Towards spectral intensity interferometry 

GEORGIY SHOULGA1 AND EREZ N. RIBAK1,* 1Department of Physics, Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel *Corresponding author: eribak@physics.technion.ac.il  

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXX 

Weuseintensityinterferometrytoimageagroupofpointsources,inacomputersimulationandlabora‐torydemonstration.Weacquiretheimageofthis‘as‐terism’bymeasuringthesecondandthirdordercor‐relationsbetweenphotonarrivaltimes,usingthreesinglephotondetectors.Toreducenoiseeffects,wedividethelightcollectorintosegments,andeachseg‐ment isdispersed into spectralbands,wherepho‐tons in each band are correlated separately. Thiscomprisesanewopticaldesignthatis(1)stableforpartiallycollimatedbeams;(2)islightefficient;and(3)isnotanywiderthantheincomingbeam.©2016OpticalSocietyofAmerica

OCIS codes: (100.3175) Interferometric imaging; (270.5585) Quan‐tum information and processing; (100.6640) Superresolution.  

http://dx.doi.org/10.1364/AO.99.099999 

1. INTRODUCTION Inthe1970'sHanburyBrownandTwissbuiltanintensityinterferome‐terandsuccessfullymeasureddiametersofaseveralbrightstars1.SincetheaverageintensitydistributionofthestaranditscoherencefunctioninthefarfieldarerelatedbyaFouriertransform2,onecanobtaintheimageofthestarbytakingtheinverseFouriertransformofthecoher‐encefunction.However,measuredintensityfluctuationsinthefarfielddomainprovideonlytheamplitudedistributionintheFourierdomain,butallphaseinformationislost.Inordertoobtainthephaseaswell,itispossibletomeasuretriplecorrelationsbetweenphotonarrivaltimes3.Widebandmeasurementsrevealthestructureofachromaticobjects,whilenarrowbandonesprovideadditionalspectro‐spatialinformation.Inthiswork,wesimulateinthecomputerandmeasureinthelaban

asterismofthreeartificialstars,detectedbythreemovablephotonsen‐sors.Weacquiretheimageoftheasterismbyfirstmeasuringthesecondand thirdorder correlation functionsbetweenphotonarrival times,thentakinganinverseFouriertransformofthecorrelationfunction.Itwasshownrecently thatmultiple‐photoncorrelationsallowre‐

solvingtheobjectbeyondthediffractionlimit4‐7,providedthefluxissuf‐ficient8, 9. However, cross‐spectral occasional correlations and singlephotoncountsmightscreenthesignal,weakasitis.Thusweproposeanopticalsystem,whichdispersesthelightintomultiplespectralbands.Thisdispersioncanbeusedtomeasurethecorrelationfunctionateachbandseparately,leadingtoanimagewithdifferentspectralregionsinasourcestructure.Asthisdispersioncanimprovethesignaltonoiseratio,weanticipatebeingabletodetectfinerdetailsofthesource.

Amplitudeinterferometry,althoughmoreefficientthanintensityin‐terferometry10,cannotreachshorterwavelengthsduetoatmosphericturbulence1.Unfortunately,ouratmosphereisopaqueintheultra‐vio‐let,whichraisestheoptionofspaceintensityinterferometrywithitslesseraccuracyrequirements10‐12.Wehavealreadystartedlaboratoryexperiments13andsimulations14.

2. RESEARCH GOAL Theultimategoalofthisresearchistobuildaspaceintensityinterfer‐

ometer(Fig.1).Suchaninterferometerwillbemadeupofasmallflotillaoflightcollectorsinaformationflight.Eachsuchcollectorisoflowopti‐calquality,sincetheeffectivetemporalbandwidthof~1GHzallowssur‐faceerrorsandpathdifferencesofafewcentimeters.Decameter‐sizecollectors,placedkilometersapart,providegooduv(farfieldplane)cov‐eragewhilenotresolvingsmallobjectdetails1.Eachsuchcollectorpro‐videsbasicspectraldispersion,toimprovethesignaltonoiseratioofthefinalimagetobecomposed(seealsoSection7).Thusafewspectralchannelsprovideeachastreamofphotonevents,whichneedstobecor‐relatedwithalltheotherstationsatsomecentralstation.Thatcentralstationcanbeatspaceorontheground,anditneedstocollectthedatastreamsfromallstations,performmultiplecorrelationsonthem,andproduceimages.

