(完整版)光电子学第2章-介质波导与光纤

Chapter2

DielectricWaveguidesandOpticalFiber2.1SymmetricplanardielectricSlabwaveguide2.2dispersionintheplanarwaveguide2.3Stepindexfiber2.4Numericalaperture2.5Dispersioninsinglemodefibers2.6Dispersion,electrical,andopticalbandwidth2.7Thegradedindexopticalfiber2.8Lightabsorptionandscattering2.9Attenuationinopticalfibers2.10Fibermanufacture

“Theintroductionofopticalfibersystemswillrevolutionizethecommunicationsnetwork.Thelow-transmissionlossandthelargebandwidthcapabilityofthefibersystemsallowsignalstobetransmittedforestablishingcommunicationscontactsoverlargedistancewithfewornoprovisionsofintermediateamplification.”—CharlesK.Kao光纤系统的引入将使通信网络产生革命。光纤系统的低传输损耗和巨大的带宽能力允许信号在很少经过或者不经过中间放大的情况下实现超远距离的传输，建立通信联系。——查尔斯高锟TerminologyWaveguide:波导FurtherPhaseChangeofφ：附加相位φ；OpticalPathLength（OPL）：光程；ConstructiveInterference：相长干涉，相干加强；DestructiveInterference：相消干涉，相干减弱；PhaseDifference(PD)：相位差；OpticalPathDifference(OPD):光程差；PolarizationStateofLightWave：光波偏振态StateofPolarization（SOP）:偏振态FieldPattern：场图ResultantWave：合成波；StandingWave：驻波；TerminologyWavevector：波矢SingleModeWaveguide(SMwaveguide):单模波导MultimodeWaveguide(MMwaveguide):多模波导NormalizedThickness:归一化膜厚NormalizedFrequency:归一化化频率V-number:V数Cut-offWavelength:截至波长TransverseElectricfieldmode:横电场模式TransverseMagneticfieldmode:横磁场模式ModeFieldWidth(MFW):模场宽度ModeFieldDiameter(MFW):模场宽度Constantphasewavefront：等相位面Tounderstandthegeneralnatureoflightwavepropagationinopticalwaveguides,wefirstconsidertheplanerdielectricslabwaveguideshowninFigure2.1,whichisthesimplestwaveguideintermsoftractableanalysis.Aslabofdielectricofthickness2aandrefractiveindexn1

issandwichedbetweentwosemi-infiniteregionsbothofrefractiveindexn2(n2<n1).Theregionofhigherrefractiveindex(n1)iscalled

coreandtheregionoflowerrefractiveindexn2sandwichingthecoreiscalledthe

cladding.

理解传输在光波导中光波的基本性质，我们首先需要考虑平面介质波导，如图2.1所示，用于分析的最简的波导。厚度为2a和折射率为n1的介质平板被反射率为n2的两个半无限大区域的包起来。折射率较高的区域成为芯，包裹芯的折射率较低的区域称为包层。Figure2.1Alightraycanreadilypropagatealongsuchawaveguide,inazigzagfashion,provideditcanundergototalinternalreflection(TIR)atthedielectricboundaries.Itseemsthatanylightwavethathasanangleofincidence

θgreaterthanthecriticalangle

θforTIR,willbepropagated.This,however,istrueonlyforaverythinlightbeamwithadiametermuchlessthantheslabthickness,2a.

Weconsidertherealisticcasewhenthewholeendofthewaveguideisilluminated,asdepictedinFigure2.1.Tosimplifytheanalysis,wewillassumethatlightislaunchedfromalinesourceinamediumofrefractiven1.Ingeneral,therefractiveindexofthelaunchingmediumwillbedifferentthann1,butthiswillaffectonlytheamountoflightcoupledintotheguide.

光线很容易沿着这样的波导传输，以z字形，如果在介质的边界上发生全反射的话。看样子似乎是，任何入射角只要大于临界角发生全反射的光波，都会被传输。然而，只有当光束的直径远小于波导厚度2a的光才能真正存在。我们考虑实际的情况，光由波导的一端注入，如图2.1所示。为了简化问题的分析，我们假设在折射率为n1中光从线光源出射。通常，发光媒质的折射率将与n1不同，但是它只影响耦合入波导中光的多少。AsshowninFigure2.2,WewilltaketheelectricfieldEtobealongx,paralleltotheinterfaceandperpendiculartoz.Therayisguidedinazigzagfashionalongtheguideaxiszbyreflectionsfromthecore-cladding(n1

