Development and Application of the Finite Difference Time Domain (FDTD) Method

Dvlopmnt and Applcaton of th Fnt Dffrnc Tm Doman (FDTD) Mthod by W Fan Submttd n partal fulflmnt of th rqurmnts for th dgr of Doctor of Phlosophy at Dalhous Unvrsty Halfax, Nova Scota Novmbr, 017 Copyrght by W Fan, 017

I ddcat ths thss to my dar wf X Zhong for all hr lov, support, and always sharng hr valuabl thoughts wth m.

Tabl of Contnts Lst of Tabls... v Lst of Fgurs... v Abstract... x Lst of Abbrvatons Usd... x Acknowldgmnts... x CHAPTER 1 INTRODUCTION... 1 1.1 Prfac... 1 1. Rsarch Background... 1 1.3 Objctvs... 3 1.4 Contrbutons of Ths Thss... 4 1.5 Organzaton of Ths Thss... 6 CHAPTER THE STABILITY OF THE FDTD METHOD AND THE UNCONDITIONALLY-STABLE WAVE-EQUATION BASED METHOD... 8.1 Introducton... 8. Matrx Formulaton Of Th FDTD mthod And Stablty Analyss... 9..1 Th Matrx Formulaton of Th FDTD mthod... 9.. Egn-Analyss of th FDTD Mthod... 11..3 Stablty Analyss of Th FDTD Mthod... 1.3 An Excpton: Can th CFL Condton Always Guarants th Stablty?... 18.3.1 Thortcal Analyss... 18.3. Vrfcaton of Th Prdctd Phnomnon... 0.3.3 Dscussons and Summars... 5.4 Th Uncondtonally Stabl Wav-Equaton Basd FDTD Mthod... 6.4.1 Th Proposd Uncondtonally Stabl Mthod... 6.4. Numrcal Exprmnts... 8.5 Summary... 31 CHAPTER 3 ANALYTIC SOLUTION OF THE FDTD METHOD... 3 3.1 Introducton... 3 3. Analytc Form Of th FDTD Soluton... 3

3.3 Th Hard Sourc Implmntaton... 35 3.4 Th Explct Analytcal FDTD Soluton In a Lossy Mdum... 37 3.5 Summary Of Analytc Soluton Of Th FDTD Mthod... 40 3.6 Numrcal Exprmnts... 4 3.6.1 Numrcal Exprmnt I: Analytc FDTD soluton of th H-shap Cavty..4 3.6. Numrcal Exprmnt II: Smulaton wth Hard Sourc n Lossy Mdum...44 3.6.3 Numrcal Exprmnt III: Smulaton of th Dlctrc Rod Structur..45 3.7 Tratmnt of Absorbng Boundary Condtons (ABC)... 48 3.8 Summary... 50 CHAPTER 4 DEVELOPMENT OF EFFECTIVE TIME REVERSAL METHOD..5 4.1 Introducton... 5 4. Th Locaton Condton for Rconstructon of Multpl Sourcs Usng Tm Rvrsal Mthod... 55 4..1 Thortc Analyss of Th Convntonal Tm Rvrsal Mthod.. 55 4.. Th Condton for Rconstructon of Multpl Sourcs... 58 4..3 Vrfcaton of th Condton... 60 4.3 Applcaton of Th Condton For Rconstructon Of Multpl Sourcs... 61 4.3.1 Th Proposd Mthod to Fnd th Sourc Locatons... 6 4.3. Numrcal Exprmnt wth th Proposd RLS Mthod... 63 4.4 Sourc Rconstructon Wth Ralstc Band-Lmtd Frquncy Doman Sgnals.64 4.4.1 Rconstructon of Flds from Band-Lmtd Frquncy Doman Rsponss or Masurmnts... 65 4.4. Th Condton for Sourc Rconstructon from Band-Lmtd Fld Rsponss... 69 4.4.3 Th RLS Mthod wth th Band-lmtd Fld Rsponss... 70 4.5 Summary... 71 CHAPTER 5 CONCLUSION... 73 5.1 Concludng Rmarks... 73 5. Rcommndaton For Th Futur Work... 74 BIBLIOGRAPHY... 76 v

APPENDIX I Drvaton of th Equvalnc btwn th CFL Condton and Maxmum Egnvalu of th FDTD Systm Matrx... 83 APPENDIX II Stablty Condton of th FDTD Mthod n Lossy Cass... 85 APPENDIX III Copyrght Prmsson... 89 v

Lst of Tabls Tabl 1 Comparson of th mthods for rsonant frquncy of dlctrc rsonator 4 v

Lst of Fgurs Fgur.1 Th four possbl locaton scnaros of pol z1 and z..14 Fgur. Largst magntud of th two pols z1 and z vrsus th corrspondng gn valu.17 Fgur.3 Angular frquncy θ1 of th gnmod vrsus th gn valu λ 17 Fgur.4. Th squar cavty wth PEC boundary, TM wav xctd..1 Fgur.5 Th tm-doman lctrc fld rcordd at th cntr of th cavty for 0000 smulaton tm stps Fgur.6 Maxmum tm-doman fld valu obtand wthn th 50000 stps of th smulatons usng monochromatc sourcs of dffrnt frquncs. Black dots ar th rsults for cas (1) (th DC sourc), blu dots for cas () (a random frquncy btwn 0 and fs/), grn dots ar for cas (3) (a natural rsonant frquncy of th cavty), rd crcls for cas (4) (th prdctd unstabl frquncs calculatd by (.6))....4 Fgur.7 Maxmum fld valus obtand wth monochromatc sourcs whos frquncs ar nar an unstabl frquncy.5 Fgur.8 Structur of th H typ mtal cavty...9 Fgur.9 Frquncy rsponss obtand wth th convntonal FDTD and th proposd mthod.30 Fgur.10 Elctrc fld dstrbuton obtand wth th convntonal FDTD smulaton (lft) and proposd mthod (rght)...31 Fgur 3.1 Structur of H typ mtal cavty: (a) th thr-dmnsonal vw, (b) th crosssctonal vw...43 Fgur 3. Fld dstrbuton rcordd at th 00,000th tm stp of proposd mthod (lft) and th convntonal FDTD mthod (rght) 43 Fgur 3.3 Rsults (rcordd at th obsrvaton pont) obtand wth th proposd and th convntonal FDTD mthods. (a) shows th fld valus obtand by th two mthods at obsrvaton nod and (b) shows thr rlatv rrors for a 10 4 stps smulaton. Th rrors ar blow 10-11.44 v

Fgur 3.4 Rsults (rcordd at th obsrvaton pont) of fld valus obtand wth th proposd and th convntonal FDTD mthods for th lossy-mdum and hard sourc cas......45 Fgur 3.5 Gomtry of th dlctrc rod rsonator n rctangular cavty 46 Fgur 3.6 PML layrs surrounds th computatonal doman; th crcls rprsnt th ntrfacng nods.....49 Fgur 4.1 A typcal tm rvrsal procss: (a) forward flds xctd by a pont sourcs; (b) wav propagaton wthn th cavty and rcordd at th output nods; (c) r-njcton of th rcordd flds that ar tm-rvrsd at th output nods; (d) r-focusng of th flds at th orgnal sourc nods at th nd of th backward propagaton.....54 Fgur 4. Two cass n whch th rconstructon of sourcs wth th tm rvrsal mthod s unsuccssful. (a) Th two orgnal sourc ampltuds (wth rd dots) ar qual. A pak s found at a sourc-fr locaton. (b)th two orgnal sourc ampltuds ar 1 and 0.3 (wth rd dots). Th sourc of ampltud 0.3 cannot b dntfd unambguously...58 Fgur 4.3 Dtrmnant of matrx (4.13) for 10000 groups of two nods. Among thm only th dtrmnant of on group has a vry small valu clos to zro. Ths group turns out to b th corrct par of sourc nod. 61 Fgur 4.4 Rsult of sourc rconstructons wth th proposd mthod. Th dots n th lft rgon of th problm doman ndcat th ght output nods. Th sourcs ar shown as th paks. (a) shows th orgnal sourc locatons and thr ampltuds. (b) prsnts th rsults obtand wth th RLS mthod 64 Fgur 4.5 Craton of th band-lmtd frquncy doman flds. Th top two fgurs ar th magntud and phas of th frquncy-doman flds rcordd n a full spctrum. Th bottom two fgurs show th band-lmtd spctrum that rsults from th rmoval of th contnts outsd th prslctd frquncy band of [k l, k h ]... 66 Fgur 4.6 Comparson btwn orgnal and th rconstructd tm rsponss. Th two fgurs on th lft ar th tm-doman sgnals and th fgurs on th rght ar th frquncy doman sgnals. Th top two fgurs show th orgnal rsponss, and th bottom two fgurs show th rsponss rconstructd wth (4.18)... 67 Fgur 4.7 Sourcs rconstructd from band-lmtd rsponss, yldng wll-dfnd paks at th orgnal sourc nods..67 v

Abstract Tm-doman numrcal mthods ar wdly appld n modrn ngnrng problms. In modlng lctromagntc structur problms, fnt-dffrnc tm-doman (FDTD) mthod s on of th most wll-known and wdly adoptd mthods du to ts algorthmc smplcty and flxblty. Th major constrant of th FDTD mthod s, n ts tratv soluton procss, that th tm stp s rstrctd by th Courant-Frdrchs-Lwy (CFL) condton. Smply to say, th fnr th spatal dscrtzaton (oftn rqurd by accuracy), th smallr tm stp that can b usd. As a rsult, th computatonal spd and ffcncy ar lmtd. In th frst half of ths thss, w analyz th FDTD mthod, rvw ts nstablty and prsnt ts gn-mod dcomposton. Basd on th fndng, w thn drvd th analytc soluton of th FDTD mthod, prsntng an altrnatv non-tratv tm-doman approach for lctromagntc problms. In th scond half of th thss, w focus on an mportant applcaton of th FDTD mthod, th computatonal tm rvrsal (TR) tchnqu, whch s an algorthm appld n nvrs sourc problms such as sourc rconstructon. Th algorthm s thoroughly nvstgatd n thory, a nw condton s prsntd for prcs sourc rconstructons, and a mathmatcal modl s dvlopd to rformulat th tm-rvrsal procss n an optmzaton mannr. Fnally, band-lmtd flds or sgnals ar ncorporatd nto th modl to mak th tm rvrsal mthod practcal. Intal numrcal xprmnts ar conductd, and th rsults dmonstrat th ffctvnss and potntals of th proposd tm-rvrsal mthod n sourc rconstructons and mcrowav structur synthss n th futur. x

Lst of Abbrvatons Usd FDTD CFL TLM MOR TR RLS PML ABC Fnt Dffrnc Tm Doman Courant-Frdrchs-Lwy Transmsson Ln Matrx Modal Ordr Rducton Tm Rvrsal Rgularzd Last Squar Prfctly Matchd Layr Absorbng Boundary Condton x

Acknowldgmnts I would lk to thank my suprvsor, Dr. Zhzhang Chn for hs constant gudanc, ncouragmnt, support and patnc. Hs nsghtful advc s vry hlpful for m to buld th path of th rsarch. Hs humor maks thos lk plasant journys. Hs broad vson n th rsarch hlpd m look at th problms and ssus from a hghr dmnson. I am vry glad to b hs studnt and b nfluncd by hs way of thnkng. I gratly apprcat th support and advc from my othr commtt mmbrs, Dr. Srgy Ponomarnko, Dr. Wllam Phllps, and Dr. Puyan Mojab of th Unvrsty of Mantoba. I want to thank Dr. Shunchuan Yang, Dr. Xaoyan Zhang, Dr. Adong Yang, Dr. Fard Jolan, Dr. Coln O Flynn, Luyun Wang and Dachuan Sun, for all thr valuabl thoughts shard wth m n our numrous dscussons and daly chattng. I am also thankful to Wan Png, Jachng Guo, Dr. Zhmng Xu, and all othr collagus at Mcrowav and Wrlss Laboratory and I am glad to hav thm asd. My thanks also go to Ncol Smth, Dr. Jason Gu, and all th faculty and staff of Dpartmnt of ECE, for thr support, hlp and warm smls. Fnally, my thanks to my famly and my parnts, for always bng thr for m. x

CHAPTER 1 INTRODUCTION 1.1 PREFACE Ths thss manly focuss on nvstgatons on tm-doman numrcal mthods of computatonal lctromagntcs; t ncluds th dvlopmnt of th fnt-dffrnc tm-doman (FDTD) mthod, and ts applcaton n nvrs sourc problms. Ths chaptr s to ntroduc th rsarch background and rvw th stat-of-th-art of FDTD mthod, as wll as th motvatons and objctvs. Th contrbutons and th organzaton of th thss wll also b prsntd. 1. RESEARCH BACKGROUND Solutons of partal dffrntal quatons (PDEs) ar much dmandd for modrn ngnrng problms. For xampl, Dffuson quaton, Posson quaton, Laplac quaton, Maxwll quatons, and Schrödngr quaton, ar wll-known PDEs that ar wdly studd and usd n varous flds. Thr ar manly two ways to solv PDE, analytc approach and numrcal approach. Th analytc approach s xplct and accurat; hnc t s prfrrd whnvr t s applcabl. Howvr, thr ar vry rar cass n whch th analytcal solutons ar obtanabl. Usually an analytcal soluton s avalabl only whn th shaps, boundars of a problm structur or doman ar rgular and smpl. For most practcal stuatons, th ffort ndd to rach th analytc soluton s too prohbtv. In short, th analytc approach s lmtd and unachvabl n solvng practcal problms. Th scond way s to us numrcal mthods. In a numrcal mthod, th frst stp s to dscrtz th problm n spac and tm, transfr th problm from a contnuous modl nto a dscrt on. By applyng numrcal approxmatons of th dffrntal oprators, th PDEs s transformd nto dscrt quatons n th dscrt doman. Thn th solutons to th dscrt quatons ar sought whch ar xpctd to th approxmatons to th analytc solutons to th orgnal contnuous PEDs. Fnt-dffrnc tm-doman (FDTD) mthod [1] s on of th most wll-known numrcal mthods to solv PDEs nvolvng tm. For lctromagntc problms, t 1