Fig.1.Schematicsetupoftheresearchgoal:a)recordingphotonarri‐valsinanumberoftelescopesinaformation,b)transmissionoftherec‐ordeddatatogroundantennas,c)correlationsofthereceivedsignals,d)calculationofthecoherencefunction,e)reconstructionoftheangulardiameterofastar. Wehavealreadyexperimentedwithfasttransmittersandreceivers,

andwithfastdataprocessors13.Duetotheexpecteddataload,itmightbeessentialtocompressthedatabetweenstations.Sincethedataaretotallyrandomandresistcompression,itisnecessarytochoosecare‐fullyoverlappingtemporalwindowsfromthedifferentstations14.

Herewedescribetheessentialnextstep:staticexperimentsinthelabor‐atory,photonstatistics,performingmeasurementsofcoherencefunc‐tionsofdifferentorders(andclosurephases),andprocessingthemtoobtainanimageofanon‐trivialobject.

3. OPTICAL SYSTEM TheschematicsetupoftheopticalsystemispresentedinFig.2.Lightproducedbya532nmgreenlaserwaspassedthroughaneutraldensityfilter(NDF)inordertoreducetheintensity,toavoiddamagetothepho‐tomultipliers(PMTs)andsignalamplifiers.Then,itwassentontoaro‐tatinggroundglass(GG)disk(Newport,100diffuser)poweredbyaDCmotor.TherotationspeedoftheDCmotorwasadjustedto500rpm,toreducethecoherenceofthescatteredlight.Thescatteredlight,thuscon‐sideredapseudo‐thermallight15,illuminated350μmpinholes,thuscre‐atingartificialstars.Thesetofpinholeswasplacedascloseaspossibleto the rotating ground glass, because the scattered light produces aspecklepatternwhoseaveragesizeshouldbelessthanareaofeachar‐tificialstar.Then,light(thespecklepattern)fromtheartificialasterismwassplittothedetectorsintheuvplane.ThecollectionareaofeachPMT(Hamamatsu,R7400UPMT)shouldbelessthantheaveragespecklesize,andthus,15μmpinholeswereplacedinfrontofeachPMTtolimittheircollectionarea.PhotoneventsfromthePMTwereaugmentedbyhighspeedamplifiers(Becker&HicklHFAC‐26DB‐10UA,poweredbyseparate12vbatteries)andtheirmutualcorrelationswasperformedinMATLAB.

Fig.2.Schematicsetupoftheopticalsystem.

Insimilarsetups3,16themeasurementsofthefulluvplaneinafarfieldweredonebyfixingthedetectorsandrotatingthesourcestarfordiffer‐entorientations.Herewefixedthesourceasterismandmeasuredtheintensitiesintheuvplanedirectlybymovingthedetectorsinafarfieldplane.Thisisclosertoasituationwherethelightcollectorsaremovingandmappingthefarfieldplane.

4. PHOTON STATISTICS Thecoherencetimeofthesourceiscontrolledbytherotationspeedofthegroundglassdisk.Theslowertherotation,thehigherthecoherencetimeandviceversa.Byappropriatechoiceofthecoherencetimerela‐tivetotherecordedlengthofthesignal(patch)wecanacquiredifferentphotonstatisticsbyplottingthenumberofpatcheswithagivennumberofphotonsasafunctionofnumberofphotonsineachpatch.Indeed,inalimitwherecoherencetimeismuchshorterthanapatchlength,pho‐tonswillobeyaPoissonstatistics.Ontheotherhand,inalimitwherecoherencetimeismuchlongerthanapatchlength,photonswillobeyaBose‐Einstein(BE)statistics.Inanycase,purePoissonstatistics(BEsta‐tistics)forphotonarrivaltimeswilltakeplaceonlyinthecasewhere

coherencetimeiszero(infinity).Therefore,therewillalwaysbeamix‐tureofthetwo.Theshorterthecoherencetime,thelargerfractionofthephotonswillobeyPoissonstatisticsandlessBEstatisticsandviceversa.Inthecurrentwork,agroundglassdiskwasrotatingataconstant

speedcorrespondingtothecoherencetimeofabout35μs.ThiswasdonebymeasuringaHWHMofthebunchingpeakaftercrosssignalcor‐relationprocess.Thesinglepatchlengthshavebeenchosentobe2μs(muchshorterthanacoherencetime,BEstatisticswereexpected),160μs (much longer than a coherence time, Poisson statisticswere ex‐pected)and35μs(aboutthecoherencetime,anequalmixtureofPois‐sonandBEstatisticswereexpected).Overall,10 equallengthpatcheswereusedtoacquireeachofthestatisticsgraphs(Fig.3).