/n2)boundaries.TheresultistheeffectivepropagationoftheelectricfieldEalongz.Thefigurealsoshowstheconstantphasewavefronts,normaltodirectionofpropagation,onthisray.ThisparticularrayisreflectedatBandthenatC.JustafterthereflectionatC,thewavefrontatCoverlapsthewavefrontatAontheoriginalray.Thewaveinterfereswithitself.UnlessthesewavefrontsatAandCinphase,thetwowillinterferedestructivelyanddestroyeachother.Onlycertainreflectionanglesθgiverisetotheconstruc-tiveinterferenceandhenceonlycertainwavescanexistintheguide.如图2.2所示，我们让电场E沿着x轴，平行于交界面，垂直于z轴。光线沿着波导z轴，通过在芯与包层边界上的反射，形成z字形的传导。结果是电场E沿着z轴有效的传输。图中表示等相位面，为垂直于光线的传输方向。光线先在B点反射，然后在C点。在C点反射之后，C点的波阵面与原光线上A点的波阵面交叠。光波发生自干涉。除非A点播阵面与C点是同相的，否则两波将相干减弱，互相破坏。只有确定的反射角导致相干加强，因此在波导中只能传输角度的光波。Itmaybethoughtthatthetreatmentaboveissomewhatartificialaswetookanarrowangleforθ.ItturnsoutthatEq.(3)canbederivedasageneralwaveguideconditionforguidedwaveswhetherweuseanarroworawiderangle,oneormultiplerays.WecanderivethesameconditionifwetaketwoarbitraryparallelraysenteringtheguideasinFigure2.3.Therays1and2areinitiallyinphase,andrepresentthesame“plane-wave”.Ray1thensufferstworeflectionsatAandB,andisthenagaintravelingparalleltoray2.Unlessthewavefrontonray1justafterreflectionatBisinphasewiththewavefrontatB’onray2,thetwowoulddestroyeachother.Bothraysinitiallystartinphase;ray1atAjustbeforereflectionandray2atA’.Ray1atB,justaftertworeflections,hasaphasek1AB-2φ.Ray2atB’hasaphasek1(A’B’).Thedifferencebetweenthetwophasemustbem(2π)andleadstothewaveguideconditioninEq.(3).上面的处理可以被认为是有一些人为假设的成分，原因为是小入射角的前提条件。等式（3）是一个基本的导波条件，无论我们使用小入射角还是大入射角，一束光还是多束光。如果我们取两束任意的平行光，输入到如图3所示的波导中，同样可以得出导波条件。初始光线1和2是同相的，用平面波表示。光线1在A点和B点经过反射，传输又与光线2平行。光线1的波阵面在B点反射之后，除非光线2中的B‘同相，否则两束光将相互破坏。两光线初始同相位，光线1在A点反射之后，光线2到达A’点。光线1在B点，经过两次反射后，相位为AB-2φ。光线2在B‘点的相位k1(A’B’)，两相位之间的差值应该是m(2π)，可以倒出（3）式的波导条件。Toobtainthewaveguideconditionandsolvethepropagationmodesforthesymmetricplanardielectricwaveguides:(1)ThewaveopticsapproachSolveMaxwell’sequations.Thereisnoapproxima-tionsandtheresultsarerigorous.(2)ThecoefficientmatrixapproachStraightforward.Notsuitableformultilayerproblems.(3)ThetransmissionmatrixmethodSuitableformultilayerwaveguides.(4)ThemodifiedraymodelmethodItissimple,butprovideslessinformation.km:transversepropagationconstantThesimplisticanalysisasembeddedinthewave-guideconditioninEq.(3)showsquiteclearlythatonlycertainreflectionanglesareallowedwithintheguidecorrespondingtom=0,1,2,….Wenotethathighermvaluesyieldlowerθm.EachdifferentmvalueleadstoadifferentpropagationconstantalongtheguidegivenbyEq.(4).Ifweweretoconsidertheinterferenceofmanyrays,asinFigure2.3,wewouldfindthattheresultantwavehasstationaryelectricfieldpatternalongthey-direction,andthisfieldpatterntravelsalongtheguide,z-axis,withapropagationconstantβm.WecanshowthisbyconsideringtheresultantofthetwoparallelraysinFigure2.3thathaveincidenceanglesθmsatisfyingthewaveguidecondition.Figure2.5Thelowestmode(m=0)hasamaximumintensityatthecenterandmovesalongz

