17 NO 9 SEPTEMBER 1999 HighDensity Integrated Optics C Manolatou Steven G Johnson Shanhui Fan Pierre R Villeneuve H A Haus and J D Joannopoulos Abstract This paper presents twodimensional 2D 64257nite difference time domain FDTD si ID: 25356
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JOURNALOFLIGHTWAVETECHNOLOGY,VOL.17,NO.9,SEPTEMBER1999High-DensityIntegratedOpticsC.Manolatou,StevenG.Johnson,ShanhuiFan,PierreR.Villeneuve,H.A.Haus,andJ.D.JoannopoulosAbstractÐThispaperpresentstwo-dimensional(2-D)nitedifferencetimedomain(FDTD)simulationsoflow-lossright-anglewaveguidebends,T-junctionsandcrossings,basedonhighindex-contrastwaveguides.Suchstructuresareessentialforthedenseintegrationofopticalcomponents.Excellentperformancecharacteristicsareobtainedbydesigningthewaveguideintersec-tionregionsaslow-resonantcavitieswithcertainsymmetriesandsmallradiationloss.Asimpleanalysis,basedoncoupled m,chosentoensuresinglemodeoperationovertheentirebandwidthoftheexcitation.Thefundamentalmodeofthesewaveguidesisstronglyguidedwitheffectiveindex2.457at1.55 m.Thesourceisa50-fsor25-fswideGaussianpulsemodulating m.Thespatialdependenceofthesourceisthefundamentalwaveguidemodeprolewhichiscalculatedalongacrosssectionofthe2-Dwaveguide(slabmode).Theelectriceldpolarizationischosenperpendiculartotheplane.Thepowertransmittedthroughawaveguidecross Waveguidebendsarebasicstructuresforopticalintercon-nectsandthereforeveryimportantcomponentsinopticalintegratedcircuits.Anyabruptdirectionalchangeindielectric1999IEEE MANOLATOUetal.:HIGH-DENSITYINTEGRATEDOPTICS Fig.1.Schematicofa90sharpwaveguidebendmadeofahighindexsurroundedbyair.Thecrosssectionswheretransmissionandre ectionarecalculatedarealsoshown.waveguidescausesarotationofthephasefrontsofthepropagatingmodes.Thisperturbationcausesmodeconversionintounguidedmodesgivingrisetoradiationloss,mostlyattheoutersideofthecorner.Proposedmethodsforreducingtheradiationlossinwaveguidebendsincludedeceleratingthephase-frontinsidetheabruptbend[4],acceleratingthephase-frontoutsideanabruptbend[5],addingamicroprisminthebendregion[6]andreplacingabendwithsuccessivebendsofsmallerangle[7].Thesestudieswereconcernedwithsmall-anglebends.Forbendsoflargeranglescornerre ectorshavebeenproposed[8].Low-loss90 -bendswithsmallradiusofcurvatureandwideanglesplittersmadeofhighindex-contrastwaveguides(Si/SiO)havebeenreportedin[9].Theirperfomancereliesonstrongconnementinthesewaveguides.Inthiswork,theperformanceof90 sharpbendsisgreatlyenhancedbyplacingaresonantcavityatthecorner.Aschematicofasimple waveguidebendisshowninFig.1.Theelectriceldinthisbendandthetransmissionandre ectionspectraareshowninFig.2(a)and(b),respectively.Duetomodemismatchatthecorneralargefractionofthepowerislosttoradiationorre ectedbackwardresultinginverypoortransmission.ThewaveguidebendisnowmodiedbyplacingaresonantstructureatthecornerasshownintheschematicofFig.3.Theideaisbasedontheprincipleofasymmetricresonatorwithtwoports.Atresonance,thetransmissioniscompletewithnore ectioniftheresonatorislossless.Ifthisideaisappliedtoindexguidedstructures,radiationlossisunavoidableleadingtononzerore ectionandreducedtransmission.Theeffectsofradiationcanbecounteractedbymakingtheexternal oftheresonatorverysmall.Thisisachievedbystrongcouplingofthewaveguidemodestotheresonatormode.Thisconceptissimplyexplainedusingcouplingofmodesintime[10].Becausethisanalysisisbasedonperturbationtheory,itcanonlyprovideaqualitativepredictioninthecaseofstrongcouplingbetweenthecavityandthewaveguidemodes.Followingtheapproachof[10],theamplitudeofthemodeinthecavityisdenotedby andisnormalizedtotheenergyinthemode.Thedecayratesofthemodeamplitudeduetothecouplingtothetwowaveguidesare and respectively,relatedtotheexternal 'sby and where istheresonancefrequency. (a) Fig.2.