Fiber-OpticCommunicationsTechnologyLecture3X.WuDepartmentofOpticalEngineeringZJUTopicsLighttravelinginfiberRectangularwaveguide:ananalogyCircularwaveguideCut-offwavelengthGroupvelocityDispersionDepartmentofOpticalEngineeringZJUInterestingObservation:

whenmultimodefibermeetssinglemodefiberReduceendseparationtoobserveoutputmodesDepartmentofOpticalEngineeringZJUEndseparationMultimodeSinglemodeDepartmentofOpticalEngineeringLightPropagationinWaveguidesMaxwell’sequationsRectangularwaveguideDeterminedbyboundaryconditionApplytoz-componentFirstfindEzandHz,andyouhavealltherestcomponentsDepartmentofOpticalEngineeringZJUNoTEM:Ez=Hz=0,thenEx=Ey=Hx=Hy=0TE:Ez=0,E-fieldlinesaxisofwaveguideTM:Hz=0,B-fieldlinesaxisofwaveguideTEandTMmodesTE:TM:DepartmentofOpticalEngineeringh2

h2=TEandTMFieldForTMmodes,l=0andm=0arenotpossible,TM11isthelowestTMmode.DepartmentofOpticalEngineeringCutofffrequencyandwavelength=+jh2=2＋k2，k＝/vIf>0,waveguideattenuationoccurs,notsupportpropagationCutofffrequencyCutoffoccursEvanescent:nopropagationPropagation:allowtraveling

DepartmentofOpticalEngineeringCutofffrequencyandwavelengthCutofffrequencyPhaseconstantWith’=√()Threewavelengths:Unboundedwavelength=v/f:freespace,noboundaryGuidedwavelengthg=/[1-(f/fc)2]1/2:Allowtotravel!Cutoffwavelengthc=v/fc

:attenuation/evanescent

Modespecific<ca/morb/nHowfieldlooksbetweenparallelplatesTMmodesm=1m=2m=3DepartmentofOpticalEngineeringWhataboutthefieldincircularwaveguidesNoneedtoproveezterm(why?)AgainyouonlyneedtofindEzandHzLaplacianequationDepartmentofOpticalEngineeringTEandTMincircularwaveguideTE:E(a,,z)=0TM:Ez(a,,z)=0DepartmentofOpticalEngineeringTE11TM01Modesinweak-boundedwaveguideDepartmentofOpticalEngineeringModalTheoryofStepIndexfiberGeneralexpressionofEM-waveinthecircularfiber:Eachofthecharacteristicsolutionsiscalledmthmodeoftheopticalfiber.

ItisoftensufficienttogivetheE-fieldofthemode.DepartmentofOpticalEngineeringModepropagationconstantThemodalfielddistributionUm(,),andthemodepropagationconstant,mareobtainedfromsolvingtheMaxwell’sequationssubjecttotheboundaryconditionsgivenbythecrosssectionaldimensionsandthedielectricconstantsofthefiber.Mostimportantcharacteristicsoftransmissionaredeterminedbythemodepropagationconstant,m（）DepartmentofOpticalEngineeringAteachfrequencyorwavelength,thereexistsonlyafinitenumberofguidedorpropagatingmodesthatcancarrylightenergyoveralongdistancealongthefiber.Eachofthesemodescanpropagateinthefiberonlyifthefrequencyisabovethecut-offfrequency,cutoff

,(orthesourcewavelengthissmallerthanthecut-offwavelength)obtainedfromcut-offconditionthatis:Tominimizethesignaldistortion,thefiberisoftenoperatedinasinglemoderegime.Inthisregimeonlythelowestordermode(fundamentalmode)canpropagateinthefiberandallhigherordermodesareundercut-offcondition(non-propagating).Multi-modefibersarealsoextensivelyusedformanyapplications.Inthesefibersmanymodescarrytheopticalsignalcollectively&simultaneously.DepartmentofOpticalEngineeringDepartmentofOpticalEngineeringAttenuationandphaseshiftintotalinternalreflectionTotalinternalreflectionisnottotal:evanescentwave:Attenuatione-y,FresnelformulasReflectanceTransmittance

