Chapter2

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

“Theintroductionofopticalfibersystemswillrevolutionizethecommunicationsnetwork.Thelow-transmissionlossandthelargebandwidthcapabilityofthefibersystemsallowsignalstobetransmittedforestablishingcommunicationscontactsoverlargedistancewithfewornoprovisionsofintermediateamplification.”—CharlesK.Kao光纖系統(tǒng)的引入將使通信網(wǎng)絡(luò)產(chǎn)生革命。光纖系統(tǒng)的低傳輸損耗和巨大的帶寬能力允許信號在很少經(jīng)過或者不經(jīng)過中間放大的情況下實現(xiàn)超遠(yuǎn)距離的傳輸，建立通信聯(lián)系?！闋査垢咤KTerminologyWaveguide:波導(dǎo)FurtherPhaseChangeofφ：附加相位φ；OpticalPathLength（OPL）：光程；ConstructiveInterference：相長干涉，相干加強(qiáng)；DestructiveInterference：相消干涉，相干減弱；PhaseDifference(PD)：相位差；OpticalPathDifference(OPD):光程差；PolarizationStateofLightWave：光波偏振態(tài)StateofPolarization（SOP）:偏振態(tài)FieldPattern：場圖ResultantWave：合成波；StandingWave：駐波；TerminologyWavevector：波矢SingleModeWaveguide(SMwaveguide):單模波導(dǎo)MultimodeWaveguide(MMwaveguide):多模波導(dǎo)NormalizedThickness:歸一化膜厚NormalizedFrequency:歸一化化頻率V-number:V數(shù)Cut-offWavelength:截至波長TransverseElectricfieldmode:橫電場模式TransverseMagneticfieldmode:橫磁場模式ModeFieldWidth(MFW):模場寬度ModeFieldDiameter(MFW):模場寬度Constantphasewavefront：等相位面Tounderstandthegeneralnatureoflightwavepropagationinopticalwaveguides,wefirstconsidertheplanerdielectricslabwaveguideshowninFigure2.1,whichisthesimplestwaveguideintermsoftractableanalysis.Aslabofdielectricofthickness2aandrefractiveindexn1

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

coreandtheregionoflowerrefractiveindexn2sandwichingthecoreiscalledthe

cladding.

理解傳輸在光波導(dǎo)中光波的基本性質(zhì)，我們首先需要考慮平面介質(zhì)波導(dǎo)，如圖2.1所示，用于分析的最簡的波導(dǎo)。厚度為2a和折射率為n1的介質(zhì)平板被反射率為n2的兩個半無限大區(qū)域的包起來。折射率較高的區(qū)域成為芯，包裹芯的折射率較低的區(qū)域稱為包層。Figure2.1Alightraycanreadilypropagatealongsuchawaveguide,inazigzagfashion,provideditcanundergototalinternalreflection(TIR)atthedielectricboundaries.Itseemsthatanylightwavethathasanangleofincidence

