96年度第二学期光纤通信导论

Chapter2PropagationofSignalsinOpticalFiberIntroductiontoOpticalNetworks12.PropagationofSignalsinOpticalFiberAdvantagesLowloss~0.2dB/kmat1550nmEnormousbandwidthatleast25THzLightweightFlexibleImmunitytointerferencesLowcostDisadvantagesandImpairmentsDifficulttohandleChromaticdispersionNonlinearEffects22.1

Light

PropagationinOpticalFiberCladdingrefractiveindex≈1.45core8~10μm,50μm,62.5μmdoped

342.1.1GeometricalOpticalApproach(RayTheory)Thisapproachisonlyapplicabletomultimodefibers.

incidentangle(入射角)refractionangle(折射角)reflectionangle(反射角)

Snell’sLaw5=>CriticalangleWhentotalinternalreflectionoccurs.let=airrefractiveindex=acceptanceangle(totalreflectionwilloccuratcore/claddinginterface)6

(2.2)7IfΔissmall(lessthan0.01)For(multimode)NumericalApertureNA=Becausedifferentmodeshavedifferentlengthsofpaths,intermodaldispersionoccurs.8InfermodedispersionwillcausedigitalpulsespreadingLetLbethelengthofthefiberTheraytravelsalongthecenterofthecoreTherayisincidentat(slowray)9Assumethatthebitrate=Bb/sBitdurationThecapacityismeasuredbyBL(ignoreloss)Foeexample,if10Foroptimumgraded-indexfibers,δTisshorterthanthatinthestep-indexfibers,becausetheraytravelsalongthecenterslowsdown(nislarger)andtheraytravelinglongerpathstravelsfaster(nissmall)11Thetimedifferenceisgivenby(ForOpticalgraded-indexprofile)and(singlemode)IfLonghaulsystemsusesingle-modefibers

122.1.2WaveTheoryApproachMaxwell’sequationsD.1D.2D.3

D.4

:thechargedensity,:thecurrentdensity:theelectricfluxdensity,:themagneticfluxdensity:theelectricfield,:themagneticfieldρ13Becausethefieldarefunctionoftimeandlocationinthespace,wedenotethembyand,whereandtarepositionvectorandtime.Assumethespaceislinearandtime-invarianttheFouriertransformofis2.4letbetheinducedelectricpolarization2.5:thepermittivityofvacuum2.6:themagneticpolarization:thepermeabilityofvacuum注意有些書Fouriertransform定義為14LocalityofResponse:andrelatedtodispersionandnonlinearitiesIftheresponsetotheappliedelectricfieldislocaldependsonlyonnotonothervaluesofThispropertyholdsinthe0.5~2μmwavelengthIsotropy:TheelectromagneticpropertiesarethesameforalldirectionsinthemediumBirefringence:Therefractionindexesalongtwodifferentdirectionsaredifferent(lithiumniobate,LiNbO,modulator,isolator,tunablefilter)15Linearity：

(ConvolutionIntegral)2.7:linearsusceptibilityTheFouriertransformofis2.8WhereistheFouriertransformof(issimilartotheimpulseresponse)isfunctionoffrequency=>Chromaticdispersion16Homogeneity:AhomogeneousmediumhasthesameelectromagneticpropertiesatallpointsThecoreofagraded-indexfiberisinhomogeneousLosslessness：NolossinthemediumAtfirstwewillonlyconsiderthecoreandcladdingregionsofthefiberarelocallyresponsive,isotropic,linear,homogeneous,andlossless.Therefractiveindexisdefinedas2.9Forsilicafibersdef17FromAppendixDFor(zerocharge)(zeroconductivity,dielectricmaterial)Fornonmagneticmaterial18Assumelinearandhomogenence19TakeFouriertransformRecall2.8Denote

c:speedoflight

(Locallyresponse,isotropic,linear,homogeneous,lossless)2.9

20

palacianoperation2.10

(freespacewavenumber)21ForCartesiancoordinatesForCylindricalcoordinatesρ.φandz

n:{a:radiusofthecoreSimilarly2.11Boundaryconditionsisfiniteandcontinuityoffieldatρ=a

References:G.P.Agrawal“Fiber-OpticalCommunicationSystem”Chapter2JohnSenior“OpticalFiberCommunications,Principlesandpractice”JohnGowar“OpticalCommunicationSystems”注意有些書在

timedomain運算有些書在frequencydomain運算

22FiberModescladdingcorexzy23

mustsatisfy2.10,2.11andtheboundaryconditions.letWhereareunitvectorsForthefundamentalmode,thelongitudinalcomponentisthepropagationconstant24

