光纤通信技术：Chapter 3 Signal Propagation in Fibers

Chapter3

SignalPropagationinFibersVocabularyChapter32SOEI,HUSTTotalinternalreflection全内反射Step-indexfiber阶跃折射率光纤Graded-indexfiber渐变折射率光纤Core-claddinginterface芯层和包层界面Modaldispersion模式色散Numericalaperture数值孔径Impulse冲激Meridionalrays子午光线Parabolic-indexfiber抛物线折射率分布光纤Refractiveindex折射率Absorptioncoefficient吸收系数Chromaticdispersion色度色散Opticalmode光模式Modeindex模式折射率Effectiveindex有效折射率Cutoffcondition截止条件Normalizedfrequency归一化频率Single-modecondition单模条件Eigenvalueequation本征值方程Birefringence双折射Propagationequation传播方程NLS非线性薛定谔方程Gaussianapproximation高斯近似

PMD偏振模色散Confinementfactor限制因子Fieldradius模场半径Attenuationcoefficient衰减系数Impurity杂质Intrinsicabsorption本征吸收Rayleighscattering瑞丽散射Miescattering米氏散射Waveguideimperfection波导不完善Macrobending宏弯Microbending微弯GVD:群速度色散Intramodaldispersion:模内色散Intermodaldispersion：模间色散Dispersionparameter：色散参数Materialdispersion：材料色散Waveguidedispersion：波导色散Sellmeierequation：塞米尔方程Zero-dispersionwavelength：零色散波长Dispersion-shiftedfiber：色散位移光纤Chapter33SOEI,HUSTDispersion-flattenfiber：色散平坦光纤Dispersiondecreasingfiber：色散渐减光纤Dispersioncompensationfiber：色散补偿光纤Dispersionslope：色散斜率Differential-dispersionparameter：微分色散参数Polarizationmodedispersion：偏振模色散Pulsebroadening脉冲展宽ChirpedGaussianpulse啁啾高斯脉冲FWHM:半高全宽Ramanscattering喇曼散射Brillouinscattering布里渊散射SRS受激喇曼散射SBS受激布里渊散射Electrostriction电致伸缩效应Brillouinshift布里渊频移Acousticphonon声学声子Inhomogeneous非均匀的Germania锗Isotropic各向同性Self-phasemodulation自相位调制Cross-phasemodulation交叉相位调制Largeeffective-areafiber大有效面积发光Phase-matchingcondition相位匹配条件Opticalphaseconjugation光相位共轭Elasticscattering弹性散射Inelasticscattering非弹性散射FiberManufacturing：光纤制作Doublyclad：双包层Depressed-claddingfiber:凹陷包层光纤Cylindricalpreform：预制棒MCVD：改进的化学汽相沉积OVD：轴外汽相沉积VAD：轴向沉积PCVD：等离子体化学汽相沉积Flamehydrolysis火焰裂解Sintering：烧结Light-dutycable轻型光缆Heavy-dutycable重型光缆Connector连接头Furnace熔炉Chapter34SOEI,HUST3.1Basicpropagationequation3.2Impactoffiberloss3.3Impactofdispersioninfiber3.4Impactofpolarizationmodedispersion(PMD)infiber3.5Impactoffibernonlinearity3.6SupplementaryChap.3SignalPropagationinFibersChapter35SOEI,HUST3.1.1FiberFundamentals

Totalinternalreflectionatthecore-claddinginterface.

Spatialmodedistribution:approximatelyGaussianSpotsizeforSMF:Single-modepropagationfor:BasicconceptsrelatedtoSMFRetrospectionforequationsderivationChapter36SOEI,HUSTMaxwell’sEquationsWaveEquationGuidedWaveEquationHelmholtz’sEquation

(x,y,z)tcylindricalcoordinatesGeneralSolutionEigenvalueEquationseparatevariablesboundaryconditionstangentialcomponentsnumericalsolvingChapter27SOEI,HUSTHelmholtzequationAfterusingthemethodofseparationofvariables,

theradialdirectionhastheform:TheFouriertransformoftheZdirection,canbegivenas:光纤光学光纤通信技术Chapter28SOEI,HUSTWhathappenstoopticalsignal?Fiberloss:limitingthetransmissiondistance(minimumlossnear1.55µm).Chromaticdispersion:limitingthebitratethroughpulsebroadening.Nonlineareffects:limitingthemaximuminputpower.Chapter39SOEI,HUST3.1.2BasicPropagationEquation∆ω=ω−ω0