θgreaterthanthecriticalangle

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

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

光線很容易沿著這樣的波導(dǎo)傳輸，以z字形，如果在介質(zhì)的邊界上發(fā)生全反射的話。看樣子似乎是，任何入射角只要大于臨界角發(fā)生全反射的光波，都會被傳輸。然而，只有當(dāng)光束的直徑遠(yuǎn)小于波導(dǎo)厚度2a的光才能真正存在。我們考慮實際的情況，光由波導(dǎo)的一端注入，如圖2.1所示。為了簡化問題的分析，我們假設(shè)在折射率為n1中光從線光源出射。通常，發(fā)光媒質(zhì)的折射率將與n1不同，但是它只影響耦合入波導(dǎo)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.如圖2.2所示，我們讓電場E沿著x軸，平行于交界面，垂直于z軸。光線沿著波導(dǎo)z軸，通過在芯與包層邊界上的反射，形成z字形的傳導(dǎo)。結(jié)果是電場E沿著z軸有效的傳輸。圖中表示等相位面，為垂直于光線的傳輸方向。光線先在B點反射，然后在C點。在C點反射之后，C點的波陣面與原光線上A點的波陣面交疊。光波發(fā)生自干涉。除非A點播陣面與C點是同相的，否則兩波將相干減弱，互相破壞。只有確定的反射角導(dǎo)致相干加強(qiáng)，因此在波導(dǎo)中只能傳輸角度的光波。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).上面的處理可以被認(rèn)為是有一些人為假設(shè)的成分，原因為是小入射角的前提條件。等式（3）是一個基本的導(dǎo)波條件，無論我們使用小入射角還是大入射角，一束光還是多束光。如果我們?nèi)墒我獾钠叫泄?，輸入到如圖3所示的波導(dǎo)中，同樣可以得出導(dǎo)波條件。初始光線1和2是同相的，用平面波表示。光線1在A點和B點經(jīng)過反射，傳輸又與光線2平行。光線1的波陣面在B點反射之后，除非光線2中的B‘同相，否則兩束光將相互破壞。兩光線初始同相位，光線1在A點反射之后，光線2到達(dá)A’點。光線1在B點，經(jīng)過兩次反射后，相位為AB-2φ。光線2在B‘點的相位k1(A’B’)，兩相位之間的差值應(yīng)該是m(2π)，可以倒出（3）式的波導(dǎo)條件。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.我們已經(jīng)看到，每一個m確定一個可能的值，在z方向上對應(yīng)一個由E(y)式描寫的特殊的傳輸波，傳輸波的βm傳輸矢量由(4)式給出。每一個傳輸波，具有不同場的圖樣，組成一個傳輸模式。整數(shù)m并成為模式的模數(shù)。光能只能通過可能存在于波導(dǎo)中的的傳輸模式來傳輸能量，如圖2.7所示。注意到光線滲透到包層中，在包層中一個明顯反射回來。因為大數(shù)值的m，θm非常小，較高的模式，更多的滲透到包層中，存在更多的反射，如圖2.7所示。Forthelowermode,m=0,whichleadstoθmbeingclosestto90?andthewaveissaidtotravelaxially.LightthatislaunchedintothecoreofthewaveguidecantravelonlyalongtheguideintheallowedmodesspecifiedbyEq.(3).Thesemodeswilltraveldowntheguideatdifferentgroupvelocities.Whentheyreachtheendoftheguidetheyconstitutetheemerginglightbeam.Ifwelaunchashort-durationlightpulseintothedielectricwaveguide,thelightemergingfromtheotherendwillbeabroadenedlightpulsebecauselightenergywouldhavebeenpropagatedatdifferentgroupvelocitiesalongtheguideasdepictedinFigure2.7.Thelightpulsethereforespreadsasittravelsalongtheguide.對于低階模，m=0,使θm接近于90度，光波是被認(rèn)為是沿軸傳輸?shù)摹０l(fā)射光進(jìn)入波導(dǎo)的芯層后，只能以(3)式（導(dǎo)波條件）決定的模式沿著波導(dǎo)傳輸。這些模式傳輸在波導(dǎo)中，具有不同的群速度。當(dāng)傳輸?shù)讲▽?dǎo)的另一端時，產(chǎn)生新的光束。如果我們發(fā)射一個短周期的光脈沖輸入到介質(zhì)波導(dǎo)中，在另一端產(chǎn)生的出現(xiàn)的光脈沖將發(fā)生加寬，原因是傳輸在波導(dǎo)中的光具有不同的群速度。因此，光脈沖沿著波導(dǎo)傳輸會被展寬。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（垂直于光路經(jīng)）的電場矢量都可以分解為E//和E┸兩個電場分量。兩個電場具有不同的相移φ//和φ┸，分別要求沿著波導(dǎo)傳輸有不同的θm。因此，對于E//和E┸具有不同的傳輸模式。與E┸相關(guān)的稱為橫電場模式，用TEm表示，因為事實上E┸垂直于傳播方向z。ThemodesassociatedwiththeE∥fieldhaveamagneticB┸,whichisperpendiculartothedirectionofapropagationandaretermedtransversemagneticfieldmodes,denotedbyTMm.ItisinterestingthatE//hasafieldcomponentparalleltothez-axis,shownasEz

,whichisalongthedirectionofpropagation.ItisapparentthatEz

isapropagatinglongitudinalelectricfield.Infreespace,itisimpossibleforsuchalongitudinalfieldtoexistbutwithinanopticalguide,duetotheinterferencephenomena,itisindeedpossibletohavealongitudinalfield.Similarly,thosewithB//,haveamagneticfieldalongzthatpropagatealongthisdirectionasalongitudinalwave.E∥有一個磁場分量B┸，與它相關(guān)的垂直于傳輸方向的模式，稱為橫磁場模式，用TMm表示。E//有一個分量平行與z軸，沿著傳輸方向,表示為Ez。

