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SolutionsManualforOptoelectronicsandPhotonics:PrinciplesandPracticesS.O.Kasap1.15March2002Chapter11.1GaussianbeamsConsidertwoidenticalsphericalmirrorsAandBthathavebeenalignedtobefocaldirectlyfaceeachotherasinFigure1Q1.Thetwomirrorsandthespaceinbetweenthem(theopticalcavity)formanopticalresonatorbecauseonlycertainlightwaveswithcertainfrequenciescanexitintheopticalcavity.ThelightbeaminthecavityisaGaussianbeam.WhenitstartsatAitswavefrontisthesameasthecurvatureofA.SketchthewavefrontsonthisbeamasittravelstowardsB,atB,asitisthenreflectedfromBbacktoA.IfR=25cm,andthemirrorsareofdiameter2.5cm,estimatethedivergenceofthebeamanditsspotsize(minimumwaist)forlightofwavelength500nm.Twoconfocalsphericalmirrorsreflectwavestoandfromeachother.FisthefocalpointandRistheradius.TheopticalcavitycontainsaGaussianbeamWavefrontSphericalmirrorOpticalcavitySphericalmirrorABRFLRFigure1Q1SolutionLetD=diameterofthemirror,fromFigure1Q1,tanθ=(D/2)/Rgivesθ=arctan(D/2R)=arctan(0.05)=0.05rad.Divergenceis2θor0.1rad.Divergence2θandspotsize2woarerelatedby242θλπ=()woanddependsonthewavelengthofinterest.Takingλ=500nm,and,24245001001641096wo==×=×−−λπθπ()()(.).mmorabout6micron.1.2RefractiveindexaConsiderlightoffree-spacewavelength1300nmtravelinginpuresilicamedium.Calculatethephasevelocityandgroupvelocityoflightinthismedium.Isthegroupvelocityevergreaterthanthephasevelocity?SolutionsManualforOptoelectronicsandPhotonics:PrinciplesandPracticesS.O.Kasap1.25March2002bWhatistheBrewsterangle(thepolarizationangleθp)andthecriticalangle(θc)fortotalinternalreflectionwhenthelightwavetravelinginthissilicamediumisincidentonasilica/airinterface.Whathappensatthepolarizationangle?cWhatisthereflectioncoefficientandreflectanceatnormalincidencewhenthelightbeamtravelinginthesilicamediumisincidentonasilica/airinterface?dWhatisthereflectioncoefficientandreflectanceatnormalincidencewhenalightbeamtravelinginairisincidentonanair/silicainterface?Howdothesecomparewithpart(c)andwhatisyourconclusion?SolutionaFromnandNgvs.λcurves,atλ=1300nmn=1.447andNg=1.462Phasevelocity:v=c/n=(3×108ms-1)/1.447=2.073××××108ms-1Groupvelocity:vg=c/Ng=(3×108ms-1)/1.462=2.052××××108ms-1Forglasses,dn/dλisnegativesothatNgnandhencevgv.Notethatvgvinamediumthatwillhaveapositivedn/dλ.Forexample,PbS,PbTe,PbSeintheregionλ=1−3.5µm.bThepolarization(theBrewster)angleisθp=arctan(n2/n1)=arctan(1/1.447)=34.65°°°°Atthisangleofincidence,r//=0,thereflectedwavehasanE-fieldonlyperpendiculartotheplaneofincidence.ThecriticalangleforTIRis,θc=arcsin(n2/n1)=arcsin(1/1.447)=43.72°°°°cForlighttravelinginglassincidentontheglass-airinterfaceatnormalincidence,rrr===−+=−+⊥//..nnnn12121447114471=0.183Thus,R=r2=(0.183)2=0.0335dForlighttravelinginairincidentontheair-glassinterfaceatnormalincidence,rrr===−+=−+⊥//..nnnn12121144711447=−−−−0.183i.e.R=r2=(−0.183)2=0.0335Thereisa180°phasechangeasrisnegative.Noticethatinbothcasestheamountofreflection(3.35%)isthesame.1.3RefractiveindexandtheSellmeierdispersionequationTheSellmeierdispersionequationisanempiricalexpressionfortherefractiveindexofglassintermsofthewavelengthλ,nGGG21221222222322321−=−+−+−λλλλλλλλλSellmeierequation(1)SolutionsManualforOptoelectronicsandPhotonics:PrinciplesandPracticesS.O.Kasap1.35March2002whereG1,G2,G3andλ1,λ2andλ3areconstants(calledSellmeiercoefficients)thataredeterminedbyfittingthisexpressiontotheexperimentaldata.TheactualSellmeierformulahasmoretermsintherighthandsummationofthesametypee.g.Giλ2/(λ2−λi2)wherei=4,5,...butthesecangenerallybeneglectedinrepresentingnvs.λbehaviorovertypicalwavelengthsofinterestandensuringthatthreetermsincludedinEq.(1)correspondtothemostimportantorrelevanttermsinthesummation.Table1Q3givestheSellmeiercoefficientsforpureSilica(SiO2)andSiO2-13.5mol.%GeO2.Writeaprogramonyourcomputerorcalculator,oruseamathsoftwarepackage(e.g.Mathcad,Matlab,Theorist,Mathview)orevenaspreadsheetprogram(e.g.Excel)toobtaintherefractiveindexnasafunctionofλfrom0.5µmto1.8µmforbothpuresilicaandSiO2-13.5%GeO2.Obtainthegroupindex,Ng,vs.wavelengthforbothmaterialsandplotitonthesamegraph.Findthewavelengthswherematerialdispersionbecomeszerointhetwomaterials.Table1Q3TheSellmeiercoefficientsforSiO2andSiO2-13.5%GeO2.Theλ1,λ2,λ3areinµm.G1G2G3λλλλ1λλλλ2λλλλ3SiO20.6967490.4082180.8908150.06906600.1156629.900559SiO2-13.5%GeO20.7110400.4518850.7040480.06427000.1294089.425478SolutionSilica;nvsλλλλ(µµµµm),thincurve;Ngvsλλλλ(µµµµm),thickboldcurve.Ngvsλλλλ(µµµµm)Minimumisaround1.3µµµµm.SiO2-13.5mol.%GeO2:nvsλλλλ(µµµµm),thincurve;Ngvsλλλλ(µµµµm),thickboldcurve.Ngvsλλλλ(µµµµm)Minimumisaround1.4µµµµm.1.4AntireflectioncoatingaConsiderthreedielectricmediawithflatandparallelboundarieswithrefractiveindicesn1,n2andn3.Showthatfornormalincidencethereflectioncoefficientbetweenlayers1and2isthesameasthatbetweenlayers2and3ifn2=√[n1n3].Whatisthesignificanceofthis?bConsideraSiphotodiodethatisdesignedforoperationat900nm.Givenachoiceoftwopossibleantireflectioncoatings,SiO2witharefractiveindexof1.5andTiO2witharefractiveindexof2.3whichSolutionsManualforOptoelectronicsandPhotonics:PrinciplesandPracticesS.O.K