The Finite Difference Time Domain (FDTD) Method Discretizing Maxwell s equation in space and time

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1 Chape 3 The Fe Dffeece Te Doa FDTD Mehod I hs chape we povde a evew of heoecal echques of FDTD ehod eploed he cue wo. Ou sulaos ae based o he well-ow fe-dffeece e-doa FDTD[] echque. The FDTD ehod s a goous soluo o Mawell's equaos ad does o have a appoaos o heoecal escos. Ths ehod s wdel used as a popagao soluo echque egaed opcs especall suaos whee soluos obaed va he Plae Wave paso PW [34] ehod cao cope wh he sucue geoe o ae o adequae soluos. FDTD s a dec soluo of Mawell s cul equaos ad heefoe cludes a oe effecs ha a soluo of he oochoac wave equao. Whle os of hese echques ae esg developed ehods he a hee s o povde a copehesve pcue of hese ehods. 3.. Dsceg Mawell s equao space ad e FDTD Mehod Ioduco Fsl oduced b Kae Yee 966 [] he FDTD appoach s based o a dec uecal soluo of he e-depede Mawell's cul equaos b usg he Fe-Dffeece echque. Ths leads o a full-wave aalss fo all wavelegh foao whou a pesupo o he aeal odel ad o he sucue. Mawell's equao hee desos 3D The e-depede Mawell's equao fo he hoogeeous aeal wh o elecc o agec cue souces ca be epessed as:. s Whee s he delecc pev s he coducv ad s he agec peeabl of he vacuu. The efacve de s gve b ω We ow we ou he veco copoes of he above equaos Caesa coodaes.

2 Ths elds he followg sse of s coupled scala equaos: 3.3c 3.3b 3.3a 4c b 4a 3. Mawell's equaos wo desos D I D sulao we assue ha he sucue beg odeled eeds o f he -deco ad he phooc devce s lad ou he - plae. The popagao s alog o. Ths assupo eoves all he devaves fo Mawell's equaos ad spls he o wo T ad TM depede ses of equaos. T ode Mawell's equaos D I he D T cases Mawell's equaos ae he followg fo: 3.5a 3.5b 3.5c I geeal s he ao copoe he sulao. TM ode Mawell's equaos D I he D TM cases Mawell's equaos ae he followg fo: 3.6a 3.6b 3.6c I geeal s he ao copoe he sulao. -

3 Yee's Algoh-FDTD equaos The Yee algoh solves boh he elecc ad agec felds Mawell's equaos e doa ad space doa b usg he fe-dffeece echque. I cees s ad copoes hee-desoal space so ha eve copoe s suouded b a cculag copoe ad eve copoe s suouded b a cculag copoe. Fe Dffeeces ad oao I geeal a space po he Caesa sse a ufo ecagula lace ca be deoed as: 3.7 whee ad ae he space seps he hee decos. Followg hs a fuco u of space ad e doa evaluaed a a dscee po he ecagula lace ca be deoed as: u 3.8 u u whee Δ s he e cee assued ufo ove he obsevao eval ad s a ege. B usg he ceeed fe-dffeece ehod ha ogall s deved fo Talo's sees epaso he paal space devave of u -deco a he fed e ca be epessed as: u u u u Ο[ ] 3.9 Thee Desoal FDTD equaos I a 3D sulao he sulao doa s a cubc bo he space seps ae ΔΔ ad Δ ad decos especvel. ach feld copoe s peseed b a 3D aa. The feld copoe posos Yee's Cell ae show Fgue 3.. Ths placee ad he oao cause he ad copoes o be eleaved a evals of space ad fo he pupose of pleeg a leapfog algoh. 3

4 Fgue 3. I a Yee cell deso A A A oe how he feld s copued a pos shfed oe-half gd spacg fo he feld gd pos []. Now we ca appl he above fe-dffeece deas oao ad feld dsplacee o acheve a uecal appoao of Mawell's equao The FDTD equao ca be we as: - 3.a - 3.b c