Fig.3.Photonstatisticshistogramsfordifferentpatchlengths:timein‐tervalismuchshorter(top),equal(middle)andmuchlonger(bottom)thanthecoherencetime.Reddottedlinesrepresentthebestfitforsta‐tistics:Bose‐Einstein(top),Poisson(bottom)andacombinationofthetwo(middle).

Ascanbeseen,incasewherethepatchlengthismuchshorter(longer)thanthecoherencetimethelargerfractionofthephotonsobeystheBE(Poisson)statistics.Incasewherethepatchlengthisequaltothecoher‐encetimethecorrespondingdistributionisanequalmixtureofBEandPoissonstatistics.

5. IMAGE RECONSTRUCTION Forreconstructionoftheobjectweusedthemeasuredcorrelations,butthelackofphaseisaseverelimitation.Phaseambiguitycanberemovedbytriplecorrelation,whichforthreesignalsf,g,hisdefinedby15

, ≡ ∗ , (1)

where and areintroducedtimedelays(aphasecosineambiguityremains,seeAppendix).Therefore,triplecorrelation,incontrasttothesecondordercorrelation,isatwodimensionalstructure,whereeachaxisrepresentsadifferenttimedelay.Themethodispresented,asde‐velopedoriginally,fortemporalcorrelations,butthesamelogicappliestoourcaseoftwo‐dimensionalspatialcorrelations.Itisverydifficultandtimeconsumingtocalculatethetriplecorrela‐

tiondirectly,pointbypoint,withthefullsetoftimedelays and .Themethodwehaveusedinsteaddoesnotcalculatethetriplecorrelation

pointbypoint,ratherwewanttoconstructthetwodimensionalstruc‐turerowbyrow.Thatis,freezingthefirsttimedelayat 0inEq.(1)weget

0, ≡ ∗ , (2)

where ∗ ∗ isjustadirectmultiplicationoffirsttwosig‐nals.Asaresult,wegetasimplecross‐correlationbetween and

,whichcanbecalculatedintheFourierdomain.Inthiswayweob‐tainthefirstrowofthetriplecorrelationgraph.Tocalculatethenextrow,weuse 1unit(0.8ns)i.e.valuesofthesecondsignalarefirstshiftedby1unitandthenmultipliedbythefirstsignal.Then,again,thecrosscorrelationwiththethirdsignalisperformed.Repeatingthispro‐cessfordifferent delays,weconstructthetriplecorrelationrowbyrow.Theresulting2Dstructurehassomeimportantfeatures:ittendstobemaximalinthedirectionswhere 0,or 0,or (Fig.4).Thisresultisobvious,sincethecorrelationtendstobehighwithinthe

coherencetime(small timedelays)anddecreasesastimedelays in‐crease18,19.ThetriplecorrelationiszeroatthetopleftandbottomrightcornersinFig.4,right,sinceattheseregionsthefirsttwosignalsarede‐layedby(atleast)halfsignallengthsandinoppositedirections,resultinginacombinedsignal ∗ whichisidenticallyzeroforallvalues,evenbeforeperformingacrosscorrelationwithathirdsignal.

Fig.4.Experimentaltriplecorrelation(left)anditstopview(right).

Evenifthismethodismuchfasterthanapointbypointcomputation,thewholetriplecorrelationisnotnecessarytoevaluatethedegreeofthethirdordercoherence.Itisenoughtocomputetherowforeitherof

0, 0,or (Fig.5).Thedegreeofthethirdordercoher‐enceisthenobtainedbynormalization,bydividingthepeakvaluebythevalueofthemaximumofthe“triangle”beneath,whichisresponsibleforallnon‐bunchedphotons.