(a)Electriceldamplitudedistributioninaplain90-bend.(b)Transmissionandre ectionspectraoftheplainbend. Fig.3.Schematicofatwo-portresonatorconnectedtothewaveguidesofthe90Thedecayrateduetoradiationlossis incoming(outgoing)wavesatthetwoportsaredenotedby and (Fig.3)andarenormalizedtothe JOURNALOFLIGHTWAVETECHNOLOGY,VOL.17,NO.9,SEPTEMBER1999 Fig.4.Cornerregionofbendmodiedbyformingasquareresonatorinsidethecorner.powercarriedbythewaveguidemode.Iftheexcitationis witha timedependence,and thenatsteadystatewehave[10] (1)and which,dueto(1)nallygive (3) (4)At there ectioniszeroandthetransmissionmaximizedif Thus,asymmetricsystem allowscompletetransmissionatresonanceprovidedthatitislossless.Thewidthofthefrequencyresponseisdeterminedby Iflossispresenttheratio thepeaktransmissionandminimumre ectionas WemodifythecornerintoaresonantcavitybyincreasingthevolumeofdielectricatthisregiontoformasquareresonatorasshowninFig.4.Fig.5(a)showstheelectriceldpatternofabendwithasquareresonatorofside m,denedintheschematic,whereitcanbeseenthattheradiationhasbeenreducedcomparedwithFig.2(a). (a) Fig.5.(a)Electriceldamplitudeinthemodiedbendforasquareofsidem.(b)Transmissionandre ectionspectraofthemodiedbend(solidline)comparedwiththeplainbend(dashedline).ComparingthespectraofthemodiedandunmodiedbendinFig.5(b)wecanseethatthere ectionhasdroppedbyanorderofmagnitudeandthepeaktransmissionhasmorethandoubledtoabout70%.Theperformanceisstillpoorbecauseofconsiderableradiationlossthatresultsinarelativelyhigh in(6),estimatedfromthetransmissionpeaktobe0.44.The and estimatedfrom(4)areabout13and30,Inordertoreducetheloss,weneedaresonantcavitythatiscapableofsteeringthewaveguidemodearoundthecornerwithverylowradiationloss.Thiscouldbeachievedbyplacingaquarter-diskatthecornerasacurvedFabry±Perotresonator.Thehighindex-contrastpermitsaverysmallradiusofcurvaturewithoutsignicantradiationloss.However,itisdesirabletoavoidcurvedinterfacesthatarehardertofabricate,soinsteadwefurthermodifythecornerbymakinga45 cut MANOLATOUetal.:HIGH-DENSITYINTEGRATEDOPTICS Fig.6.Schematicofasquareresonatoratthebendwitha45-cutatitsoutercorner.attheoutercornerofthesquareasshownintheschematic(Fig.6).Inthiscongurationthestronglyguidedwaveguidemodeundergoestotalinternalre ectionatthe45 cutandisguidedaroundthemodiedcornerbytheouterwalls.Thevolumeofdielectricinthestructuredeterminesthewavelengthofmaximumtransmissionandthedepth ofthecutaffectsthebandwidthoftheresponse.Inthenumericalsimulationswehaveobservedthatasthedepthofthecutisincreasedthetransmissionandthebandwidthincreaseuptoacertainpoint.Furtherincreaseofthedepthofthecutresultsinnarrowerbandwidthandslightlylowerpeaktransmission.Inordertokeepthewavelengthofmaximumtransmissionaround1550nm,thesideofthesquaremustbemadelargertocompensateforthedielectrictakenoutbythecut.Inmostofthecaseswehaveexamined,thebestperformanceintermsofmaximumtransmissionandbandwidth,isachievedwhenthedepthofthe45 