Ez明顯是一個縱向傳輸?shù)碾妶?。在自由的空間中，像這樣的縱向電場是不可能存在的。但由于干涉現(xiàn)象，存在于波導(dǎo)中的場，確實可能是一個縱向場。類似的，對于B//同樣有一個縱向的磁場沿著z軸方向傳輸。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專業(yè)詞匯選編RayPath:光程PropagationConstantβ:傳輸常數(shù)βPropagationMode：傳輸模式GroupVelocityvg:群速vgDispersionDiagram:色散圖ModalDispersion(IntermodalDispersion):模式色散（模間色散）IntermodeCoupling:模式耦合WaveguideDispersion：波導(dǎo)色散MaterialDispersion：材料色散IntramodeDispersion：模內(nèi)色散NormalizedIndexDifference:歸一化折射率差專業(yè)詞匯選編MeridionalRay：子午光線SkewRay:斜光線HelicalPath:螺旋路徑WeaklyGuidingFibers：弱導(dǎo)光纖LinearlyPolarized：線偏振Fundamentalmode：基模NormalizedPropagationConstant：歸一化傳輸常數(shù)NumericalAperture：數(shù)值孔徑MaximumAcceptanceAngle：最大受光角TotalAcceptanceAngle：總受光角Monochromatic:單色的（單頻的）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光纖的色散特性3.2.1色散的概念當(dāng)日光通過棱鏡或水霧時會呈現(xiàn)按紅橙黃綠青藍(lán)紫順序排列的彩色光譜。這是由于棱鏡材料（玻璃）或水對不同波長（對應(yīng)于不同的顏色）的光呈現(xiàn)的折射率n不同，從而使光的傳播速度不同和折射角度不同，最終使不同顏色的光在空間上散開。自然光的色散光脈沖中的不同頻率或模式在光纖中的群速度不同，這些頻率成分和模式到達(dá)光纖終端有先有后，使得光脈沖發(fā)生展寬，這就是光纖的色散，如圖所示。色散一般用時延差來表示，所謂時延差，是指不同頻率的信號成分傳輸同樣的距離所需要的時間之差。色散引起的脈沖展寬示意圖模式色散：不同模式不同傳輸速度（僅多模光纖有）材料色散：不同頻率不同折射率波導(dǎo)色散：不同頻率不同模場分布偏振模色散：不同偏振態(tài)不同傳輸速度色散分類t1t2t3t4脈沖展寬導(dǎo)致接收端無法將相鄰的脈沖分開，從而導(dǎo)致誤碼。因此，射散特性限制了光纖的傳輸容量。色散描述方式信號分量的群速率是頻率/波長的函數(shù)：即不同的頻率分量間存在群時延差。信號在傳輸了距離L，頻率分量w經(jīng)歷的延時為：假設(shè)輸入脈沖的譜寬Dw不太寬，那么脈沖展寬的多少可以由下式?jīng)Q定：群速度色散(GVD)（ps2/km)通常光源的譜寬用Dl來表示。根據(jù)w和l之間的關(guān)系代入DT中，那么可以得到：其中D(l)稱為色散系數(shù)(單位長度群延遲差):

ps/(km·nm)3.2.2模式色散

多模光纖中不同模式的光束有不同的群速度，在傳輸過程中，不同模式的光束的時間延遲不同而產(chǎn)生的色散，稱模式色散。所謂模式色散，用光的射線理論來說，就是由于軌跡不同的各光線沿軸向的平均速度不同所造成的時延差。1.階躍型光纖中的模式色散 在階躍型光纖中，傳播最快的和最慢的兩條光線分別是沿軸線方向傳播的光線①和以臨界角θc入射的光線②，如圖3.6所示。因此，在階躍型光纖中最大色散是光線①和光線②到達(dá)終端的時延差。圖3.6階躍型光纖的模式色散DL為兩種模式的光程差。2.漸變型光纖中的模式色散 在漸變型光纖中合理地設(shè)計光纖折射率分布，使光線在光纖中傳播時速度得到補(bǔ)償，從而模式色散引起的光脈沖展寬將很小。3.2.3材料色散

由于光源的不同頻率（或波長）成分具有不同的群速度，在傳輸過程中，不同頻率的光束的時間延遲不同。由于材料折射率隨光信號頻率的變化而不同，光信號不同頻率成分所對應(yīng)的群速度不同，由此引起的色散稱為材料色散。

一般情況下，材料色散往往是用材料色散系數(shù)這個物理量來衡量，材料色散系數(shù)定義為單位波長間隔內(nèi)各頻率成份通過單位長度光纖所產(chǎn)生的色散。光纖的折射率是波長的函數(shù)n(l)，則不同的波長的傳播函數(shù)b不同：