combns th fnt dffrnc algorthm wth th spcfc lattc/grd pattrn ntroducd by Y n 1966. It was furthr dvlopd by Taflov who namd t n 1980 []. Snc thn, applcatons and rsarch ntrsts n ths mthod hav xplodd as can b sn n mllons of publcatons drctly rlatd to FDTD [3-6] so far. In th past four dcads, du to FDTD s smplcty, robustnss, and xplctnss, th rang of applcatons of FDTD has bn xpandd ovr a vry wd spctrum[6]. Howvr, for th xplct tm-doman mthod lk FDTD mthod, th maxmum tm stp s rstrctd by th Courant-Frdrchs-Lwy (CFL) condton [7], whch s rlatd to th smallst cll sz n a dscrtzd spatal doman. It s th major constrant of FDTD n ts tratv soluton procss whr th choc of tm stp s rstrctd. Th fnr th spatal dscrtzaton s (oftn rqurd by accuracy), th smallr tm stp that has to b usd. Hnc, whn mult-scal structurs or gomtry fn dvcs ar to b modld, dscrtzd grd or cll sz must b small to captur th dtal of gomtrs; so dos th tm stp. Th rstrcton mposs a lmt on computatonal spd and ffcncy; somtms t can mak th computatonal tm prohbtvly long. If th CFL condton s not satsfd, solutons of th FDTD mthod (as wll as th othr xplct tm-doman mthods) wll bcom unstabl and dvrg as th tm marchs, ladng to numrcal xplosv rsults. To addrss th stablty ssu, n rcnt yars, a numbr of uncondtonally stabl schms hav bn dvlopd to rmov th dpndnc of tm stp on spac stp n th FDTD mthod. For xampl, th altrnatvly-drcton-mplct (ADI) FDTD mthod was dvlopd [8-1]. Wth th mplctnss, th sz of th tm stp can b mad ndpndnt of cll szs. Thrfor, an arbtrarly larg tm stp can b usd wth th only constrant on th tm stp bng modlng accuracy. Othr mplct uncondtonally stabl FDTD mthods hav bn dvlopd too n th past dcads; thy nclud th locally-on-dmnsonal (LOD) FDTD mthods [13-15], th mult-stag splt FDTD (SFDTD) [16, 17] and Crank Ncolson (CN) FDTD mthods [18-1]. All of ths mthods ar mplct mthods that mathmatcally rqur a systm matrx soluton. To lss xtnt of th rmoval of th stablty condton, othr mthods hav also bn proposd to rlax th stablty condton and nlarg th tm stps. A spatal fltrng tchnqu has bn dvlopd to xtnd th stablty lmt of th xplct FDTD

mthod [-5] n lctromagntcs. Th tm stp s succssfully xtndd byond th CFL lmt, whl th fltrng procss rqurs ntnsv computaton. Th caus of th nstablty s nvstgatd wth th gn matrx thory and an uncondtonal FDTD schm s ntroducd by modfyng th convntonal FDTD lap-frog tratv procss [6, 6]. In ths mthods, th tratv march-on-tm procdurs ar rqurd and th CFL condtons ar nhrntly prsnt n th FDTD modl. On th othr hand, as on of th mportant applcatons of th FDTD mthod or tm-doman numrcal mthods n gnral, computatonal tm-rvrsal (TR) tchnqus hav bn studd for solutons of nvrs sourc problms n acoustcs, lctromagntcs and othr aras [7-31]. To undrstand TR procss, consdr a two-stp procdur: frst sgnals (or flds) mttd by sourcs ar propagatd, rcordd at pr-slctd output locatons n a soluton doman, whch can b namd as forward propagaton; thn n a backward propagaton, th rcordd sgnals ar rvrsd n tm and r-njctd at th output locatons nto th problm doman. Ths procdur s th tm rvrsal mthod. As long as th soluton doman s rcprocal, ths r-njctd sgnals (or flds) xprnc th sam propagaton condtons (.g. multpath, rflctons, rfractons) as th forward propagatng sgnals, rsultng n fld focusng or paks around th orgnal sourc locatons. Robust and smpl to mplmnt, th TR mthods hav drawn much attnton of th nvrs problm communty. In computatonal lctromagntcs, TR mthods hav bn formulatd and mplmntd usng th Transmsson Ln Matrx (TLM) mthod [3-34], and th Fnt Dffrnc Tm Doman (FDTD) mthod [35]. In spt of th progrss of th rcnt yars n th ara, thr ar stll major challngs and ssus wth th TR tchnqus dvlopd so far for practcal applcatons. For xampl, th sourc locatng by pak dntfcaton may not work wll n crtan cass, spcally whn multpl sourcs xst; and th rlvant thory has not bn dvlopd. Also, th sgnals of full-spctrum ar mostly not avalabl for th nvrs opraton and th TR procss. All ths lmtatons motvat us for th furthr dvlopmnt of th TR tchnqus as on of th major applcatons of th FDTD mthod or tm doman numrcal mthods n gnral. 1.3 OBJECTIVES 3

Th objctv of th thss s to fnd nw paths and formulatons for th FDTD mthod and th applcatons n lght of th challngs and ssus dscrbd n th prvous scton. As th rsult of my work, ths thss achvs th followng spcfc objctvs of lmtd scop: 1) prform dtald analyss of th convntonal FDTD mthod and thorzs ts stablty problm from th aspct of a dscrt systm for a tm-doman mthod; ) ovrcom constrant of CFL lmt and dvlop a wav-quaton basd uncondtonally-stabl schm for th FDTD solutons; 3) drv th analytcal form of th FDTD soluton; 4) dvlop th thortcal modl for th tm rvrsal mthod; 5) drv a condton for prcs sourc rconstructons wthout fals rsults; 6) rformulat and dvlop th TR procss nto an optmzaton problm for ffctv sourc dntfcatons and rconstructons; and 7) propos an xtracton mthod for ncorporatng band-lmtd sgnals or rsponss nto th TR procss for practcal applcatons. 1.4 CONTRIBUTIONS OF THIS THESIS Ths thss prsnts th analyss of th ssntals of th FDTD mthod. By consdrng FDTD mthod as a dscrt systm wth nput and output, th stablty ssu of FDTD mthod s studd n a succnct mannr. Basd on th analyss, a pcular nw phnomnon s frst obsrvd and rportd: th CFL condton may not always guarant th stablty of FDTD mthod (and othr tm-doman mthods). Wth th stablty studs, th caus of numrcal nstablty s dntfd. By rmovng th unstabl mods contand n th FDTD solutons, an uncondtonally stabl wav-quaton FDTD mthod s dvlopd. Numrcal xprmnts ar conductd to vrfy th uncondtonal stablty. Analytc form of FDTD solutons s drvd for th frst tm. Soluton of a FDTD modl of a structur at any tm stp can thn b obtand drctly wthout th tratv procss lk that of th convntonal FDTD smulaton. Dtals of th analytcal FDTD 4

soluton n dffrnt scnaros of hard sourcs, lossy mdum, and non-unform dlctrc matrals ar dscussd and vrfd. Tratmnt of mor complx scnaro lk absorbng boundary condtons ar also dscussd and a pathway s provdd for futur studs. Th tm rvrsal mthod s an mportant applcaton of th fnt-dffrnc tmdoman mthod to solv nvrs sourc problms. In ths thss, dtald thortcal modls and analyss of tm rvrsal n both tm-doman and frquncy doman ar prsntd for th frst tm. Thy show that th convntonal TR focusng rls on pak dntfcaton somtms fals. To addrss ths, a nw mathmatcal condton for rconstructon of multpl sourcs ar dvlopd. To apply th condton as a drct sourc locaton solvr, a nw mthod, th rgularzd last squar tchnqu, s formulatd for th TR procss; t can prsnt th accurat dntfcaton of th numbrs, locatons and ampltuds of th sourcs. Sgnals usd n th TR mthod rqurs full-spctrum nformaton so far. Howvr, n practc, sgnal nformaton s oftn band-lmtd. To ovrcom th problm, n ths thss, an xtracton mthod that allows th ncorporaton of th band-lmtd sgnals or fld rsponss nto th TR mthod s dvlopd. It maks th TR mthod practcal and usful. Th abov work, as a part of ths thss, has bn publshd n [36-40] and anothr two paprs ar currntly accptd for publcaton. As th frst author n th paprs lstd blow, ths PhD canddat s rsponsbl of th thortc dvlopmnts, numrcal vrfcatons, organzng and wrtng of ths paprs. [1] W. Fan, Z. D. Chn, and S. Yang, A wav quaton basd uncondtonally stabl xplct FDTD mthod, n Proc. 015 IEEE MTT-S Intrnatonal Confrnc on Numrcal Elctromagntc and Multphyscs Modlng and Optmzaton (NEMO), 015. [] W. Fan, Z. Z. Chn, and S. C. Yang, "On th Analytcal Soluton of th FDTD Mthod," IEEE Transactons on Mcrowav Thory and Tchnqus, vol. 64, pp. 3370-3379, Nov 016. [3] W. Fan, Z. Chn and W. J. R. Hofr, "Sourc Rconstructon From Wdband and Band-Lmtd Rsponss by FDTD Tm Rvrsal and Rgularzd Last Squars," IEEE Transactons on Mcrowav Thory and Tchnqus, vol. 65, no. 1, pp. 4785-4793, Dc. 017. 5

[4] W. Fan and Z. Chn, "A nw tm rvrsal mthod wth xtndd sourc locatng capablty," In Proc. 017 IEEE MTT-S Intrnatonal Mcrowav Symposum (IMS), Honololu, HI, 017, pp. 704-706. [5] W. Fan and Z. Chn, "A condton for multpl sourc rconstructons wth th tmrvrsal mthods," In Proc. 016 IEEE MTT-S Intrnatonal Mcrowav Symposum (IMS), San Francsco, CA, 016, pp. 1-4. 1.5 ORGANIZATION OF THIS THESIS Chaptr prsnts th dtald analyss of th FDTD mthod. By applyng th matrx thory and th z-transform, t rvals th root caus of th nstablty of th FDTD mthod. Basd on th analyss, t s dscovrd that th CFL condton dos not always nsur th stablty of Th FDTD mthod as provd by numrcal xprmnts. Furthrmor, an uncondtonally stabl wav-quaton basd tratv mthod for th FDTD solutons s proposd. Numrcal xprmnts vrfy th uncondtonal stablty and ffctvnss of th proposd mthod. Chaptr 3 dvlops th analytc soluton of th FDTD mthod. For th frst tm, th analytc form of FDTD soluton s xplctly rprsntd, whch maks possbl to drctly obtan FDTD smulaton rsults at any tm gvn wthout th convntonal tratv march-on-tm procss. Th analytc soluton s mathmatcally drvd and numrcally vrfd. Bsds, notabl cass such as hard sourc, lossy mdum and nonunform dlctrc matrals ar consdrd and varants of th analytc solutons ar also prsntd. Futur drctons of handlng of absorbng boundary condtons ar also provdd. Chaptr 4 ddcats to th applcaton of tm-doman mthods to th tm rvrsal (TR) mthod. Th TR tchnqu s appld n solvng th nvrs sourc problms such as sourc rconstructons. Howvr, th convntonal tm rvrsal mthod s lmtd n practcal applcatons for possbl fals rsults. Dtald thortcal analyss of th tm rvrsal n both tmporal and frquncy doman ar prsntd for th frst tm. A nw condton s dvlopd for rconstructon of multpl sourcs wth 6

rmoval of th possblty of th fals rsults. A mathmatcal formulaton s thn dvlopd for th TR procss for prcs sourc rconstructons. To mak th TR procss practcal, an xtracton mthod s dvlopd that can xpand th band-lmtd sgnals or fld rsponss to th full-spctrum sgnals for ncorporatons nto th TR procss. Ths dvlopmnts lay th foundatons for makng th TR mthod practcal for sourc rconstructons and synthss n th nar futur. Chaptr 5 concluds th rsarch n th thss and prsnts th futur drctons. 7

CHAPTER THE STABILITY OF THE FDTD METHOD AND THE UNCONDITIONALLY-STABLE WAVE-EQUATION BASED METHOD.1 INTRODUCTION Many numrcal mthods hav bn dvlopd to solv Maxwll s quatons for lctromagntc structur problms. Among thm, th xplct tm-doman numrcal mthods such as th FDTD mthod [1, ], has attractd partcular ntrst for thr capablty of modlng transnt rsponss. Bsds, th xplct mthods ar normally fr of matrx solutons and thr solutons rflct physcal vnts n ts natural tmporal dmnson, provdng a wdband soluton wth a sngl run of smulaton. It also has advantags n modlng nonlnar phnomnon wthout much dffculty. Howvr, for th xplct tm-doman mthods, th tm stp sz s rstrctd by th Courant-Frdrchs-Lwy (CFL) condton [3, 7], whch placs an uppr lmt of th tm stp of ths xplct numrcal mthods; t hnc rstrcts computatonal spds and ffcncy. If a tm stp s chosn largr than th CFL lmt, th tm-doman solutons of ths mthods wll bcom unstabl and dvrgnt as thy march on tm. Th lmt dpnds on th szs of lmnts or grds whch dscrtzs solutons domans and proprts of th mda to b modld. Th smallr th lmnts (or fnr th numrcal grds) ar, th smallr th lmt s. As a rsult, th CFL condton may caus long, somtms prohbtvly long, smulaton tm du to th small tm stp that has to b takn. Extnsv rsarch fforts hav bn mad rcntly n crcumvntng th CFL condton by dvlopng mplct FDTD mthods [8, 9, 11, 41] or rlaxng or vn rmovng th CFL-causd nstablty [6, 4]. For xampl, spatal fltrng s usd to rmov unstabl solutons by fltrng out th hghr-ordr componnts n spatal doman [3-5, 4]. In ths chaptr, w wll show that th FDTD (and othr tm-doman mthods) s a dscrt systm wth sngularty pols n ts mpuls rspons; th CFL condton only nsurs that th poston of th pols wll not rsult n nstablty for th mpuls rspons 8

but not ncssarly th FDTD solutons at all th tms. By applyng th gn matrx analyss, th pols ar found to b rlatd to th gnvalus of th FDTD systm matrx; and th nstablty s causd by th gnvalus whos valus ar largr than 4. In addton, larg gnvalus and thr corrspondng gnmods ar found to corrsponds to hgh frquncy componnts of th FDTD solutons. By rmovng th hgh-frquncy gnmods that caus th nstablty, a wav-quaton basd FDTD soluton s dvlopd to solv lctrc and magntc flds xplctly wthout numrcal nstablty. Th dtals ar prsntd n th followng sctons.. MATRIX FORMULATION OF THE FDTD METHOD AND STABILITY ANALYSIS..1 Th Matrx Formulaton of Th FDTD mthod Consdr Maxwll s quatons n an sotropc and losslss mdum of prmttvty ε and prmablty μ: H Ñ E= -µ, t E Ñ H= + J, t (.1) n whch E rprsnts th lctrc fld, H rprsnts th magntc fld. J s th lctrcal currnt. Wth rplacmnts of th drvatvs wth thr cntral fnt dffrnc countrparts, quaton (.1) s dscrtzd n both spac and tm. Th rsults ar th march-on-n-tm formulatons of th FDTD mthod: 1 1 H[ n+ ] = H[ n- ]-DtDEE[ n], (.a) 1 Dt 1 E[ n+ 1] = E[ n] +D tdhh[ n+ ]- J [ n+ ]. (.b) whr D t s th dscrtzng tm stp, 1 [ n - ] H s a column vctor whos lmnts ar magntc fld at all th magntc fld grd ponts (or nods) at tm nstant 9

