On the Numerical Solutions of Two Dimensional Maxwell s Equations

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1 ues oea ceces : 8-88 I -9 IOI Pubcaos O e umeca ouos of Two mesoa Mawe s uaos.. weam M.M. Kae a.t. Moame epame of Maemacs acu of ceces Cao Uves Ga gp epame of Maemacs acu of ceces Bea Uves Bea gp epame of Basc ceces Moe Acaem fo geeg a Tecoog Cao gp Absac: I s pape wo ffee meos ae sue o sove umeca me epee Mawe s euaos wo mesos. Te meos ae e scouous Gae Meo GM a e Vaaoa Ieao Meo VIM. scouous Gae meo s ese fo ffee umbe of aguao eemes ffee me evas a ffee oes of bass fucos. Compasos bewee e souos of Mawe s euaos usg GM a VIM ae pesee. I s fou a VIM s va fo age me bu GM ovecome s pobem. Ke wos: scouous Gae meo me-oma mawe s euaos wo mesos vaaoa eao meo ITROUCTIO Mawe s euaos ae oe of e mpoa moes ffee fes; escbes eecomagec peomea suc as cues mco ao a aa waves. I s we ow a Mawe s euaos ae a o sove aaca ca be sove aaca o fo smpe omas suc as a spee a a fe ccua ce []. umeca meos fo e Mawe s euaos ae usua efee o as Compuaoa ecomagec CM. Te moeg of ssems ae vovg eecomagec waves s ow we oe oug e souo of e me oma Mawe euao o space g. uc sems wee sove w ma ffee meos e fs meo fo e umeca smuao of me epee eecomagec waves e e ffeece Tme oma Meo TM was popose []. I s cea a TM ow ue s smpc a effcec bu s ffcu o geeae o usucue o-caesa gs a suffe fom e accuae epeseao of e souo o cuve bouaes []. Moeove TM as accuac mao fo seco oe accuac sevee ms e ab o coec epese wave moo ove og saces uess e g s pobve fe []. Ma ffee pes of meos ave bee popose wc ae base o usucue gs a ca ea w compe geomees e e eme Tme oma Meos TMs [4 5]. Tee ae wo ffcues appea we usg e saa e eme Meo M. s e meo geea use o a goba cofomg mes a mea a mes wou agg coecg oes o msmac of mes pos aog ea bouaes. eco ow we ca epese coe sguaes [4 6]. Te scouous Gae Meos GM meos [7-] ae base o scouous fe eeme spaces. I s eas ae eemes of vaous pes a sapes egua o-cofomg meses a eve oca vag pooma egee. Moeove cou s wea efoce acoss mes efaces b ag suabe bea foms e so-cae umeca fues o e saa vaaoa fomuaos []. e o eaea meses usg Lagage poomas [ ] o o eaea meses usg poucs of Lagage poomas []. O e oe a Vaaoa Ieao Meo VIM [4-] s popose b J.. e [6 7] as a mofcao of a geea Lagage mupe meo. Ts ecue poves a seuece of fucos wc coveges o e eac souo of e pobem. I as bee sow a s poceue s a powefu oo fo sovg vaous s of pobems. Ts ecue soves e pobem wou a ee o sceao of e vaabes eefoe s o affece b comp uao ou off eos a oe s o face w ecess of age compue memo a me. Aso s ecue poves e souo of e pobem a cose fom we e mes po ecues suc as e fe ffeece meo pove e appomao a mes pos o. Ts pape s ogae as foows: I seco e moe pobem a e agom of souo ae Coespog Auo:... weam epame of Maemacs acu of ceces Cao Uves Ga gp 8

2 ues oea c. : 8-88 ouce. I seco VIM s ouce o oba e appomae souo of Mawe s euaos. Te as seco 4 gves a scusso of ou esus. GM OR T MOL PROBLM Le us cose e wo mesoa vacuum Mawe s euaos; wc s ow as Tasvese Magec fom TM [9]; efe Ω [ ] [ ] [ T]. wee ae e compoes of e magec fe a s e eecc fe -eco. uemoe s cae magec pemeab a s cae eecc pemv. o e boua coos assume a e wa of e cav s pefec eecca coucg suc a e agea compoe of e eecc fe vases a e wa.e. a e was aso e a coos of e pobem ae pace sceao: a Assume a e compuaoa oma Ω s compose of K ooveappg -smpces o eemes. Ω Ω K wee s a sag se age a e aguao s assume o be geomeca cofomg; fo eampe Ω s appomae b a pecewse ea pogo w eac e segme beg a face of age. b Assume a e oca souo of e veco of e fe [] T a e oca fu ca be efe: p. wee p s e mumesoa Lagage poomas efe b some g pos a efe as P p spa j ; j p s m s wee m ae aba cosas. o smpc e us we Mawe s euao e fu fom [9]: w a Q.. T [ ] Q e e I e foowg we epa bef e basc ea of GM o sove. e fs pa we pese e space sceao a e seco pa we pese e me sceao. 8 c To mpeme e umeca appomao mup euao. w es fuco e egag b pas wce s ea us o. * [ ] ˆ..4 wc epese e sog fom of GM fo pobem.. Assume a e oca souo s epesse as p p ˆ ψ.5