Fig.5.Slicesofthetriplecorrelationfunctionfor 0(left), 0(middle)and (right).

6. SIMULATIONS AND LABORATORY MEASUREMENTS Anartificiallymadeasterismofthreestarswasmeasuredinthelabora‐torywiththreesinglephotondetectors.Suchanasterismwasamask

withthreepinholes,withnon‐redundantdistancesbetweeneachtwocomponentsofthetriplestarsystem(Fig.6,right).Themaskwasmadebydrillingthree350μmholesinasheetofaluminum.ThenweranaMATLABsimulationtocomputetheintensitydistributionofthesameasterisminthefarfieldandfromitreconstructedanimageintwocases:withandwithoutphaseinformation.TheintensitydistributioninthefarfieldwascalculatedbyFouriertransformofthemask(orcouldbedonebyusingFraunhoferdiffractionintegral).TheimagereconstructionwasobtainedbyaninverseFouriertransformappliedtotheuvimage(Fig.6,left)containingtheamplitudeandphaseateachpoint(Fig.6,right)andtotheuvimagewithabsolutevaluesoftheamplitudeateachpoint(Fig.6,center),thuslosingthephaseinformation.Alreadyfromthelat‐terreconstructionwecandeducethattheobjectconsistsofthreepin‐holes, but the orientation of these three pinholes (the phase infor‐mation)willbesuppliedbythedegreeofthethirdordercoherence.

Fig.6.MATLABsimulation:intensitydistributioninthefarfield(left),imagereconstructionwithout(middle)andwith(right)phaseinfor‐mation:aperfectreconstruction.Thebottompanelsarethenegativesofthetopones,toemphasizethefaintparts.TheexperimentitselfwassetupasdepictedinFig.2.ThreePMTswereplacedinsuchamannerthatPMT1andPMT3werefixedintheirplacesthroughout thewholeexperimentwithaconstantbaselineof~0.15mm.WemovedPMT2intheuvplane(perpendiculartotheopticalaxis)frompointtopoint,producingnewbaselinesbetweenitandPMT1andPMT3,andrecordedthesignalsatallthreePMTs.TherecordingsignalsforPMT2weretakenat0.15mmstepsbetween‐1.5mmand1.5mm.Thisway,ineachdirection21differentbaselineswereproduced,result‐inginagridof21×21measurementsmatrix(overall,441measurementpoints).Finally,fromtherecordedsignalsateachpoint,wecalculatedthedegreesofthesecondandthirdordercoherence(seeAppendix).Fig.7presentsgreyscaleresultsofthedegreeofthesecondordercoherenceateachpoint(21×21matrix).

Fig.7.Degreeofthesecondordercoherenceγ(2)intheuvplane:PMT1‐PMT2 (left), PMT2‐PMT3 (middle)andPMT1‐PMT3 (fixedbaseline,right).Asexpected, ,thedegreeofthesecondordercoherencebetweenPMTs1‐3isratherconstant,sincethesePMTswerefixedataconstantbaseline.Moreover, and areverysimilar,differingonlybyasmallshift,causedbythebaselinebetweenPMT1andPMT3.Therecon‐structedimage(withoutphaseinformation)isshowninFig.8.Asex‐pected,alreadyatthispointthestructureofthemask(threecompo‐nents)canbededucedfromthesixweeksatellites,butnotitsorienta‐tion.Asseen,thePMT1‐PMT3imagereconstruction(Fig.8,right)leadstonoadditionalinformationduetotheirfixedbaseline.Sincetheeffec‐tiveapertureofthedetectorswastentimessmallerthantheirspacing,wepaddedthedataatthebottomrowinFig.8toachievealesspixelatedimage.

Fig.8. Imagereconstructionwithoutphaseinformation.Top:PMT1‐PMT2 (left), PMT2‐PMT3 (middle) and PMT1‐PMT3 (right). Imagesizesare21×21.Bottom:thesameafterydirectionbiasremoval,andzeropaddingoftheresultsdepictedinFig.7.Imagessizesare441×441.Thesquarepixelsizesare119μm(top)and5.67μm(bottom).