1 t = ( n- ) D t, E[ n] lctrc fld grd ponts at tm nstant t s a column vctor whos lmnts ar lctrc fld at all th = nd t, E rprsntng th fnt-dffrnc form of oprator D s th matrx of dmnson Nh N 1 µ - Ñ and H dmnson N Nh rprsntng th fnt-dffrnc form of oprator tm stp stppng ndx, numbr of magntc fld nods. D s th matrx of N s th numbr of th lctrc fld nods and 1 - Ñ. n s th N h s th Eq. (.) s th lap-frog form of th convntonal FDTD formulaton. As can b sn, th lctrc and magntc fld ar solvd n an altrnatng mannr half a stp apart. Eq. (.) can b furthr smplfd by substtutng (.b) nto (.a) and vc vrsa. Th rsults ar th dscrtzd wav quatons for lctrc or magntc flds, rspctvly: Dt æ 1 1 ö E[ n+ 1] + E[ n-1] - E[ n] +D t DHDEE[ n] =- ç J[ n+ ]-J[ n- ], (.3a) è ø 3 1 1 1 Dt 1 H[ n+ ] + H[ n- ]- H[ n+ ] +D t DEDHH[ n+ ] =- DEJ [ n+ ]. (.3b) Th numbrs of unknown quantts of (.3a) and (.3b) ar N and N h, rspctvly. Th abov two quatons hav th smlar form. Thrfor, thy hav th sam CFL condton, whch s also th CFL condton of th FDTD formulaton (.) (snc (.3) s drvd from (.)). W tak lctrc fld wav quaton (.3a) for our analyss. It can b smply rwrttn as E[ n+ 1] + E[ n-1] - E[ n] + ME[ n] = x [ n] (nput). (.4) whr M=Dt DHDE whch s th fnt-dffrnc form of oprator 1 Dt( µ ) - Ñ Ñ. [] n =- -1 D t( [ n+ 1/] - [ n-1/] ) x J J whch rprsnts th sourc trm. In th cas of known boundary condtons, t may also nclud boundary condton trms mposd. 10

.. Egn-Analyss of th FDTD Mthod Dnot th Matrx M as th systm matrx of Th FDTD mthod. It has a dmnson of N N, N s th numbr of th lctrc fld nods. It s a ral symmtrc matrx as wll as sm-postv dfnt [6, 6]. Thrfor, ts gnvalus ar non-ngatv ral numbrs and ts gnvctors ar lnarly ndpndnt of ach othr. As a rsult, N gnvctors of M form a complt soluton spac for th FDTD lctrc fld solutons. In othr words, any FDTD lctrc fld solutons can b rprsntd by a combnaton of ths gnvctors. Λ = dag[ l l... l... l ] Dnot 1 N as th gnvalu matrx of M wth ts dagonal lmnts bng th gnvalus, V = [V 1 V... V... V ] as th assocatd N gnvctor matrx of N N lmnts whos columns ar gnvctor V of N 1 lmnts that s assocatd wth gnvalul. W can thn xpand a FDTD fld soluton and ts sourc at th n-th tm stp n trms of th gnvctors as follows: E[] n = V a[] n (.5a) x[] n Vb[] n = (.5b) an [ ] [ a a... a... a ] T = and bn [ ] = [ b1, n b, n... b, n... bn, n] ar th T whr 1, n, n, n N, n xpanson coffcnt column vctors at th n-th tm stp for E [ n] and x[ n ]; lmnt a and n, n, b ar th tm-stp dpndnt xpanson coffcnts. Equaton (.5) ndcats that any FDTD solutons and thr sourcs can b xpandd n trms of th spatally nvarant gnmods wth tm-dpndnt xpanson coffcnts. Substtuton of (.5) nto (.4) lads to: Van [ + 1] + Van [ -1]- Van [ ] + MVan [ ] = V bn [ ]. (.6) 11

Snc MV= VΛ, Van [ + 1] + Van [ -1]- Van [ ] + VΛan [ ] = V bn [ ]. (.7) By lft-multplyng -1 V to both sds of (.7), w obtan an [ + 1] + an [ -1]- an [ ] + an [ ] = bn [ ] Λ. (.8) Snc Λ s a dagonal matrx, quaton (.8) can b fully dcomposd for vry lmnt a of vctor a[n]: n, a + a -( - l ) a = b, for n = 1,, 3,... (.9) n, + 1 n, -1 n, n, whr n rprsnts th n-th tm stp of th FDTD march on tm. From (.9), w can s that th soluton of th FDTD mthod can b dcomposd n trms of gn spatal gnmods wth xpanson coffcnts a. If xpanson n, coffcnt a dvrgs as n (or tm) ncrass, th corrspondng gnmod, as a part n, of th FDTD soluton, wll dvrg and mak th FDTD soluton dvrgnt and unstabl. Not that th gnvctor or gnmods ar nvarant wth tm, whl th gnvalus hav a proportonal rlatonshp wth th squar of th tm stp chosn. In th followng scton, w wll fnd how th gnvalus nflunc xpanson coffcnts and gnmods and hnc th stablty of th FDTD solutons...3 Stablty Analyss of Th FDTD Mthod Eq. (.9) can b consdrd as a srs of dscrt scalar sub-systms that can b solvd rcursvly. Such dscrt sub-systms can thn b analyzd ffctvly wth th Z-transform [43]. 1

For smplcty and wthout loss of gnralty, t s assumd that th ntal condtons ar a, a,0 a,1 0 = = = and thy ar appld hncforth. By applyng th - n unlatral Z-transformaton to both sds of (.9), th followng quatons ar obtand: za z z A z A z A z B z -1 [] + [] - [] + l [] = [], A[] z z H[] z = =, B z z z [] + 1 -(-l) (.10) whr Az [] = Za {, n} can b consdrd as th rspons or output functon, B[] z = Z{ b } can b consdrd as th sourc or nput functon. H [ z ] s thn th, n transfr functon (as usd n th crcut thory) or mpuls rspons (as rfrrd n communcatons thory) n th Z-doman; th lattr nam of mpuls rspons s usd hncforth n ths thss. Equaton (.10) has two pols whch mak ts dnomnator zro. Dnot ths pols as z 1 and z. Th tm-doman mpuls rspons s th nvrs Z-transform of (.10): hn Z H z -1 [ ] { ( )} =. (.11) Th two pols of dscrt systm dscrbd n (.9), z 1 and z, dtrmns whthr h [ ] n wll dvrg or not and hnc th stablty of th FDTD systm tslf. Fgur. shows th four scnaros whr z 1 and z ar locatd dffrntly and thy ar analyzd blow. 13

Fgur.1 Th four possbl locaton scnaros of pol z 1 and z Scnaro I (Fgur.a): z,1 z, 1 Equaton (.10) bcoms: = = (or l = 0) z H[] z = ( z -1). (.1) Th corrspondng mpuls rspons s: h [ ] 1 n = n- wth h[0] = h[1] = 0. (.13) It mans that th mpuls rspons of th FDTD dscrt systm s ncrasng lnarly wth tm n magntud and hnc s unstabl. Not that (.13) can b provn to b a soluton of (.9) by drctng substtutng t nto (.9) although (.13) s a dvrgnt soluton n tm. Scnaro II (Fgur.b): z,1 = z, =- 1( l =4) Equaton (.10) s thn: z H[] z = ( z + 1). (.14) Corrspondngly th mpuls rspons s 14

hn n-1 [ ] ( 1) ( n 1) = - - wth h[0] = h[1] = 0. (.15) It mans that th soluton of th FDTD dscrt systm s oscllatory but ncrasng n magntud lnarly wth tm and hnc s unstabl. Scnaro III (Fgur.c):,1 z > 1.0 and z, < 1.0 (or l > 4.0) Equaton (.10) has on pol z lyng outsd th unt crcl and anothr on,1 z, nsd th unt crcl. Th nvrs of (.10) prsnts th tm-doman mpuls rspons: wth h[0] = h[1] = 0, z,1 1 1 hn [ ] = ( z ) + ( z ) n-1 n-1,1, ( z,1 -z, ) ( z, -z,1) (.16) l - + l -4l l - + l -4l j =- = p (.17a) z, 1 l -- l -4l jp = = (.17b) z,1 Snc l - + l -4l z,1 = > 1.0, th mpuls rspons (.16) s dvrgnt and th FDTD mthod s unstabl. wth f s By carfully xamnng th tmporal frquncs of th pols, w hav: f,1 = p 1 fs. pdt = Dt = (.18) 1 = bng th tmporal samplng frquncy of th FDTD systm. In othr D t words, th pols that caus th nstablty carrs a frquncy that s on half of th FDTD samplng frquncy. Furthr mathmatcal analyss shows that l > 4.0 corrsponds to th cas whr th CFL condton s not satsfd (s Appndx I). 15

Scnaro IV (Fgur.d):,1, z = z = 1.0(or 0< < 4.0) Fg..1d shows th postons of th two pols, z 1 and z ; thy ar conjugat to ach othr and ar on th unt crcl. pols ar: Th mpuls rspons can stll b rprsntd by (.16) but th two conjugat l n whch th angls ar j ( - l) 4 1, ± j l -l z q 1, = = (.19a) 4l - l q 1 = arctan( ), - l q =-q 1. (.19b) Analyss n Appndx I shows that t corrsponds to th cas whr th CFL condton s satsfd. Mathmatcally, t mans that th FDTD systm s stabl snc z = z = 1.0.,1, To furthr xplor th rlatonshps btwn th pols and th gnvalus, Fgur. and Fgur.3 plot th rlatonshps of pols and corrspondng gnvalus, showng th pols magntuds(largst of th two) and angls as functon of th gnvalus. Th angl of th pol s quvalnt to th angular frquncy of th gnmod. 16

Fgur. Largst magntud of th two pols z 1 and z vrsus th corrspondng gn valu Fgur.3 Angular frquncy θ 1 of th gnmod vrsus th gn valu λ It can b sn thatl =4 s th cut-off pont for both th two fgurs. From Fgur., th largst magntud of th two pols wll b kpt at 1 whn th gnvalu s smallr than 4. Wth gnvalu largr than 4, th largst magntud of pol wll b ovr 1. 17

Instablty occurs n such a cas. In Fgur.3, t rvals that th frquncy of th gnmod also has a monotonc mappng rlatonshp whn ts gnvalu smallr than 4. Largr gnvalu corrsponds to hghr frquncy, and vc vrsa. Whn th gnvalu s largr than 4, th angular frquncy of th gnmod wll qual to π. In summary of th abov four scnaros, t can b sn that th gnvalu of th FDTD systm matrx ssntally dtrmns th stablty. Th FDTD soluton can b stabl only f all th gnvalus (or th maxmum gnvalu) of M satsfs: l ( M) < 4 max (.0) In Appndx I, t s provn that th CFL condton s a ncssary condton of (.0) and thus th numrcal stablty of th FDTD modl tslf s nsurd. Morovr, th gnvalu also dtrmns th tmporal frquncy of th corrspondng gnmods. Egnmods wth largr gnvalu ar of hghr frquncy n th FDTD soluton, whl th gnmods wth smallr gnvalu rprsnts th low-frquncy componnts. Snc th nstablty s causd by th gnvalus whch ar largr than 4, on can say that th FDTD nstablty s causd by hgh-frquncy gnmods of th FDTD soluton. Th abov fndng s vry mportant. In practc, th spatal and tmporal dscrtzaton n Th FDTD mthod rsult n numrcal dsprson [44, 45]. It mans that th actual hgh frqunt componnts of lctromagntc flds cannot b accuratly modlld by th FDTD mthod. In othr words, th hgh frquncy componnts of th FDTD modls rprsnt numrcal artfacts rathr than ts physcal vnts. Thrfor, rmoval of ths hgh-frquncy componnts n a FDTD modl s not harmful to th accuracy of th FDTD soluton. Basd on ths fact, w wll propos an uncondtonally stabl wav-quaton basd FDTD mthod n scton.5. Although 0<= l <=4 can mak th FDTD systm stabl, t s only th ncssary condton not suffcnt. In th followng subscton, a furthr nvstgaton s conductd..3 AN EXCEPTION: CAN THE CFL CONDITION ALWAYS GUARANTEES THE STABILITY?.3.1 Thortcal Analyss 18

In ths scton, w focus th dscusson on th scnaro IV of scton.3. whr 0< l <4. As Fg..1d shows, th FDTD dscrt systm hav two conjugat pols on th unt crcl, z1 and z, corrspondng to th -th gnmod. Th mpuls rspons of th -th gnmod n th z-doman can b xprssd as z H[ z] = ( z-z )( z-z ) 1 Th angls of th two pols ar functons of th corrspondng gnvalu l 4l - l q 1 = arctan( ) - l q = p -q 1 (.1) (.) As dscussd n th prvous scton, th scnaro corrsponds to th cas of a losslss FDTD systm wth th CFL condton satsfd. As convntonally blvd, th FDTD soluton wll b stabl. Howvr, w hav dscovrd that n such a cas, th FDTD systm may not b stabl whn crtan nput s chosn. Consdr a monochromatc nput xn [ ] = cos[ nq 1]. Notc that th angl frquncy of th nput concds wth on of th pol angl of th FDTD systm. Th z-doman ntrprtaton of th sourc nput s: X[ z] = Z( x[ n]) = Z(cos[ nq ]) 1 zz ( - cos q 1) = z - zcosq + 1 1 zz ( - cos q 1) = ( z-cosq - jsn q )( z- cosq + jsn q ) zz ( - cos q 1) = ( z-z )( z-z ) 1 1 1 1 1 Th output y[n] of (.1) for th FDTD soluton s thn: (.3) 19

yn Z YZ Z X ZH Z -1-1 [ ] = ( [ ]) = ( [ ] ( )) = Z z ( z- cos q ) ( ) -1 1 ( z-z 1) ( z-z) A A A A = Z ( + + + ) -1 1 3 4 ( z-z 1) ( z-z) ( z-z 1) ( z-z) = A( z ) + A ( z ) + An( z ) + An( z ) n n n n 1 1 3 1 4 (.4) Not that last two trms ar multplcaton of oscllaton trm ( z 1) n and lnarlyncrasng trm n. Th trm n wll thn lad to th uncontrolld ncras output as tm marchs (or n ncrass). In othr words, monochromatc nput at spcfc angular frquncs wll rsult n a scond-ordr pol of th output n th Z-doman, causng th output to b dvrgnt and unstabl. Mor gnrally, any nput, whos Z-doman xprsson has a pol concds wth any pol of th FDTD systm wll rsult n unstabl FDTD solutons. Snc th angular frquncy q1s dtrmnd by gnvalu l of th FDTD systm matrx M as shown n (.), th frquncs causng unstabl FDTD solutons ar vntually dtrmnd by th FDTD systm. In summary, th CFL condton, f satsfd, only guarants th FDTD systm tslf s stabl. Th FDTD solutons ar also dpndnt on th nput or th xctatons. If th xctatons or th nputs contan componnts whos frquncs concd wth th gn frquncs of th FDTD systm, t s stll possbl to gnrat dvrgnt FDTD solutons. To th author s bst knowldg, ths phnomnon has not bn rportd so far..3. Vrfcaton of Th Prdctd Phnomnon In th abov scton, an unrportd cas whr th CFL condton dosn t guarant th stabl FDTD soluton s shown. To vrfy ths, a smpl but typcal numrcal xampl s consdrd: a two-dmnsonal squar cavty wth th prfct conductng (PEC) boundary. It s smulatd wth th FDTD mthod. 0

