光学相干理论的应用

CHAPTERFOURCHAPTERFOURApplicationsofopticalcoherencetheoryOlgaKorotkovaa,GregGburbaDepartmentofPhysics,UniversityofMiami,CoralGables,FL,UnitedStatesbDepartmentofPhysicsandOpticalScience,UNCCharlotte,Charlotte,NC,UnitedStatesDedicationThisarticleiswritteninthememoryofProfessorEmilWolf.Theideasheintroducedandthecollaborationshefosteredcontinuetohaveanimpacttoday.OurarticleisaspecialtributetoEmilWolfwho,inadditiontoestablishingthemajortheoreticalconceptsofthisfield,hasalsocontributedtothedevelopmentitsapplications,includingdiffractiontomography,beamshaping,andclassicalimaging,amongothers,andhaspromotedendlessideasforothertechnologiessuchasfree-spacecommunicationsandX-raycrystallography.ContentsIntroduction 44Coherencefundamentals 45Astronomy 48Intensityinterferometry 53Ghostimaging 55Opticalcoherencetomography 57Tomography 60Beampropagationinnaturalturbulentmedia 62Partiallycoherentimaging 67Specklemitigationincoherentopticalsystems 70Inertialconfinementfusion 75Beamshaping 78Trappingandmanipulation 81Coherencebeyondlightwaves 83Electromagneticcoherence 85Sunlightcoherenceandphotovoltaics 88Concludingremarks 90References 92AbstractOverthelastcentury,classicalopticalcoherencehasdevelopedfromavaguelyrelatedconceptsintoastandingalongbranchofopticsand,morebroadly,ProgressinOptics,Volume65 #2020ElsevierB.V. 43ISSN0079-6638 Allrightsreserved./10.1016/bs.po.2019.11.004PAGE104OlgaKorotkovaandGregGburPAGE104OlgaKorotkovaandGregGburPAGE103ApplicationsofopticalcoherencePAGE103Applicationsofopticalcoherencetheoryelectromagnetics,thathasresultedinanumberofgroundbreakingdiscoveriescerningthenatureoflight,itsevolutionandinteractionwithmatter.Whilethetheoret-icaldevelopmentsofthisfieldhavebeenwelldocumentedinanumberofexcellentmonographsandreviewarticles,itsapplicationshaveneverbeenproperlysummarized.Inthisreviewwecoverbroadlyemployed,currentlydeveloping,andyetuntappedpracticaloutcomesofopticalcoherencetheoryusedinotherfieldsofscience,technol-ogy,andmedicine.Keywords:Coherence,Opticalcoherencetomography,Imaging,Speckle,ElectromagneticsIntroductionoftoofaanditwastheoftoofasofononoftheoftheandaButEmilWolfintroducedtwoimportantideasinthe1950sthatmaybeconsideredfoundationalpillarsofopticalcoherencetheory.Thefirstofpillarsistheemphasisonopticsintermsofobservablequantities(Wolf,1954):thatopticscan,andshould,bebasedonquantitiesthatcanbemeasured,likeintensitiesandcorrelationfunctions,ratherthanfieldswhichoscillatefastinthevisiblerangetobedirectlyobserved.Thesecondofthesepillarsistherecognitionthatthecorrelationfunctionsoflightalsosatisfytheirwaveequations,nowreferredtoastheWolfequations(Wolf,1955),showsthatthestatisticalpropertiesoflightfollowpredictablelawsthatbeusedtoimproveapplications,orcreatenewonesentirely.Wemayarguethattheuseofopticalcoherencetoimproveexistingapplicationsordevelopnewoneshasbecomeanothermajorcomponentofthesubject,takingadvantageofthenowwell-establishedphysicalthatWolfintroduced.Thoughtheearlyyearsofopticalcoherencefocusedonunderstandingthephysicsoffluctuatingfields,therehasanincreasingemphasisontailoringthesefluctuationstoimprovesensing,imaging,communications,opticaltrapping,andotherpracticaltasks.However,therehasnottodatebeenacomprehensivereviewofapplications,andthisarticleaimstoprovideone.Webeginbysummarizingkeyresultsfromclassicalopticalcoherencetheory,andthenreviewavarietyofapplications.Somearewell-established,evenpredatingtheformaltheoryofopticalcoherence,andothersareisingbutstillworksinprogress.Itisworthnotingthateachapplicationsomewhatdifferentcoherenceneeds,andresearchersinthedifferenthavedevelopedtheirownpartiallycoherentsources.Ourstudyofthecationsofcoherencetheoryisthereforealsoastudyofthemethodswhichpartiallycoherentsourcescanbesynthesized.CoherencefundamentalsWebeginwithastochasticcomplexscalarwavefieldU(r,t)thatsatisfiesthewaveequation,2 1∂2Uðr,tÞrUðr,tÞ-c2 ∂t2 ¼0, (1)whereristhepositionvector,tisthetime,andcisthevacuumspeedoflight.WecharacterizethecoherencepropertiesofthefieldbythemutualcoherencefunctionΓ(r1,r2,τ),definedasΓðr1,r2,τÞ¼hU*ðr1,tÞUðr2,t+τÞi, wherethebracketsh⋯irepresentatimeaverageoranensembleaverage,andtheasteriskrepresentsthecomplexconjugate.Theergodichypothesisistypicallyassumed,inwhichcasethetwoaveragesareequivalent.Thefieldisalsotakentobestatisticallystationary:thatis,itsstatisticalpropertiesareindependentoftheoriginoftime.Thisappliestomoststeady-statelightsources,suchasaCWlaser,astar,oralightbulb.Atruestatisticallystation-arysourceisindependentoftheoriginoftimeforcorrelationfunctionsoforders;weusethereducedassumptionofstatisticallystationaryinthesense,aforwhichitisonlyassumedthattheaverageofthefieldisindepen-dentoftimeandthatthemutualcoherencefunctiononlydependsonthetimedelayτ.FromthedefinitionofthemutualcoherencefunctionitfollowsthataverageintensityI(r)ofthefieldisgivenbyIðrÞ¼Γðr,r,0Þ: (3)aAmongEmilWolfandhiscolleagues,“inthewidesense”hasbeenabitofarunningjoke,oftensaidinthesamesenseas“very,veryapproximately,”andusedforalmostanything,inopticsorineverydaylife.EvenEmil’swifeMarlieswoulduseitfromtimetotime.Withthis,wemayrewritethemutualcoherencefunctioninthesuggestiveform,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓðr1,r2,τÞ¼ Iðr1Þ Iðr2Þγðr1,r2,τÞ, whereγ(r1,r2,τ)iscalledthecomplexdegreeofcoherenceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγðr1,r2,