cutissuchthattheshapeoftheresonantstructureofFig.6isroughlyaquarterofanoctagon.InFig.7(a)and(b),weshowtheelectriceldamplitudeandthespectrumforaquarter-octagonwith m, mwhichresultedfromthesquareofFig.4(b)bymakinga45 cutandincreasingthesizetokeepthecenterwavelengththesame.Thetransmissionisabove98%withamaximumof98.6%,andthere ectionisontheorderof10 overabandwidthofmorethan120nm.Forcomparison,wealsoshowtheeldamplitudeandspectrumofaquarter-diskofradius0.68 minFig.8.Thesizeofthequarter-diskwaschosentogiveawavelengthresponseascloseaspossibletothatofthequarter-octagon.Fromtheeldpatternsweseethatthemodeisguidedmostlybytheouterwallsfollowingalmostthesamepathinbothcases.Anintegernumberofhalfwavelengths(inthiscase3)ªtºinthemodiedcorneraswewouldexpectforaFabry±Perotresonator.Intheabovecasesthestrongcouplingintothecavityleadstoverylowªloadedº andverywidetransmissionbandwidths,sotheresonantnatureisnotobvious.Infact,theshapeofthestructurewehaveusedcombinestheapproachesof[4]and[8].Resonantbehaviorcanbeseenmoreclearlyifwemakethecavitylarger.Fig.9(a)and(b)showstheeldamplitudeforcavitieswith m, mand m, m,respectively(roughlyquarteroctagons),thatsupporthigherordermodesinthebandwidth (a) Fig.7.(a)Electriceldamplitudeinthemodiedbend,form.(b)Transmissionandre ectionspectraofthemodiedbend(solidline)comparedtotheplainbend(dashedline).ofinterest.Theelectriceldamplitudedistributionsshowclearlythatthereisenergystorageinthecavity,andthatthemodepatternisnotwellmatchedtothewaveguidemode.InFig.9(c)thetransmissionspectrumtakenoverawiderbandwidthshowsthatthesestructuressupportpairsofbroadresonancescloselyspacedinwavelengththattendtomergeintooneverybroadpeak,asinthecaseofthelargercavityofFig.9(b).Similarbehaviorissuggestedalsobytheshapeandwidthofthetransmissionspectrumfor m,showninthesameplot.III.T-JAwaveguidecomponentusefulforpowersplittingorcom-biningisaT-junctionwhichisessentiallya -branchof90 half-angle.Moststudiesintheliteraturehavebeenconcerned JOURNALOFLIGHTWAVETECHNOLOGY,VOL.17,NO.9,SEPTEMBER1999 (a) Fig.8.(a)Quarter-diskresonantcavityatthecornerwithradius0.68(b)Comparisonofthewavelengthresponseforquarter-diskofradius0.68(solidline)andquarter-octagonofm(dashedline).withthedesignofsmall-angle -branchesthatrequirelargedevicelengthstoachievealow-losstransitiontothetwoarms.Indexorheighttaperinghavebeenusedforadiabaticmodeevolution[11],[12],andintegratedmicroprismshavebeenproposedforphase-frontcompensationinaschemeanalogoustothatemployedforlossreductioninbends[13].Thestructurespresentedintheprevioussectionforthelow-lossbendcanbeusedwithsmallmodicationsinaT-junction.WeconsiderrstaplainT-junctionshownintheschematic(Fig.10).TheelectriceldamplitudeofaT-junctionandthetransmissionandre ectionspectraareshowninFig.11(a)and(b).Duetosymmetrythetransmittedpowerisequallydistributedinthetwooutputs,therefore,thetotalpowertransmittedintothetwoportsisplottedtoshowtheanalogywiththe90 -bend. (a)(b) Fig.9.