Fgur.4. Th squar cavty wth PEC boundary, TM wav xctd. A 41x41 vnly-spacd grd s usd to dscrtz th cavty. Th numbr of unknown lctrc fld ponts s thn 151. Th matrx M and ts gnvalus and gnvctors can b obtand usng gn-solvrs of varous mathmatcal softwar packags. A pont lctrc fld sourc s put nsd th problm ara, hr chosn to b at th cntr, xctng a monochromatc snusodal sgnal nto th spac. Th sgnal can b dscrbd as sn[ nq] = sn[ n( p fdt)] q = p fdt Th angl frquncy s chosn qual to (.5) q = arctan 4l - l - l (.6) 1

n whch th λ s on of th gnvalu of th FDTD systm matrx. By ths sourc sttng, th nput wll gnrat pols n Z-doman, who wll concd wth th pols of th FDTD systm. Th smulaton s prformd for 0000 stps. Th rcordd tm-doman fld valus dvrg as shown n Fg..5; Fgur.5 Th tm-doman lctrc fld rcordd at th cntr of th cavty for 0000 smulaton tm stps. Although Fg..5 shows th soluton xploson, on may stll quston whthr t s th rsult of th nstablty as prvously dscrbd snc th cavty s losslss and monochromatc sgnal s contnuously njctd nto th FDTD computatonal doman. Anothr suspcon s that th dvrgng tm-doman soluton may b causd by th natural rsonancs of th cavty whch can produc larg fld valus. In th followng paragraphs, w wll addrss th concrns by carryng varous numrcal xprmnts. Mor spcfcally, dffrnt angl frquncs ar chosn for th monochromatc sourcs to xct th FDTD numrcal grds. Th frquncs ar chosn from as follows, rspctvly: (1) 0, whch s a DC sourc;

() A random valu btwn 0 and f s /, whr f = 1/ Dts th tmporal samplng frquncy; (3) th natural rsonant frquncs of th cavty calculatd by (.7) s f c 1 æmp ö ænp ö = ç + ç p µ è a ø è b ø Dt æmp ö ænp ö qc = πd tfc = ç + ç µ è a ø è b ø ; (.7) n whch a and b ar th szs of th cavty, rspctvly. m and n rprsnt th mod ndcs of th TM(m,n) mod; and (4) th unstabl frquncs prdctd by (.6) Wth ach of th abov four sourcs, FDTD smulaton s prformd ovr 50000 tm stps. Maxmum tm-doman lctrc fld valus ar rcordd for ach cas. Fgur.6 shows th rsults. 3

Fgur.6 Maxmum tm-doman fld valu obtand wthn th 50000 stps of th smulatons usng monochromatc sourcs of dffrnt frquncs. Black dots ar th rsults for cas (1) (th DC sourc), blu dots for cas () (a random frquncy btwn 0 and f s /), grn dots ar for cas (3) (a natural rsonant frquncy of th cavty), rd crcls for cas (4) (th prdctd unstabl frquncs calculatd by (.6)). It can b sn from Fgur.6 th only n cas (4), abnormally hgh valus of lctrc fld ar obsrvd, whch ndcats th nstablty of FDTD smulaton occurs. As dscrbd abov, ths frquncs ladng to unstabl tm-doman fld solutons ar just th unstabl frquncs dtrmnd by gnvalus of M through (.6). Othr frquncs, ncludng th natural rsonant frquncs of th cavty, do not ntroduc uncontrollabl larg fld valus (or numrcal nstablty) n th tsts, unlss thy happn to concd wth th unstabl frquncs ash shown for cas (4). Ths xprmnt shows that th abnormally hgh valus ar nthr rsultd from contnuous monochromc sgnals nor natural rsonancs of th cavty; thy cam from th numrcal nstablty dscussd bfor. 4

Anothr ntrstng xprmnt s to tst how snstv th systm s to th cours frquncy whn t gts clos to th unstabl frquncs. By njctng monochromatc sourcs whos frquncy s nar th unstabl frquncy, maxmum tm-doman lctrcal fld valus ovr 0000 smulaton stps rcordd and shown blow n Fg..7. As th frquncy approachs th unstabl frquncy, fld valu bcoms largr and largr, vntually rachng uncontrollabl hgh numbrs; th nstablty occurs. Fgur.7 Maxmum fld valus obtand wth monochromatc sourcs whos frquncs ar nar an unstabl frquncy. From Fgur.7, t can b sn that f th sourc frquncy s % off th unstabl frquncy, th FDTD solutons ar normal wthout th nstablty ssu. As a rsult, whn th nstablty occurs whn th CFL condton s satsfd, th sourc frquncy nds to b movd away from th unstabl frquncy by mor than %..3.3 Dscussons and Summars From th abov analyss, w can hav th followng conclusons: 1) Th FDTD solutons may b consdrd as consstng of many gnmods, th sz of Matrx M,.. th numbr of unknown lctrc fld nods to b solvd for. 5

) Th gnvalus assocatd wth th gnmods ar proportonal to th squar of th FDTD tm stps. Whn th gnvalu s smallr than or qual to 4, th assocatd gnmod s stabl. Whn th gnvalu s largr than 4, th assocatd gnmod s unstabl. 3) Th qualty of CFL condton corrsponds to th stuaton whr th largst gnvalu s qual to 4. Whn th CFL condton s satsfd, th FDTD solutons contan only th stabl mods and whn th CFL condton s not satsfd, th FDTD solutons contan both th stabl and unstabl mods. If th unstabl mods can b rmovd, th FDTD solutons can rman stabl. 4) Evn whn th CFL condton s satsfd, th FDTD solutons may bcom unstabl f th sourcs or xctatons hav th sam pols as thos of th gnmods. For a problm of a larg physcal doman that has a larg numbr of lctrc fld nods, th sz of M can b hug so s th numbr of th pols of th FDTD systm. Th pols may b clustr dnsly togthr and t s hardr for th sourcs or xctatons to avod havng th pols of th FDTD systms, nstablty may occur mor frquntly..4 THE UNCONDITIONALLY STABLE WAVE-EQUATION BASED FDTD METHOD As mntond bfor, gnvctors of M do not chang wth th tm stps chosn but gn valus do. Gvn a tm stp, gnvalus can b dtrmnd, and so can th stablty of th corrspondng gnvctors. In othr words, a stabl FDTD soluton can b obtand by rmovng th unstabl gnvctors n a FDTD soluton. It wll b laboratd blow..4.1 Th Proposd Uncondtonally Stabl Mthod Th frst stp of th proposd mthod s to drv th gnvalus and gnvctors of th FDTD systm matrx whch s spars. Th stabl gnmods ar actually th gnvctors of M that hav th gnvalus of smallr than 4. Drct solutons of gnvalus and gnvctors of M s not practcal f th sz of th FDTD systm s larg. Thr ar varous mthods (.g., powr mthod, Lanczos Itraton, Arnold Itraton) and 6

softwar packags to fnd gnvalus and gnvctors (mods) of a matrx accuratly and ffcntly. Among thm, th mplctly rstartd Arnold algorthm [46] s vry robust and ffcnt and t s adoptd by sophstcatd softwar lk Matlab. Altrnatvly, a smpl procdur, as prsntd n [6, 47], can b appld to fnd a lmtd numbr of physcally mportant stabl gnvctors. Th procdur starts from a tral convntonal FDTD smulaton of a lmtd numbr of marchng tratons, say from n=1 to k. Th tm stp s chosn to satsfy th CFL condton. At ach tm ncdnc, th FDTD soluton s orthonormalzd wth prvous soluton vctors and stord n a matrx S. By applyng th followng ordr rducton procss, R S M S = (.8) T k* k k* N N* N N* k n whch N s th numbr of unknown lctrc fld nods. Snc k s chosn much smallr than N.. k<< N, R k * k s a small matrx and a common gn-solvr could b appld to t and fnd th gnvalus and gnvctors at lttl cost. Dnot th corrspondng gnvctors found as M, dnotd as V, can b found as: V R. Thn th physcally mportant gnvctors of V S V = (.9) N*1 N* k Rk*1 Th procdur s trmnatd whn th nw soluton vctor has nglgbl componnt orthogonal wth th spac dfnd by V: T En ( )-VV En ( ) En ( ) -3 10 (.30) 3 10 - s a prscrbd rror that can also b othr small valus. Suppos that a tm stp s chosn, for nstanc, p tms of th CFL lmt. Thn th stabl mods (dnotd as V s ) can b slctd from th gnvctors and xprssd as: { Vs} { V l< 4/ p } = (.31) 7

In othr words, th unstabl gnvctors ar dscardd. Th FDTD soluton of lctrc flds can thn b projctd on th sub-spac { V } xprssd n th quaton blow: ' s s composd of ths stabl mods E= VV E (.3) s Anothr thng whch should b pad attnton s not all gnmods hav th sam wghts or ar physcally mportant n th soluton. Som gnmods may hav nglgbl vn zro wght n th soluton. Th wghts of th gnmods or gnvctors n th soluton ar dtrmnd by th sourcs or xctatons n a lnar mdum; thy can b obtand wth th nnr products of th gnvctors wth th sourcs or gnvctors. Anothr way to fnd th wghts of th gnmods s to run a short run of th FDTD soluton and th wghts of th gnmods can b drvd by nnr product of th FDTD soluton obtand and th gnmods. Th mods that hav nglgbl wghts can b gnord. By dong ths, th numbr of stabl mods n th FDTD solutons can b furthr rducd and th computaton wll bcom mor ffcnt. Basd on th abov analyss, w propos th stabl FDTD soluton as follows: [1] Gvn a tm stp, apply th procdur (.8) to (.31) and fnd th stabl and physcally mportant mods of M, { V s }. [] Run ntal two stps of th FDTD smulaton wth a tm stp whch could b largr than th CFL lmt. [3] Expand th FDTD solutons n trms of (.5) to projct th FDTD soluton of lctrc flds to th sub-spac dfnd by { V } a [0] and a [1] for ach stabl gnmod. s. Drv th xpanson coffcnt [4] Us (.9) to fnd all th coffcnts a [n] and (.5) to obtan lctrc flds. [5] Fnd magntc flds from th obtand lctrc flds through Maxwll s quatons..4. Numrcal Exprmnts 8

In th followng paragraphs, th proposd mthod s tstd wth a practcal cas. An H shap cavty s consdrd, as shown n Fgur.8. TM wavs nsd th cavty ar smulatd and obsrvd. Du to th narrow gap n th cavty, fn grds ar mployd to obtan good modlng accuracy. As a rsult, wth th FDTD mthod, th uppr lmt of th tm stp, as rstrctd by th small spac grd sz, s small; th smulaton wll tak a rlatvly long tm and rlatvly larg mmory. Fgur.8 Structur of th H typ mtal cavty. Wth th proposd mthod, w can choos a larg tm stp, apply th soluton stps dscrbd abov, and obtan a stabl soluton. A unform numrcal grd s usd wth a cll sz of 1mm 1mm. A Gaussan puls s xctd at th cntr of th cavty. Fgur.9 and Fgur.10 show th numrcal rsults. By takng th dscrt Fourr transform of th solutons, th frquncy doman solutons ar obtand and shown n Fgur.9. As can b sn, th rsults of th proposd mthod agr wll wth thos of th convntonal FDTD mthod xcpt hgh-frquncy componnts; ths s xpctd snc th hgh frquncy componnts corrspond to unstabl mods and ar dscardd wth th proposd mthod. 9

Fgur.9 Frquncy rsponss obtand wth th convntonal FDTD and th proposd mthod. Fgur.10 shows th fld dstrbutons obtand wth th convntonal FDTD mthod and th proposd mthod. Th frst on of Fgur.10 s th normalzd fld dstrbuton at th 100,000 th tm stp, obtand wth th convntonal march-on-tm rcursv FDTD smulaton. Th tm stp s 11.18ns whch s 0.5 of th CFL lmt. Th scond on s th fld dstrbuton at th 5,000 th tm stp, obtand wth th proposd mthod. Th tm stp was 44.7ns whch s two tms of th CFL lmt and four tms of th tm stp usd wth th convntonal FDTD smulaton. In othr words, 5,000 tm stps of th proposd FDTD mthod rprsnt th sam physcal tm of 100,000 tm stps of th convntonal FDTD smulaton. Th dstrbutons ar bascally th sam xcpt that th hgh-ordr mods assocatd wth th hgh frquncs ar dscardd wth th proposd mthod. 30

Fgur.10 Elctrc fld dstrbuton obtand wth th convntonal FDTD smulaton (lft) and proposd mthod (rght)..5 SUMMARY In ths chaptr, basd on th gn analyss of th FDTD mthod, th root caus of th nstablty s rvald and xpland usng th concpt of dscrt systms and Z- transformaton. Th rlatonshp btwn th CFL condton and th stablty s rvald. An ntrstng phnomnon s found that th CFL condton dos not always guarant th stablty of th FDTD solutons, f spcfc monochromatc sourc s njctd nto th systm. Numrcal xampls vrfd th fndngs. Basd on th abov analyss, w propos an uncondtonally stabl wav-quaton basd FDTD mthod; n t, lctrc and magntc flds ar solvd n a dcoupld and smplfd mannr. Numrcal FDTD solutons can thn b xpandd by a lmtd numbr of stabl gnmods. Each gnmod and corrspondng gnvalu can b calculatd at O(N ) complxty. Onc th gnmods ar found, th soluton can b smply obtand wth th dffrnc quatons of coffcnts nstad of tm-marchng tratons. Numrcal xampls ar prsntd to vrfy th ffctvnss of th proposd mthod. Bcaus of that th prncpls and dvlopmnts ar wthout much of thortcal rstrctons, dscussons and formulatons prsntd n ths chaptr can b gnralzd and xtndd to othr tm doman numrcal mthods. 31

CHAPTER 3 ANALYTIC SOLUTION OF THE FDTD METHOD 3.1 INTRODUCTION In Chaptr, w hav analyzd th FDTD mthod and proposd an altrnatv nw FDTD soluton procss. Mor spcfcally, w hav systmatcally nvstgatd th FDTD formulaton and dvlopd th wav-quaton basd dscrt systm. W hav transformd th systm nto a srs of dscrt scalar sub-systms and apply th Z- doman solutons for stablty analyss. Th fnal FDTD solutons ar thn obtand by solvng th dffrnc quatons of th xpanson coffcnts. In ths chaptr, as a furthr dvlopmnt and sgnfcant xtnson of th analyss of th prvous chaptr, w drv th analytc solutons of th FDTD mthod that do not rqur th convntonal tratv march-on-n-tm soluton procss. Th FDTD soluton can thn b drctly obtand at any gvn tm dsrd. Othr dtals prtnnt to th analytc solutons, ncludng sourc mplmntatons, tratmnt of lossy mda, ar also prsntd. Fnally, w numrcally vrfy th proposd mthod and conclud that th analytc approach can srv as an altrnatv way to th solutons of th convntonal FDTD mthod. 3. ANALYTIC FORM OF THE FDTD SOLUTION In prvous chaptr, w hav prsntd th wav-quaton basd FDTD mthod n ts matrx form, whch s E[ n+ 1] + E[ n-1] - E[ n] + ME[ n] = x [ n] (nput) (3.1) whr matrx M s th FDTD systm matrx as dfnd bfor. Egn-analyss s prformd on (3.1). By xpandng th lctrc fld soluton vctor E[n] and th nput trm x[n] on th gnvctors of matrx M, E[ n] = Va[ n] x V V x V x -1 [ n] = b[ n] or b[ n] = [ n] = ' [ n] (3.) 3