F(x,y)representsspatialprofileofthefibermode

Chapter310SOEI,HUSTexpandβL(ω)isaTaylorseriesaroundω0:

Chapter311SOEI,HUSTPulseenvelopeisobtainedusing

Chapter312β1

termcorrespondstoaconstantdelayexperiencedbyapulseasitpropagatesthroughthefiber.Sincethisdelaydoesnotaffectthesignalqualityinanyway,itisusefultoworkinareferenceframemovingwiththepulse.Thiscanbeaccomplishedbyintroducingnewvariablest’andz’ast’=t−β1z

andz’=z.Thusthebasicpropagationequationcanbegivenas:

SOEI,HUST(G.P.Agrawal,“NonlinearFiberOptics”,the4thversion,2009,pp34,ElsevierPteLtd.)NonlinearcontributionChapter313Nonlineartermcanbewrittenas

SOEI,HUST

Sothebasicpropagationequationcanalsobewrittenas:Chapter314Ifthethird-orderdispersiveeffectscanbeneglected,i.e.β3=0,wewillobtain:Ifwesetα=0,weobtaintheso-calledNLSequation:3.1.3NonlinearSchrodingerEquation

Chapter315Chap.3SignalPropagationinFibers3.1Basicpropagationequation3.2Impactoffiberloss3.3Impactofdispersioninfiber3.4Impactofpolarizationmodedispersion(PMD)infiber3.5Impactoffibernonlinearity3.6SupplementarySOEI,HUSTChapter3163.2.1SourcesofFiberLossLossSourcesMaterialabsorption(silica,impurities,dopants)Rayleighscattering(variesasλ-4)Bendingloss(macroandmicro-bending)SOEI,HUSTChapter217SOEI,HUSTG.657FiberChapter318

AttenuationCoefficientPinPoutSOEI,HUST

Canbederivedfromthebasicpropagationequationα:AttenuationCoefficient,1/km3.2.2LossCompensationChapter319Fiberlossmustbecompensatedfordistance>100km.AOEOrepeatercanbeusedforthispurpose.Opticalbitstreamisfirstconvertedintoelectricdomainandthenregeneratedwiththehelpofanopticaltransmitter.Itbecomesquitecumbersomeandexpensivebecauseofthewavelengthdependence.Alternativesolution:usingopticalamplifiers.SOEI,HUSTChapter320Severalkindsofopticalamplifiersweredeveloped.Erbium-dopedfiberamplifiersRamanamplifiersSemiconductoropticalamplifiersOpticalamplifiersaredividedintotwocategories:lumpedanddistributedamplifiers.AllmodernWDMsystemsemployopticalamplifiers.TheyamplifymultipleWDMchannelssimultaneously.Amplifiercanbecascadedandenableustotransmitoverdistancesaslongas10,000km.SOEI,HUSTChapter3213.1Basicpropagationequation3.2Impactoffiberloss3.3Impactofdispersioninfiber3.4Impactofpolarizationmodedispersion(PMD)infiber3.5Impactoffibernonlinearity3.6SupplementarySOEI,HUSTChap.3SignalPropagationinFibersChapter2223.3.1ImpactofDispersioninFiberInter-symbolinterference!!!Bit2Bit1Bit2Bit1Bit2Bit1SMFSOEI,HUSTChapter323

SOEI,HUST

dispersionparameterdispersionslopeChapter3243.3.2PulseBroadeningChirpedGaussianPulseChirpedGaussianPulseatz=0:

SOEI,HUST

Chapter325UsingtheFourier-transformmethod,generalsolutionis

SOEI,HUST

Pulsespectrum:Itisrelatedtothebandwidthoftheopticalsignal,notcorrespondingtotheopticalsource!26 •Opticalfieldatadistancezisfoundtobe:

Chapter3MaintainingGaussianshapebutthewidth,chirpandamplitudewillbechangedbythefactorSOEI,HUSTPulseBroadening(Forbriefness,β3=0)Thewidthchangedto:Thechirpchangedto:

Chapter327Broadeningdependsonthesignofβ2C.Unchirpedpulsebroadensbyafactor

SOEI,HUSTChapter3283.3.3DispersionInducedlimitationsSOEI,HUSTVω

=2σωσ0isadimensionlessparameterL:fiberlengthσ0:RMStemporalwidthoftheinputGaussianpulse:dispersion-introducedbroadeningσω:RMSspectralwidthofthesourceConsideringthecombinedimpactsofβ2,β3,Candsourcespectralwidth:Note:Ifβ3isconsidered,theaforementionedintegralcanstillbeperformedinclosedformintermsofanAiryfunction.However,thepulsenolongerremainsGaussiananddevelopsatailwithanoscillatorystructure.SuchpulsescannotbeproperlycharacterizedbytheirFWHM.ApropermeasureofitistheRMSwidth.

