光学相干和光子统计.docx
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光学相干和光子统计
外文文献:
OpticalCoherenceandPhotonStatistics
Thefieldofoptics,afterseemingtohavereachedasortofmaturity,isbeginningtoundergosomerapidandrevolutionarychanges.Thesechangesareconnectedwiththingswhichwehave,asamatterofprinciple,knownaboutformanyyears,buttheextenttowhichwecouldputourknowledgeintopracticehas,untiljustafewyearsago,beenextremelylimited.Thustheelectromagneticcharacteroflightwaveshasbeenfamiliarknowledgesincethelastcentury.Avastbodyoftheoryandtechniqueconcerningthegenerationofelectromagneticwaveshasbeenbuiltupduringtheseyears,butvirtuallyallofithasdealtwithradiofrequencyfields.Lightwavesofcourse,areofthesameelectromagneticcharacterasradiowaves.Butbecausetheonlywayswehadofgeneratingtheminthepastwereextremelyclumsy(inasenseweshallpresentlydiscussatsomelength)therehasbeenverylittleoccasionuntilrecentlytoapplytheinsightsofradio-frequencytheoryinoptics.Asimplephysicalreason,asweshallsee,liesatthebottomofthis:
allofthetraditionaltypesofopticalsourcespossessacertainchaoticqualityincommon.Theyarewhataradioengineerwouldrefertoasnoisegenerators,andallofthedelicateandingenioustechniquesofopticsareexercisesintheconstructiveuseofnoise.Theinventionoftheopticalmaserliasremovedthisbarrierwithalmostasinglestroke.Itallowsustopresumethatwewillsomedaybeabletocontrolfieldsoscillatingatopticalorhigherfrequencieswiththesamesortofprecisionandversatilitythathavebecomefamiliarinradiofrequencytechnology.
Anotherrecentchangeisthedevelopmentofdetectorswhichrespondstronglytoindividualquantaoflight.Thesehavepermittedustoexplorethecorpuscularcharacterofopticalfields.Allofthetraditionalopticalexperimentshavenotonlydealtwithextremelycrudesources,buthavepaidverylittleattentiontothedetectionofindividuallightquanta.Thedetectorsusedweretopicallysensitiveonlytosubstantialnumbersofphotonsandwerequiteslowinactionsothatwemeasuredonlyintensitieswhichhadbeenaveragedoverrelativelylongperiodsoftime.Thenewlightdetectorsenableustoaskmoresubtlequestionsthanjustonesaboutaverageintensities;wecanforexample,askquestionsaboutthecountingofpairsofquanta,andcanmakemeasurementsoftheprobabilitythatthequantaarepresentatanarbitrarypairofspacepoints,atanarbitrarypairoflimes.
Iftheinstrumentationinopticshasmadelongstridesinthedirectionofdealingwithphotons,itisworthmentioningthattheinstrumentationintheradiofrequencyfieldisleadinginthatdirectionaswell.Theenergiesofradiofrequencyphotonsareextremelysmall,muchsmallerthanthethermalfluctuationenergykT(T=noisetemperature~roomtemperatureformostamplifiers).Therehasconsequentlynotbeenmuchneedinradiofrequencytechnologytodatetopayattentiontothecorpuscularstructureofthefield.Therecentinvention,however,oflownoiseamplifiers,suchasthemicrowavemaser,hasloweredthenoisetemperatureofthedetectingdevicetosuchadegreethatwithfurtherprogressitseemsnotimpossiblethatindividualphotonsmaybedetected.So,eveninthemicrowaveregion,thereisnowacertainamountofattentionbeingpaidtothecorpuscularstructureoflight.
Itisinteresting,inanycase,toinvestigatethecorpuscularnatureofelectromagneticfields,becauseitwillsettheultimatelimitationtothepossibilityoftransmittinginformationbymeansoffields.Wewillnotdiscussinformationtheoryinthisarticle,butwewillhavesomethingstocaywhicharerelatedtonoisetheory.Noisetheoryistheclassicalformofthetheoryoffluctuationsoftheelectromagneticfieldandisquitenaturallyrelatedtothetheoryofquantumfluctuationsofthefield.Allofthesesubjectsfallunderageneralheadingwhichwemightcallphotonstatistics.Coherencetheorytoo,isproperlyspeaking,arathersmallareaofthesamegeneralsubject.Itspurposeissimplytoformulatesomeusefulwaysofclassifyingthestatisticalbehaviouroffields.
Theproblemtowhichweshalladdressourselvesinthisarticleistheconstructionofafairlyrigorousandgeneraltreatmentoftheproblemsofphotonstatistics.Thereisnoneed,indoingit,tomakeanymaterialdistinctionbetweenradiofrequencyandopticalfields(orbetweentheseandX-rayfieldsforthatmatter).Apartoftheformalism,thatwhichhastodowiththedefinitionofcoherence,issuggestedinfactasawayofunifyingtheratherdifferentconceptsofcoherence,whichhavecharacterizedtheseareasinthepast.