Γr,r,ττÞ三 ð12ÞffiffiffiffipffiτÞ三 ð12Þ1 212Thisquantitycanbeshowntobeconstrainedbytheinequality0jγ(r1,r2,τ)j1,01Theofγ(r1,τ)istoofr1r2aτ.Atheoftwoinandbeas2jUðr1,tÞ+Uðr2,t+2

Ir+Ir¼ðÞ ð2+ffiffiffiffiffiffiffiffiReγr1,¼ðÞ ð2whereRestandsfortherealpart.Asalreadynoted,afoundationalresultofcoherencetheoryistheequations,whichfollowfromthewaveequationandthedefinitionofmutualcoherencefunction.Theyarewrittenas2 1∂2Γðr1,r2,τÞr1ð1,2,τÞ-c2 τ2 ¼0, (7)2 1∂2Γðr1,r2,τÞr2ð1,2,τÞ-c2 τ2 ¼0, (8)iwherer2representstheLaplacianwithrespecttori,withi¼1,2.Theseequationsarestraightforwardtoderive,andthederivationtakesonlyailines;nevertheless,theyweremetwithsomesurpriseandresistanceoriginallyintroduced.bTheWolfequationsdemonstratethatthestatisticalpropertiesoflight,intheformofthemutualcoherencefunction,alsoagateasawave.However,itistobenotedthatneithertheintensitynorcomplexdegreeofcoherencethemselvessatisfyawaveequation.Inopticalphysics,calculationsaretypicallydonewithmonochromaticwaves,forsimplicity.Incoherencetheory,ItisalsoconvenienttobWhenWolffirstpresentedhisderivationtoMaxBorn,Bornreplied,“Wolf,youhavealwaysbeensuchasensiblefellow,butnowyouhavebecomecompletelycrazy!”Aftersomethought,Bornacceptedtheresult(Wolf,1983).inthespace–frequencydomain,usingthecross-spectraldensityW(r1,r2,definedasthetemporalFouriertransformofΓ(r1,r2,τ),11∞Wðr1,r2,ωÞ¼2π∞-