(a),(b)Electriceldamplitudeinlargercavitiesthatrevealstheenergystorage(a)m(b)(c)Transmissionandre ectionspectracomparedform(solidline),1m(dashedline)and1.2m(dottedline). Fig.10.SchematicofaplainT-junctionmadeofahighindexwaveguidesurroundedbyair.Thecrosssectionswheretransmissionandre ectionarecalculatedarealsoshown.PlacingaresonantcavityatthejunctionandtreatingtheT-junctionasathree-portsystem(Fig.12),theanalysisofSectionIIcanbeextendedbydeningincoming(outgoing) andaddingadecayrateforthethirdport MANOLATOUetal.:HIGH-DENSITYINTEGRATEDOPTICS (a) Fig.11.(a)ElectriceldamplitudeinaplainT-junction.(b)Totaltrans-missionandre ectionspectraoftheplainT-junction. Excitingport1with atfrequency wehave Theoutgoingwavesaregivenbyexpressionssimilarto(2)anddueto(7)nallybecome (8) (9) From(8)wecanseethatinordertotransfertheinputpowerintoports2and3withoutre ectionatresonance,thedecay Fig.12.Schematicofathree-portresonatorconnectedtothewaveguidesoftheT-junction. Fig.13.ModiedT-junctionforimprovedperformance,resultingfromtwolow-loss90-bendsplacedback-to-back.ratesmustsatisfy Theratioofthepoweratthetwooutputportsisthen soif thepowerisequallysplitbetweenthetwoports.Notethatfornegligibleloss,condition(11)isinagreementwiththewellknowntheoremofscatteringmatrixtheoryaccordingtowhichitisimpossibletomatchallthreeportsofareciprocallosslessthree-portsystem.Thuswecannotmakeare ectionless branchwiththree-foldsymmetry ThechoiceoftheresonantstructuretobeplacedatthewaveguideintersectionfollowsfromregardingtheT-junctionastwo90 -bendsthatlookinoppositedirections,placedbacktoback.AsshowninFig.13,themodiedT-junctionemploysacavitythatismadeupoftwostronglycoupledcavities,eachcapableofdirectingawaveinoneofthetwodirectionsaswehaveshowninSectionII.TheperformanceofthemodiedT-junctionisimprovedbymakingtheverticalsideofthecavityshorterthanthehorizontal.Furtherimprovementofthetransmissionandre ectionresponseisobtainediftheanglebetweenthecutandthesymmetryaxisofthestructureismadeslightlysmallerthan45 .Theelectriceldamplitudeforthebestcase,with m, m, m, m,isshowninFig.14(a)andthespectruminFig.14(b).Wecanseethatthebehaviorofthisstructureisremarkablysimilartothelow-loss90 -bend.Themaximumtotaltransmissioniscloseto99%andtheminimumre ection JOURNALOFLIGHTWAVETECHNOLOGY,VOL.17,NO.9,SEPTEMBER1999 (a) Fig.14.(a)Electriceldamplitudeinalow-lossT-junctionwithm.(b)Transmissionandre ectionspectraofthemodiedT-junction(solidline)comparedwiththeplainT(dashedline).ontheorderof10 .Inanalogywiththebend,largerresonantstructuressupporthigherordermodesinthebandwidthofinterestandthetransmissioncurvesshowpairsofcloselyspacedbroadresonances.Itisintrerestingtoseehowthepowerdistributionatthetwoportschanges,iftheinputwaveguideisdisplacedfromthesymmetryplaneofthecavity.Fig.16showsthetransmissionandre ectionfordifferentdisplacements thewaveguideaxiswithrespecttothesymmetryplaneofthecavity,denedintheschematicofFig.15.ThesymmetricTthatwestartfromhas m, m,and m.Bymovingthewaveguidetotheleft,thetransmissionattheleftport(port2)increases.Althoughthetotalstructureisnolongersymmetric,theconstraint(11)isstillroughlysatisedasissuggestedbythefactthatthere ectionremainslowevenforallthedisplacementsfromthecenterandthetotaltransmittedpowerneverfallsbelow95%.Theeldamplitudeinthecasethatthewaveguidewallcoincideswiththesymmetryplaneoftheresonantstructure nm)isshowninFig.17(a).Thegoodperformanceofthisasymmetricstructuresuggeststhatitispossibletomakealow-lossbranchingstructurebykeepingtherightpartthe