' ' uz ( ', ρ ') = k ρ dk z 2π φ P π z

32 Spctral rprstatio (EM cas) df Q G d d d d d jk[ rf ( αθ,, β) + r' f ( α', θ', β')] c ( l) = [( α α'), φφ, '] siθsi θ' θ θ' α α' C C C C α α' θ θ' ASYMPTOTIC ANALYSIS (I) f ( α, θ, β ) = si β siθ cosα + cos β cosθ = cosγ (') (') (') (') (') (') (') (') (') (') cosγ = 1 Th dsird statioary phas coditio is achivd at 1 cosθ cosβ 1 cos θ 'cos β ' cosα = cos α ' = siθ si β si θ 'si β ' Two mor coditio ar rquird

33 ASYMPTOTIC ANALYSIS (II) I th asymptotic aalysis, th structur of th spctrum (spctral sythsis) rquirs. υ = ( α α') idpdt of θθ, ' This is achiv d by θ = θ '( aalytic cotiuatio of th diffractio co). Th valu of υ = ( α α') at th statioary poit is: cos 1 cosβcos β' = si βsi β' ( α α' ) (Rubiowicz) Also si( α α') = j cos β cos β' si βsi β' 2 si θ = si βsi β '

34 ITD SOLUTION FOR WEDGE PROBLEM F d c d (,, ') 0 E '( Q') jkr F β D υφφ β ( Q') = d = F 0 (,, ') E '( Q') φ h υφφ D φ 2π r ( ) Dh, (,, υ φφ') = D[, υ Φ ], + D[, υ Φ + ] d ± ± ± D[, υ Φ ] = d [, υ Φ ] + d [, υ Φ ] [, υχ] = π χ si 1 2 υ π χ cos cos ITD statioary phas poit xplicitly satisfis rciprocity cosυ = φ' S β' Q' 1 cosβcos β' si β si β' β Φ ± φ π =φ φ' z wll-bhavd at ay aspct, icludig obsrvatio poits lyig o th axis of th local caoical wdg.

35 ITD vs. UTD Wh a statioary phas coditio has b wll stablishd, th itgratio of ITD icrmtal filds smoothly blds ito th fild prdictd by th UTD. UTD ITD

36 Far-zo pattr of a circular disc ITD vs. MOM, UTD

37 Far-zo pattr of a circular disc PTD vs. MOM, UTD

38 TWO ALTERNATIVE APPROACHES Th GO ray fild is wll-bhavd withi its domai of xistc, th total fild is E E F ggo = + l E = Ε + E E E ggo i po po go d d ( Q') dl Th GO ray fild rgim has ot b wll stablishd or wh a augmtatio of th PO fild is dsird E diff E utd E i po d po = E + E + F ( Q') F d ( Q') dl l It is mor covit from umrical poit of viw E E = E E fw ptd gtd po d Not: th frig cotributios ar diffrt from thos of PTD, oly th trmiology is th sam

39 ITD FRINGE FORMULATION (I) E E F = + s po f l ( Q') dl S P Wll-bhavd at ay aspct Tds to rmdy th typical orciprocity of PO. A localizd icrmtal, frig fild cotributio is obtaid as Q' l z F f ( Q') F f ( Q') = F d ( Q') F po ( Q') c c d Th icrmtal PO dg-diffractd fild F po d ( Q') is obtaid by applyig th ITD localizatio procss to a local caoical problm with PO currts.

40 PO EDGE-DIFFRACTED DIFFRACTED FIELD Th PO dg-diffractd fild is obtaid by itgratio of th icrmtal PO d-poit cotributios F po ( ) Q s E po d = F l po d ( Q') dl po F d ( Q') Thy ar foud by applyig th ITD localizatio procss to th local caoical problm of a pla half-lit ad half-shadowd

41 INCREMENTAL PO EDGE-DIFFRACTED DIFFRACTED FIELD F F E ( Q) po po po jkr po β D11 (,, υφφ') D12 (,, υφφ') β ' l d ( Q') = po = po F 0 22 (,, ') Eφ '( Ql ) φ D υφφ 2π r D D D po 11 po 22 po 12 d [, υ Ω= ] d1 (;, υφφ') = d[, υφ ] [, υφ ] + 1 d1 (;, υφφ') = d[, υφ ] + [, υφ ] (;, υφφ') = cosθ 1 siω 2cosυ + cosω Φ ± =φ φ' S φ' β' Q' y β φ P x cos θ = sg( z + z') 1 si βsi β' ITD statioary phas poit cosυ = 1 cosβcos β' si β si β' z