Γðr1,r2,

-iωτ

dτ, (9)whereωistheangularfrequency.Inthisform,thecross-spectraldensitybereadilyshowntosatisfyapairofHelmholtzequations,2 2r1Wðr1,r2,ωÞ+kWðr1,r2,ωÞ¼0, (10)2 2r2Wðr1,r2,ωÞ+kWðr1,r2,ωÞ¼0, (11)¼wherekω/c.¼Thespectraldensity(orspectralintensity)ofthefieldatfrequencyωgivenbytheequal-positionvalueofthecross-spectraldensity,Sðr,ωÞ¼Wðr,r,ωÞ: (12)Itistobenotedthat,fromEq.(9),wemaythensaythatthespectralatapointrisgivenbythetemporaltransformofthefunctionΓ(r,r,τ);thisisintheWiththeofspectralwemayalsowritetheffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWðr1,r2,ωÞ¼ Sðr1,ωÞ Sðr2,ωÞμðr1,r2,ωÞ, whereμ(r1,r2,ω)iscalledthespectraldegreeofcoherence,andmaybedefinedμðr1,r2,

Wr,r,ωωÞ¼ ð12 Þ 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiωÞ¼ ð12 Þ 12Aswiththecomplexdegreeofcoherence,thespectraldegreeofcoherencesatisfiesaninequality:0jμ(r1,r2,ω)j1.Thelimitjμj¼0isincoherent,andjμj¼1iscoherent.Thephysicsofthecross-spectraldensityisdifficulttodeducefromEq.(9),asitistheFouriertransformofacorrelationfunction.Butingroundbreakingpaperin1982,Wolfshowedthatthecross-spectraldensitycanitselfbeexpressedasacorrelationfunction,intheformWðr1,r2,ωÞ¼hU*ðr1,ωÞUðr2,ωÞiω, (15)ωwhereh⋯iωrepresentsanaverageoveraspeciallyconstructedensembleofmonochromaticfieldsU(r,).Thisensembleisnotarealensemble,ωbutamathematicalone;however,itisalwayspossibletointroducesuchanensembleforanycross-spectraldensity.Furthermore,thecross-spectraldensityisaHermitian,nonnegativedefinitefunction.WolfshowedthatitcanthereforebewritteninMercer-typeexpansion(Mercer,1909)oftheform,nWð1,2,Þ¼λnωÞ*1,ωϕnr2,ω, (16)nnwhereλn(ω)20representtheeigenvaluesofthecross-spectraldensityϕn(r,ω)theorthonormaleigenfunctions,whichsatisfyaFredholmintegralequation,DZWðr1,r2,ωÞϕnðr1,ωÞdNr1¼λnðωÞϕnðr2,ωÞ: (17)DTheontheofDandtheNofandthesumnmaybeorandonetheistobetheofancalandtheisisasthemodeasitisanofentIthasbeentofortheofandThissectionhassummarizedsomeofthekeyquantitiesusedinthedescriptionofpartiallycoherentlight.MoreinformationcanbefoundMandelandWolf(1995),BornandWolf(1999),andWolf(2007).AstronomyTheearliestfieldinwhichcoherenceeffectswereusedtomakemeasurementsisastronomy,andinfactthesemeasurementslongpredatetheformaltheoryofopticalcoherence.Wenote,inparticular,theuseinterferometrytodeterminethesizeofstellarobjectsthatcanotherwisenotberesolvedbyordinarytelescopes.TheideawasfirstputforthbyMichelson(1890a)in1890,inalecturetitled“Measurementbylightwaves.”Init,Michelsonnotedthatopticaldevicessuchastelescopesandmicroscopesareusedforthreetasks:Resolvingsmall,closelypackedobjects,(2)Imagingthestructureofobjects,and(3)Determiningtheprecisepositionoftheseobjects.For3rdcase,heobservedthattheinterferencepatternproducedbylightthesourceisthemosthelpful,andthatbyblockingthecentralapertureofanimagingdeviceandleavingonlytwoslitsontheextremeedges,onecananinterferometricmeasureofpositionthatissuperiortowhatcanachievedbythedeviceundernormaloperation.Inessence,hesuggestsingthetelescopeintoaYoung-typeinterferometerfordeterminingchangesinphaseduetoposition.Michelson(1890b)elaboratedontheseideasthatsameyearinapapertitled,“OntheApplicationofInterferenceMethodstoMeasurements,”andprovidedcurvesforthevisibilityofthefringesonewouldexpectfromanextendedsource.Remarkably,henoopticalcoherencetheorytoworkwith,butthroughsimpleshowedhowthevisibilitycurvesoflinear,circular,ordoublesourcesdependonthesourcesizeandgeometry,allowingtheroughstructureastellarobjecttobededucedfrominterference.HealsoproposedtheconceptofwhatwouldbecomeknownastheMichelsonstellarinterferom-eter,doingawaywiththelensofatelescopeentirelyandinsteadcollectinglightusingwidelyseparatedmirrors.Ashenoted(Michelson,1890b),Thus,whileitwouldbemanifestlyimpracticabletoconstructobjectivesmuchlargerthanthoseatpresentinuse,thereisnothingtopreventincreasingthedis-tancebetweenthetwomirrorsoftherefractometertoevententimesthissize.Therealizationofthisplanwouldtakeseveraldecadestoaccomplish.But1920,MichelsonworkedwithGeorgeElleryHale(Michelson,1920)performmeasurementswiththe40-in.refractingtelescopeatObservatoryandthenthe60-and100-in.reflectingtelescopesatWilsonObservatory.Thetwoaperturemethodwasusedonthetelescopes,andsuccessfulmeasurementsweremadeofCapellaintheconstellationofAuriga,whichhadbeenshownthroughspectroscopicmethodstobedoublestar.Anderson(1920)presentedthedetailedresultsofthatmeasure-mentthesameyear.Thenextyear,MichelsonandPease(1921)performedmeasurementswiththefirstcustom-builtMichelsonstellarinterferometer,usingittoestimatethesizeofthestarBetelgeuse.TheiroriginalsketchofinterferometerisshowninFig.1.Michelson’smethodwasfinallyputonarigoroustheoreticalbasisZernike(1938),whoalsointroducedtheconceptofthedegreeofcoher-ence.WeexplainthemethodusingthevanCittert–Zerniketheoremthespace–frequencydomain,asfollows.eTheevolutionofthecross-spectraldensityfromtheplanez¼0toaplanez>0maybemodeledusingFresnelpropagation,intheformeWðr,r,z,ωÞ¼1ZZWλ2λ2