:BesselfunctionsThetransversecomponentsForcylindricalsymmetryofthefiberIngeneral,wecanwrite(AppendixE)25WhereThemultimodefibercansupportmanymodes.Asinglemodefiberonlysupportsthefundamentalmode.Differentmodeshavedifferentβ,suchthattheypropagateatdifferentspeeds.=>modedispersion(Wecanthinkofa“mode”asonepossiblepaththataguidedraycantake)26Forafiberwithcoreandcladding,ifawavepropagatingpurelyinthecore,thenthepropagationconstantisλ:freespacewavelengthThewavenumberSimilarlyifthewavepropagatingpurelyinthecladding,thenThefibermodespropagatepartlyinthecladdingandpartlyinthecore,soDefinetheeffectiveindexThespeedofthewaveinthefiber=27Forafiberwithcoreradiusa,thecutoffconditionis

:normalizedwavenumber

RecallV↓whena↓and△↓Forasinglemodefiber,thetypicalvaluesarea=4μmand△=0.00328Thelightenergyisdistributedinthecoreandthecladding.2930SinceΔissmall,asignificantportionofthelightenergycanpropagateinthecladding,themodesareweaklyguided.Theenergydistributionofthecoreandthecladdingdependsonwavelength.Itcauseswaveguidedispersion(differentfrommaterialdispersion)(AppendixE)Forlongerwave,ithasmoreenergyinthecladdingandviceversa.31AmultimodefiberhasalargevalueofVThenumberofmodesForexamplea=25μm,Δ=0.005V=28at0.8μmDefinethenormalizedpropagationconstant(ornormalizedeffectiveindex)mode32PolarizationTwofundamentalmodesexistforallλ.Othersonlyexistforλ<λcutoff,Linearlypolarizedfield:Its

directionisconstant.Forthefundamentalmodeinasingle-modefiber333435Fibersarenotperfectlycircularlysymmetric.Thetwoorthogonallypolarizedfundamentalmodeshavedifferentβ=>Polarization-modedispersion(PMD)Differentialgroupdelay(DGD)Δτ=Δβ/w

～typicalvalueΔτ=0.5100km=>50psPracticallyPMDvariesrandomlyalongthefiberandmaybecancelledfromansegmenttoanothersegment.Empirically,Δτ

～0.1-1Someelementssuchasisolators,circulators,filtersmayhavepolarization-dependentloss(PDL).362.2LossandBandwidth3738RecallTakethebandwidthoverwhichthelossindB/kmiswithafactorof2ofitsminimum.80nmat1.3μm,180nmat1.55μm=>BW=35THz39Erbium-DopedFiberAmplifiers(EDFA)operateinthecandLbands,FiberRamanAmplifiers(FRA)operateintheSband.AllWavefibereliminatestheabsorptionpeaksduetowater.40412.2.1BendinglossAbendwithr=4cm,loss<0.01dBr↓loss↑2.2.3ChromaticDispersionDifferentspectralcomponentstravelatdifferentvelocities.Materialdispersionn(w)Waveguidedispersion,differentwavelengthshavedifferentenergydistributionsincoreandcladding=>differentβ,kn2<β<kn1422.3.1ChirpedGaussianPulses