可以得到傳播了L后波長l所經(jīng)歷的群延時(材料色散)為：式中：Δλ為光源的譜線寬度，即光功率下降到峰值光功率一半時所對應(yīng)的波長范圍；L是光纖的傳播長度。Dm為材料色散系數(shù)。減小材料色散方法：選擇譜寬窄的光源，采用較長的工作波長。3.2.4波導(dǎo)色散單模光纖只有約80%的光功率在纖芯中傳播，20%在包層中傳播的光功率其速率要更大一些。這種由于光纖波導(dǎo)結(jié)構(gòu)引起的色散稱為波導(dǎo)色散。假設(shè)纖芯和包層的折射率與波長無關(guān)，而且折射率差D=(n1-n2)/n1非常小，傳播函數(shù)b近似等于：可以得到傳播了L后波長l所經(jīng)歷的群延時為：其中V為歸一化頻率。進(jìn)一步可以得到波導(dǎo)色散導(dǎo)致的脈沖展寬：其中Δλ為光源的譜線寬度，即光功率下降到峰值光功率一半時所對應(yīng)的波長范圍；L是光纖的傳播長度。Dw(λ)為波導(dǎo)色散系數(shù)波導(dǎo)色散系數(shù)一般為負(fù)值例：令n2=1.48，D=0.2%，從左圖可以看出當(dāng)V=2.4時，有：因此可以算出在1320nm處，波導(dǎo)色散為：2.4隨V變化的曲線標(biāo)準(zhǔn)單模光纖總的模內(nèi)色散1320一般來說材料色散的影響大于波導(dǎo)色散：|Dm|>|Dw|波導(dǎo)色散特性取決于光纖的特性，如：芯徑a，相對折射率差以及折射率分布等，因此可以通過改變光纖特性來改變其色散特性。色散的改變主要集中在零色散波長的位移和色散平坦兩方面。3.2.5偏振模色散（極化色散）偏振模色散（PMD）也稱為極化色散。由于光信號的兩個正交偏振態(tài)在光纖中有不同的傳播速度而引起的色散稱偏振模色散。偏振模色散本征光纖雙折射隨機(jī)的偏振模耦合雙折射的光通信器件

偏振模色散產(chǎn)生的原因+外界的擠壓光纖的彎曲、扭轉(zhuǎn)外界環(huán)境溫度的變化等

EDFA，F(xiàn)BG，DCFIsolators,Couplers,Filtersetc.偏振模色散(PMD)PMD受環(huán)境(如振動、溫度、應(yīng)力等)影響非常顯著，跟模內(nèi)色散相比具有不穩(wěn)定性和突發(fā)性。因此，PMD補(bǔ)償?shù)碾y度比較大，關(guān)于補(bǔ)償?shù)姆椒壳吧袩o定論。PMD對傳輸?shù)挠绊慞MD特點

一般采用兩偏振模的群時延差(DifferentialGroupDelay)來表示偏振模色散的大小兩偏振模之間的模式耦合隨波長和時間隨機(jī)變化，所以偏振模色散是一個統(tǒng)計量，并隨時間而變化當(dāng)光纖很長時，PMD呈現(xiàn)為麥克斯韋分布。這個分布可以是一組相同的光纖在同一波長處測量的結(jié)果，也可以是一根光纖在同一波長處但不同時間的測量結(jié)果，或者是同一光纖在不同波長處的測量結(jié)果PMD表征方式由于統(tǒng)計特性，群時延差(DGD)很難用于實際測量通常采用以下幾種方式來定義PMD值：群時延差的平均值(MeanDifferentialGroupDelay)(ps)群時延差的平均值系數(shù)傳輸時間的均方差(RMSDGD，約為1.08)實際測量的群時延差值可能比群時延差的平均值大或小許多PMD對光通信系統(tǒng)的影響按照國際標(biāo)準(zhǔn)技術(shù)規(guī)范小組的觀點：為保證PMD導(dǎo)致的系統(tǒng)功率代價在1dB以下，偏振模色散的群時延差的平均值必須小于一比特周期的十分之一(10psfora10Gb/ssystem)。當(dāng)大于這一規(guī)定值時，需對系統(tǒng)偏振模色散進(jìn)行補(bǔ)償！比特率(Gb/s)允許的DGD平均值(ps)MeanDGD系數(shù)為0.1ps/