As drvd n th prvous chaptr, (3.1) can fnally turn nto th dffrnc quaton of coffcnt a for ach gnvctor, as xprssd by (.9): a + a -( - l ) a = b, for n = 1,, 3,... (3.3) n, + 1 n, -1 n, n, (3.3) s th dffrnc quaton for coffcnt a whch can b solvd tratvly. It can also b consdrd as a dscrt systm, whos rspons can b drvd usng th Z- transform: za z z A z A z A z B z -1 [] + [] - [] + l [] = [], A[] z z H[] z = =. B z z z [] + 1 -(-l) Th mpuls rspons n th tm doman s 1 1 hn [ ] = ( z ) + ( z ) n-1 n-1,1, ( z,1 -z, ) ( z, -z,1) (3.4) (3.5) As dscussd, whn th CFL condton s mt, th gnvalus of matrx M s wthn th rang (0,4), whch nsurs th systm pols on th unt crcl n th Z-doman. Th two pols, z 1 and z, ar conjugat to ach othr and ar on th unt crcl, whch can b xprssd as Th angls ar j ( - l) 4 1, ± j l -l z q 1, = =. (3.6) æ 4l - l ö q 1 = arctan, ç - l è ø q =-q. 1 Corrspondngly, th mpuls rspons (3.5) for ths stabl systm can b xprssd as (3.7) 33

(3.1). 1 1 h[ n] = ( z ) + ( z ) n-1 n-1,1, ( z,1 -z,) ( z, -z,1) 1 = [cos( n- 1) q + sn( n-1) q] snq 1 + [cos( n-1) q -sn( n-1) q] -snq sn[( n -1) q] = wth h[0] = h[1] = 0 snq (3.8) Equaton (3.8) s th stabl mpuls soluton of th FDTD systm as dscrbd by Consdr now Kronckr mpuls xctaton that has non-zro valu at tm T ncdnc n=1,.., b[1] = [ b1,1 b,1... b,1... bn,1] s not zro and th xctaton at othr tm nstancs bn [ ] n= s zro. In othr words, at ntal tm n=1, vry gn mod,3,... V s xctd wth an ampltud of b. Thn,1 and th ntal ampltud of V s B() z = b, (3.9),1 a = Z { B( z) H ( z)} = Z { b H ( z)} -1-1 n,,1 = bhn ( ) = b,1,1 As a rsult, by (3.3), th FDTD soluton s: sn[( n -1) q] q éa1, n ù ê... ú ê ú E[ n] = V a[ n] = V êa ú n, = [V 1a1, n... V a, n... V N a,] N n ê ú ê... ú êa ú ë N, û = [V b h( n)... V b h( n)... V b h ( n)] 1 1,1 1,1 N 1,1 N 1 1 1,1,1 N 1,1 1 (3.10) sn[( n-1) q ] sn[( n-1) q ] sn[( n -1) qn ] = [V b... V b... V b ] snq snq snq N (3.11) 34

In most practcal cass, th sourc sgnal has a duraton. Consdr that th sourc has a duraton of squntal M mpulss. In such a cas, bcaus th FDTD systm s a lnar systm, th fld soluton as a rsult of xctaton b at th k-th tm stp can b k, consdrd th sam as that du to b n (3.11), xcpt dlayd by k tm stps and a,1 chang of ampltud from b,0 to b k,. Th ovrall FDTD solutons wll b th sum of th rsponss du to all th sourc ampltuds of from b,0 to b k,. In othr words, th full FDTD solutons ar thn: M M é ù é sn( n- k) q1 ù b1, 1( ) ( ) 1, ( ) kh n k u n k b k u n- k ê å - - ú ê å 1 k 1 snq ú k= = 1 ê ú ê ú ê... ú ê... ú ê M ú ê M ú sn( n- k) q E[ n] = V ê b, ( ) ( ) V bk, u( n k) kh n-k u n- k ú= ê - ú å ê k= 1 ú ê å k= 1 snq ú ê ú ê ú ê... ú ê... ú ê M ú ê M sn( n- k) q ú N êåbn, ( ) ( ), ( ) khn n-k u n-k b ú êå N k u n- k ú ë k= 1 û êë k= 1 snq1 úû (3.1) In th abov quaton, functon un ( ) s th unt functon that s untary at n ³ 0 and zro othrws. Equaton (3.1) s th xplct analytcal xprsson of th FDTD soluton. Th FDTD soluton s a functon of dscrt tm stp n and can b obtand drctly at any tm nstant n as long as th gnvalus and vctors of M ar known or found n advanc. 3.3 THE HARD SOURCE IMPLEMENTATION Analyss abov s basd on th soft sourc sttngs. A soft sourc mans that th sourc s smply addd to th currnt fld valu at th sourc nods, whch can b consdrd as a smpl nput of th systm. For th cass wth th hard sourcs, on th othr hand, th sourc nods ar assgnd wth crtan fld valus, or boundary nods ar mposd wth fxd boundary condtons; th sourc fld valus ar not addtv to th currnt fld valus but mposd on thm. 35

Consdr th convntonal FDTD lap-frog schm 1 1 H[ n+ ] = H[ n- ] -DtD EE[ n] 1 E[ n+ 1] = E[ n] +D td HH[ n+ ] n whch matrxs D E and D H follows th sam dfnton as thos of (.). (3.13) Now dcompos th lctrc fld vctor E[ n ] of th whol problm rgon nto two sts: th fld valu on th hard sourc (or forcd boundary) nods E[ s n ], and thos on othr lctrc nods E[ u n] whos valu ar unknown and nd to b solvd. Th dcomposton can b wrttn as ée[ u n] ù E[ n] = ê E[ s n] ú ë û Dnot th numbr of unknown lctrc fld nods as, N u (3.14), th numbr of sourc and known-boundary nods as N, and s, N h as th numbr of magntc fld nods. Corrspondngly, matrx D E and D H can b rwrttn nto two blocks wth rspct to th dcomposton of lctrc fld vctor E[ n ]. Th matrx D E can b parttond as D [D D ] = (3.15) E E, u E, s n whch D Eu s th matrx of, Nh N, rprsnts, u 1 µ - Ñ opraton on th unknown lctrc nods for updatng th magntc fld. D Es s th matrx of, Nh N,, s rprsntng th sam opraton on lctrc fld nods (and forcd boundary nods.) Smlarly, th matrx D H can b dcomposd as n whch D Hu s th matrx of sz, N, D H édhu, ù = ê D ú ë Hs, û u. (3.16) 1 N, rprsnts th dscrtzd - Ñ oprator at th poston of unknown magntc nods for updatng th lctrc fld. h 36

D Hs s of sz, Ns, sourc and forcd boundary nods. N, rprsntng th sam oprator for updatng th lctrc hard h Basd on ths notatons, th updat quatons (3.13) can b rwrttn as 1 1 ée[ u n] ù H[ n+ ] = H[ n- ] -DtéD, D, ë Eu Esùûê E[ s n] ú ë û ée[ 1] E[ ] D u n+ ù é u n ù é Hu, ù 1 ê = +D tê H[ n+ ] E[ n 1] E[ n] D ú + ú ê ú ë s û ë s û ë Hs, û (3.17) Snc th fld valus at th hard sourc nods or forcd boundars ar known or spcfd, updat of E s ar not ndd and (3.17) can b smplfd as: 1 1 H[ n+ ] = H[ n- ] -DtDEu, E u[ n] -DtD Es, E s[ n] 1 E[ u n+ 1] = E[ u n] +D tdh, uh[ n+ ] (3.18) Th updat of lctrc fld s now prformd on E u only. (3.18) can b furthr rducd to th wav quaton by substtutng th xprsson for H nto th xprsson for E u : E[ n+ 1] + E[ n-1] - E[ n] +D t D D E[ n] =-D t D D E[ n] (3.19) u u u H, u E, u u H, s E, s s Apparntly, th abov wav quaton has th form smlar to th FDTD quatons of (3.1) for th soft sourcs. Th dffrncs ar that (3.19) updats unknown lctrc fld E u only and th hard sourc flds (ncludng th forcd boundary condtons) E s s transformd nto an quvalnt soft sourc trm -D t D D E [ n] on th rght hand Hs, Es, s sd of th quaton. Th rsultng quaton fts wll wth th form of (3.1). By gnanalyss of th nw systm matrx Dt D D Hu, Eu, can b drvd wth th smlar format of quaton (3.1). of (3.19), th xplct analytc soluton 3.4 THE EXPLICIT ANALYTICAL FDTD SOLUTION IN A LOSSY MEDIUM 37

In th cass of lossy mdum, Maxwll quatons bcom whr s s th conductvty of th mdum. H Ñ E =-µ t E Ñ H= se+ + J t, (3.0) By followng th sam FDTD dscrtzaton procss of (.1)-(.4) (as appld n th abov sctons n ths chaptr), th rcursv dscrt lctrc wav quaton n th lossy mdum can b obtand as æ sdtö æ sdtö ç 1+ [ n+ 1] + ç 1 - [ n-1] -( - ) [ n] = [ n] è ø E è ø E M E x (3.1) Th quantts ar dfnd n th sam way as thos dfnd at (.4) n th losslss cas. Agan, by xpandng E [ n] and x[ n ] n trms of th gnvctors of M, (3.1) bcoms a srs of rcursv quatons: æ sdtö æ sdtö ç 1+ a[ n+ 1] + ç 1 - a[ n-1] -( - l) a[ n] = b[ n] è ø è ø for = 1,,3... Applcaton of th unlatral Z-transform to th abov quaton lads to æ sdtö æ sdtö ç ç è ø è ø -1 1+ zy[ z] + 1 - z Y[ z] - Y[ z] + ly[ z] = X[ z] Y [ z] 1 z H[ z] = = X[ z] æ sdtö æ sdtö ç 1+ 1 ( l ) ç - è ø - z - z+ è ø æ sdtö æ sdtö ç 1+ ç 1+ è ø è ø (3.) (3.3) (3.4) Stablty of ths systm dpnds on th pols of th transfr functon abov. Dtald analyss (s Appndx II) wll show that n lossy cass, 0< l < 4 for all th gnvalus s stll th ncssary condton for stabl FDTD solutons, whras l > 4 wll lad to unstabl FDTD solutons. 38

Now assum that 0< l < 4. Analytcal solutons to th rcursv quatons of (3.3) can b obtand. Thr ar two cass to consdr: Cas 1: s Dt 0< l < 4 and l - 4l + > 0 Th two pols ar ral numbrs that can b xprssd as: z,1or th mpuls rspons of ths systm s = s Dt - l ± l - l + sdt + 4 n n,1, 1 1 n-1 n-1 h[ n] = ( z,1 -z, ) æ sdt ö z 1,1 - z, ç + è ø sdt + 1 n-1 n-1 = ( z,1 -z, ) æ sdt ö 1 s Dt ç + l - 4l + è ø = z - z s Dt l - 4l + (3.5). (3.6) Th abov soluton can b vrfd by substtutng t nto (3.) wth th ntal condtons of h[0] = h[1] = 0. Snc magntuds of,1 (s Appndx II), h [ n] wll b dampd to zro as tm progrsss. Cas : s Dt < - + <. 0< l 4 and l 4l 0 z and z ar smallr than 1, Th two pols ar conjugat complx numbrs that can b xprssd as z,1or = s Dt - l ± j 4l -l - sdt +. (3.8) 39

Thr angls and magntuds ar æ s Dt ö ç 4l -l - q 1 = arctanç ç - l ç è ø q =-q 1 Th mpuls rspons of ths systm s s Dt 4 - z,1 = z, = z = < 1 sdt + 1 1 n-1 n-1 h[ n] = ( z1 -z ) æ sdt ö z 1,1 - z, ç + è ø 1 1 = z - jq - jq æ sdt ö z ( ) 1 - ç + è ø 1 n-1 sn[( n -1) q] = z æ sdt ö snq 1 ç + è ø n j( n-1) q - j( n-1) nq ( ). (3.9). (3.30) Snc z and,1 z ar conjugat complx numbrs wth magntud smallr than 1,, th FDTD mpuls rspons wll b oscllatng and dcayng to zro as tm progrsss (.., n ncrass). hn [ ] = ar obtand, th FDTD soluton can b found through Onc 1,,..., N smlar procdur as that for th losslss cas (3.10~3.1). 3.5 SUMMARY OF ANALYTIC SOLUTION OF THE FDTD METHOD From th analyss and rsults prsntd n chaptr two, solutons of th FDTD mthod can b xpandd n trms of fxd gnmods wth tm-dpndnt xpanson 40

coffcnts. Instad of solvng th tm-dpndnt coffcnt numrcally, th xplct analytcal xprssons of th coffcnts can b obtand by solvng th dscrt systm usng th Z-transformaton and so can th FDTD solutons at any tm ncdnc. Mor spcfcally, th stps to obtan th FDTD solutons can b summarzd blow: 1) Choos tm stp D t, construct th dscrt systm n th wav quaton form of (3.1) and obtan matrx M (whch s th dscrtzd fnt-dffrnc form of oprator Dt( µ ) - 1 0 Ñ Ñ ). W could choos th tm stp whch satsfs th CFL condton to obtan th analytc soluton formulaton. Th choc dos not affct th ffcncy of th mthod, snc th soluton at any tm ncdnc wll b solvd analytcally wthout tratv procss. ) Apply known ffcnt algorthms or softwar packags (.g., [16]) to fnd physcally mportant gnvctors (or gnmods) wth th corrspondng gn valus 0< l < 4. 3) For any gvn tm ncdnc n, comput (3.1) to fnd mod ampltud bn [] for ach mod. 4) Us (3.) to obtan lctrc flds E [ n]. 5) Fnd magntc flds from th obtand lctrc flds through Maxwll s quatons (.1). Compard wth th convntonal FDTD soluton approach, th abov proposd mthod nvolvs gn-dcomposton to systm matrx M whch s a spars and symmtrc matrx. Thrfor, th computatonal ffcncy of th proposd mthod s vry much dpndnt on th algorthms usd to xtract th gnvalus and gnvctors. Varous mathmatcal algorthms hav bn dvlopd for ffcnt gn-dcomposton of a ral, spars, sm postv-dfntv and symmtrc matrx. Aftr gndcomposton s don, analytcal FDTD solutons ar obtand at almost no addtonal computatonal costs. Comparsons btwn th proposd FDTD soluton approach and th convntonal FDTD tratv mthod ar not asy to mak du to th dffrnt solutons paths. Howvr, two obvous advantags of th proposd mthod ovr th convntonal approach can b sn n thr on of th two stuatons: 1) whn an FDTD 41

soluton rqurs a long traton such that th traton tm xcds th tm n xtractng th gnvalus and gnvctors of th stabl mods, and ) whn th FDTD solutons of a structur nd to b stord or rcordd. In addton, snc only lctrc or magntc fld s dalt wth, th proposd approach uss lss mmory than th convntonal FDTD soluton approach whr both lctrc and magntc flds ar smultanously solvd. Although th abov analyss ar prformd on th FDTD mthod, thy can b xtndd to othr march-on-tm numrcal mthods snc rcnt studs hav ndcatd that numrcal mthods ar of th sam natur n thr solutons and can b unfd wth th sam numrcal mathmatcal framwork [48, 49]. In th nxt scton, numrcal xprmnts ar carrd out usng th proposd mthod. 3.6 NUMERICAL EXPERIMENTS To vrfy th proposd analytcal FDTD soluton prsntd abov, two structurs wr computd as xampls. Th frst s an H shap cavty as shown n Fg. 3.1 and th scond on s th dlctrc rod whch s shown latr. Th cavty s chosn bcaus t s not only a smpl structur wth radly avalabl analytcal solutons for comparsons but also mbods multpl fld scattrngs and rflctons of lctromagntc wavs from th cavty walls. If a mthod or an approach s ncorrctly formulatd, such multpl wavs wll b wrongly smulatd, appar n th rsults, and can b asly dntfd. In othr words, cavts prsnt sold and convncng tsts of a numrcal mthod. Th xprmnts ar laboratd and dscrbd as follows. 3.6.1 Numrcal Exprmnt I: Analytc FDTD soluton of th H-shap Cavty Th frst xprmnt s prformd on an H-shap cavty. It s shown n Fgur 3.1. A unform numrcal cll sz of 1 mm 1 mm 1 mm was usd to dscrtz th cavty. A Gaussan puls was xctd at th cntr of th cavty. TM wavs ar smulatd and obsrvd for 5 10 stps. Th tm stp sz was chosn qual to th CFL lmt. For 4

comparsons, both th proposd and convntonal FDTD soluton approachs wr carrd out. Fgur 3.1 Structur of H typ mtal cavty: (a) th thr-dmnsonal vw, (b) th crosssctonal vw. Fgur 3. and Fgur 3.3 show th fld dstrbutons obtand wth th two mthods. Th dffrncs btwn th rsults ar normalzd to th largst valus of th transnt flds. As can b sn, th fld dstrbutons ar vsbly th sam and th rlatv dffrncs btwn th rsults obtand wth th two mthods ar xtrmly small, namly at th nos lvl. 43