Fig.15.AsymmetricT-junctionresultingfromdisplacingtheinputwave-guidefromthesymmetryaxisofthecavity. Fig.16.Powerdistributionandre ectionattheportsofanasymmetricT-junctionfordisplacementoftheinputwaveguideintherangefrom0to100nm.sameandreplacingtheleftbyacontinuationoftheinputwaveguide.TheelectriceldamplitudeforthiscaseisshowninFig.17(b).Thetransmissionandre ectionspectraofthetwoasymmetricstructuresareverysimilaraswecanseeinFig.17(c).Therefore,properchoiceofthecavitydimensionsallowstotapoffacertainpercentageofthepowerfromastraightwaveguidewithlowlossandre ection.IV.WAVEGUIDEWaveguidecrossingsareimportantcomponentsofanop-ticalintegratedcircuitastheyallowtheinteractionofmany MANOLATOUetal.:HIGH-DENSITYINTEGRATEDOPTICS (a)(b) Fig.17.Electriceldamplitudein(a)asymmetricTwith=100and(b)branchingstructureresultingfromasymmetricT.(c)Transmissionandre ectionspectraofbranchingstructure(solidline)andasymmetricT(dashedline).waveguideswithinalimitedspace.Ideallythewavelaunchedinoneoftheintersectingwaveguidesmustbefullytrans-mittedforwardwithoutanypowerescapingsidewaysandwithnegligiblere ectionandradiationloss.Modicationofthewaveguideintersectionforcrosstalkreduction,hasbeenemployedelsewhere,e.g.,[14],butforshallowanglecrossingsandwithapenaltyinthroughput.Thecrosstalkatrightanglecrossingsisverysmallwhenthewaveguidesaremanywavelengthswidebutcloseto10%forwaveguidesontheorderoftheopticalwavelenghasisthecaseinhighindex-contraststructures.Ahighindex-contrastwaveguidecrossingandtherelevantcrosssectionsforthecalculationoftransmission,crosstalkandre ection,areshowninFig.18.TheelectriceldamplitudeandthewavelengthresponseareshowninFig.19(a)and(b),respectively.Althoughmostoftheinputpowerisdirectedforward,nearly10%oftheinputpowerescapessidewaysleadingtounwantedcrosstalk.Thewaytoachievethedesiredperformanceisbasedagainonthemodicationoftheinter-sectionintoaresonantcavity.Asexplainedin[2],thecavity Fig.18.Schematicofaplaincrossingofhighindexwaveguidesinair,showingalsothecrosssectionswheretransmission,crosstalkandre ectionarecalculated. (a) Fig.19.(a)Electriceldamplitudeinaplainwaveguidecrossing(b)Transmissionandcrosstalkspectraofthewaveguidecrossing.musthavefourforldsymmetryanditssymmetryplanesmustcoincidewiththewaveguideaxes.Thecavitymustsupportonlytwodegeneratemodesinthefrequencyrangeofinterest, JOURNALOFLIGHTWAVETECHNOLOGY,VOL.17,NO.9,SEPTEMBER1999 Fig.20.Schematicofafour-portresonantcavityplacedatthewaveguideonemodewithoddsymmetrywithrespecttothehorizontalwaveguideaxisandevensymmetrywithrespecttotheverticalwaveguideaxis,andonemodewiththeoppositesymmetries.Undertheseconditions,eachresonantmodewillcoupletothemodeofjustonewaveguideandwillbeorthogonaltothemodeoftheperpendicularwaveguide.Thus,excitationbythemodeoftheverticalwaveguidewillleavetheotherwaveguideunexcitedleadingtofulltransmissionwithzerore ection(asdiscussedinSectionII,forlosslesscavity),andnocrosstalk.Thisschemehasbeentestedin[2]byFDTDsimulationsincrossesmadeofphotoniccrystalwaveguidesandhighindexcontrastwaveguideswithairholes,yieldinguptoeightordersofmagnitudereductionincrosstalk.Wecanuseanalternativedecompositionintosymmetricmodestoexplainthisideawithcouplingofmodesintime.Thecavityattheintersectionisregardedasafour-portsystem(Fig.20),thatsupportstwomodeswiththesameresonant