5 3.a - 3.b - 3.c D T Wave FDTD equao The D copuaoal doa s show Fgue 3.. The space seps he ad decos ae Δ ad Δ especvel. ach esh po s assocaed wh a specfc pe of aeal ad coas foao abou s popees such as efacve de ad dspeso paaees - Fgue 3. Nuecal epeseao of he D copuaoal doa 5

6 I he D T sulao ach feld s epeseed b a D aa ad coespodg o he D esh gd show Fgue 3.. The dces ad accou fo he ube of space seps he X ad Z decos especvel. The locao of he felds he esh s show Fgue 3.3. Fgue 3.3 Locao of he T felds he copuaoal doa a 3.3b 3.3c D TM wave FDTD equao I he D TM sulao ach feld s epeseed b a D aa ad coespodg o he D esh gd gve Fgue 3.. The locao of he TM felds he copuaoal doa follows he sae phlosoph ad s show Fgue

7 Fgue 3.4 Locao of he TM felds he copuaoal a 3.4b c Space sep ad e sep The fudaeal cosa of he FDTD ehod s he sep se boh fo he space ad e. Space ad e seps elae o he accuac uecal dspeso ad sabl of he FDTD ehod. Ma efeeces ad boos dscuss hese pobles [45]. I geeal o esue ha he esuls ae accuae ad have a low uecal dspesve he esh se s ofe quoed as " cells pe wavelegh" eag ha he sde of each cell should be λ/ o less a he hghes fequec shoes wavelegh. Noe ha FDTD ehod s voluec copuaoal ehod so ha f soe poo of he copuaoal space s flled wh peeable aeal we us use he wavelegh he aeal o deee he au cell se. The followg equao s fo he suggesed esh se 7

8 u λ 3.5 a whee a s he au efacve de value he copuaoal doa. Oce he cell se s deeed he au se fo he e sep edael follows he Coua-Fedchs-Lev CFL codo. Fo a 3D FDTD sulao he CFL codo s: v 3.6 whee v s he speed of he lgh edu. Fo a D sulao he above CFL codo ca be splfed as: v 3.7 FDTD Bouda codo The bouda codos [89] a he spaal edges of he copuaoal doa us be caefull cosdeed. Ma sulaos eplo a absobg bouda codo ha elaes a ouwad popagag eeg ha pges o he doa boudaes. Oe of he os effecve s he pefecl ached lae PML [8] whch boh elecc ad agec coducves ae oduced such a wa ha he wave pedace eas cosa absobg he eeg whou ducg eflecos. Peodc bouda codos PBC Peodc bouda codos PBC [9] ae also poa because of he applcabl o PBG sucues. Thee ae a ube of vaaos o he PBC bu he all shae he sae coo head: he bouda codo s chose such ha he sulao s equvale o a fe sucue coposed of he basc copuaoal doa epeaed edlessl all desos. PBCs ae os ofe eploed whe aalg peodc sucues. Peodc bouda codos ae que useful whe wog wh peodc sucue pes. A peodc bouda spulaes ha a feld whch leaves he bouda o oe sde of he doa should eee he doa o he oppose sde. Ths ca be epessed aheacall as ep whee he sucue s assued o be peodc alog he coodae wh peod ad a phase dffeece. The peod s defed b he legh of he doa 8