Fgur 3. Fld dstrbuton rcordd at th 00,000th tm stp of proposd mthod (lft) and th convntonal FDTD mthod (rght). Fgur 3.3 Rsults (rcordd at th obsrvaton pont) obtand wth th proposd and th convntonal FDTD mthods. (a) shows th fld valus obtand by th two mthods at obsrvaton nod and (b) shows thr rlatv rrors for a 10 4 stps smulaton. Th rrors ar blow 10-11. 3.6. Numrcal Exprmnt II: Smulaton wth Hard Sourc n Lossy Mdum Th scond xprmnt s to tst th proposd schm for hard sourc and n lossy mdum sttngs. Th smulaton s prformd wth th convntonal FDTD schm and th proposd mthod for comparsons. In ths xprmnt, th sam cavty structur as that of Fgur 3.1 s smulatd but nstad of ar, lossy mdum of conductvty σ of 0.1 S/m s flld n th cavty. Th sourc s placd at th cntr of th cavty and s st to b a hard sourc, whch forcs th lctrc fld to b 1V/m at th sourc. Th tm stp sz was chosn qual to th CFL lmt. Th lctrc fld at th obsrvaton pont s rcordd for ach tm ncdnc n. Comparson of th rsults s shown n blow: 44

Fgur 3.4 Rsults (rcordd at th obsrvaton pont) of fld valus obtand wth th proposd and th convntonal FDTD mthods for th lossy-mdum and hard sourc cas. As shown n Fgur 3.4, t can b sn that th two solutons agr wll wth ach othr. Th dffrncs ar blow 10-11, showng that th drvd analytcal FDTD solutons for lossy-mdum and hard sourc ar corrct. 3.6.3 Numrcal Exprmnt III: Smulaton of th Dlctrc Rod Structur In th thrd numrcal xprmnt, a dlctrc rod rsonator n a mtallc rctangular cavty s consdrd as shown n Fgur 3.5. Th paramtrs of th dlctrc rsonator ar chosn n accordanc wth th modl usd n [50, 51] : a=.536 cm, b=.536 cm, h=0.6985 cm and l=.5718 cm. Th support of th dlctrc rod s assumd to hav dlctrc constant of 1. Th rod has a dlctrc constant of 38. In th xprmnts, th sz of th structur a b l s dscrtzd nto a numrcal grd of 6 6 6 and 31 31 3, rspctvly. 45

Fgur 3.5 Gomtry of th dlctrc rod rsonator n rctangular cavty. Th lctrc fld wav quaton for ths dlctrc-loadd structur can b found as: E E E ME x (3.3) 1 [ n+ 1] + [ n-1] - [ n] + [ r] - [ n] = [ n] (nput) n whch [ε r ] s th dagonal matrx of dlctrc constant. Th dlctrc constant s a valu whch s a ral numbr and largr than 1. Compard wth (3.1), apparntly, [ε r ] -1 M s not ncssarly sm-postv dfnt bcaus t s not ncssarly a symmtrc matrx. Thrfor, th mthod proposd n th prvous dscusson cannot b appld drctly. To addrss ths, (3.3) s rformulatd and a varaton of Maxwll wav quaton for non-unform dlctrc mdum cass s dvlopd: E[ n+ 1] + E[ n-1] - E[ n] + ME[ n] = x[ n] E= E x = x 1/ 1/ [ r], [ n] [ r] [ n] é Dt ù M = [ r] Ñ Ñ [ ] ë û -1/ -1/ ê ú r 0u0, (3.33) Wth ths varaton, M s a ral symmtrc matrx whch nsurs that ts gnvctors form a complt st of th bass functon n th soluton spac. Th gn- 46

analyss can thn b carrd out on (3.33) nstad of (3.1) and E s solvd followng th stps proposd n scton III. Dscrt Fourr Transform s prformd on E va E= [ ] r 1/ - Eto fnd th rsonanc frquncs. Th rsults ar compard wth th masurd rsults [50, 51] and ar prsntd n Tabl I for two dffrnt damtrs of th dlctrc rod. Th rsults obtand wth th two dffrnt mthods show vry good agrmnt, whch vrfs th proposd mthod. Tabl 1 Comparson of th mthods for rsonant frquncy of dlctrc rsonator. Sz Grds Rsonant Frquncs f(ghz) R (cm) t (cm) Masurd FDTD mthod Proposd mthod 1.7551 0.5893 6 6 6 4.136 4.171 4.171 1.98 0.646 31 31 3 3.76 3.696 3.696 As a short summary, th four xampls abov vrfy that th proposd mthod provds th xact soluton of th convntonal FDTD mthod n a dffrnt soluton path. Th proposd mthod appls th gn xpanson of th dscrt FDTD systm and thrfor ts succssful applcatons hng on havng an ffcnt and ffctv gnsolvr. Snc a larg porton of th gnvalus and thr assocatd gnvctors rprsnts hgh-frquncy mods or unstabl mods, thy can b dntfd and rmovd bcaus thy do contrbut accurat solutons as a rsult of nhrnt FDTD hgh numrcal dsprson rrors at hgh frquncs ; only a rlatvly small numbr of th gnvalus of small valus nd to b found. In gnral, thy account for 10%-0% of all th gnvalus. Many ffcnt tchnqus hav bn dvlopd to fnd ths gnvalus and thr gnvctors, such as th modl-ordr-rducton tchnqu dscrbd n [6] and th combnd FDTD-tral-smulaton and modl-ordr-rducton tchnqu prsntd n [47]. In our cas, w us th on prsntd n [6]. W obtan th smlar computatonal spdup of roughly sx tms n fndng th ndd gnvalus and thr assocatd mods n comparson wth th convntonal FDTD mthod at th sam soluton accuracy lvl. 47

3.7 TREATMENT OF ABSORBING BOUNDARY CONDITIONS (ABC) Th abov numrcal xprmnts ar prformd n a closd structur. In opn structurs, th absorbng boundary condtons (ABCs) ar always ndd to truncat th modlng domans whl lmnatng th rflctons nducd by th truncaton. Among varous absorbng boundary schms, th prfctly-matchd layrs (PML) s consdrd to hav th bst prformanc of all and s wdly usd. Th xtnson or ncorporaton of th proposd analytcal mthod to th absorbng boundary condtons ncludng PML has bn found non-trval. It rqurs thorough analyss lk thos prsntd n th abov agan. W ar currntly workng on t and xpct to rport th rsults n our futur publcatons. Nvrthlss, t s worth to mnton that a straghtforward and practcal tchnqu to dal wth ABCs s to us th approach dscrbd n [6]: a soluton doman s dvdd nto th ABC rgons and th man rgon contanng th structurs to b solvd for. In th ABC (.g., PML) rgons, th flds ar computd n th convntonal FDTD mannr. In th man rgon, th proposd mthod s appld but wth th fld valus at th ntrfac nods btwn th two rgons actng as soft sourcs. Blow ar our ntal formulatons of th PMLs wth th proposd analytcal mthod. Tak Brngr s fld-splt PML formulatons [5, 53] for TE wav as an xampl. Th govrnng quatons n th PML rgon can b xprssd as: é ù ê [-s y] [ ] [ ] y y ú [ 0] E ê ú é ù é x ù ê úé Ex ù ê [ ] [ ] [ ] [ 0] E s ú ê ú ê - x - - y x x úê E ú y ê ú ê ú = ê úê ú. (3.34) ê [ u ] ú t êh ú ê êh ú [- ] [-s ] ú 0 zx zx ê ú ê ú mx ê ú ë [ u0] û H ê zy x ú ë û ëhzy û ê ú ê[ ] [-s my] ú êë y úû Th bracktd trms ar th corrspondng paramtrs n matrx form. (3.34) s much mor complx than (.) bcaus of th ntroducton of ansotropc and lossy trms n th PML formulatons. It wll lad to a gnralzd gnvalu (GEV) systm, l Ax= Bx (n whch λ and x ar gnralzd gnvalus and gnvctors of th 48

systm). Onc th stabl gnmods(vctors) ar solvd, th smlar rsults may b obtand. An altrnatv approach to ncludng th PML s to dvd a problm doman nto two rgons: th PML rgon and th man rgon n whch th structur to b studd s contand. Bcaus th PML rgon s fcttous and should not contan nformaton about th structur studd, th abov dscrbd mthods ar only to th man rgon. In th PML rgon, smulaton can b don wth th convntonal FDTD mthod. Intractons btwn th two rgons occur at th ntrfacng nods only. To th man rgon, th nfluncs du to th PML rgon can b consdrd as th quvalnt nput sourcs at ths ntrfacng nods. In anothr word, soft sourcs ar usd to count for th PML ntrfacs. Fgur 3.6 PML layrs surrounds th computatonal doman; th crcls rprsnt th ntrfacng nods. Mor spcfcally, lctrcal fld n th man rgon can b dvdd nto éem ù En [ ] = ê E ú ë û, (3.35) 49

In whch E ar th flds at th ntrfacng nods and th E m rprsnts th flds n th man rgon. Th wav quaton (3.1) can b appld to th man rgon drctly, whch s now wrttn as éem[ n+ 1] ù éem[ n] ù éem[ n-1] ù éem[ n] ù ê - + + M = x[ n] E[ n+ 1] ú ê E[ n] ú ê E[ n-1] ú ê E[ n] ú ë û ë û ë û ë û, (3.36) ém M ù = ê ú ë û m-m m- M M -m M -, (3.37) n whch matrx M m-m, M m-, M -m, and M - ar th sub-matrcs of systm matrx M for th man rgon, wth szs of E m, E, rspctvly. Also, th fld quantts at th ntrfacng nods ar nvolvd n th calculatons of th PML layr. Th fld valus at th ntrfacng nods du to PML can b consdrd as nput sourcs (.. th soft sourc trms) n th fld updat procss of th man rgon. (3.36) can b xpandd as E [ n+ 1] - E [ n] + E [ n- 1] + M E [ n] + M E[ n] = x [ n] m m m m-m m m- m E [ n+ 1] - E [ n] + E [ n- 1] + M E [ n] = x [ n] -M E[ n] m m m m-m m m m-. (3.38) Th fnal quaton abov has th form smlar to quaton (3.1) and (3.19). Th flds n th man rgon can thn b solvd by th procdur dscrbd bfor. Th nflunc from th PML rgon to th man rgon s contand n th trm - M E[ n] and bcom part of th nput sourc trm. In rfrnc [8], a smlar absorbng tratmnt, whch dvds th whol rgon nto a man computaton rgon and a ABC rgon; thy ar thn solvd sparatly. Furthr work along ths ln s ndd whch s byond th scop of ths thss du to th lmtaton of th tm. m- 3.8 SUMMARY In ths chaptr, basd on th gn analyss of th FDTD formulaton, w hav drvd th analytcal solutons for th FDTD mthod and r-analyzd th stablty 50

condton of th FDTD mthod n trms of mpuls rsponss. Basd on our analyss, th numrcal soluton of th FDTD can b xpandd n trms of spatally tm-nvarant gnmods wth th tm-varyng xpanson coffcnts a. Th tratv march-on-n- n, tm procss of th FDTD can b rplacd by drctly solvng th xpanson coffcnt a. As a rsult, an altrnatv approach to solvng th FDTD mthod s n, dvlopd whr th FDTD soluton can b obtand at any tm stp wthout rcursv march on tm. Ths nw approach allows th us of xstng gn solvrs and may prsnt tm-savng n som cass whr long smulaton tm s rqurd. Mor sgnfcantly, th analytcal forms of th FDTD solutons prsnt th possblty of applcatons of advancd sgnal procssng tchnqus as wll as storng of structural mpuls rsponss aftr pr-computng. Thr prlmnary numrcal xampls ar gvn to vrfy th thory and th ffctvnss of th proposd approach. Th work opns anothr horzon n obtanng and usng FDTD solutons. 51