Duetothefourfoldsymmetry,eachmodecanbechosentocoupleequallyintoallfourports.Theiramplitudesaredenotedby and andthedecayratesintoeachportby and ,respectively.Withinput ,atsteadystatewehave Thedesiredbehaviorofthecrossingisachievedifthetwomodescoupleinphaseintoports1and2andoutofphaseintoports3and4.Then,assumingforsimplicityrealcouplingcoefcientsandretainingonlytherelativephasebetweenthetwomodes,theamplitudesoftheoutgoingwavesare (13) (a)(b) Fig.21.Possibleimplementationsofacavityattheintersectionwith(a)one,(b)two,and(c)three,pairsofcutsoneachofthecrossingwaveguides.which,dueto(12),give (14) (15) (16) (17)At there ectionandthetransmissionsidewaysarezeroandtheincomingsignalisfullytransmittedforward.If,in theoutputsatports3and4remainzeroatallfrequencies.Thenthefrequencyresponseattheports1and2isessentiallythatofthesymmetrictwo-portresonantsystemofSectionIIwithdecayrate ateachport.Notethat and arethesumandthedifference,respectively,ofthemodesusedintheanalysisof[2].SimpleimplementationsofacavityattheintersectionareshowninFig.21.Thenumericalsimulationsrevealthedependenceoftheresponseondifferentdesignparameters.Inthesimplecase(a),theresonancefrequencyismorestronglyaffectedbythesizeoftheblockatthecenter.Thewidthofthegroovesdeterminesthecouplingintothecavityand MANOLATOUetal.:HIGH-DENSITYINTEGRATEDOPTICS Fig.22.Wavelengthresponseforthethreemodiedcrossingscomparedwiththeplaincrossing(solidline):onepairofcuts(dashedline),twopairsofcuts(dottedline),andthreepairscuts(dashed-dottedline).inasmallerdegreetheresonancewavelength;widergroovesresultinweakercouplingandthereforenarrowerresonanceandtheresonanceshiftstoshorterwavelengthduetotheloweringoftheaveragedielectric.Similareffectsoccurbymakingmorecuts.Anadditionalpairofcutsasincase(b)yieldsnarrowerresonance,andfurtherreductionofthecrosstalk.Thepeaktransmissionisalsoincreasedbecausetheradiationlossisgreatlyreducedinthecavity.Furtherincreaseofthenumberofcuts,asin(c),lowerstheaveragedielectricandshiftstheresonancetoshorterwavelengths.Moreoever,assumingsameradiationlossasin(b),theincreaseintheexternal duetoweakercouplingleadstoaslightdecreaseofthepeaktransmission.Thebestcompromiseforthisparticularcongurationiswhentwocutsaremade.Thepeaktransmissionforthiscaseis96%andthecrosstalklessthan10 .ThecomparisonofthespectraofallthreecaseswiththeplaincrossisshowninFig.22for m, mand nm.The associatedwiththecouplingateachoneofthefourports,and relatedtoradiationlosses,areestimatedfrom(15)for Wendapproximately: (b) and(c) Theeldpatternsforcases(a)and(b)inFig.23,showaclearreductionofcrosstalkwithadditionalV.DWehaveshownthatwaveguidecomponentsforopticalin-terconnectionswithexcellentperformancecharacteristicscanbedesignedbyplacingalow- resonantcavityatthewave-guideintersection,aspredictedbycouplingofmodesintime (a) Fig.23.Electriceldamplitudeforacrosswith(a)oneand(b)twopairsofcuts.analysis.Thesuccessofthisconceptreliesonthestronglightconnementinhighindex-contrastwaveguides,whichallowstheuseofcavitiesontheorderoftheguidedwavelengthandwithcouplingtoradiationmuchlowerthanthecouplingtothewaveguide.Wehaveillustratedtheoperationalprinciplesby2-Dnumericalsimulationsofacollectionofexamplestructuresmadeofhighindex-contrastsingle-modewaveguides.Theconceptpresentedhereisverygeneralandcouldbeappliedusingavarietyofresonantstructures,thereforeourresultsdo