9 alog he specfed coodae. Peodc boudaes ca ol be appled o a cobao of he hee coodaes bu cao be appled o each bouda dvduall. Sec Sec bouda codos assue ha he feld s sec abou oe o oe plaes of efleco ad ae also ow as eve bouda codos. Ths s useful whe boh he sucue ad he sulaed felds have se abou oe o oe plaes of efleco. Ths ca be epessed aheacall as - whee s he specfed coodae abou whch he feld s o be assued sec. The poso of he efleco plae s gve b he doa defo. Sec bouda codos ca be appled o a cobao of he hee coodaes as well as each of he s boudaes dvduall. A-Sec A-Sec bouda codos assue ha he feld s a-sec abou oe oe oe plaes of efleco ad ae also ow as odd bouda codos. The feld s heefoe se o a he bouda ad he felds o boh sdes of he bouda ae assued o be of oppose sg. Ths ca be useful fo sec sucues whch suppo a asec feld fo sace a-sec ode - whee s he specfed coodae abou whch he feld s o be assued a-sec. The poso of he efleco plae s gve b he doa defo. A-sec bouda codos ca be appled o a cobao of he hee coodaes as well as each of he s boudaes dvduall. Ipu Wave The FDTD uecal schee elds he soluo of a al value pobles The algoh eques he al feld ecao ha wll be popagaed hough he copuaoal doa. The Ipu Wave us coa fou ds of he foao: Ipu wave popagao deco Ipu wave foao e doa 3 Ipu wave pofle asvese plae 4 Polaao. Toal-Feld/Scaeed-Feld TF/SF esuls fo aeps o eale a wave popagao a desed deco. The copuaoal doa s sepaaed o wo sub-egos he oal feld ego ad he efleced feld ego. The plae sepaag hese egos s called he cde feld pu plae see Fgue

10 Fgue 3.5 Toal/Refleced feld foulao I he Toal Feld Rego he wavegude sucues of ees ae desged. The eaco bewee he cde feld ad he wavegude sucue ae place hs ego. Ths s wh he Toal Feld Rego coas foao fo boh he cde ad scaeed efleced waves. I he Refleced Feld Rego he geoe s ufo ad he popagag waves ae peseed b he felds efleced fo he Toal Feld Rego. Thee ae o obecs hs ego ad he sgal wll o be efleced bac o he oal feld ego. Couous wave CW o pulsed ecaos ca be used. You ca cosde he cde feld as beg geeaed b a flashlgh locaed o he cdece plae facg he Z deco. Befoe sag he sulao he flashlgh s ued off ad he feld values he whole copuaoal doa ae equal o eo. The flashlgh s swched o a ad lluaes ol he Toal Feld Rego. If he ecao schee s pefec hee should o be a lgh deeced b a obseve locaed he Refleced Feld Rego uless hee ae soe obsacles whch would geeae he eflecos. The cde wave ca be geeaed b specfg he eac feld dsbuo o he cde plae a each e eval. The TF/SF eques he specal eae fo FDTD equao he Ipu plae. Ipu Wave foao e doa Two ds of e doa pu wave foaos ca be sulaed he FDTD. Oe s fo he sgle wavelegh sulao-he Couous Wave CW pu. The ohe s fo he wde bad sulao-he Gaussa Modulaed Couous Wave GMCW. I geeal s called Pulsed Ipu. 3

11 CW cao I CW ecao he e depedece of he cde feld s a sgle fequec susodal fuco. Fo eaple he cde feld has he followg fo: c c AT F c s ω θ 3.8 whee A s he feld aplude AF c s he asvese feld dsbuo a he cde plae locao c. The al phase offse θ s he phase dffeece bewee pos he cdece plae. Ths offse ca be adused o defe he deco of he cde feld. ω π/λc s he fequec of he pu wave. I he CW case he opcal wave aalog popagaes ul eaches he saoa sae evewhee he copuaoal wdow. Gaussa Modulaed Couous Wave GMCW cao Pulsed cao Fo pulsed ecaos he cde feld has he fo: c whee c AF X Z c s ω θ c c AT F c s ω θ T ep off s he pulse evelope fuco _ ff s he e offse ad _ s he pulse wdh paaee. Fo pulsed ecaos he e seppg coues ul he desed lae-e pulse espose s obseved a he feld pos of ees. Gaussa Tasvese ecao Fo Gaussa Tasvese ecao he asvese feld has a Gaussa pofle ha ca be calculaed fo he followg equao: ep[ X ]ep[ Y ] 3. 3