withapropagationconstantofβ0.Thereisapropagatingevanescentwaveinthecladdingneartheboundary.Figure2.6illustratesthefieldpatternsforthefirstthreemodes,m=0to2.Noticethattherayshavebeenshowntopenetratethecladding，andreflectedfromanapparentplaneinthecladding.Wehaveseenthateachmleadstoanallowedθmvaluethatcorrespondstoaparticulartravelingwaveinthez-directionasdescribedbyEq.(8)withaparticularwavevectorβmasdefinedbyEq.(4).Eachofthesetravelingwaves,withadistinctfieldpattern,Em(y),constitutesamodeofpropagation.Theintegermidentifiesthesemodesandiscalledthemodenumber.ThelightenergycanbetransportedonlyalongtheguideviaoneormoreofthesepossiblemodesofpropagationasdepictedinFigure2.7.Noticethattherayshavebeenshowntopenetratethecladding，andreflectedfromanapparentplaneinthecladding.Sinceθmissmallerforlargerm,highermodesexhibitmorereflectionsbuttheyalsopenetratemuchmoreintothecladdingasschematicallydepictedinFigure2.7.我们已经看到，每一个m确定一个可能的值，在z方向上对应一个由E(y)式描写的特殊的传输波，传输波的βm传输矢量由(4)式给出。每一个传输波，具有不同场的图样，组成一个传输模式。整数m并成为模式的模数。光能只能通过可能存在于波导中的的传输模式来传输能量，如图2.7所示。注意到光线渗透到包层中，在包层中一个明显反射回来。因为大数值的m，θm非常小，较高的模式，更多的渗透到包层中，存在更多的反射，如图2.7所示。Forthelowermode,m=0,whichleadstoθmbeingclosestto90˚andthewaveissaidtotravelaxially.LightthatislaunchedintothecoreofthewaveguidecantravelonlyalongtheguideintheallowedmodesspecifiedbyEq.(3).Thesemodeswilltraveldowntheguideatdifferentgroupvelocities.Whentheyreachtheendoftheguidetheyconstitutetheemerginglightbeam.Ifwelaunchashort-durationlightpulseintothedielectricwaveguide,thelightemergingfromtheotherendwillbeabroadenedlightpulsebecauselightenergywouldhavebeenpropagatedatdifferentgroupvelocitiesalongtheguideasdepictedinFigure2.7.Thelightpulsethereforespreadsasittravelsalongtheguide.对于低阶模，m=0,使θm接近于90度，光波是被认为是沿轴传输的。发射光进入波导的芯层后，只能以(3)式（导波条件）决定的模式沿着波导传输。这些模式传输在波导中，具有不同的群速度。当传输到波导的另一端时，产生新的光束。如果我们发射一个短周期的光脉冲输入到介质波导中，在另一端产生的出现的光脉冲将发生加宽，原因是传输在波导中的光具有不同的群速度。因此，光脉冲沿着波导传输会被展宽。ThephasechangeatTIRdependsonthepolarizationoftheelectricfield,anditisdifferentforE⊥

andE//.Thesetwofieldsrequiredifferentanglesθm

topropagatealongthewaveguide.Anyotherfielddirection(perpendiculartothepathofray)canberesolvedtohaveelectricfieldcomponentsalongE//andE┸.Thesetwofieldsexperiencedifferentphasechanges,φ//andφ┸,andconsequentlyrequiredifferentanglesθmtopropagatealongtheguide.WethereforehaveadifferentsetofmodesforE//andE┸。ThemodesassociatedwithE┸(orEx)aretermedtransverseelectricfieldmodes,denotedbyTEm,becauseE┸isactuallyperpendiculartothedirectionofpropagation,z.任何其他方向（垂直于光路经）的电场矢量都可以分解为E//和E┸两个电场分量。两个电场具有不同的相移φ//和φ┸，分别要求沿着波导传输有不同的θm。因此，对于E//和E┸具有不同的传输模式。与E┸相关的称为横电场模式，用TEm表示，因为事实上E┸垂直于传播方向z。ThemodesassociatedwiththeE∥fieldhaveamagneticB┸,whichisperpendiculartothedirectionofapropagationandaretermedtransversemagneticfieldmodes,denotedbyTMm.ItisinterestingthatE//hasafieldcomponentparalleltothez-axis,shownasEz