Oncewehadtheamplitudesoftheobject’stransform,weneededtoaddphasestothem.Essentiallytheorientationoftheimageismissing.Aftercalculationofthethirdordercoherenceusingphaseclosurewewereabletoconstrainthephasevalues.Thisconstraintwasappliedtothetheoretical21×21matrixofphasevaluesatthepossibleorientations.Thismatrixofphasevalueswascombinedwiththematrixofamplitudevaluesderivedfromthedegreeofthesecondordercoherencematrix(Fig.7,left)usingtheSiegertrelation20,producingthefullintensitydis‐tribution(amplitudeandphase)inafarfield(Fig.9).

Fig.9.Imagereconstructionwithphaseinformation:MATLABsimula‐tion(left)andexperimentalresults(right).

7. SPECTRAL INTENSITY INTERFEROMETRY Tillnow,thesourcewasconsideredasquasi‐monochromatic.Realstarsarenotmonochromatic,andmeasurementusuallyinvolvesaband‐passfilter.Thisway,allphotonsatallotherwavelengthsarelost.AlreadyHanburyBrown1pointedoutthatadditionalspectralchannelsincreasethefinalsignal,despitethefactthattheysplittheincomingphotonsontomanydetectors.Sinceintensityinterferometryhaslowsignaltonoiseratio,itisimportanttomeasureallincomingphotons,regardlessoftheirwavelength.Onesolutionistodividetheincomingraysbyconsecutivecolor filters intomultiplebeams, eachcontaining adesired rangeofwavelengths,eachtobedetectedandcorrelatedseparatelywithotherbeamsofthesamecolors.Anothersolutionistodispersetheincomingraysintoacontinuousspectrum,andtoproceedsimilarly.Herewepro‐poseonesuchopticaldesign.Intensityinterferometryisusedtomeasuretheintensityfluctuations,

andallthatisrequiredisphotonshittingthedetector.Itdoesnotdependonthewaya(bunched)photonhasarrivedaslongasitwaswithinthecoherencetimeofthesource.Therefore,animagingsystemisnotre‐quiredanditisenoughtojustcollectthephotons,forexampleusingso‐lar collectors21,22. Similarly, Cherenkov telescopes measure photonsfromatmosphericscintillations23.Radiotelescopescanreach100me‐tersindiameter,andhavesurfaceprecisionofmillimeters(enoughforintensity interferometry if reflective in thevisible regime). Forverylargelightcollectors,weconsideramulti‐inputscenario:theapertureissegmentedoptically. Eachcollectorsectorisimagedontoaseparatechannel,andthesechannels’detectorsarecorrelatedintheuvplane.Theresultsaresomewhatsimilartopupilplaneinterferometers,exceptthatwemeasureγ(2)insteadofγ(1).Welosesensitivitytolightbutgainimmunitytoaberrations,spectralresolution,redundantbaselines,etc23.Hereweproposethateachaperturesectorisnotimageddirectlyon

asingledetector,butinsteadhasadispersionelementinfrontofanareadetector.However,low‐qualityopticsalsomeanthatlightarrivingatthedispersionelement,beitadiffractiongratingoraprism,isnotwellcol‐limated.Thiscanbeduetolow‐qualityoptics,atmosphericturbulence,vibrations,gravitysag,spacedeployment,orotherfactors.Thuswecan‐notusethetechnologyofintegralfieldunits(IFU)whichdisperseper‐fectlycollimatedlight.Suchadispersingopticaldesignshouldthus(1)bestableforraysthat

arenotcompletelycollimated;(2)collectasmuchlightaspossible;and(3)theopticalsystemdimensionsdonotexceedtheareaoftheincom‐inglight,enablingthecreationofatwo‐dimensionalarrayofsuchopti‐calsystemmodules.Thislastrequirementallowsustotilethepupilplane(oranimage

thereof)withanarrayofspectralintensitydetectors.Inaddition,wearealwaysinterestedinacompactsystem,especiallywhenweconsiderspace‐basedinstrumentation.Suchanarraycanbeaslargeasasingleopticaldish(≳5m)andbeusedtoimageobjectsatloworhighresolu‐tionwithouttheneedfordetectorsdisplacementorskyrotation.In‐deed,suchanarraywillprovideahugenumberofdifferentbaselinesonitsown,andbeingstatic,itreducestherequirementsonprecisebaselinemeasurements.