42 ITD FRINGE FORMULATION FOR A LOCAL HALF-PLANE F F E ( Q') f f f jkr f β D11(,, υφφ') D12(,, υφφ') β ' c ( Q') = f = f F 0 22(,, ') Eφ '( Q') φ D υφφ 2π r D D D d f f f + 11 υφφ = d2 υφ d2 υφ f f f + 22 υφφ = d2 υφ + d2 υφ f 12(;, υφφ') = cosθ f 2 (;, ') [, ] [, ] (;, ') [, ] [, ] [, υ Ω= ] cos( Ω 2) ( υ ) + ( Ω ) 1 2cos 2 si 2 Φ ± =φ φ' cos θ = sg( z + z') 1 si βsi β'

43 FRINGE ITD vs. PTD ITD frig augmtatio tds to rmdy th typical o-rciprocity of currt basd mthods

44 INCREMENTAL THEORY OF DIFFRACTION Circular Cylidrs

45 INCREMENTAL SCATTERED FIELD CONTRIBUTION Total fild (scalar formulatio) t i s Ψ ( P) = Ψ ( P) +Ψ ( P) Scattrd fild s Ψ = ψ ( P) ( Q ) dl l l Th icrmtal fild may b dducd from a appropriat local caoical problm tagt at Q l. To this purpos, w d to fid th covit fild rprstatio for th local caoical problm Ψ s c ( P) = ψ c( zdz ) Th, at high-frqucy, it is assumd that ψ( Ql) = ψ c( z) z= 0

46 LOCAL CANONICAL PROBLEM Caoical solutio (spctral sythsis) 1 Ψ = Ψ s j( φ φ ) s c(, rr ) (, rr ) 8π j = h 1, h ( ) ( r,r ), ( ) (2) ( ) (2) z ( ) jk z z = c kρa H kρ H kρ dkz 2π ψ ρ ρ G ( ρρ,, ) h, '' k ρ whr c ( k a) ( h, ) '' ρ = J H J H '' ( kρ a) (2) '' kρ a ( ) '' '( kρ a) (2) '' kρ a ' ( ) soft b.c. ( ) hard b.c. ( h) k = k ρ ( k z ) ad '' 2 '' 2

47 THE APPROPRIATE SPECTRUM FUNCTIONS (I) Soft boudary coditios J ( k a) H ( k a) 2 c ( k a) ( k a) ( k a) '' (1) '' '' ρ ρ '' '' ρ = = + = + σ (2) '' (2) '' ρ σ ρ H ( kρa) H ( kρa) for k ρ ral H H '' ( ka ρ ) σ = (1) (2) ( k a) ρ ( k a) ρ = H H ( k a) (1) '' ρ ( k a) (2) '' ρ j2 ϑ ( k a) ρ jϑ ( k a) ( ka ) ρ σ ρ = Thus, th typical -th trm i th itgrad of th xact solutio is h, '' 1 '' '' (2) '' (2) '' G ( ρρ,, kρ) = 1 + σ( kρa) σ( k a) ρ H ( kρρ) H ( kρρ ) 2

48 THE APPROPRIATE SPECTRUM FUNCTIONS (II) This lads to th usful rprstatio 1 G k = G k + G k 2 (1) (2) ( ρ, ρ', ρ) ( ρ, ρ, ρ) ( ρ, ρ, ρ) whr G ( ρ, ρ, k ) = H ( k ρ) H ( k ρ ) (1) (2) (2) ρ ρ ρ (2) (2) (2) G ( ρρ,, kρ) = σ ( kρa) H ( kρρ) σ ( kρa) H ( kρρ ) jϑ ( k a) ( ka ) ρ σ ( k ρ = ρ ral)

49 INCREMENTAL CONTRIBUTION Usig it i th ITD-FT covolutio procss lads to ψ ψ ψ (1) (2) ( Ql) = ( Ql) + ( Ql) by ivrs FT () i 1 () i () i j( kzz kz z ) = U ( ) ( ) 2 kρ U kρ dkzdkz (2 π ) ψ ρ ρ with 1 () i ( ) () (2) ( ) () 1 U ( k ρ ρ ) = H ( ) ; ( ) ( ) k ρ ρ i = σ ( ) 2 k ρ a (Rciprocity)

50 ELECTROMAGNETIC CASE I th lctromagtic cas for z- dirctd dipol illumiatio ad obsrvatio ( h, ) Ψ EM Vctor Pottials ( Ez, Hz) ( k kz ) ζ = Ψ 2 2 ( h, ) Thus, ( s s dez, ζ dhz ) is obtaid aftr rplacig i () i ψ U ( k ρ ρ ) () i ( ) () by k U ( k ) ( ) ( i) ( ) ( ) ρ ρ ρ

51 HIGH-FREQUENCY EXPRESSIONS (I) () i ψ I ordr to fid tractabl xprssios for first w us C C C C θ θ α α 1 H k = dα (2) jkρ ρcosα jα jπ 2 ( ρ ρ) π C α Nxt, collctig trms of th product of th two spctrum fuctios yilds th four fold spctral itgral rprstatio for ach s s de, ζ dh as ( z z ) [ ( θα, ) r ( θ, α )] h, h, j( α α ) jk r ( ρ ) ( ρ ) si si + σ kaσ ka θ θ dα dα dθ dθ I th sphrical spctral domai k = kcos θ, C ( j, π + j ) () () z η ( ) () () () () () () () () () r ( θ, α ) = r si β siθ cos α ( + / ) cos β cosθ Stadard ITD form.