ð0,r0,ωeik1r02ik2r02d2r0d2r0,12ðzÞ

01

2z 1 2z 2

1 2(18)MM1M2M3bM4dc0 5 101520aFEETFig.1OriginalillustrationofMichelson’sstellarinterferometer.FromMichelson,A.A.,&Pease,F.G.(1921).Measurementofthediameterofalpha-Orionisbytheinterferometer.ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica,7,143–146.whereW0(r10,r20,ω)theinthesourceWethatthesourceisspatiallyincoherentandmaybeasWωÞ¼C2S0ðr01,-whereδ(2)isthetwo-dimensionaldeltafunction,S0representsthespectraldensity,andCisaparameterwithunitsoflengthfordimensionalconsistency.Onsubstitutionandintegration,wehaveWðr1,r2,z,ωÞ¼2e2z12S0ðr,z12dr: CWðr1,r2,z,ωÞ¼2e2z12S0ðr,z12dr: ðλzÞThecross-spectraldensityobservedatanyplaneisthereforeproportionaltheFouriertransformofthesourcespectraldensity.Foracircularsourceradiusawithconstantspectraldensity,theFouriertransformmaybereadilyevaluatedandthespectraldegreeofcoherenceμ(r1,r2,z,ω)takesonthesimpleform,Jkr1-r21μr,r,z,ωÞJkr1-r21