Chapter229SOEI,HUSTOpticalSourceswithaLargeSpectralWidth,ex.LEDnon-zero-dispersionwavelength,neglectingβ3

Chapter230SOEI,HUSTzero-dispersionwavelength,β2=0OpticalSourceswithaSmallSpectralWidth,ex.LDnon-zero-dispersionwavelength,neglectingβ3

Chapter231SOEI,HUSTzero-dispersionwavelength,β2=0

Chapter3323.3.4DispersionCompensationBasicidea:Compensatedispersionalongfiberlinkinaperiodicfashionusingfiberswithoppositedispersioncharacteristics.

SOEI,HUSTConditionforperfectdispersioncompensation(onlyforβ2

):Chapter233SOEI,HUSTD’=-100ps/(nm·km)L2=?①SMF②DCFD=16ps/(nm·km)L1=50kmABCllonglshortExample:DispersioncompensationDSFandDCFDispersionslopecompensation(forβ3)?ProblemChapter234SOEI,HUST某光纤通信线路使用线宽很窄的单纵模激光器作为光源，发射机输出光功率为0.5mw，线路工作速率为10Gbit/s。假定光纤的色散系数D＝16ps/(nmkm)@1550nm且只考虑二阶群速度色散的影响。若光纤的损耗为0.2dB/km，接收机的灵敏度要求为-22dBm，请问此时该线路是何种因素受限（损耗或色散）？

Chapter3353.1Basicpropagationequation3.2Impactoffiberloss3.3Impactofdispersioninfiber3.4

Impactofpolarizationmodedispersion(PMD)infiber3.5Impactoffibernonlinearity3.6SupplementarySOEI,HUSTChap.3SignalPropagationinFibers363.4.1BirefringenceinFibersGeometricbirefringence:smalldeparturesincylindricalsymmetryduringmanufacturing(fibercoreslightlyelliptical).Stressbirefringence:anisotropicstressonthefibercoreduringmanufacturingorcablingofthefiber.

Chapter3SOEI,HUSTStateofpolarization(SOP)ofopticalsignaldoesnotremainfixedinpracticalopticalfibers.Itchangesrandomlybecauseoffluctuatingbirefringence.Chapter3373.4.2PolarizationModeDispersion

Wheninputpulseispolarizedalongaprincipalaxis,itsSOPdoesnotchange.Wheninputpulseisnotpolarizedalongaprincipalaxis,itsenergyisdividedintotwopolarizationmodes.Twoorthogonallypolarizedcomponentsofthepulseseparatealongthefiberbecauseoftheirdifferentgroupvelocities.SOEI,HUSTAphenomenonknownaspolarization-modedispersion(PMD)inducespulsebroadening.Chapter3383.4.3PMDinPMFDelay∆τ

(differentialgroupdelay,DGD)inthearrivaloftwocomponentsisgivenby

ForaPMF,∆β1∼1ns/km.

SOEI,HUSTChapter3393.4.4PMDinSMFSOEI,HUSTConventionalfibersexhibitmuchsmallerbirefringence(∆n=10−7).Birefringencemagnitudeanddirectionsofprincipalaxeschangerandomlyatalengthscale(correlationlength)lc~10m.Chapter3

SOEI,HUST

40

Chapter3413.4.5High-orderPMD

DGDfluctuateswiththewavelengthoflight.MeasuredvaluesofDGDvaryrandomlyfrom2pstomorethan

30psdependingonwavelengthoflight.SOEI,HUSTChapter3423.1Basicpropagationequation3.2Impactoffiberloss3.3Impactofdispersioninfiber3.4Impactofpolarizationmodedispersion(PMD)infiber3.5Impactoffibernonlinearity3.6SupplementarySOEI,HUSTChap.3SignalPropagationinFibersChapter3433.5.1FiberNonlinearitySOEI,HUST