CHAPTER 4 DEVELOPMENT OF EFFECTIVE TIME REVERSAL METHOD 4.1 INTRODUCTION Sourc locatng has bn a topc of rsarch and dvlopmnt n acoustc, lctromagntcs and othr aras. Varous mthods hav bn dvlopd to locat sourcs n a tm-nvarant nvronmnt or doman. Many of thm us th tm-of-arrval or phas for th sourc locatons, whch may not ncssarly us all th nformaton n th fullwav rcvd sgnals or flds. Envronmntal ffcts oftn ntrfr th locatng procss and dgrad locatng accuracy. Th tm rvrsal (TR) mthod [7, 8] has bcom a choc for sourc dntfcatons n many applcatons. Thr ar two stps wth th tm rvrsal tchnqu: forward propagaton (smulaton) and backward propagaton (smulaton). In th forward propagaton, sgnals, for xampl lctromagntc flds mttd by unknown sourcs, propagat and ar thn rcordd at prslctd output nods of a doman. Thn n th backward smulaton, th rcordd flds ar rvrsd n tm and r-njctd back nto th propagaton doman for th sam prod. Th fld paks wll b formd and obsrvd at th nd of th rvrsd sgnal propagaton and thy ndcat th orgnal sourc locatons. Robust and smpl to mplmnt, th TR mthods hav drawn much attnton rcntly. For xampl, [7, 8] apply TR for acoustc analyss; [54] uss th TR tchnqu to locat arthquak sourcs. [9, 30, 3, 55] apply th TR procss to lctromagntc sourc locatons. Wth a convntonal sourc locatng tchnqu, multpath s consdrd as an advrs ffct aganst th locaton accuracy. Th tm rvrsal s th tchnqu whch can utlz th multpath for prformanc mprovmnts for targt dtcton and/or sourc localzaton. It xplots th rcprocty of th propagaton mdum and th tm-rvrsal nvarant natural of th wav quaton for fld focusng. It procsss th multpath cops of th transmttd sgnals n th mdum, as nducd by rflcton, rfracton and multpl scattrngs, n a constructv mannr, rsultng n th mprovmnt of th focusng rsoluton. Th tm rvrsal tchnqu has succssfully bn appld and ts capablty has bn dmonstratd [7-3][54-55]. Snc thn, th tm-rvrsal mthods hav also bn appld to lctromagntc structurs. Th wd-rang applcatons hav spurrd 5

xtnsv rsarch n ths ara. Th work prsntd n ths thss focuss on th tmrvrsal mthods wth tm-doman lctromagntc modlng. Fgur 4.1 shows a typcal computatonal tm rvrsal smulaton mployng th FDTD mthod. Th problm doman undr consdraton s a cavty whch provds a multpath-rch nvronmnt. Fgur 4.1a & 4.1b shows th sourc xctaton and th fld propagaton wthn th cavty n th forward propagaton. Th flds ar thn rcordd at th slctd output nods for a gvn lngth of tm (or a fxd numbr of tm stps). Thy ar thn r-njctd nto th cavty at th output locatons. Fgur 4.1c shows th rcordd flds that ar r-njctd back nto th cavty. Th r-njctd flds wll thn propagat back nto th cavty and produc a tm rspons at vry nod n th FDTD grd, ncludng at th orgnal sourc nods. Ths s th rvrs or backward smulaton procss. If th duraton of th backward propagaton s th sam as that of th forward propagaton, th flds wll gnrally form maxma at th orgnal sourc nods and th locatons of th sourcs ar dntfd. Fgur 4.1d shows th rsults of th tm rvrsal procss n whch th sourc s dntfd by th fld pak. 53

Fgur 4.1 A typcal tm rvrsal procss: (a) forward flds xctd by a pont sourcs; (b) wav propagaton wthn th cavty and rcordd at th output nods; (c) r-njcton of th rcordd flds that ar tm-rvrsd at th output nods; (d) r-focusng of th flds at th orgnal sourc nods at th nd of th backward propagaton. Th fgur abov shows good focusng ffct at th orgnal sourc locaton wth th tm rvrsal; th sourcs locaton can b dntfd asly. Thr ar ssus wth th convntonal tm rvrsal mthod. Frst, sourcs ar smply locatd by pak dntfcaton. In th followng sctons, t can b shown by xampls and n thory that th pak valus may not always prsnt th tru locatons of th sourcs. whn multpl sourcs xst smultanously, ntrfrnc may occur among th sgnals xctd by ths sourcs, ladng to complxty and challngs n pak dntfcaton and locatng. 54

Scondly, n th smulatons so far, full-spctrum mpulss ar consdrd. In practc, only lmtd-band of th rcordd sgnals ar avalabl by masurmnts. Thrfor, t s much dsrabl to dvlop a TR mthod that accommodat th sgnals of lmtd-bands. Th motvaton of th work prsntd n th followng sctons s to addrss th abov two ssus and dvlop nw mthods and tchnqus whnvr propr and ndd. 4. THE LOCATION CONDITION FOR RECONSTRUCTION OF MULTIPLE SOURCES USING TIME REVERSAL METHOD 4..1 Thortc Analyss of Th Convntonal Tm Rvrsal Mthod In ths scton, wthout loss of gnralty, thortcal analyss of a typcal computatonal tm rvrsal s prformd. Thn, an rgodc ar-flld cavty boundd by prfctly lctrc conductors (PECs) [33, 56] s consdrd as a numrcal xampl and th fnt-dffrnc tm-doman (FDTD) mthod s mployd for smulatons. Wthn th cavty, multpl sourc and output (or rcvr) nods ar chosn and an lctrc fld mpuls xctaton s appld at th sourc nods. Frst, w consdr th forward propagaton (smulaton). Suppos that J sngl mpuls xctatons x j (n) ar njctd nto th cavty at J sourc nods. Mathmatcally, thy can b xprssd as: x [ n] = a d[ n], j= 1,..., J. (4.1) j j a j s th ampltud of th mpuls njctd at th j-th sourc nod at th tm stp n. δ s th Kronckr or unt mpuls functon and n s th tm ndx or tm stp wth th total numbr of tm stps bng N. Now consdr I prslctd output (or rcvr) nods. Th sgnals or flds rcordd at ach of thm can thn b xprssd as: y[ n] = a h [ n], = 1,..., I, n= 0,1,..., N -1 j j j= 1 J å. (4.) 55

whr hj[ n] s th mpuls rspons rcordd at th -th output nod du to th unt mpuls xctaton at th j-th sourc nod. If th transmsson ln matrx (TLM) mthod s appld, hj[ n ] s dfnd as an lmnt of th Johns matrx [57, 58]. Assum that th propagaton mdum s rcprocal. Th rspons at th -th nod du to th unt mpuls xctaton at th j-th nod s thn th sam as th rspons at th j- th nod du to th unt mpuls xctaton at th -th nod. In th subsqunt backward propagaton, th fld rsponss dscrbd by (4.) that wr rcordd at th -th nod, ar rvrsd n tm: r y [ n] = y[ N -1- n] = a h [ N -1-n] j j j= 1 J å. (4.3) Ths nvrs rspons, whn r-njctd nto th problm doman at th -th output nod, producs th followng output at th j-th sourc nod: r r S [ n] = h [ n] Ä y [ n] = h [ n] Äy [ n] j j j n å r = h [ m] y [ n- m], n= 0... N-1 m= 0 j. (4.4) Th total flds or sgnals at th j-th sourc nod wll b th sum of th rsponss du to th r-njctons at all th output nods: I I n r j å j åå j = 1 = 1 m= 0 S [ n] = S [ n] = h [ m] y [ n-m] I n J åå = h [ m] a h [ N -1- n+ m] j j ' j ' = 1 m= 0 j' = 1 I J n ååå = h [ m] a h [ N -1- n+ m] = 1 j' = 1m= 0 J I n åå j j ' j ' å å = a ( h [ m] h [ N -1- n+ m]) j ' j j ' j= 1 j' = 1 m= 0. (4.5) n whch m=0,..,n-1 s th dummy tm ndx for th convolutonal shft of tm nstancs. j ' s th ndx of sourc nods whch w hav ntroducd to dstngush t from opratons nvolvng th ndx j. At th last stp of th backward propagaton,.. n = N-1, (4.5) bcoms 56

I J N-1 åå å S [ N - 1] = a ( h [ m] h [ N -1- N + 1 + m]) j j ' j j ' = 1 j' = 1 m= 0 I N-1 I J N-1 å å å å å = a ( h [ m] h [ m]) + a ( h [ m] h [ m]) j j j j ' j j ' = 1, j= j' m= 0 = 1 j' = 1, j' ¹ j m= 0 I I J N-1 å å å å = ar [0] + a( h[ mh ] [ m]) j hjhj ' j j ' = 1 = 1 j' = 1, j' ¹ j m= 0 Th frst trm n th rght-hand sd (RHS) of (4.6), N-1 N-1 h [0] [ ] [ ] [ ] 0 jh = j j j = j ³ m= 0 m= 0 R h m h m h m. (4.6) å å, (4.7) s th aggrgaton of auto-corrlaton of h j [n] whch tnds to hav a rlatvly larg valu; t lads to th spatal and tmporal fld focusng or pak at th j-th sourc locaton. Th scond trm n th RHS of (4.6) s a summaton of cross-corrlatd trms gnratd by dffrnt sourcs. It rprsnts th rspons rcordd at on sourc causd by othr sourcs from th forward and backward propagaton. Snc ths sgnals ar not corrlatd, th scond trm has n gnral a rlatvly low valu n comparson wth th frst trm. Thrfor, (4.6) s n gnral domnatd by th frst trm and wll prsnt larg valus at th orgnal sourc locatons. Thn, th usual way to dntfy th sourc nod locatons (or to rconstruct th sourcs) s to fnd th hghst paks at th fnal tm of th backward propagaton. In th abov analyss, w assum that all h j [n] ar dffrnt from ach othr so that th auto-corrlaton occurs only at th orgnal sourc locaton j. For th cas wthout rgodc proprty, fals paks can also ars du to dgnrscnc of th fld rspons or symmtrs. h j [n] can b th sam for dffrnt valus of j and th rconstructon ylds paks at all thos fals postons. Evn n th rgodc cass, somtms, th scond trms or cross trms of (4.6) may stll bcom larg, dstroy th fld focusng, and ntrfr wth th abov pak dntfcaton procss; n othr words, th auto-corrlaton trms may not b suffcntly larg to b clarly dntfabl. In such a stuaton, th sourc rconstructon may fal f on rls only on th dntfcaton of th pak valus. 57

To llustrat th unsuccssful cas, xprmnts ar prformd on th cavty as shown n Fgur 4.. As sn, th sourcs ar dffcult to dntfy du to th fals paks rsultng from th larg uncorrlatd trms at non-sourc locatons (or du to nsuffcnt magntud of th paks at th orgnal sourc locatons). Th sourc rconstructon dos not work n ths cas. Ths phnomnon s also found n crtan cass modld by TLM mthod. Fgur 4. Two cass n whch th rconstructon of sourcs wth th tm rvrsal mthod s unsuccssful. (a) Th two orgnal sourc ampltuds (wth rd dots) ar qual. A pak s found at a sourc-fr locaton. (b)th two orgnal sourc ampltuds ar 1 and 0.3 (wth rd dots). Th sourc of ampltud 0.3 cannot b dntfd unambguously. In th nxt scton, followng up on th abov analyss, w drv a quanttatv condton for th xact dntfcaton of th sourc nod locatons vn whn th pak dntfcaton approach fals, or th fld focusng dos not occur. Th condton rqurs nthr th knowldg of th propagaton nvronmnt nor th dntfcaton of fld paks. 4.. Th Condton for Rconstructon of Multpl Sourcs In ths scton, w drv a mathmatcal condton for th accurat dntfcaton of multpl sourc-nod locatons. 58

Frst, w sum th squars of th fld rspons sampls (.. (4.)) rcordd at th -th output nod durng th forward propagaton: N-1 N-1 J åy [ n] = å( åajhj[ n]) n= 0 n= 0 j= 1 = N-1 J J ååå ( aa h[ nh ] [ n])) n= 0 j= 1 j' = 1 j j ' j j '. (4.8) Thn, unlk n th backward propagaton of th convntonal tm-rvrsal mthod whr th tm-rvrsd flds or sgnals rcordd ar rjctd nto th problm doman at all th output nods smultanously, w prform I sparat backward propagatons. Durng ach smulaton w r-njct only a sngl tm-rvrsd rspons nto a sngl output nod, say th -th nod, at whch that rspons has bn rcordd; w dnot th procss as th -th backward propagatons. At th fnal tm stp of th -th backward transmsson.. n=n-1, th fld rspons at th j-th sourc s: S [ N - 1] = S [ n] = ( h [ n] Ä y [ n]) r n= N - 1 j n= N -1 j = ( h [ n] Ä a h [ N -1-n]) N-1 j j ' j ' n= N -1 j '1 = = ( h [ n] Ä a h [ N -1-n]) = = å j j ' j ' n= N -1 j '1 = j j ' j ' m= 0 j' = 1 N -1 å m= 0 J J J h [ m] a h [ m] h [ m] y[ m] j å å å By comparng (4.8) and (4.9), w fnd that N-1 J n= 0 j= 1 j. (4.9) åy [ n] = å ajs [ N -1]. (4.10) j (4.10) stablshs th rlatonshp btwn th nrgy of rcvd sgnal y [n] (whch s rvrsd and r-transmttd) wth th fnal stat valu at th sourcs n th backward propagaton. Now w now rformulat (4.10) as: 59

J å j= 1 as j j j = 1, n whch S j = N -1 S [ N -1] å n= 0 y [ n]. (4.11) Th condton (4.11) must b satsfd at ach sourc locaton. In fact, s th total sgnal nrgy rcvd or rcordd at th -th output nod durng th forward propagaton. S [ N - 1] s th magntud of th sgnal rcordd at th j-th sourc locaton j at th last tm stp of th -th backward propagaton. N -1 å n= 0 y [ n] 4..3 Vrfcaton of th Condton A numrcal xprmnt was prformd to vrfy th condton (4.11). Wthout loss of gnralty, th FDTD mthod s appld to smulat a PEC rgodc cavty wth two sourcs and thr output locatons (.., J= and I=3). Th ampltuds of th two sourcs ar chosn to b 0. and 1, rspctvly. Frst, th forward propagaton s run and lctrc fld valus or sgnals ar rcordd at ach output nod. Th sgnals ar tm-rvrsd. Thn thr sparat backward propagatons ar run and n ach of thm, only on of th thr output nods s xctd wth th tm-rvrsd sgnal t has rcordd. Th fld dstrbuton at th nd of ach backward propagaton s rcordd. Aftr th backward propagatons, w chck whthr condton (4.11) s satsfd for any group of two nods n th soluton doman. If on of th groups dos satsfy (4.11), t wll rprsnt th actual sourcs. Mor spcfcally, w randomly group th nods nto pars and chck f th followng condton s satsfd for all j: j j 1 1 as 1 as + =. (4.1) j=1, and 3 rprsnt th backward transmssons wth th tm-rvrsd sgnals rnjctd nto ach of th 3 output locatons, rspctvly. 60

In ralty, chckng condton (4.1) can b mad quvalnt and asy by xamnng whthr th dtrmnant of matrx (4.13) blow s zro. If th dtrmnant s zro, th condton s satsfd and th sourc locatons ar found. és ê ês ê ês ë 1 1 1 S 1 S 3 3 1 S -1ù ú -1ú ú -1ú û. (4.13) Fgur 4.3 shows th dtrmnant valus of (4.13) for 10,000 par group of sourc canddats. It appars that only on group among ths 10,000 ylds a dtrmnant that s sgnfcantly smallr than all th othrs by about ght ordrs of magntud. Ths partcular group turns out to corrspond to th orgnal sourc locatons usd n th forward propagaton. Onc th sourcs ar corrctly locatd, thr ampltuds ar asly solvd wth quaton (1). Fgur 4.3 Dtrmnant of matrx (4.13) for 10000 groups of two nods. Among thm only th dtrmnant of on group has a vry small valu clos to zro. Ths group turns out to b th corrct par of sourc nod. 4.3 APPLICATION OF THE CONDITION FOR RECONSTRUCTION OF MULTIPLE SOURCES 61