12 Whee s he cee po whee he Gaussa bea has he pea value. X ad Y ae he half wdhs. Maeal Model [67] Oe of he a advaages of he FDTD appoach s he lac of appoaos fo he popagag feld-lgh s odeled s full chess ad cople. The ohe sgfca advaage s he gea vae of aeals ha ca be cossel odeled wh he FDTD coe. ee we ae a bef evew of soe of he a aeal popees ha ca be hadled. Loss deleccs Befoe poceedg wh a oe dealed descpo should be ephased ha he fac ha he e doa all he felds ae eal quaes. Thus accoug fo loss s possble ol hough a o-eo coducv s of he edu: s ω - ω ω eff.3 ω ee we have assued ha e ad ω coespods o e-o-fequec doa Foue asfo. The eal ad aga pa of he pev ca be epessed hough he eal ad aga pa of he Iaga efacve de: Re I -κ κ κ - ω/ e 3.4 Ths aes he efacve de appoach ad he coducv appoach equvale. Sellee equao odel I he Sellee equao odel he aeal cosa efacve de fo he use specfed wavelegh s calculaed b he Sellee equao: A A λ M λ Γ λ 3.5 Whee A s he Segh λ s he esoa wavelegh Γ he dapg coeffce ad 3

13 λ s he wog wavelegh. Noe ha hee ae wo applcaos fo he Sellee equao odel he FDTD. Oe s used o calculaed he cosa efacve de fo a aeal. I s he cosa efacve de coupled o he Mawell's equao ad o he Sellee equao coupled o Mawell's equao so such a sulao he aeal sll does o have he dspesve popees. The secod applcao fo Sellee equao s o asfe he Sellee odel o Loe odel ha wll be coupled o Mawell's equao. I such a sulao he aeal wll have he dspesve. Loe Dspeso aeal B Loe dspeso aeals we ea aeals fo whch he fequec depedece of he delecc pev ca be descbed b a su of ulple esoace Loea fucos: N N χ Gω ω G ω Γ ω ω whee ω G Γ ae he esoa fequeces s elaed o he osllao seghs s he dapg coeffce χ - s s he pev a fe fequec s he pev a ω 3.6 I he lossless case quao 3.6 s decl elaed o he Sellee equao whch he hee esoaces ca be peseed as: Aλ Aλ Aλ A G 3 χ λ λ λ λ λ λ I he loss case he Sellee equao ca be we a geealed fo accoug fo a o-eo dapg coeffce Γ as well as fo asoop he dspeso popees: λ λ λ ω A A A λ Γ λ λ λ Γ λ λ λ Γ λ λ λ λ λ Thee ae dffee was o plee quao 3.6 o he FDTD foals. ee we cosde he so-called polaao equao appoach he sgle esoace case. I uses he 33

14 delecc suscepbl fuco: χ ω χ ω ω Γω ω 3.9 ad he elao bewee he polaao ad he elecc feld. Tag he Foue asfo of he las equao leads o he followg dffeeal equao: P P Γ ω P χ ω 3.3 Dude odel aeal Dude odel s aohe dspeso effec. I os cases sulaes he oble eal o he suface plasa. Maeals wh Dude dspeso ae opcall lea aeals fo whch he fequec depedece of he delecc pev ca be descbed b he Dude Dspeso elao: ω p ω Γω ω 3.3 whee ω p s he plasa fequec ad Γs he dapg ae. The above equao ol descbes he dspeso elao fequec doa. Because FDTD s a e doa ehod he Dude odel us be asfeed o he e doa ad solved e doa. Fo Dude aeal he coespodg Mawell's equaos ae: J J ΓJ ω p 3.3 The FDTD schee fo Dude aeal ca be deved fo he above equaos b usg he fe-dffeece echque. 34