52 HIGH-FREQUENCY EXPRESSIONS (II) Th icrmtal lctromagtic scattrd fild is, ( s s) π jγ s (, h) z z c jkr j de, ζdh = c ( β, β, a) siβsiβ r r ( h, ) ( β, β, a) = H J () ()(2) ( ka si βsi β ) ( ka si βsi β ) jkr ITD statioary poit 1 cosβcosβ θ = θ ; cos( α + α ) = si βsi β cosβcosβ si θ = si βsi β ; γ s = cos si βsi β wll-bhavd at ay icidc ad obsrvatio aspcts

53 THE DYADIC SCATTERING COEFFICIENT s i de ( P) s β D ( β, β, φ, φ' a) 0 Eβ = s s i de ( P) 0 h ( β, β, φ, φ' a) φ E D φ jkr 4π r D s (,, ) 4 j( φ φ ) jπ jγ s (, h) (,, ) ( h, ) β β a = j c β β a = c ( h, ) ( β, β, a) = H J (,) (2)( ) ( ka si βsi β ), ( ka si βsi β ) wll-bhavd at ay icidc ad obsrvatio aspct, icludig β = 0 β = 0 Th xpctd trasitioal bhavior of th fild is rcostructd by umrical itgratio of th icrmtal cotributios alog th curvd axis of th actual cylidrical cofiguratio.

54 NUMERICAL RESULTS Rsults from simulatios ar compard with a MoM solutio (Fko ) Straight uiform cylidr azimuthal sca x = 6 λ r = 8 λ L = 20 λ ka = π /5 ka = 2π /5 ka = 3π /5 CPU tims ka = π /5 ka = 2π /5 ka = 3π /5 ITD 1.1 s ( av = 3.6) 2.3 s ( av = 3.8) 4 s ( av = 4) MoM 15.3 s s s

55 NUMERICAL RESULTS Rsults from simulatios ar compard with a MoM solutio (Fko ) Straight taprd cylidr azimuthal sca x = 8 λ r = 10 λ L = 20 λ a mi = 0.05 λ a max = 0.4 λ CPU tims ITD MoM 1.2 s ( av = 5.5) 17 mi 6 s

56 NUMERICAL RESULTS Rsults from simulatios ar compard with a MoM solutio (Fko ) Circular torus lvatio sca a = 0.1 λ R T = 8 λ z = 5 λ r = 15 λ caustics CPU tims ITD MoM 2.2 s ( av = 3.7) 2 mi 25 s

57 INCREMENTAL THEORY OF DIFFRACTION Doubl Edg Diffractio

58 ITD FORMULATION FOR DOUBLE EDGE DIFFRACTION E dd 21 ( P)

59 ITD FORMULATION FOR DOUBLE EDGE DIFFRACTION E dd 21 ( P) ( ) ( ) (, ) E Q E Q F Q Q dl = = i d d l

60 ITD FORMULATION FOR DOUBLE EDGE DIFFRACTION E dd 21 ( P) ( ) ( ) (, ) E Q E Q F Q Q dl = = i d d l E 1 d ( ) F (, ) P = PQ dl d 2 l

61 ITD FORMULATION FOR DOUBLE EDGE DIFFRACTION E dd 21 ( P) E E d ( ) = F (, ) P PQ dl d 1 l d ( ) = F (, ) P PQ dl d 2 l E dd ( P) F (,, ) = dd 21 l l P Q Q dl dl

62 ITD FORMULATION FOR DOUBLE EDGE DIFFRACTION dd ( ) = F ( ) E P PQ,, Q dldl dd 21 l l (,, ) (,, ) ( υ, φ, φ ) ( γ ) ( υ, φ, φ ) ( ) ( ) dd i F P Q jks jks 2 Q1 E Q β β = D M 12 D dd i Fφ P Q 2 2 Q 1 Eφ Q πs 2πs 12 2 i d d Eβ E β cosγ si γ E β ( γ ) = M 12 = i d d E E si γ12 cosγ φ 12 E 2 φ 1 φ1