,whichisalongthedirectionofpropagation.ItisapparentthatEz

isapropagatinglongitudinalelectricfield.Infreespace,itisimpossibleforsuchalongitudinalfieldtoexistbutwithinanopticalguide,duetotheinterferencephenomena,itisindeedpossibletohavealongitudinalfield.Similarly,thosewithB//,haveamagneticfieldalongzthatpropagatealongthisdirectionasalongitudinalwave.E∥有一个磁场分量B┸，与它相关的垂直于传输方向的模式，称为横磁场模式，用TMm表示。E//有一个分量平行与z轴，沿着传输方向,表示为Ez。

Ez明显是一个纵向传输的电场。在自由的空间中，像这样的纵向电场是不可能存在的。但由于干涉现象，存在于波导中的场，确实可能是一个纵向场。类似的，对于B//同样有一个纵向的磁场沿着z轴方向传输。QuestionsandProblems2.3TEandTMModesindielectricslabwaveguide

(p.98)Consideraplanardielectricguidewithacorethickness20

m,n1

=1.455,n2=1.440,lightwavelengthof1.3

m.GiventhewaveguideconditionandtheexpressionsforphasechangesandinTIRfortheTEandTMmoderespectively,Usingagraphicalsolutionfindtheangle

forthefundamentalTEandTMmodesandcomparetheirpropagationconstantsalongtheguide.QuestionsandProblems2.5Dielectricslabwaveguide(p.98)

ConsideradielectricslabguidethathasathinGaAslayerofthickness0.2

mbetweentwoAlGaAslayers.TherefractiveindexofGaAsis3.66andthatofAlGaAsthelayersis3.40.Whatisthecut-offwavelengthbeyondwhichonlyasinglemodecanpropagateinthewaveguide,assumingthattherefractiveindexdoesnotvarygreatlywiththewavelength?Ifaradiationofwavelength870nm(correspondingtobandgapradiation)ispropagatingintheGaAslayer,whatisthepenetrationoftheevanescentwaveintotheAlGaAslayers?Whatisthemodefielddistanceofthisradiation?

Chapter2

DielectricWaveguidesandOpticalFiber2.1SymmetricplanardielectricSlabwaveguide2.2dispersionintheplanarwaveguide2.3Stepindexfiber2.4Numericalaperture2.5Dispersioninsinglemodefibers2.6Dispersion,electrical,andopticalbandwidth2.7Thegradedindexopticalfiber2.8Lightabsorptionandscattering2.9Attenuationinopticalfibers2.10Fibermanufacture专业词汇选编RayPath:光程PropagationConstantβ:传输常数βPropagationMode：传输模式GroupVelocityvg:群速vgDispersionDiagram:色散图ModalDispersion(IntermodalDispersion):模式色散（模间色散）IntermodeCoupling:模式耦合WaveguideDispersion：波导色散MaterialDispersion：材料色散IntramodeDispersion：模内色散NormalizedIndexDifference:归一化折射率差专业词汇选编MeridionalRay：子午光线SkewRay:斜光线HelicalPath:螺旋路径WeaklyGuidingFibers：弱导光纤LinearlyPolarized：线偏振Fundamentalmode：基模NormalizedPropagationConstant：归一化传输常数NumericalAperture：数值孔径MaximumAcceptanceAngle：最大受光角TotalAcceptanceAngle：总受光角Monochromatic:单色的（单频的）Thepropagatingmodesthatexistinaslabwaveguidearedeterminedbythewaveguidecondition.Eachchoiceofmfrom0toitsmaximumvalueresultsinonedistinctsolutionandonepossiblepropagationconstantβm.Weseethateachmodepropagateswithadifferentpropa-gationconstantevenifilluminationisbymonochromaticradiation.ExaminationofFigure2.7givestheimpressionthattheaxialrayhastheleastreflectionsandthereforeseemstoarrivemorequicklythanahigher-moderay.LowordermodeHighordermodeCladdingLightpulset0tSpread,DtBroadenedlightpulseIntensityIntensityAxialHighermodeszigzagmorealongtheguideandappeartohavelongerraypaths.However,therearetwoimportantwrongimpressionswiththisview.First,whatisimportantisthegroupvelocityvgalongtheguide,thevelocityatwhichtheenergyorinformationisthetransported.