Wehavealreadyremarkedthatopticalexperimentshaveonlyrarelydealtwithindividualphotons.Muchthesameobservationcanbemadeforopticaltheoryaswell.Ifthephotonhastosucharemarkabledegreeremainedastrangertoopticaltheorysomejustificationforthatfactsurelyliesinthegreatsuccessofthesimple,wavemodelsintheanalysisofopticalexperiments.Suchmodelsareusuallyspokenofasbeingclassicalincharactersincetheyproceedtypicallyfromsomekindofanalogytoclassicalelectromagnetictheoryandpayaslittleattentiontothecorpuscularcharacteroftheradiationastheexperimentalarrangementwillpermit.
Figure1
Intheseapproachesonetalkstypicallyaboutsomekindof“opticaldisturbancefunction”whichisassumedtoobeythewaveequationandperhapscertainboundaryconditionsaswell.Thefunctionmayrepresentthecomponentsofthoelectricvectororpossiblyotherfieldquantitiessuchasthevectorpotential,orthemagneticfield.Inmanyapplicationsinfactonedoesnotneedtobeveryspecificaboutwhatitreallydoesordoesnotrepresent.
LetusconsidertheYounginterferometer(Fig.1)inordertoillustratetheelementaryapproacheswearediscussing.Aplane,quasi-monochromaticwavecomingfromapointsourceσimpingesonthescreenΣwithtwoparallelslitsatthepositionsP1andP2.
ThetwowavesemergingfromtheslitsgiverisetoaninterferencepatternonthescreenΣ,whichwecanoftenseewiththeunaidedeye.Thesimplestwayofpredictingtheformoftheinterferencepatternistoignorethevectorcharacteroftheelectromagneticfieldandintroduceascalarfieldwhichispresumedtodescribethe“opticaldisturbance”.Wethentrytofindafunction.whichsatisfiesthewaveequationtogetherwithasetofboundaryconditionswhichwetaketorepresenttheeffectofthescreenΣ.Thatproblem,asyouremember,isingeneralagooddealtoodifficulttobesolvedexactly,anditiscustomarytomakeanumberofsimplifyingapproximationssuchasdealingverycrudelywiththeboundaryconditions,andmakinguseofHuygens’principle.Bythesefamiliarmethodswereachasimpleevaluationofthefielddistribution.onthescreenΣ..
Ofcourse,ifwearetopredicttheformoftheinterferencepattern,wemustatsomestagefacethequestionofattachingaphysicalinterpretationtothefields.Themostfamiliarapproachistoregardφ(r,t)asarealfieldandtoidentifyit,perhaps,withoneofthecomponentsoftheelectricfieldvector.Theexperimentalfringepatternisthenpredictedquiteaccurately,asweallknow,ifthelightintensityonthescreenisidentifiedwithφ2,thesquareofouropticalfield.Theidentificationpossessesthejustification,fromthestandpointofclassicaltheory,thatthePoyntingvector,whichtellsustheenergyflux,isindeedquadraticinthefieldstrength.Inspiteofthisevidentsupporttheidentificationisnotauniqueone,however;itpaystoolittleattentiontothewayinwhichthelightisdetected.
Letussupposethatthelightintensityismeasuredbyusingaphotoncounteratthepositionofthescreen.Wethenaskhowwemaypredicttheresponseofthecounterasitisusedtoprobethepattern.Althoughtheuseofthewaveequationtofindthefieldamplitudedidnotintroduceanydistinctionsbetweentheclassicalandthequantumtheoreticalapproachestothediffractionproblem,theuseofaphotoncounterasadetectordoesintroduceadistinction.Thephotoncounterisanintrinsicallyquantummechanicalinstrument.Itsoutputisonlypredictableintermsofstatisticalaveragesevenwhenthestateofthefieldisspecifiedprecisely.Ifwearetopredictthisaverageresponsewemustberathermorespecificthanwehavethusfarbeenaboutthefieldwhichthecounterseesandwemusttreatthedetectionmechanisminafullyquantummechanicalway.Whatwefindwhenwedothesethingsisthatthecountermaybemoreaccuratelythoughtofasrespondingtoacomplexfieldφ+ratherthantherealfield.,andashavinganoutputproportional,nottoφ2,butto|φ+|2,(Thedistinctionisnotatrivialonephysically,sinceinamonochromaticfieldφ2oscillatesrapidlyinmagnitudewhile|φ+|2remainsconstant.)Oncethisanswerisknownitcanbeusedasacruderuleforbypassingtheexplicitdiscussionofthedetectionmechanisminapplicationstootherdetectionproblems.
Theuseofsuchrulesasameansofavoidingtheexplicituseofquantummechanicshasseveraltimesbeencalledthe“semi-classicalapproach”.Whileapproachesofthistypeclearlyneedaruleofsomesorttobridgethegapbetweentheirdescriptionsofthewaveandparticlebehaviorsofphotonstheymayremainperfectlycorrectapproachesinaquantummechanicalsenseaslongastherulehasbeenchosencorrectly.Thefactthatamistakenformofthisruleliasbeenusedrepeatedlyin“semi-classical”discussionsisagoodindicationthatthefullyquantummechanicaldiscussionisnotentirelybesidethepoint.
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