: (21)12 2z1

kzr1-r2aInthisexpression,a=z�sinðθÞ,whereθistheangularsizeofthesource.ThefirstzerooftheBesselfunctionJ1(u)appearswhenu¼3.83;bymea-suringtheseparationj1-2jatwhichinterferencefringesfirstdisappear,isthenpossibletodeterminethesourcesize.Asimilarcalculationcandoneinthetimedomain,inwhichtheequaltimedegreeofcoherencej(r1,r2,z)replacesμ(r1,r2,z,ω),andthewavenumberkisreplacedbythemeanwavenumber.Stellarinterferometrywasimplementedfrom1998to2006attheObservatory,operatingatinfraredwavelengths;the85mbaselinebetweenthetwinKecktelescopeshadthepotentialforextremelyhighresolution.However,fundingissueskepttheinterferometerfrombeingfullypleted,andtheprojectwasputonholdin2006.Forvisiblelight,Michelson’sdesignislimitedbytheneedtodirectlycombinetheopticalsignals.Astheseparationofthemirrorsbecomeslarger,atmosphericeffectsstarttodistortthephasesofthefields,washingoutdelicateinterferencepatterns.Forthisreason,long-baselineinterferometricexperimentstendtoapplyintensityinterferometry,discussedinthesection.However,thereweretwoplanstoputMichelsoninterferometersinspace,wheretheatmospherewouldnotbeafactor.TheStarLightmission(Blackwoodetal.,2003)wouldhaveusedtwoseparatecraftsastheinter-ferometer,withabaselinethatcouldvaryfrom35to125m.Unfortunatelythismission,andthelaterSpaceInterferometryMission(SIM)(KahnAaron,2003),werebothdefundedbeforelaunch.ej2-1Michelson’sstrategyusesquasimonochromaticlight,andforstellarsourcesisthereforeonlytakingadvantageofpartoftheavailablespectrum.In1995,James,Kandpal,andWolfdemonstratedthatthespectralpropertiesoflightcanalsobeemployedinabroadbandversionofMichelson’stech-nique.Intheirmethod,theymadethreeassumptionsaboutthesource,allreasonableforastronomicalobjects:(1)Itisaquasihomogeneoussecondarysource,(2)Itobeysthescalinglaw(Wolf,1986),and(3)Thenormalizedspectrumofthesourceisconstantthroughoutitsdomain.ThelightfromthesourceisassumedtopassthroughaYoung-typeinterferometer(forthe-oreticalcalculations,effectivelythesameasaMichelsonstellarinterferome-ter),anditsspectrumismeasuredatasinglepointontheobservationscreen.Followingthepreviousanalysis,thespectraldegreeofcoherenceμej2-10μ12ωÞ¼

z0ðÞ

: (22)IncontrastwiththeMichelsonstellarinterferometer,wefixthepinholesep-arationandinsteadconsiderthechangesinthespectrumattheobservationplane.ThisquantityisgivenbyS,ωÞ¼2S1ÞðÞ1+μ12ωÞjcos½β12ðÞ+ðR2-R1=cg:(23)HereS(1)representsthespectrumatrwhenonlyonepinholeisopen,representsthephaseofμ12(ω),andRjisthedistancefromthejthpinholetheobservationscreen.Becauseμ12(ω)dependsonthesourcestructureandthefrequency,spectrumwillpossessoscillationsthatcanbeusedtodeterminedetailsthesource.Fig.2illustratesanexample,inwhichthesourceistakentobedoublestar.Thefastmodulationisduetostellarseparation,andtheslowmodulationisduetostellarsize.ThismethodwasdemonstratedexperimentallyforasinglesourceVicalvi,Spagnolo,andSantarsiero(1996)usingaslitilluminatedbyastenlamp,andexcellentagreementwiththeorywasfound.ThisspectralinterferencemethodwastestedusingactualstarlightbyKandpalet(2002),andthemeasuredsizesofstarswereinexperimentalagreementknownvaluesdeterminedbyothermeans.Modulationduetosourcesize:Modulationduetoseparation:Interferencefringes:2Modulationduetosourcesize:Modulationduetoseparation:Interferencefringes:S(PS(P,w)2S(1)(P,w)10