Chirped:frequencyofthepulsechangeswithtime.Causeofchirp:directmodulation,nonlineareffects,generatedonpurpose.(soliton)43AppendixE,orGovindP.Agrawal“Fiber-OpticCommunicationSystems”2ndEdition,JohnWiley&Sons.Inc.PP47~51AchirpedGaussianpulseatz=0isgivenbyTheinstantaneousangularfrequency44k=ThechirpfactorDefine:Thelinearlychirpedpulse:theinstantaneousangularfrequencyincreasesordecreaseswithtime,(k=constant)NoteSolvewiththeinitialcondition(E.7)Weget(E.8)A(z,t)isalsoGaussianpulse45BroadeningofchirpedGaussianpulsesTheyhavethesameofbroadeninglength.Note46InFig2.9,,itistrueforstandardfibersat1.55μmLetbethedispersionlengthIf,dispersioncanbeneglected47Forkβ2<0Forβ2>0,highfrequencytravelsfaster=>thetailtravelsfasters=>compression=>makeβk<0,LDincreases↓decreasesincreases↑forcertainz(Fig2.10)48Noteβ2>0,k<0β2k<0492.3.2ControllingtheDispersionProfileDef:ChromaticdispersionparameterD=D=DM+DwThestandardsinglemodefiberhassmallchromaticdispersionat1.3μmbutlargeat1.55μm50At1.55μmlossislow,andEDFAiswelldeveloped.DispersionbecomesanissueWehavenotmuchcontroloverDM,butDwcanbecontrolledbycarefullydesignedrefractiveindexprofile.Dispersionshiftedfibers,whichhavezerodispersionin1.55μmband51522.4NonlinearEffectsForbitrate≦2.5Gb/s,powerafewmwLinearAssumptionisvalidNonlineareffectappearsforhighpowerorhighbitrate≧10Gb/sandWDMsystemsThefirstcategoryrelatestotheinteractionoflightwavewithphonons(molecularvibrations)-RayleighScatteringStimulatedBrillouinScattering(SBS)StimulatedRamanScattering(SRS)53Thesecondcategoryisduetothedependenceoftherefractiveindexontheintensity-self-phasemodulation(SPM)four-wavemixing(FWM)SBSandSRStransferenergyfromshortλ(pump)tolongλ(stokeswave)Scatteringgaincoefficient,g,ismeasuredinmeter/wattandΔf.●SPMinduceschirpingInaWDMsystem,variationofndependingontheintensityofallchannels.=>YieldsCross-phasemodulation(CPM)=>interchannelcrosstalk54FWM,552.4.1EffectiveLengthandAreaThenonlineareffectdependsonfiberlengthandcross-section.(forlonglink)56Inadditionnonlineareffectintensity572.4.1StimulatedBrillouinScattering(SBS)ThescatteringinteractionoccurswithacousticphononsoverΔf=15MHz,at1.55μm,stokesandpumpwavespropagateinoppositedirections.Ifspacing>20MHz=>noeffectsondifferentchannelsPs(0)Pp(L)pumpingSBS58592.4.3StimulatedRomanScattering(SRS)SRSwilldepleteshortwavepowerandamplifierlongwave.602.4.4PropagationinaNonlinearMediumInanonlinearmedium,FourierTransferisnotapplicable.Whentheelectricalfieldhasonlyonecomponent,wecanwriteandasthescalarfunctions,and.AppendixF,containshigherordertermslinearpolarizationnonlinearpolarization61Becauseofsymmetry,andThenonlinearresponseoccurslessthan100x10-15sec.Ifthebitrateislessthan100Gb/s,then:thethird-ordernonlinearsusceptibilityindependentoftimeForsimplicity,assumethatthesignalsaremonochromaticplanewavesisconstantintheplaneperpendiculartothedispersionofpropagationInWDMsystemswithnwavelengthsattheangularfrequencies0622.4.5Self-phaseModulation(SPM)Becausenisintensity–dependent=>inducesphaseshiftproportionaltotheintensity=>createschirping=>pulsebroadeningItissignificantforhighpowersystems.Considerasinglechannelcase63RecallforlinearmediumNow,wehavetomodifyasWeget=>Phasechangeswithintensitypropagationconstantchangeswith64

,whosephasechangesas,thisphenomenonisreferalasself-phasemodulation(SPM)TheintensityoftheelectricalfieldTheintensity-dependentrefractiveindexisThenonlinearindexcoefficient

insilicafiberWetakeforexampleBecauseapulsehasitsfinitetemporalextent=>ThephaseshiftisdifferentindifferentpartsofthepulseTheleadingedgeshavepositivefrequencyshiftThetailingedgeshavenegativefrequencyshift=>SPMcausespositivechirping652.4.6SPM-inducedchirpforGaussianPulsesConsideranunchirpedpulsewithenvelopewhichhasunitpeakamplitudeand-widthT0=1,andthepeakpowerP0=1DefinethenonlinearlengthasIflinklength≧LNL=>nonlineareffectissevere66FromAppendixE,(E.18)AfterpropagationLdistance,TheSPM-inducedphasechangeisTheinstantaneousfrequencyisgivenbyReferences:AppendixE,and(Arg97)67682.4.7Cross-PhaseModulationInWDMsystems,theintensity-dependentnonlineareffects(phaseshift)areenhancedbyothersignals,thiseffectisreferredtoascross-phasemodulation(CPM)ConsidertwochannelsRecall692w1+