Fig.10.Opticaldisperserdesignforspectralintensityinterferometer.Itconsistsofturbulenceaberratingzone(a),plano‐convexlenses(b,c,g),aconcavemirror(e),aflatmirror(f),aprism(d)andadetector(h).Wetestedanumberofdesignsfordispersionofnearly‐collimated

light,andweshowhereanopticaldesign(performedwithZEMAX)whichmeetstheaboverequirements(Fig.10).

Fig.11.Detectorirradiancebycolor,foradeviationangleof0°,i.e.com‐pletelycollimatedbeam(left),and(toright)deviationanglesof0.1°,0.2°,0.3°,0.4°and0.5°.Thethirdconditionismet,sincethedimensionsofthedesigndonot

exceedthedimensionsofthecollectionareaofthere‐imagedcollectoraperturesegment(elementb).Inordertotestthefirstcondition,atur‐bulencevolume(a)wasinsertedattheentranceoftheopticalsystem.Suchavolumedisplaceseachrayfromcollimationbyarandomanglebetweenzeroandamaximumdeviationangle(Fig.11).Totestthesecondcondition,wecalculatedthepercentageofinitial

raysthathitadetectorfordifferentdeviationangles(Fig.12).Sinceab‐errationsandatmosphericturbulencearemostlybelow0.1°~0.2°,wecanconcludethatsuchasystemcanbestablenotonlyforbadweatherconditions,butalsoinlargetelescopesorlightcollectorswithmirrorde‐formations.

Fig.12.Fractionofrayshittingthedetectorasfunctionofturbulenceorsurfaceaberrationangle.

Tocheckhowthearea,illuminatedbyphotonsofthesamewavelength,increasesasafunctionofdeviationangle,wetakesimilarimagesaspre‐sentedabove,butacquiredforfourwavelengths:400nm,500nm,600nmand700nm,atnegligiblebandwidth.Thenweplottheirradianceasafunctionofthelateralaxisofeachpic‐ture(Fig.13).Inotherwords,wemeasuretheverticalsmearofthedis‐persedspotsinFig.11.Aswecansee, thenumberofdetectorsthatmeasuredifferent spectral bands canbe evaluated according to themaximumdeviationangleallowedinthesystem.

Fig.13.Irradiancealongthedispersionaxisofthedetectorforphotonswithwavelengthof400nm,500nm,600nmand700nm.Deviationfromcollimationismarkedineachpanel.Forhighturbulenceorlargetelescopesurfaceerrors(0.20deviation

angle)aboutfivedifferentspectralbandscanbemeasuredsimultane‐ously.Forlowerdeviationangles(e.g.spacetelescopes)thenumberofsuchbandscanbeincreasedsignificantly.Thefullwidthsathalf‐maxi‐mum(FWHM)oftheabovecurvesasafunctionofthedeviationangle(fromperfectcollimation)arepresentedinFig.14.

8. CONCLUSIONS Insummary,inlaboratorytestsofpotentialspaceintensityinterferom‐etryinstrumentation,wewereabletoreconstructanimageofatriplestarfromtwodimensionalmeasurementsofthesecondandthirdordercoherencefunctions,byaddingclosurephaseinformation.Therecon‐structedimage,althoughmadeofonlyafewpixels,showsarathergoodagreementwiththesourceinsimulation.Forthecaseofasinglelargelightcollectorofaloweropticalquality,weproposetoperformspectralintensity interferometrybetweenthecollectorsegments. Wedevel‐opedanopticaldesignforadispersionsystem,whichistoleranttonon‐collimatedincomingrays.Thisallowsreconstructionoftheimagebyin‐tensityinterferometryfordifferentwavelengths.Alternativelyiftheob‐ject ismonochromatic,weimprovethequalityof the imagewithouthavingtopayforthecolorsplittingamongmultipledetectors.

Fig.14.FWHMofthecurvesinFig.13asafunctionofdeviationangle.Funding:TheIsraeliMinistryofScience(inpart).