1 2 3 4 5w(s–1)¼¼xFig.2Thespectrumproducedbyadoublestar,eachofangularradiusα31-8andwithangularseparationΔ3107,withpathdifference10mandbaseline5m.AdaptedfromJames,D.F.V.,Kandpal,H.C.,&Wolf,E.(1995).Anewmethodfordetermin-ingtheangularseparationofdoublestars.TheAstrophysicalJournal,45,406¼¼xIntensityinterferometryOneareawhereMichelson-typeinterferometryhasbeenusefulisinradioastronomy,wherebaselinescanberealizedoverkilometers.Radioantennascanconveythereceivedoscillatingsignallongdistancesovercables,allowinginterferencepatternstoberecordedelectronically.Intheearlydaysofradioastronomy,however,itwasthoughtthatbaselinesofhundredsoreventhousandsofkilometerswouldbenecessarytoproperlyresolvethesizeofradiostars.Thephasestabilityofthesignalswouldbedif-ficultorimpossibletomaintainoversuchdistances,andadifferentapproachwouldberequired.Intheearly1950s,HanburyBrownandTwiss(1954)introducedthemethodofintensityinterferometry,inwhichthefluctuatingintensitiesoftheradiosignals,ratherthanthefields,wouldberecordedwithsquarelawdetectorsandcorrelated.Asimplifieddescriptionofhowthisworksbeginswiththeinstantaneousintensityofthefieldinthetimedomain,Iðr,tÞ¼U*ðr,tÞUðr,tÞ, andtheinstantaneousvariationofintensityfromthemean,ΔIðr,tÞ¼Iðr,tÞ-hIðr,tÞi: Ifonelooksatthecorrelationofintensitiesattwopointsinspaceandtime,onegetstheexpressionhΔIðr1,tÞΔIðr2,t+τÞi¼hIðr1,tÞIðr2,t+τÞi-hIðr1,tÞihIðr2,t+τÞi:(26)Itistobenotedthat,inthisexpression,theaverageintensityisindependentoftime,i.e.hI(r,t)i¼I(r),asinEq.(3).Thefirstintensitycorrelationfunc-tionontheright-handsideofthisequationisafourth-orderfieldcorrelationfunction.UndertheassumptionthatthelightsourcesatisfiesGaussianstatistics,whichisvalidfornaturalsources,itmaybewrittenas2hIr1,tIð2,t+τÞi¼hIð1,tÞiIð2,tÞi+jð1,r2,τÞj:(27)Onsubstitution,wereadilyfindthatthenormalizedintensitycorrelationgivenby212hΔIðr1,tÞΔIðr2,t+τÞi¼jγðr,r,τÞj2: (28)12hIðr1,tÞihIðr2,tÞiThenormalizedintensitycorrelationissimplygivenbythesquaredabsolutevalueofthecomplexdegreeofcoherence.Becausethefirstzeroofthisdegreeofcoherencecanbeused,asintheMichelsonstellarinterferometer,todeterminethesizeofasource,intensitycorrelationscanalsobeusedsuchsourcemeasurements.Thefirstexperimentaltestofthemethodwasdonein1952,2beforethepublishedtheoreticalanalysisofHanburyBrownandBrown,Jennison,andGupta(1952)measuredthesizesofradiosourcesJodrellBank,inManchester.HanburyBrownandTwissnextturnedtolaboratoryexperimentsdeterminewhetherintensitycorrelationscanbemeasuredinopticalsignals(Brown&Twiss,1956),andtheirpositiveresultmetwithmuchresistance.Whereasradiowavedetectorsmaybethoughtofasclassicalwavedetectors,photodetectionisaninherentlyquantumprocess,andtherewasconcernthattheshotnoiseofphotondetectionwouldoverwhelmthedesiredfluctuations.Infact,earlyexperimentsbyandVarga(1954)andBrannenandFerguson(1956)foundnointensitycorrelationsatAdetailedanalysisoftheirexperimentsbyBrownandTwiss(1956),ever,determinedthattheirexperiments,asdesigned,wouldneedtorun1011yearsand1000years,respectively,tofindasignal.cWiththevalidityofthetechniqueestablished,BrownandTwiss(1956)performedaninitialtestatJodrellBankbymeasuringtheangulardiameterofthestarSiriusA.Themeasuredvaluewas0.0068000.000500theacceptedvalueis0.005936000.00001600Intheearly1960s,theUniversityofManchesterandUniversityofSydneycollaboratedtobuildastellarintensityinterferometerattheNarrabriObservatoryinNewSouth(Brown,Davis,&Allen,1967),formeasurementsofstellardiameters.Ofthemanyobservationsmadethere,weonlynotethemeasurementZetaPuppisin1969(Davis,Morton,Allen,&Brown,1970).Thestudyofintensitycorrelationshashadapplicationsbeyondastron-omy,notablyinthestudyofthestatisticsofquantizedfields.Forexample,Hennyetal.