Secondisthatthehighermodespenetratemoreintothecladdingwhererefractiveindexissmallerandthewavestravelfaster.(2)Equation(2)considersonlytwoextrememodes,thelowestandthehighest,anddoesnotconsiderwhethersomeintermediatemodescanhavegroupvelocitiesfallingoutsidetherangec/n1

toc/n2

(considertheslopedω/dβ

inFigure2.10).Neitherdoesitconsiderhowthelightenergyisproportionedbetweenvariousmodes.Takingn1=1.48(core)andn2=1.46(cladding),wefindΔτ/L≈6.7×10-11sm-1or67nskm-1.Ingeneral,intermodaldispersionisnotashighasindicatedbythisestimateduetoan“intermodecoupling”thediscussionofwhichisbeyondthescopeofthisbook.ThespreadΔτinEq.(2)isbetweenextremesofthebroadenedoutputlightpulse.InoptoelectronicswearefrequentlyinterestedinthespreadΔτ1/2

betweenthehalfintensitypointsthatissmallerthanthefullwidth.ThedeterminationofΔτ1/2

dependsonthetemporalshapeoftheoutputlightpulse,butasafirstorderapproximation,whenmanymodesarepresent,wecantakeΔτ1/2≈Δτ.yE(y)CladdingCladdingCorel2>l1l1>lcw2<w1w1<wCut-offvg1yvg2>vg1Thus,agreaterportionofthelightenergyiscarriedbythecladdinginwhichthephasevelocityishigher.Longerwavelengthpropagatefaster,eventhoughbythesamemode.Thisiscalledwaveguidedispersioninasmuchasitresultsfromtheguidingpropertiesofthedielectricstructureandithasnothingtodowiththefrequency(orwavelength)dependenceoftherefractiveoftheguide,thedisper-sioncanalsobestateasduetothewavelengthdependenceoftheV-number.Thehigherthewavelength(lowerthefrequency),thegreaterthepenetrationofthefieldintothecladding,asdepictedinFigure.bmwSlope=c/n2Slope=c/n1TE0wcut-offTE1TE2Therefractiveindexoftheguidematerialwillalsodependonthewavelengthandthusmodifytheω-βm

behaviorinFigure.Thechangeinthegroupvelocityofagivenmodeduetothen-λdependencealsogivesrisetothebroadeningofapropagatinglightpulse.Thisiscalled

materialdispersion.

Thusbothwaveguideandmaterialdispersionacttogethertobroadenalightpulsepropagatingwithinagivenmode.Combineddispersioniscalledintramodedispersion.QuestionsandProblems2.7Dielectricslabwaveguide(p.100)

3.2光纤的色散特性3.2.1色散的概念当日光通过棱镜或水雾时会呈现按红橙黄绿青蓝紫顺序排列的彩色光谱。这是由于棱镜材料（玻璃）或水对不同波长（对应于不同的颜色）的光呈现的折射率n不同，从而使光的传播速度不同和折射角度不同，最终使不同颜色的光在空间上散开。自然光的色散光脉冲中的不同频率或模式在光纤中的群速度不同，这些频率成分和模式到达光纤终端有先有后，使得光脉冲发生展宽，这就是光纤的色散，如图所示。色散一般用时延差来表示，所谓时延差，是指不同频率的信号成分传输同样的距离所需要的时间之差。色散引起的脉冲展宽示意图模式色散：不同模式不同传输速度（仅多模光纤有）材料色散：不同频率不同折射率波导色散：不同频率不同模场分布偏振模色散：不同偏振态不同传输速度色散分类t1t2t3t4脉冲展宽导致接收端无法将相邻的脉冲分开，从而导致误码。因此，射散特性限制了光纤的传输容量。色散描述方式信号分量的群速率是频率/波长的函数：即不同的频率分量间存在群时延差。信号在传输了距离L，频率分量w经历的延时为：假设输入脉冲的谱宽Dw不太宽，那么脉冲展宽的多少可以由下式决定：群速度色散(GVD)（ps2/km)通常光源的谱宽用Dl来表示。根据w和l之间的关系代入DT中，那么可以得到：其中D(l)称为色散系数(单位长度群延迟差):

ps/(km·nm)3.2.2模式色散

多模光纤中不同模式的光束有不同的群速度，在传输过程中，不同模式的光束的时间延迟不同而产生的色散，称模式色散。所谓模式色散，用光的射线理论来说，就是由于轨迹不同的各光线沿轴向的平均速度不同所造成的时延差。1.阶跃型光纤中的模式色散 在阶跃型光纤中，传播最快的和最慢的两条光线分别是沿轴线方向传播的光线①和以临界角θc入射的光线②，如图3.6所示。因此，在阶跃型光纤中最大色散是光线①和光线②到达终端的时延差。图3.6阶跃型光纤的模式色散DL为两种模式的光程差。2.渐变型光纤中的模式色散 在渐变型光纤中合理地设计光纤折射率分布，使光线在光纤中传播时速度得到补偿，从而模式色散引起的光脉冲展宽将很小。3.2.3材料色散