exampleDepartmentofOpticalEngineeringMoreaboutmodesTruemodesNaturalmodesTE,TMLinear-polarizedmodesLPmodesThreetypesofmodesGuided:

internallyreflectedREPETIVELY!:2krequirementRadiation:notinternallyreflected+no2krequirement(continuum)LeakynotrealmodesMoreaboutmodesinfiberDepartmentofOpticalEngineeringLinearlypolarizedModes:LPlmDepartmentofOpticalEngineeringFormedbyexactmodesTE,TM,HE,andEHl,mdescribeE-fieldprofile2lfieldmaximaaroundfibercorecircumferencemfieldmaximaalongradialdirectionSixLPmodesDepartmentofOpticalEngineeringNumberofmodesv.s.V-numberDepartmentofOpticalEngineeringModepropagationasafunctionoffrequencyModepropagationconstant,mn(),isthemostimportanttransmissioncharacteristicofanopticalfiber,becausethefielddistributioncanbeeasilywrittenasafunctionof.Inordertofindamodepropagationconstantandcut-offfrequenciesofvariousmodesoftheopticalfiber,firstwehavetocalculatethenormalizedfrequency,V,definedby:DepartmentofOpticalEngineeringPropagationconstantasafunctionofnormalizedfrequencyforafewofthelowest-ordermodesDepartmentofOpticalEngineeringWhyhigh-ordermodeshavelower/k?Cutoffwavelength:twoexpressionsForanymode:Therangeoverwhichmodemnwillpropagate0<<c,mn

ForafibertooperatesinglemodeSetV=2.405,findc(LP11)c(LP11)<DepartmentofOpticalEngineeringCutoffwavelength:twoexpressionsIllustrationAssumea=b=10m:Modem=1,n=0:<c(1,0)=a/1=10mModem=1,n=1:<c(1,1)=a/2=7.1m

<…c(m>1,n>1)…<c(1,1)<c(1,0)DepartmentofOpticalEngineeringKeepinthisdomain->singlemodeoperationSingle-modeoperationDepartmentofOpticalEngineeringTwo!Whatisagain?ExampleSinglemodeoperationat0=1.55m:n1=1.5,=0.002radiusa<6.254mn1=1.4572,=0.0042:a<6.4767

DepartmentofOpticalEngineeringDepartmentofOpticalEngineeringGroupvelocityandpowertransportGroupandphasevelocityforE0cos(t-z)Phasev=/=c/n;puresinewave(orwavelet),notmeasurableGroupvg=/,signalandpowertravelingvPowertransportanddistributionMultimodefiberV=40:Pclad/Ptotal=2%SinglemodefiberV=2.405:Pclad/Ptotal=40%DepartmentofOpticalEngineeringPowerconfinement&cutoffconditionLimitingthenumberofmodesVMethodstoreduceV:Asolution:weakboundingCutoffwavelengthDepartmentofOpticalEngineeringDispersioninmultimodefibers:generalTwotypesofdispersion:intermodalandintramodalTotaldispersionPulsewidth:

Fullwidthathalfthemaximum(FWHM)(fig.4.19)Opticalbandwidth:

Half-power(intensity)0.5PowerfrequencyrangeElectricalbandwidth:

0.707Powerfrequencyrange

DepartmentofOpticalEngineeringDispersioninmultimodefibers:intermodalIntermodaldispersion:1storderapproximation:

80ns/km(v.s.30ns/kmtruevalue)Problematicassumptions:AllmodesexcitedEachmodecarriesequalpowerModesdoesnotinteractwithoneanotherAllmodestravelatthesamevelocitynisindependentonDepartmentofOpticalEngineeringDispersioninmultimodefibers:intramodal

(chromatic,material)Sellmeierequations

Sellmeierequationsareessentiallyempiricalfitstotheactualrefractiveindexofamaterial,usingtheresultfor'forLorentzoscillatorsasbasisfunctionsDepartmentofOpticalEngineeringDepartmentofOpticalEngineeringSellmeierEquationforFusedSilicawithdopingModel:Three-polewithpolesatapproximately9.9m,116nmand68nmDepartmentofOpticalEngineeringDepartmentofOpticalEngineeringDispersioninmultimodefibers:intramodal

(chromatic,material)Propagation