References 1. R. H. Brown, The Intensity Interferometer (Taylor & Francis, London, 

1974). 2. A. Labeyrie, S. G. Lipson, and P. Niesenson, An Introduction to Optical 

Stellar Interferometry (Cambridge, 2006). 3. T. Sato, S. Wakada, J. Yamamoto and J. Ishii, "Imaging System Using an 

Intensity Triple Correlator". Appl. Opt. 17, 2047‐2052 (1978). 4. S. Oppel, T. Buettner, P. Kok and J. von Zanthier, “Superresolving Mul‐

tiphoton Interferences with In‐dependent Light Sources”, Phys. Rev. Lett. 109, 233603 (2012). 

5. M. E. Pearce, T. Mehringer, J. von Zanthier, and P. Kok, “Precision esti‐mation of source dimensions from higher‐order intensity correlations”, Phys. Rev. A 92 043831 (2015). 

6. A. Kellerer, “Beating the diffraction limit in astronomy via quantum clon‐ing”, Astron. Astroph. 561, A118 (2014), Corrigendum, Astron. Astroph 582, C3 (2015). 

7. A. Kellerer and E. N. Ribak, “Beyond the diffraction limit via optical ampli‐fication”, Optics Letters 41, 3181‐3184 (2016). 

8. T. Wentz and P. Saha, “Feasibility of observing Hanbury Brown and Twiss phase”, Mon. Not. R. Astr. Soc. 446, 2065–2072 (2015). 

9. P. D. Nuñez and A. D. de Souza, “Capabilities of future intensity interfer‐ometers for observing fast‐rotating stars: imaging with two‐ and three‐telescope correlations”, Mon. Not. R. Astr. Soc. 453, 1999–2005 (2015). 

10. A. Ofir and E. N. Ribak, “Offline, Multidetector intensity interferometers – I. Theory”, Mon. Not. R. Astron. Soc. 368, 1646‐1651 (2006a). 

11. A. Ofir and E. N. Ribak, “Offline, Multidetector intensity interferometers – II. Implications and Applications”, Mon. Not. R. Astron. Soc. 368, 1652‐1656 (2006b). 

12. I. Klein, M. Guelman, and S. G. Lipson, “Space‐Based Intensity Interfer‐ometer”, Applied Optics 46, 4237‐4242 (2007). 

13. E. N. Ribak, P. Gurfil and C. Moreno, “Spaceborne intensity interferome‐try via spacecraft formation flight”, SPIE 8445‐8 (2012). 

14. E. N. Ribak and Y. Shulami, “Compression of intensity interferometry sig‐nals”, Experimental Astrono‐my 41, 145‐157 (2016).  

15. W. Martienssen and E. Spieler, "Coherence and Fluctuations in Light Beams". Am. J. Phys. 32, 919‐926 (1964). 

16. D. Dravins, T. Lagadec, and P. D. Nuñez. "Long‐Baseline Optical Intensity Interferometry". Astronomy & Astrophysics 580, A99 (2015). 

17. A. W. Lohmann and B. Wirnitzer, "Triple correlations," Proc. IEEE 72, 889‐901 (1984). 

18. Bartelt, A. Lohmann, and B. Wirnitzer, "Phase and amplitude recovery from bispectra," Appl. Opt.  23, 3121‐3129 (1984). 

19. A. Lohmann, G. Weigelt, and B. Wirnitzer, "Speckle masking in astron‐omy: triple correlation theory and applications," Appl. Opt.  22, 4028‐4037 (1983).   

20. A. J. F. Siegert, MIT Radiation Laboratory Report 463 (1943). 21. E. N. Ribak, A. Laor, D. Faiman, S. Biryukov, N. Brosch, “Converting 

PETAL, the 25m solar collector, into an astronomical research facility”,  SPIE 4838‐172 (2002). 

22. C.  Barbieri , D. Dravins , T. Occhipinti , F. Tamburini , G. Naletto , V. Da deppo , S. Fornasier , M. D'Onofrio, R. A. E. Fosbury , R. Nilsson and H. Uthas, “Astronomical applications of quantum optics for extremely large telescopes”, Journal of Modern Optics 54, 191‐197 (2007). 

23. D. Dravins, S. LeBohec, H. Jensen and P. D. Nunez, “Optical Intensity In‐terferometry with the Cherenekov Telescope Array”, Astroparticle Phys‐ics 43, 331‐347 (2013). 

