foueieroptics课件cxcoherenceofopticalwav

ChapterXCoherenceofopticalwaves,PickedfromJosephW.Goodman,Statisticaloptics,WileyClassicsLibraryEditionPublished2000,Reasonsforstudyingcoherence,StatisticalpropertiesoflightplayanimportantroleCompletestatisticalmodelistoocomplexSecond-orderaverages-coherencefunctionsarecommonlysatisfactoryenough,Energyspectraldensity,Ifu(t)isaFouriertransformablefunction,thenEnergyspectraldensityofu(t),Powerspectraldensity,Ifu(t)isnotFouriertransformablebutthetruncatedfunctiondoeshaveatransformPowerspectraldensityofu(t),SpectralDensityofaRandomProcess,LetrandomprocessU(t)consistsofsamplesu(t),togetherwithameasureoftheirprobabilitiesEnergyandpowerspectraldensities,Timeautocorrelationfunction,Givenasingleknowntimefunctionu(t),thetimeautocorrelationfunctionofu(t)isdefinedby,Statisticalautocorrelationfunction,GivenarandomprocessU(t),itsstatisticalautocorrelationfunctionisdefinedby,Autocorrelationfunctionpowerspectraldensity,Foraprocessthatisatleastwide-sensestationary,theautocorrelationfunctionandpowerspectraldensityformaFouriertransformpair,Theimportanceofautocorrelationfunctions,ItoffersanexperimentalmeansforultimatelydeterminingthepowerspectraldensityofthesignalItprovidesananalyticmeansforcalculatingthepowerspectraldensityofarandomprocessmodeldescribedonlyinstatisticalterms,Cross-correlation,Time-averagecross-correlationfunctionCross-correlationfunctionoftworandomprocessesU(t)andV(t),Cross-spectraldensityfunctions,Cross-correlationfunctioncross-spectraldensity,Forjointlywide-sensestationaryrandomprocessesU(t)andV(t),Monochromaticcomplexsignals,Real-valuedmonochromaticwaveThecomplexrepresentationofthissignalPhasoramplitude,Differencebetweenspectraofrealandcomplexforms,Fu(r)(t)=Fu(t)=,Fromu(r)(t)tou(t),wedoublethestrengthofthepositivefrequencycomponentandentirelyremovethenegativefrequencycomponent,Nonmonochromaticcomplexsignals,Real-valuednonmonochromaticsignalu(r)(t)withFouriertransformFollowingexactlythesameprocedureusedinthemonochromaticwecanrepresentu(r)(t)bycomplexu(t),u(t)iscalledtheanalyticsignalrepresentationofu(r)(t),Fourierintegralrepresentationofu(t),u(r)(t)u(t),Hilberttransform,Cauchyprincipalvalue,TheintegraltransformationisknownastheHilberttransformofu(r)(t),Importantpropertiesoftheanalyticsignal,Constructionofanalyticsignal,Hilberttransformingfilter,Signalwithnarrowbandspectrum,Real-valuedfunctionu(r)(t)Nonmonochromaticbutnarrowband,Narrowbandslowlyvaryingsignal,SuchasignalmaybewrittenintermsofaslowlyvaryingenvelopeA(t)andaslowlyvaryingphase(t)Correspondinganalyticsignal,ComplexEnvelopesorTime-VaryingPhasors,Complexenvelopeisdefinedbythistime-varyingphasoramplitudeForanysignal(widebandornarrowband),wecanwritetheanalyticsignalrepresentationintheform,Twotypesofcoherence,TemporalcoherenceAbilityofalightbeamtointerferewithadelayed(butnotspatiallyshifted)versionofitselfAmplitudesplittingSpatialcoherenceAbilityofalightbeamtointerferewithaspatiallyshifted(butnotdelayed)versionofitselfWavefrontsplittingTemporal+spatialcoherencemutualcoherencefunction,Temporalcoherence,Analyticsignalu(P,t)hasacomplexenvelopeA(P,t)withafinitebandwidthFinitetimedurationA(P,t)remainsrelativelyconstantduringtheintervalA(P,t)andA(P,t+)arehighlycorrelated,orcoherentCoherencetime,MichelsonInterferometer,Patternofinterference,Phenomenaofinterference,AtzeropathlengthalargecentralpeakintheinterferogramThemirrorisdisplacedfromthezero-delaypositiondropinthefringedepthTherelativedelaygrowslargeenoughtheadditionofelementaryfringesisnearlytotallydestructive,MathematicalDescriptionoftheExperiment,IntensityincidentonthedetectorcanbewrittenaswhereK1andK2arerealnumbersdeterminedbythelossesinthetwopathsandu(t)istheanalyticsignalrepresentationofthelightemittedbythesource,Expansionofthisexpression,Autocorrelationselfcoherence,OpticalintensityAutocorrelationfunctionisknownastheselfcoherencefunctionoftheopticaldisturbance,Intensityondetector,Inthisabbreviatednotationwewritethedetectedintensityas,Complexdegreeofcoherence,NormalizedversionoftheselfcoherencefunctionThedetectorintensityisgivenby,CosineformofID,ComplexdegreeofcoherencecanbewritteninthefollowinggeneralformAssumingK1=K2=K,wecanexpresstheinterferogramas,Theformulationistypicalofthestructureoftheinterferogram,Inthevicinityofzerorelativepathlengthdifference,h0Theinterferogramconsistsofafullymodulatedcosine,Changeofinterferogramwithh,WhenpathlengthdifferencehisincreasedAmplitudemodulationFringesmaysufferaphasemodulationVisibilityofasinusoidalfringe,Relationofvisibilityandcomplexdegreeofcoherence,Lookbackontemporalcoherence,2hvisibilityrelativecoherenceofthetwobeamsVisibility0pathlengthdifferenceexceedsthecoherencelengthorrelativetimedelayexceedsthecoherencetimeTheconceptoftemporalcoherencehastodowiththeabilityoftworelativelydelayedlightbeamstoformfringes,AutocorrelationfunctionPowerspectraldensity,Letacomplex-valuedrandomprocessU(t)haveautocorrelationfunctionU(t)WecalltheFouriertransformofU(t)powerspectraldens