光电子学第2章-介质波导与光纤

Chapter2DielectricWaveguidesandOpticalFiber2.1SymmetricplanardielectricSlabwaveguide2.2dispersionintheplanarwaveguide2.3Stepindexfiber2.4Numericalaperture2.5Dispersioninsinglemodefibers2.6Dispersion,electrical,andopticalbandwidth2.7Thegradedindexopticalfiber2.8Lightabsorptionandscattering2.9Attenuationinopticalfibers2.10Fibermanufacture“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.Aslabofdielectricofthickness2aandrefractiveindexn1issandwichedbetweentwosemi-infiniteregionsbothofrefractiveindexn2(n2n1).Theregionofhigherrefractiveindex(n1)iscalledcoreandtheregionoflowerrefractiveindexn2sandwichingthecoreiscalledthecladding.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.ItseemsthatanylightwavethathasanangleofincidenceθgreaterthanthecriticalangleθforTIR,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.Notsuitableformultil