4.3.1 Th Proposd Mthod to Fnd th Sourc Locatons In th abov xampl, w assum that w hav a pror knowldg of th numbr of th sourcs bng two. In practc, howvr, t s not known. A squntal tral-andrror approach can b mployd to apply th condton to sourc rconstructon. Mor spcfcally, w may frst start to assum th numbr of th sourcs to b 1 and prform th tst dscrbd abov; f th condton s satsfd, th sourc locaton s found. If not, w thn assum th numbr of th sourcs s and prform th tst, and so on. In thory, ths approach wll allow us to fnd th sourc locatons. In ralty, howvr, ts computatonal xpns wll b prohbtv, spcally for an lctrcally larg structur. Hnc, an ffcnt mthod s dsrd to apply th condton for sourc rconstructon. In th followng paragraphs, w propos th rgularzd last squar (RLS) mthod. Assum M locatons as th possbl sourc locatons. Basd on th condton (4.10), consdr that at non-sourc locatons th ampltuds should b 0; that s, (4.10) can b rwrttn as a condton that ampltuds of all ths assumd M sourcs must satsfy: éa1 ù ê ú N -1 ê ú é ù 1 1 y1 [ n] és1... S ù M ê ú êå ú n= 0 ê ú ê ú ê ú ê......... ú ê... ú = ê... ú ês... S ú ê ú ê ú ë û ê ú ê ú êa ú ë M û I I N -1 1 M I* M ê ú å yi [ n] êën = 0 úûi *1 M*1, (4.14) whr th lmnt S m s th fld valu rcordd at th assumd m-th sourc locaton at th nd of th -th backward transmsson. a m, m=1 M, s th ampltud of th mpuls xctaton at th assumd m-th sourc locaton. Snc a m s th ampltud of th mpuls xctaton at th sourc locaton, t wll hav a non-zro valu at a tru sourc locaton and zro at all othr non-sourc locatons. 6

In othr words, th xact solutons of (4.14) can automatcally yld tru sourc locatons and rmov th non-sourc locatons by dntfyng non-zro lmnts of [a m ]. Equaton (4.14) rprsnts I quatons wth M unknowns. Usually, M s much largr than I. Hnc th quaton s undrdtrmnd. It can thn b solvd n th last squar (LS) mannr. Morovr, th smallst numbr of non-zro lmnts a m s th constrant to th LS systm, ladng to a rgularzd last squar (RLS) formulaton,.. th soluton a m s spars. To fnd th spars soluton, a L-1 rgularzaton trm s addd to th undrdtrmnd quaton. Th formulaton can thn b xprssd as: N é ù mnmz ( [ S][ a] - ê yj [ n] ú + l a) n= 1 ë å (4.15) n whch λ s th rgularzaton factor chosn mprcally. Th constrant quaton (4.15) s to fnd th sparsst soluton vctor, whch can b solvd numrcally by applyng a soluton algorthm such as LASSO [59-61]. Th rsultng ampltud coffcnts [a m ] that ar non-zro dntfy th sourc locatons. 4.3. Numrcal Exprmnt wth th Proposd RLS Mthod û In th followng numrcal xampl, w choos thr sourc nods and sx output nods wthn th cavty doman. W thn apply th proposd RLS mthod by slctng 1500 nods wthn an ara that s assumd to contan th thr sourc nods. M s thus qual to 1500. Usng th algorthm dscrbd n [59], w comput th ampltuds a j ; th valu of th rgularzaton factor λ s chosn mprcally to b btwn 0.001 and 0.0. Th optmal choc of λ s a subjct of futur rsarch. Th rsults of th computaton ar shown n Fgur 4.4. As can b sn, at only thr nod locatons, a j s non-zro; thy turn out to b th corrct sourc locatons. Th computd sourc ampltuds ar also clos to th tru valus. If ndd, th xact ampltuds can b found by solvng (10) for th dntfd sourc locatons. Thrfor, ths xampl provs th ffctvnss of th proposd mthod. 63

Fgur 4.4 Rsult of sourc rconstructons wth th proposd mthod. Th dots n th lft rgon of th problm doman ndcat th ght output nods. Th sourcs ar shown as th paks. (a) shows th orgnal sourc locatons and thr ampltuds. (b) prsnts th rsults obtand wth th RLS mthod. As a summary of ths scton, w hav frst dvlopd a condton for rconstructon of multpl sourcs. It s vrfd by th numrcal xprmnts. Th condton can b usd as a valdaton of th possbl sourcs dntfd. In othr words, whn w dntfy th sourcs usng th convntonal tm rvrsal procss, th condton can b usd as a tst to s f th sourcs dntfd ar ndd tru. Nxt, basd on th proposd condton, w hav dvlopd a RLS schm to drctly calculat th possbl sourc locatons by solvng th undrdtrmnd quaton. Th rsult shows th ffctvnss and rlablty of th proposd mthod vn n th cass that th convntonal tm rvrsal has dffculty n dntfyng all sourcs. 4.4 SOURCE RECONSTRUCTION WITH REALISTIC BAND-LIMITED FREQUENCY DOMAIN SIGNALS W hav so far usd tm doman mpuls rsponss of nfntly larg frquncy band, whch yld hgh spatal rsolutons. Howvr, n ralstc stuatons, th fld rsponss at output nods ar oftn masurd or rcordd n frquncy doman wthn a 64

lmtd frquncy band. Drct applcaton of rgular frquncy-to-tm transformaton tchnqus,.g. nvrs Fourr transform, to convrt ths band-lmtd frquncydoman fld rsponss nto thr tm doman countrparts, most lkly lad to noncausal complx tm-doman sgnals; thy cannot b usd n th tm rvrsal procss. Thrfor, t s ncssary to dvlop a mthod to xtract causal tm-doman rsponss from band-lmtd frquncy doman masurmnts so thy can b usd for th tm rvrsal sourc rconstructon. In th followng subsctons, w dvlop an xtracton mthod spcfcally for th tm-rvrsal mthod. 4.4.1 Rconstructon of Flds from Band-Lmtd Frquncy Doman Rsponss or Masurmnts To bttr xplan th mthod, w us agan th numrcal xprmnt of th rgodc cavty of Fgur 4. and Fgur 4.4. Th frquncy doman sgnals at th output nods s frst obtand by th dscrt Fourr transform (DFT) of th tm-doman rsponss rcordd at th pr-slctd output nods. At th -th output nod, th rcordd tm-doman sgnal or fld s xprssd by (4.). Its frquncy doman corrspondnt s Yk [ ] = DFTyn ( [ ]), k= 0,1,,..., N- 1 (4.16) whr DFT s th dscrt Fourr transform, k s th frquncy stp ndx whch rprsnts th frquncy pont of kd f = k / Dtwth D t bng th FDTD tm stp. To mulat th band-lmtd fld rsponss, w rmov th frquncy contnts outsd a prslctd frquncy band of [ kdf, k D f] such that th followng bandlmtd rspons s usd now for th tm rvrsal sourc rconstructons: Fgur 4.5. [ ], [, ] c ìyk w Î kl kh Y [ k] = í î0, w Ï[ kl, kh] l h. (4.17) Th graphcal rprsntatons of th band-lmtd rsponss (4.17) ar shown n 65

Fgur 4.5 Craton of th band-lmtd frquncy doman flds. Th top two fgurs ar th magntud and phas of th frquncy-doman flds rcordd n a full spctrum. Th bottom two fgurs show th band-lmtd spctrum that rsults from th rmoval of th contnts outsd th prslctd frquncy band of [k l, k h ]. If th nvrs Fourr transform s drctly appld to (4.17) (whch rprsnts th band-lmtd sgnals), th rsultng tm doman rspons bcoms complx and noncausal; hnc, t cannot b usd n th tm rvrsal procss. To construct th usabl tm doman quvalnt rspons, w propos to approxmat th band-lmtd rspons as follows: kh c c k c y [ n] = Y ( k) cos( p n +Ð Y ( k)), n= 0,1,,..., N -1 N å. (4.18) k= k l It s a summaton of monochromatc cosn functons. Each cosn functon has a frquncy of kd f = k / D t, an ampltud of Y c c j (k) and a phas Ð Y ( k) (4.18) s a ral tm-doman squnc., rspctvly. By takng th DFT of (4.18), w can show that t has xactly th sam valus as Y [k] wthn th band of ntrst[ kdf, k D f]. Fgur 4.6 shows th tm and frquncy l h doman rprsntaton of (4.18) and compars t to th orgnal tm rspons (4.16). 66

Fgur 4.6 Comparson btwn orgnal and th rconstructd tm rsponss. Th two fgurs on th lft ar th tm-doman sgnals and th fgurs on th rght ar th frquncy doman sgnals. Th top two fgurs show th orgnal rsponss, and th bottom two fgurs show th rsponss rconstructd wth (4.18). Aftr (4.18) s computd for ach output nod, t s tm-rvrsd and r-njctd nto th cavty, and th backward smulaton s run by followng th procdur dscrbd n Scton II. At th fnal stp of th backward smulaton, th fld dstrbuton shows clar paks wthn th structur as Fgur 4.7 shows. Fgur 4.7 Sourcs rconstructd from band-lmtd rsponss, yldng wll-dfnd paks at th orgnal sourc nods. Mor numrcal xprmnts wr prformd wth dffrnt chocs of sourcs and output nods as wll as dffrnt lmtd frquncy bands. Th rsults ar all smlar and unambguous. Th rason can b thortcally xpland as follows. 67

Smlar to (4.9), th rspons S [ N - 1] at th j-th sourc locaton at th nd of th -th backward smulaton s: j N -1 c j j n= 0 S [ N 1] h [ n] y [ n] - =å (4.19) (4.19) shows that th fnal stat valu of backward transmsson s th corrlaton product of constructd sgnal and th corrspondng mpuls rspons. On th othr hand, basd on Planchrl thorm, th corrlaton product of th tm doman sgnals s quvalnt to thr conjugat-complx multplcaton n frquncy doman. That s, whr 1 N-1 N-1 c c * åhj[ n] y [ n] = å Hj[ k]( Y [ k]). (4.0) n= 0 N k= 0 Aftr furthr manpulaton, w hav H [ k] = DFT( h [ n]), j c c Y [ k] = DFT( y [ n]). 1 S [ N 1] H [ k] a ( H [ k]) N -1 j c - = å j å j ' j ' N k= 0 j' kh J k h 1 1 * å j j å å j ' j j ' N k= k, j= j' N j', j¹ j' k= k = a H [ k] + a H [ k] H [ k] l j l * (4.1) (4.) Th frst trm on th rght-hand sd s th summaton of th postv numbrs whch aggrgat and form a pak valu. Th scond trms of th rght-hand sd rprsnt th cross-trms btwn dffrnt sourcs and thy wll not accumulat constructvly n most cass; thrfor, thy do not ntrfr wth th pak valus. Snc th addton of th frst trm only taks plac at th sourc nods, focusng occurs at th sourc locatons, vry much lk what s dscrbd for convntonal TR focusng n Scton 4.3. In short, (4.) provds th thortcal foundaton for th proposd mthod n whch a band-lmtd rspons s usd to rconstruct ts sourcs wth th tm-rvrsal mthod. 68

4.4. Th Condton for Sourc Rconstructon from Band-Lmtd Fld Rsponss In th prvous sctons, w propos a mthod to rconstruct sourcs from bandlmtd fld rsponss va tm-rvrsal and fld pak dntfcaton. Lk that dscrbd n Scton II, th mthod should work most of th tm but may fal n som spcal cass. Hr w drv th sourc locatng condton that dos not rqur th pak dntfcatons for th sourc locatons wth th band-lmtd rsponss. From (4.19) w hav N -1 c S [ N - 1] = h [ n] y [ n] j n= 0 J J N-1 c j j - = å jå j j= 1 j= 1 n= 0 å as[ N 1] a h[ ny ] [ n] = = å N-1 n= 0 j= 1 N -1 n= 0 j J åå å c ah[ ny ] [ n] c y[ n] y [ n] j j, (4.3) n whch S j [N-1] s th fld rspons rcordd at th j-th sourc nod, at th nd of th - th backward smulaton durng whch th rcvd sgnal at th -th output nod s rvrsd and r-njctd nto th doman at th -th output nod. From our prvous analyss, w hav 1 y[ n] y [ n] Y[ k] Y [ k] N-1 N-1 c * c = n= 0 N k= 0 å å (4.4) whr Y c [k] s dfnd by (4.17). From th band-lmtd proprty of Y c [k], w can gt 1 1 N N-1 N-1 N-1 * c c c YkY [ ] [ k] = Y [ k] = y [ n] k= 0 N k= 0 n= 0 å å å (4.5) Th last qualty n th abov quaton s obtand wth Pasrval s thory. From (4.4) and (4.5), 69

Basd on (4.3) and (4.6), w rach N-1 N-1 c c åy[ n] y [ n] = å y [ n]. (4.6) n= 0 n= 0 N-1 J c y [ n] = ajsj[ N -1] n= 0 j= 1 å å (4.7) Smlar to (4.10), (4.7) shows th rlatonshp btwn th nrgy of transmttd sgnal and fnal stat valu at sourc locatons n th backward propagaton. Th only dffrnc wth (4.10) s that hr w us th constructd sgnal y c [n] n th backward propagaton. (4.7) stablshs th condton for th band-lmtd flds that prmt th locatng of th sourcs wthout dntfcaton of fld paks. 4.4.3 Th RLS Mthod wth th Band-lmtd Fld Rsponss Followng th procdur usd n scton II, th undrdtrmnd quaton s constructd basd on (4.7) and solvd wth th RLS procss and th constrant of mnmum non-zro numbr of coffcnts a j. A numrcal xampl wth four sourcs and ght output nods has bn computd, wth M=1500 slctd. Th rsults ar shown n Fgur 4.8. It s sn that th sourcs ar rconstructd xactly at thr orgnal locatons, and thr ampltuds ar vry clos to th orgnal valus. 70

Fgur 4.8 Sourc rconstructon from th band-lmtd fld rsponss by applyng th sourc locatng condton (4.7). Th locatons of th ght output nods ar ndcatd by th dots n th lft rgon of th problm doman. Th sourcs ar dntfd by th paks. (a) shows th orgnal sourcs, (b) shows th rconstructd sourcs. Th rctangular boxd ara s th doman whr M=1500 nods ar usd as sourc locaton canddats for sttng up th undrdtrmnd systm for sourc rconstructons. Th abov analyss and numrcal xprmnts dmonstrat that th proposd mthod to rconstruct mpulsv sourcs from band-lmtd rsponss s vabl. It maks th proposd tm-rvrsal mthod applcabl to ral-world problms. 4.5 SUMMARY In ths chaptr, w prsnt a thortcal analyss of sourc rconstructon usng th tm rvrsal mthod wth numrcal mthods lk FDTD. W also dvlop a ncssary condton for rconstructng multpl sourc locatons and vrfy t wth both thortcal analyss and numrcal xprmnts. Th condton dos not nvolv Grn s functons or dntfcaton of fld pak valus. Furthrmor, w dvlop a mthod to procss th band-lmtd flds or sgnals avalabl n ralty so that thy can b usd wth th tm rvrsal tchnqu for multpl sourc rconstructons. Agan, thortcal analyss and numrcal xprmnts ar provdd to support and vrfy th mthod. 71