(1999)usedintensitycorrelationstostudythestatisticsofbeamofelectrons,notingthedifferencesbetweenthisfermionicandabosonicbeamofphotons.Quiterecently,Hongetal.(2017)usedHanburyBrown–TwissinterferometrytostudythepropertiesofsinglecWenotetheseresultsnottocriticizetheauthors,buttopointoutthenaturalperilinherentininves-tigatinganypoorlyunderstoodphenomenoninphysics.photonsinanoptomechanicalresonator.Thesearejustapairofexamples;ingeneral,intensitycorrelationshavealsoplayedafundamentalroleinunderstandingthenonclassicalpropertiesoflight.ItisworthnotingthattherehasbeenrecentinterestinextendingHanburyBrown–Twisstypemeasurementstoincludepolarizationeffects.See,forinstance,Liu,Wu,Pang,Kuebel,andVisser(2018)andKuebelandVisser(2019).GhostimagingGhostimagingisatechniqueforimageformationbymeansofintensitycorrelations,instrikingcontrastwithothermethodswhichimageswithintensity,phaseorpolarizationstate.ItmaybeconsideredmoreelaborateformoftheHanburyBrownandTwissinterferometricmethodfordeterminingstellardiameters.Thebasicprincipleofghostinginvolves(i)splittingtheilluminationintotwobranches:onenotinter-actingwiththeobjectbutbeingdetectedbyacamerawithhighspatialresolution,andtheotherpassingthroughtheobjectbutbeingimagedintoabucket(single-pixel)photodetectorand(ii)correlatingthetwooutputs.Termghostcapturesthepeculiarnatureoftheimageformationmechanisminwhichneitherdetector’sintensityoutputcarriestheinformationtheobjectbyitself.ThefoundationofghostwassetbyKlyshko1988b)andthefirstwasoutinSergienko,Klyshko,andShihandPittman,Shih,andThereliedonentangledpairsproducedbyparametricandpho-usedinforscanningoftheentireThus,ghostimagingwasasapurelyquantumuntilitwasbymeansofexistinginasourcebyBennink,andBoydFig.3shows(A)thefirstsetupofghostimagingwithclassiclightand(B)thepro-ducedghostimageoftheURoflabel(fromBenninketal.,Gatti,Brambilla,Bache,andLugiato(2004)developedthetheoryghostimagingwithincoherentclassicallightandcompareditvianumericalsimulationswithimagingbasedontheentangledphotonpairs.InCaiZhu(2005)(seealsoCaiandWang(2007)),theeffectofsourcepartialcoherenceontheghostimagequalityandvisibilitywasaddressed:A BFig.3Ghostimagingwithclassiclightsource.(A)Theexperimentalsetup,and(B)formedimage.FromBennink,R.S.,Bentley,S.J.,&Boyd,R.W.(2002).“Two-photon”coin-cidenceimagingwithaclassicalsource.PhysicalReviewLetters,89,113601.anincreaseofsourcecoherencetheimagequalitydecreases(butimageibilityincreases).GhostimagingwithtwistedpartiallycoherentlightdiscussedinCai,Lin,andKorotkova(2009)whereitwasshownthatanincreaseofthetwistfactor,theimagequalityreduces.InTong,Cai,Korotkova(2010)andShirai,Kellock,Se,dg)thetech-niquewasextendedtotheelectromagneticdomainandtheeffectsofpartialpolarizationonimagequalityandvisibilitywerediscussed.Itwasthatthetrendintheimagevisibilitydependsonthedefinitionofthetromagneticdegreeofcoherence.Shapiro(2008)introducedtheideaofcomputationalghostimagingusingaspatiallightmodulatorforcontrollablerandomizationoftheillumination’sintensity.Thisallowedforeliminationofthebeamsplitterandtheimagingdetector.Hencetheonlydetectorleftinthesystemisthesingle-elementdetectormeasuringthe