100Gandbeyond40GLinearityNonlinearity10GSEForlong-haulsystems：NoiseaddedbytheamplifierchaindegradestheSNRandrequireshighlaunchedpowers.Nonlineareffectsaccumulateanddistortthebitstream.FiveMajorNonlinearEffectsarepossibleinopticalfibers:StimulatedRamanScattering(SRS)，StimulatedBrillouinScattering(SBS)，Self-PhaseModulation(SPM)，Cross-PhaseModulation(XPM)，Four-WaveMixing(FWM)Chapter344SOEI,HUSTImpairmentsonopticalsignalsChapter345TransmittedwaveformReceivedwaveformLowopticalpowerHighopticalpowerSMFafterdispersioncompensationThenonlinearitiesinsilicafibers:Stimulatedscattering:intensitydependentgainorloss,SRSandSBS.NonlinearKerreffect:intensitydependentphaseoftheopticalfield,SPM,XPMandFWM.SOEI,HUSTChapter346SOEI,HUST1.Nonlinearphaseshiftpropagationconstantnonlinearrefraction3.5.2Self-PhaseModulationChapter347Pulsepropagationinsideanopticalfiberisgovernedby

SOEI,HUST

U(z,t)satisfiestheNLSequationChapter348

SOEI,HUST

InthelimitChapter349OEI,HUST

2.FrequencychirpNewfrequencycomponentsaregeneratedcontinuouslyandbroadenspectrumofthebitstream.Chapter350

SOEI,HUSTDispersiveandnonlineareffectscanactonbitstreamsimultaneously.3.SolitonsChapter3513.5.3Cross-PhaseModulationNonlinearrefractiveindexseenbyonewavedependsonthe

intensityofothercopropagatingchannels.Nonlinearindexfortwochannels:

SOEI,HUSTTotalnonlinearphaseshiftformultiplechannels:XPMinducesanonlinearcouplingamongchannels.XPMisamajorsourceofcrosstalkinWDMsystems.1.Nonlinearphaseshift2.CoupledNLSequationsChapter352

SOEI,HUST3.LimitationonchannelpowersChapter353

Assumeinputpoweristhesameforallchannels.

Maximumvalueofphaseshiftoccurswhen1bitsinallchannels

overlapsimultaneously.Allowedpowerlevelreducestobelow1mWfor>10channels.

SOEI,HUSTChapter3543.5.4Four-WaveMixingFiberExample:thetwoinputsignalsarelocatedat

1and

2,andtraverseafiberoflength(L).Theoutputisfourdifferentsignalslocatedat1,

2,21-2,and22-1.FWMisnonlinearKerrprocessthatgeneratephasemodulationtoallinvolvedsignals.SOEI,HUSTChapter355Easilytorealizephase-matchingatzero-dispersionwavelength!SOEI,HUSTFormorecommoncase:Chapter356Input(a)andoutput(c)opticalspectraforequallyspaced

channelsInput(b)andoutput(d)opticalspectrainthecaseofunequal

channelspacings.SOEI,HUSTHowtosolvealltheaboveproblemsinDWDMapplication?largeeffectiveareanon-zerodispersion-shiftedfiberisavailable!!!Chapter257SOEI,HUSTRayleighScattering:ElasticscatteringnottogeneratenewfrequencyRamanScattering:InelasticscatteringPumpingphoton→stokesphoton+opticalphononBrillouinScattering:

InelasticscatteringPumpingphoton→stokesphoton+acousticphononbothdirections,frequencyshiftandgainbandwidtharealllarge.backwarddirectioninSMF,frequencyshiftissmall,andgainbandwidthisnarrow.3.5.5StimulatedScattering1.ScatteringChapter258SOEI,HUSTInter-channelinterferenceintimedomainTiltinfrequencydomain2.ImpairmentsChapter259SOEI,HUSTRamangainspectrumoffusedsilicaatλp=1μm

CanopticalamplifierorfibersensingutilizeRamanscattering?Stokesphoton3.SRSenergylevelsparticipatingintheSRSprocess.Chapter260SOEI,HUSTThresholdChapter361Brillouin-gainspectrameasuredusinga1.525-μmpumpforthreefiberswithdifferentgermaniumdoping(a)silica-corefiber;(b)depressed-claddingfiber;(c)dispersion-shiftedfiber.Verticalscaleisarbitrary.Brillouingainspectrumisquitenarrow(<50MHz).Multiplepeaksareduetotheexcitationofdifferentacousticmodes.AndeachacousticmodepropagatesatadifferentvelocityVAandthusleadstoadifferentBrillouinshift.SOEI,HUS