由于光源的不同频率（或波长）成分具有不同的群速度，在传输过程中，不同频率的光束的时间延迟不同。由于材料折射率随光信号频率的变化而不同，光信号不同频率成分所对应的群速度不同，由此引起的色散称为材料色散。

一般情况下，材料色散往往是用材料色散系数这个物理量来衡量，材料色散系数定义为单位波长间隔内各频率成份通过单位长度光纤所产生的色散。光纤的折射率是波长的函数n(l)，则不同的波长的传播函数b不同：

可以得到传播了L后波长l所经历的群延时(材料色散)为：式中：Δλ为光源的谱线宽度，即光功率下降到峰值光功率一半时所对应的波长范围；L是光纤的传播长度。Dm为材料色散系数。减小材料色散方法：选择谱宽窄的光源，采用较长的工作波长。3.2.4波导色散单模光纤只有约80%的光功率在纤芯中传播，20%在包层中传播的光功率其速率要更大一些。这种由于光纤波导结构引起的色散称为波导色散。假设纤芯和包层的折射率与波长无关，而且折射率差D=(n1-n2)/n1非常小，传播函数b近似等于：可以得到传播了L后波长l所经历的群延时为：其中V为归一化频率。进一步可以得到波导色散导致的脉冲展宽：其中Δλ为光源的谱线宽度，即光功率下降到峰值光功率一半时所对应的波长范围；L是光纤的传播长度。Dw(λ)为波导色散系数波导色散系数一般为负值例：令n2=1.48，D=0.2%，从左图可以看出当V=2.4时，有：因此可以算出在1320nm处，波导色散为：2.4随V变化的曲线标准单模光纤总的模内色散1320一般来说材料色散的影响大于波导色散：|Dm|>|Dw|波导色散特性取决于光纤的特性，如：芯径a，相对折射率差以及折射率分布等，因此可以通过改变光纤特性来改变其色散特性。色散的改变主要集中在零色散波长的位移和色散平坦两方面。3.2.5偏振模色散（极化色散）偏振模色散（PMD）也称为极化色散。由于光信号的两个正交偏振态在光纤中有不同的传播速度而引起的色散称偏振模色散。偏振模色散本征光纤双折射随机的偏振模耦合双折射的光通信器件

偏振模色散产生的原因+外界的挤压光纤的弯曲、扭转外界环境温度的变化等

EDFA，FBG，DCFIsolators,Couplers,Filtersetc.偏振模色散(PMD)PMD受环境(如振动、温度、应力等)影响非常显著，跟模内色散相比具有不稳定性和突发性。因此，PMD补偿的难度比较大，关于补偿的方法目前尚无定论。PMD对传输的影响PMD特点

一般采用两偏振模的群时延差(DifferentialGroupDelay)来表示偏振模色散的大小两偏振模之间的模式耦合随波长和时间随机变化，所以偏振模色散是一个统计量，并随时间而变化当光纤很长时，PMD呈现为麦克斯韦分布。这个分布可以是一组相同的光纤在同一波长处测量的结果，也可以是一根光纤在同一波长处但不同时间的测量结果，或者是同一光纤在不同波长处的测量结果PMD表征方式由于统计特性，群时延差(DGD)很难用于实际测量通常采用以下几种方式来定义PMD值：群时延差的平均值(MeanDifferentialGroupDelay)(ps)群时延差的平均值系数传输时间的均方差(RMSDGD，约为1.08)实际测量的群时延差值可能比群时延差的平均值大或小许多PMD对光通信系统的影响按照国际标准技术规范小组的观点：为保证PMD导致的系统功率代价在1dB以下，偏振模色散的群时延差的平均值必须小于一比特周期的十分之一(10psfora10Gb/ssystem)。当大于这一规定值时，需对系统偏振模色散进行补偿！比特率(Gb/s)允许的DGD平均值(ps)MeanDGD系数为0.1ps/

km光纤的传输距离（km)2.540<1600001010<10000402.5<6253.2.6光纤总色散

光纤的总色散为：

τM模式色散，τm材料色散，τW波导色散。单模光纤一般只给出色散系数D，其中包含了材料色散和波导色散的共同影响。三种光纤的总色散:1300nm最优化光纤色散平坦光纤色散位移光纤Chapter2