3 ψ p wee { } ues oea c. : 8-88 s a geue wo mesoa pooma bass of oe a p s e umbe of ems e oca epaso wc eae o e oe of e pooma w e eao p. e Te we ave..4 a.5 a afe some eaageme we ge: p p j. j j j p * ˆ.6 j j wc s e sem scee fom of Mawe s euaos wee e mass ma M j j a ca be we as M J I T J v v j.7 wee suc a a j j j j j j j j a e eeme of ege egao ma j ˆ j J I j j.8.9 a w be cacuae e same wa e e sffess ma fo moe eaes fo asfomao a cacuao see [7 ]. Aso o compue e suabe fu we efe Z ± ± Z ± ± w bouaes ˆ ˆ Aso e sffess ma ˆ ˆ Z Z Te e peag boua em ow efe as; ˆ [[ ]] ˆ ˆ [[ ]]. ˆ Z Z ˆ [[ ]] αˆ ˆ [[ ]] ˆ α. f ow e oca semscee sceme of Mawe s euaos geg afe sovg usg GM a fom. a. ae gve b:.a JM ˆ [ ] α ˆ [[ ]] [ ] [[ ]] [ ] JM [ ] ˆ α ˆ [ ] [ ] α JM ˆ ˆ.b.c Tme sceao: To compee e semscee fomuao..a-..c use a epc aw soage Rug- Kua meo o egae me w p sages p s e oe of e pooma of e bass fucos. Te me sep use e compuao s ae eusca o be CL p m ma c wee c 84

4 ues oea c. : 8-88 Tabe : Te mamum eo of e umeca souo a ffee vaues of p a ffee mes T p 4 p 8 p p A e e-6.4e e-4.4e e e e e e e-6.98e-7 A e-8.768e-.85e e e-.4e-..586e e-.6e-.8.74e-7.596e e- A e e e-.445e e-4.97e e-.4656e-.56e e-.666e e- 9.65e e- 5.e-.7e Appomae Appomae g. : umeca souo fo usg GM a p a g. : umeca souo fo usg GM a p a 8 s e wave spee e - eeme; s e eg mamum amee of e eeme; a CLp coo pca aes vaues of /p. Te umeca esus fo sovg Mawe s euao usg e GM ae sow Tabe. Ts abe peses e mamum eo of e souo fo ffee mes umbe of eemes K a e oe of pooma p. UMRICAL IMPLMTATIO O VIM I s seco we app VIM o oba e appomae souo of. w e a a coos. Accog o VIM [6] we ca cosuc e foowg eao fomua: Appomae g. : umeca souo fo usg GM a 5 p a

5 ues oea c. : Appomae ac Z g. 4: Te umeca souo usg VIM a.4 afe 6 eao ef. ac souo fo g ~. ~ ~ wee a ae geea Lagage mupes wc ca be efe opma va vaaoa eo. Te seco em o e g-a se. s cae e coeco a e subscp eoes e - oe appomao. Ue a suabe esce vaaoa assumpo.e. ~ ~ a ~ s cosee as a esce vaao we ca assume a e above coecoa fucoa ae saoa.e. ~ δ ~ δ a ~ δ e e Lagage mupes ca be efe []. Te successve appomaos a of e souo w be ea obae upo usg e obae Lagage mupes a suabe seecve fucos a. Cacuag vaao w espec o a we ca oba e foowg ffeea euaos:. Te Lagage mupes eefoe ca be efe as:. ubsug e efe mupe. o. esus e foowg eao fomua.4 We ca sa w e gve a appomao usg a coos a b e fomua.4 some appomae souos ae se beow:

6 ues oea c. : 8-88 Tabe : Te mamum eo of e umeca souo a ffee vaues of mes T Ma eo T Ma eo s ms m m m m s ms s m s s m s 87 s m s s m s s m s m s m s s m s ece e souo s ma be obae usg: m s ms s m s s m s m.5 Te beavo of e appomae souo of e VIM a e eac souo s gve g. 4 a.4. Te umeca esus fo sovg Mawe s euao usg e VIM ae sow Tabe. Ts abe peses e mamum eo of e souo fo ffee mes. COCLUIO A RMARK I s wo e scouous Gae me oma meo a VIM ae use o su e umeca souo of Mawe s euaos womesos. Te GM s appe fo ffee umbe of eemes ffee me a ffee oe of poomas. ee we e moes mus be sma eoug fo eas souo f beg age gve moe oscaos fo sma moes gve ve goo accuac aso mus be ege o fco. Aoug VIM s a ve smpe meo a as ma avaages suc as ees o sceao space a me vaabes o ee o sove ea o oea ssem of euaos bu fom e obae umeca esus Tabe we fou a VIM s va fo age me a GM ovecome suc s pobem. RRC. Aesso U.. Tme oma Meos fo e Mawe uaos. ocoa sseao Roa Isue of Tecoog ocom.. Tafove A Compuaoa ecoamcs: Te e-ffeece Tme- oma Meo Aec ouse Boso.. ee K umeca souo of a boua vaue pobems vovg Mawe s euao soopc mea. I Tas. Aeas Popagae 4: mes A. a P. Jo 997. e eemes a mass umpg fo Mawe s euaos: Te case umeca Aass C.R. Aca c. Pas 4: Moma. a. Abuasa 5. Appcao of e s vaaoa eao meo o emo euao. Caos oos a acas 7: Mo P.. e eme Meo fo Mawe s uao. Ofo Uves ew o.

7 ues oea c. : Cocbu B. G.. Kaaas a C.W. u. s couous Gae Meos. Teo Compuao a Appcaos. Vo. of Lecue oes Compuaoa cece a geeg pge-veag Be. 8. Kög M. K. Busc a J. egema. Te scouous Gae me-oma meo fo Mawe s euaos w asoopc maeas. Poocs a aosucues-uameas a 9. Goe M.J. A.. Be a. coa 6. Ieo pea scouous Gae meo fo Mawe s euaos: L-om eo esmaes. Pep o. 6- ovembe 6. eave J.. a T. Wabuo 8. oa scouous Gae Meos: Agom Aass a Appcaos pge.. Wabuo T. a J. esave. oa g-oe meos o usucue gs. I: Tme oma souo of Mawe's euaos. J. Compu. Ps. 8 : as. a. Lae. A g-oe ocofomg scouous Gae meo fo me-oma eecomagec. Joua of Compuaoa a Appe Maemacs 4: Wabuo T. a J. esave 4. g-oe oa scouous Gae meos fo e Mawe egevaue pobem. Pos. Tas. Ro. oc. Loo e. A 6: Abass T.A. M.A. -Taw a. Zoe 7. ovg o-ea paa ffeea euaos usg e mofe vaaoa eao Paé ecue. Joua of Compuaoa a Appe Maemacs 7: Abou M.A. a A.A. oma 4. Vaaoa eao meo fo sovg Buge s a coupe Buge s euaos. J. of Compuaoa a Appe Maemacs : e J Vaaoa eao meo-a of o-ea aaca ecue: some ea mpes. Ieaoa Joua of o-lea Mecacs 4: e J Vaaoa eao meo fo ea ffeea euaos. Comm. o-lea c. ume. muao : weam.. a M.M. Kae 7. Vaaoa eao meo fo oe mesoa o-ea emo -easc Caos oos a acas : weam.. a M.M. Kae. Covegece of vaaoa eao meo appe o o-ea coupe ssem of paa ffeea euaos. I. J. of Compue Ma. 87 5: -.. weam.. M.M. Kae a R.. A-Ba 8. o-ea focusg Maaov ssems b vaaoa eao meo a Aoma ecomposo meo. Joua of Pscs: Cofeece ees 96: -7.. as.. Lae a. Rape 8. eveopme of a o-cofomg scouous Gae meo o smpe meses fo eecomagec wave popagao. ou Ieaoa Cofeece o Avace Compuaoa Meos geeg ACOM.. sgos L ffeea uaos a e Cacuus of Vaaos. Tasae fom e Russa b G. aovs M Moscow. Appcaos 8:

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