JOURNALOFLIGHTWAVETECHNOLOGY,VOL.17,NO.9,SEPTEMBER1999nothavethecharacterofanoptimized,nalizeddesign.Thenumericalsimulationsshowedthatalthoughthepeaktransmis-sionissensitivetosmallsizechanges,thewidebandwidthoftheresponseallowsacertainfarbricationtolerance.Becauseweuseopendielectricstructuresthatemployindexguiding,loweringtheindexcontrastdeterioratestheperformanceduetoincreasedcouplingtoradiation.Thesizeofthewaveguidesandresonatorsisstronglydependentonthematerialsystemandfurtherinvestigationisrequiredtorevealpossiblescalinglaws.Anotherissue,characteristicofhighindex-contrastsystems,isthepolarizationdependencewhichisexpectedtostronglyaffectthedesignandtheperformance.Anaccurateestimationofthepolarizationdependenceaswellastheleakageofradiationintothesubstraterequires3Dsimulationsforamorerealisticdesign.TheauthorswouldliketoacknowledgethatthenumericalsimulationswereperformedontheNSFSanDiegoSuper-computerCenter'sCrayT90.[1]A.Mekis,J.C.Chen,I.Kurland,S.Fan,P.R.Villeneuve,andJ.D.Joannopoulos,ªHightransmissionthroughsharpbendsinphotoniccrystalwaveguides,ºPhys.Rev.Lett.,vol.77,no.18,pp.3787±3790,[2]S.G.Johnson,C.Manolatou,S.Fan,P.R.Villeneuve,J.D.Joannopou-los,andH.A.Haus,ªEliminationofcrosstalkinwaveguideintersec-Opt.Lett.,vol.23,no.23,pp.1855±1857,Dec.1998.[3]T.Doi,A.Iwata,andM.Hirose,ªAnalysisanddesignoflow-lossandlowmode-shiftintegratedopticalwaveguidesusingnite-differencetime-domainmethod,ºIEICETrans.Electron.,vol.E80-C,no.5,pp.625±630,1997.[4]E.G.Neumann,ªReducingradiationlossoftiltsindielectricopticalElectron.Lett.,vol.17,no.5pp.369±371,1986.[5]T.Shiina,K.Shirashi,andS.Kawakami,ªWaveguide-bendcongu-rationwithlow-losscharacteristics,ºOpt.Lett.,vol.11,no.11,pp.736±738,1986.[6]H.B.Lin,J.Y.Su,P.K.Wei,andW.S.Wang,ªDesignandapplicationofverylow-lossabruptbendsinopticalwaveguides,ºIEEEJ.QuantumElectron.,vol.QE-30,pp.2827±2835,Dec.1994.[7]H.F.Taylor,ªLossesatcornerbendsindielectricwaveguides,º,vol.16,no.3,pp.711±716,1977.[8]K.Ogusu,ªTransmissioncharacteristicsofopticalwaveguidecorners,ºOpt.Commun.,vol.55,no.3,pp.149±153,1985.[9]J.S.Foresi,D.R.Lim,L.Liao,A.M.Agarwal,andL.C.Kimerling,ªSmallradiusbendsandlargeanglesplittersinSOIwaveguides,ºProc.,vol.3007,p.112±118,1997.[10]H.A.Haus,WavesandFieldsinOptoelectronics.EnglewoodCliffs,NJ:Prentice-Hall,1984.[11]M.H.Hu,J.Z.Huang,R.Scarmozzino,M.Levy,andR.M.OsgoodJr.,ªAlow-lossandcompactwavguide-branchusingrefractiveindexIEEEPhoton.Technol.Lett.,vol.9,pp.203±205,Feb.1997. ,ªTunableMach-Zehnderpolarizationsplitterusingheight-IEEEPhoton.Technol.Lett.,vol.9,pp.773±775,June1997.[13]H.B.Lin,R.S.Cheng,andW.S.Wang,ªWide-anglelow-losssingle-modesymmetricY-juntions,ºIEEEPhoton.Technol.Lett.,vol.6,July[14]M.G.Daly,P.E.Jessop,andD.Yevick,ªCrosstalkreductionininter-sectingribwaveguides,ºJ.LightwaveTechnol.,vol.14,pp.1695±1698,July1996.C.Manolatou,photographandbiographynotavailableatthetimeofStevenG.Johnson,photographandbiographynotavailableatthetimeofShanhuiFan,photographandbiographynotavailableatthetimeofpub-PierreR.Villeneuve,photographandbiographynotavailableatthetimeofH.A.Haus,photographandbiographynotavailableatthetimeofpublication.J.D.Joannopoulos,photographandbiographynotavailableatthetimeof