DielectricWaveguidesandOpticalFiber2.1SymmetricplanardielectricSlabwaveguide2.2dispersionintheplanarwaveguide2.3Stepindexfiber2.4Numericalaperture2.5Dispersioninsinglemodefibers2.6Dispersion,electrical,andopticalbandwidth2.7Thegradedindexopticalfiber2.8Lightabsorptionandscattering2.9Attenuationinopticalfibers2.10FibermanufactureStepIndexFiberStep-indexFiber:

Fiberthathasauniformindexofrefractionthroughoutthecorethatisastepbelowtheindexofrefractioninthecladding.

Figure2.12StepIndexFiberWerecallthattheplanarwaveguideisboundedonlyinonedimensionsothatreflectionsoccuronlyinthey-direction.Therequirementofconstructiveinterferenceofwavesthenleadstotheexistenceofdistinctmodeseachlabeledbym.ThecylindricalguideinFigure2.12isboundedintwodimensionsandthereflectionsoccurfromallthesurfaces,i.e.fromasurfaceencounteredalonganyradialdirectionr;alongaradialdirectionatanyangleφtothey-axisinFigure2.12.Sinceanyradialdirectioncanberepresentedintermsofxandy,reflectioninbothxandydirectionsareinvolvedinconstructiveinterferenceofwavesandwethereforeneedtwointegers,landm,tolabelallthepossibletravelingwavesorguidedmodesthatcanexistintheguide.阶跃折射率光纤让我们回忆一下,平面波导的截面只在一维受到限制，因此反射只发生在y方向。因此，相干加强的光波引起的不同的导模，用m表示。如图2.12所示的柱形波导，镜面在二维都受到限制，而反射发生在所有的表面，也就是说，沿着任何半径r的方向的任何光线相遇的表面；图2.12中所示的沿着半径r与y轴成任意角度φ的方向。由于任何半径方向能够用坐标x和y表示，在x和y方向上光波的反射同样要求相干加强的条件，因此我们需要两个整数l和m标记波导中存在所有可能的传输波或者导模。StepIndexFiberInastepindexfiberbothmeridionalandskewraysgiverisetoguidedmodes(propagatingwaves)alongthefiber,eachwithapropagationconstantβ

alongz.Guidemodesresultingfrommeridionalray

TEorTMtypeasinthecaseoftheplanarwaveguide.Skewrays,ontheotherhand,giverisetomodesthathavebothEzandBz

(orHz)componentsandarethereforenotTEorTMwaves.TheyarecalledHEorEHmodesasbothelectricandmagneticfieldcanhavecomponentsalongz.Theyarecalledhybridmodes.Itisapparentthatguidedmodesinastepindexfibercannotbeaseasilydescribedasthoseintheplanarguide.阶跃折射率光纤在阶越折射率光纤中，沿着光纤的方向子午光线和斜光线引起导模，他们沿z方向的传输常数为β。与平面波导的情况相同，子午光线的导模是TE或者TM波。另一方面，斜光线导致的模式具有Ez和Bz(Hz)的分量，因此不是TE或者TM波。他们称为HE或者EH模，因为沿着z具有电场和磁场的分量。他们被称为混合模式。在阶越折射率光纤中导模不象平面波导中描述的那样简单。StepIndexFiberGuidedmodesinastepindexfiberwithΔ<<1(calledweaklyguidingfibers)aregenerallyvisualizedbytravelingwavesthatarealmostplanepolarized.Theyhavetransverseelectricandmagneticfields(EandBareperpendiculartoeachotherandalsotoz),analogoustofielddirectionsinaplanewavebutthefieldmagnitudesarenotconstantintheplane.Theseswavesarecalledlinearlypolarized(LP)andhavetransverseelectricandmagneticfieldcharacteristics.AguidedLPmodealongthefibercanberepresentedbythepropagationofanelectricfielddistributionE(r,φ)alongz.Thisfielddistribution,orpattern,isintheplanenormaltothefiberaxisandhencedependsonrandφbutnotonz.Further,becauseofthepresenceoftwoboundaries,itischaracterizedbytwointegers,landm.ThepropagatingfielddistributioninanLPmodeisthereforegivenbyElm(r,φ)andwerepresentthemodeasLPlm。Thus,anLPlmmodecanbedescribedbyatravelingwavealongzoftheform,阶跃折射率光纤在Δ<<1（被称为弱导光纤）的阶跃折射率光纤中，导模通常用线偏振传输波来表示。它具有横向的电场或者磁场（E和B相互垂直，且同样垂直于z）。它具有与平面电磁波类似的场的方向，但是幅度不恒定。这些波被称为线偏振（LP）的，具有横电场和横磁场特性。沿着光纤的LP导模可以表示为沿z轴电场分布的传输。在垂直于光纤轴的平面内电场分布，或者场图，与r和φ有关，但是与z无关。并且，由于存在二维边界，需要用两个整数l和m表示。在LP模中传播的电场分布通过Elm(r,φ)给出，可以用LPlm表示。TheintegerslandmarerelatedtotheintensitypatterninaLPlmmode.Therearemnumberofmaximaalongrstartingformthecorecenterand2lmuberofmaximaaroundacircumferenceasapperentinFig.2.14.Further,withintheraypicture,lrepresentstheextentofhelicalpropagation,oftheamountofskewraycontributiontothemode.Inthefundametalmodethisiszero.Moreover,misdirectlyassociatedwiththereflectionangleθoftheraysasintheplanarguide.整数l和m与LPlm模式的强度图相联系。m是从芯的中心起沿着r方向的最大数，2l是围绕周线的最大数。并且在光线图中，l表示螺线传输的宽度，斜光线贡献给模式的数量。在基模中m为零。此外，m直接与光线的反射角相联系，与平面波导中的情况类似。Chapter2