15 Dspesve Nolea aeals I geeal he olea behavo s due o he depedece of he polaao P o he appled elecc feld. Assug a soopc dspesve aeal Mawell's equaos ae: Whee D L NL D P P P L Γ P L P D ω P L χ G ω L P epeses he lea polaao geeal P L L 3.33 s he dspesve polaao whch s coolled b Loe odel quao 3.33 ad deoes he olea polaao. L P D P Pos Daa Aalss Dsceed Foue Tasfo DFT ad Fas Foue Tasfo FFT FDTD s e doa sulao ehod. I ca oba all he desed specal esposes hough a oe-e sulao. To oba he specal espose ou us use he DFT ad FFT ad Aalss. Dsceed Foue Tasfo obas a sgle wavelegh espose fo a e sees. whee s s he e doa espose N s he e seps ube ad ω s he agle fequec. Whe DFT s ug he sulao obas he fequec doa espose fo he cee wavelegh ol whle DFT fo he obsevao po aea ad le povdes he specal espose fo a sees of use-specfed waveleghs. Fas Foue Tasfo uses he adoal fas Foue asfo schee o oba a specal espose fo he eo fequec o he cuoff fequec. The fequec doa saplg sep s N. I geeal he saplg fequec sep fo FFT s copaable o he eesed wavelegh due o he fac ha FDTD equed e sep s ve sall. So he FFT esuls a have lage eos ha ha of he DFT esuls. Bu FFT s uch fase ha DFT. 35

16 Powe calculao ad Pog veco Fo he -deco popagao wave. The oal powe he - plae ca be dvded o wo powe values: -deco polaed -deco popagao powe P- ad -deco polaed -deco popagao powe P-. The coespodg foulas ae: polaao polaao Toal powe P powe powe P Z X P P P Z Re Re Re s s dd s * dd * * dd Re s dd * 3.34 Whee he cap do eas he cople value whch coes fo he DFT calculao ad he supescp sa eas he cople cougae value. The -deco Pog fo a po - plae s S * * 3.35 Pog veco s a cople value. I FDTD ol he aplude s show. 3. Advaages vesus dawbacs of he FDTD ehod The FDTD appoach has seveal e advaages ove all of he pecedg ehods as well as seveal dawbacs. The leapfog e seppg echas used s full eplc hee b copleel avods he pobles assocaed wh sula hs ehod scale as N N beg he ube of eal space dsceao pos. The ehod fuhe poses o esco o he pe of souces used ae o plae wave bu fo quau do po souce o Gaussa beas. Moe poa s he fac ha b usg FDTD ehod we ae able o accou fo he feess of he sucue all 3-D. Moeove he ehod allows fo he eplc eaao of he e develope of M waves he sucue ad heefoe s he bes sued algoh fo vesgag wave gudg echass ad cav couplg. 36

17 oweve hee ae soe ahe seous dawbacs o hs ehod. Fo eaple o calculae he assso ad efleco coeffce a Gaussa fequec pulse s lauched o he plae pefoed o hese values ad copoe of he pog veco pepedcula o he coespodg value of a efeece edu o eld he assso ad efleco coeffces. Ths s oe edous ad sesve o eos ha he Tasfe Ma Mehod. Refeeces. Basc FDTD eadgs []Yee K. S. "Nuecal soluo of al bouda value pobles volvg Mawell's equaos soopc eda" I Tasacos o Aeas ad Popagao []. J. D. Joaapolous R. D. Meade ad J. N. W Phooc Csals Moldg he Flow of Lgh Pceo Uves Pess 995. [3] K. Saoda Opcal Popees of Phooc Csals Spge-Velag. [4]Chu S. T. Chaudhu S.K. "A fe-dffeece e-doa ehod fo he desg ad aalss of guded-wave opcal sucues" Joual of Lghwave Techolog [5]Taflove A. agess S. "Copuaoal lecodacs: The Fe-Dffeece Te-Doa Mehod" Secod edo Ahech ouse Boso.. Maeal odels [6]Zolows R. W. "Icopoao of coscopc aeal odels o FDTD appoach fo ulafas opcal popagao" I Tasacos o Aeas ad Popagao [7] Lag T. Zolows R. W. "Dspeso effecs o gag-asssed oupu couples ude ula-fas pulse ecaos" Mcowave ad Op. Tech. Le Pefecl Mached Lae PML bouda codos [8] Béege J. P. "A pefecl ached lae fo he absopo of elecoagec waves" Joual of Copuaoal Phscs Gede S. D. "A asoopc pefecl ached lae absobg eda fo he ucao of FDTD laces" I Tasacos o Aeas ad Popagao [9] Taflove A. "Advaces Copuaoal lecodacs The Fe-Dffeece Te-Doa Mehod" Aech ouse Boso Ch