km光纖的傳輸距離（km)2.540<1600001010<10000402.5<6253.2.6光纖總色散

光纖的總色散為：

τM模式色散，τm材料色散，τW波導(dǎo)色散。單模光纖一般只給出色散系數(shù)D，其中包含了材料色散和波導(dǎo)色散的共同影響。三種光纖的總色散:1300nm最優(yōu)化光纖色散平坦光纖色散位移光纖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.階躍折射率光纖讓我們回憶一下,平面波導(dǎo)的截面只在一維受到限制，因此反射只發(fā)生在y方向。因此，相干加強(qiáng)的光波引起的不同的導(dǎo)模，用m表示。如圖2.12所示的柱形波導(dǎo)，鏡面在二維都受到限制，而反射發(fā)生在所有的表面，也就是說，沿著任何半徑r的方向的任何光線相遇的表面；圖2.12中所示的沿著半徑r與y軸成任意角度φ的方向。由于任何半徑方向能夠用坐標(biāo)x和y表示，在x和y方向上光波的反射同樣要求相干加強(qiáng)的條件，因此我們需要兩個整數(shù)l和m標(biāo)記波導(dǎo)中存在所有可能的傳輸波或者導(dǎo)模。StepIndexFiberInastepindexfiberbothmeridionalandskewraysgiverisetoguidedmodes(propagatingwaves)alongthefiber,eachwithapropagationconstantβ

alongz.Guidemodesresultingfrommeridionalray

TEorTMtypeasinthecaseoftheplanarwaveguide.Skewrays,ontheotherhand,giverisetomodesthathavebothEzandBz

(orHz)componentsandarethereforenotTEorTMwaves.TheyarecalledHEorEHmodesasbothelectricandmagneticfieldcanhavecomponentsalongz.Theyarecalledhybridmodes.Itisapparentthatguidedmodesinastepindexfibercannotbeaseasilydescribedasthoseintheplanarguide.階躍折射率光纖在階越折射率光纖中，沿著光纖的方向子午光線和斜光線引起導(dǎo)模，他們沿z方向的傳輸常數(shù)為β。與平面波導(dǎo)的情況相同，子午光線的導(dǎo)模是TE或者TM波。另一方面，斜光線導(dǎo)致的模式具有Ez和Bz(Hz)的分量，因此不是TE或者TM波。他們稱為HE或者EH模，因為沿著z具有電場和磁場的分量。他們被稱為混合模式。在階越折射率光纖中導(dǎo)模不象平面波導(dǎo)中描述的那樣簡單。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,階躍折射率光纖在Δ<<1（被稱為弱導(dǎo)光纖）的階躍折射率光纖中，導(dǎo)模通常用線偏振傳輸波來表示。它具有橫向的電場或者磁場（E和B相互垂直，且同樣垂直于z）。它具有與平面電磁波類似的場的方向，但是幅度不恒定。這些波被稱為線偏振（LP）的，具有橫電場和橫磁場特性。沿著光纖的LP導(dǎo)?？梢员硎緸檠貁軸電場分布的傳輸。在垂直于光纖軸的平面內(nèi)電場分布，或者場圖，與r和φ有關(guān)，但是與z無關(guān)。并且，由于存在二維邊界，需要用兩個整數(shù)l和m表示。在LP模中傳播的電場分布通過Elm(r,φ)給出，可以用LPlm表示。TheintegerslandmarerelatedtotheintensitypatterninaLPlmmode.Therearemnumberofmaximaalongrstartingformthecorecenterand2lmuberofmaximaaroundacircumferenceasapperentinFig.2.14.Further,withintheraypicture,lrepresentstheextentofhelicalpropagation,oftheamountofskewraycontributiontothemode.Inthefundametalmodethisiszero.Moreover,misdirectlyassociatedwiththereflectionangleθoftheraysasintheplanarguide.整數(shù)l和m與LPlm模式的強(qiáng)度圖相聯(lián)系。m是從芯的中心起沿著r方向的最大數(shù)，2l是圍繞周線的最大數(shù)。并且在光線圖中，l表示螺線傳輸?shù)膶挾龋惫饩€貢獻(xiàn)給模式的數(shù)量。在基模中m為零。此外，m直接與光線的反射角相聯(lián)系，與平面波導(dǎo)中的情況類似。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):材料色散系數(shù)WaveguideDispersionCoefficient(Dw):波導(dǎo)色散系數(shù)GroupDelay(τg):群延遲ProfileDispersion:剖面色散ChromaticDispersionCoefficient(Dch):色散系數(shù)DispersionShiftedFiber(DSF):色散位移光纖PolarizationDispersion:偏振色散(polarization-modedispersion,PMD):偏振模式色散DispersionFlattenedFiber(DFF):色散平坦光纖Isotropic(anisotropic):各向同性（各向異性）TerminologyWavelengthMultiplexing:波長復(fù)用(WavelengthDivisionMultiplexing,WDM):波分復(fù)用DenseWavelengthMultiplexing(DWDM):密集波分復(fù)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