DielectricWaveguidesandOpticalFiber2.1SymmetricplanardielectricSlabwaveguide2.2dispersionintheplanarwaveguide2.3Stepindexfiber2.4Numericalaperture2.5Dispersioninsinglemodefibers2.6Dispersion,electrical,andopticalbandwidth2.7Thegradedindexopticalfiber2.8Lightabsorptionandscattering2.9Attenuationinopticalfibers2.10FibermanufactureChapter2

DielectricWaveguidesandOpticalFiber2.1SymmetricplanardielectricSlabwaveguide2.2dispersionintheplanarwaveguide2.3Stepindexfiber2.4Numericalaperture2.5Dispersioninsinglemodefibers2.6Dispersion,electrical,andopticalbandwidth2.7Thegradedindexopticalfiber2.8Lightabsorptionandscattering2.9Attenuationinopticalfibers2.10FibermanufactureTerminologyMaterialDispersionCoefficient(Dm):材料色散系数WaveguideDispersionCoefficient(Dw):波导色散系数GroupDelay(τg):群延迟ProfileDispersion:剖面色散ChromaticDispersionCoefficient(Dch):色散系数DispersionShiftedFiber(DSF):色散位移光纤PolarizationDispersion:偏振色散(polarization-modedispersion,PMD):偏振模式色散DispersionFlattenedFiber(DFF):色散平坦光纤Isotropic(anisotropic):各向同性（各向异性）TerminologyWavelengthMultiplexing:波长复用(WavelengthDivisionMultiplexing,WDM):波分复用DenseWavelengthMultiplexing(DWDM):密集波分复用CommunicationChannel:通信信道Ngn500700900110013001500170019001.441.451.461.471.481.49Wavelength(nm)RefractiveindexnandthegroupindexNgofSiO2(silica)glassasafunctionofwavelength.Figure1.7GroupVelocityGroupIndex

Dm=0,atλ=1.27μmWhenSiO2isdopedwithGeO2toincreasethenforthecore,theDmvs.λshiftsslightlytohigherwavelengthsDwdependsontheguidegeometry,itisthereforepossibletoshiftthezerodispersionwavelengthbysuitablydesigningtheguide.Dchpassesthroughzeroatacertainwavelength,λ0.InFig.2.18,chromaticdispersioniszeroataround1320nm.Byreducingthecoreradiusandincreasingthecoredoping,λ0canbeshiftedto1550nmwherelightattenuationinthefiberisminimal.Suchfibersarecalleddispersionshiftedfibers.Althoughchromaticdispersion,Dch,passesthroughzero,thisdoesnotmeanthattherewouldbenodis-persionatall.First,weshouldnotethatDchcanbemadezeroforonlyonewavelength,λ0,notateverywavelengthwithinthespectrum,Δλ,ofthesource.Further,oth