24. P.H. van Cittert, “Die Wahrscheinliche Schwingungsverteilung In Einer Von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene“. Physica 1.1‐6 201‐210 (1934). 

25. F. Zernike, “The Concept of Degree of Coherence and its Application to Optical Problems“. Physica 785‐795 5.8 (1938). 

26. J. Nam and J. Rubinstein, “Numerical reconstruction of optical surfaces” J. Opt. Soc. Am. A 25, 1697‐1709 (2008). 

 

Appendix: 2nd and 3rd order coherence functions Weconsiderafinitequasi‐monochromaticlightsourceSofsizeb,locatedatdistancezfromtheobservationplane(Fig.15).

Fig.15.SchemeforvanCittert–Zerniketheorem.AccordingthevanCittert‐Zerniketheorem24,25,aspatialcorrela‐tionfunctionofthefirstorderΓ(1)(D)isaFouriertransformoftheradiationintensitydistributionofthesourceJ(ξ)

Γ Γ | | . (3)

InthiscasethefrequencyωisaspatialfrequencykD/z.Thenor‐malizeddegreeofspatialcoherenceis

. (4)

Generalizing,wecanwriteasecondordercorrelationfunc‐tionas

Γ , , ⟨ , , ⟩ ⟨ ∗ , ∗ , , , ⟩ .

(5)

Afternormalizationwegetthesecondordercoherence

, , ⟨ ∗ , ∗ , , , ⟩

⟨ ∗ , , ⟩⟨ ∗ , , ⟩

⟨ , , ⟩⟨ , ⟩⟨ , ⟩

(6)

Thisisageneralformula.Inthecasewherer1=r2,wegetthedegreeofthesecondordertemporalcoherenceγ(2)(τ),andwhereτ =0wegetthedegreeofthesecondorderspatialcoherenceγ(2)(r1,r2).WecancalculatethenumeratorinEq.(6)Error!Referencesource

notfound.evenfurther⟨ , , ⟩

⟨ ∗ , ∗ , , , ⟩ ⟨ ∗ , , ⟩⟨ ∗ , , ⟩ ⟨ ∗ , , ⟩⟨ ∗ , , ⟩

⟨ , ⟩⟨ , ⟩ |⟨ ∗ , , ⟩|

⟨ , ⟩⟨ , ⟩ Γ , ,

(7)

Normalizationgivesusaconnectionbetweenafirstorderandasecondordercoherence20 , , 1 , , (8)

Bymeasuring wecanobtain andfromitcalculatetheautocorrelationofthesource.Inordertoreconstructthesourcecompletelythephaseofγ(1)isrequiredaswell.Thephaseinfor‐mationcanberecoveredfromthetriplecorrelationfunction.Usingthenotation , ≡ fortheintensityIandamplitudeV,

andΓ Γ e forthespatialcorrelationfunctionof

thefirstorder,thespatialtriplecorrelationcanbeevaluatedas

⟨ ⟩ ⟨ ∗ ∗ ∗ ⟩ ⟨ ∗ ⟩⟨ ∗ ⟩⟨ ∗ ⟩ ⟨ ∗ ⟩⟨ ∗ ⟩⟨ ∗ ⟩ ⟨ ∗ ⟩⟨ ∗ ⟩⟨ ∗ ⟩ ⟨ ∗ ⟩⟨ ∗ ⟩⟨ ∗ ⟩ ⟨ ∗ ⟩⟨ ∗ ⟩⟨ ∗ ⟩ ⟨ ∗ ⟩⟨ ∗ ⟩⟨ ∗ ⟩

⟨ ⟩⟨ ⟩⟨ ⟩ ⟨ ⟩|⟨ ∗ ⟩| ⟨ ⟩|⟨ ∗ ⟩| ⟨ ⟩|⟨ ∗ ⟩|

Γ Γ Γ

Γ Γ Γ

⟨ ⟩⟨ ⟩⟨ ⟩ ⟨ ⟩ Γ ⟨ ⟩ Γ ⟨ ⟩ Γ

2 Γ Γ Γ cos

(9)

Normalizing,weobtainthethirdorderdegreeofspatialcoher‐ence

1

2 cos ,(10)

wherethephaseinthecosineargumentistheclosurephasesinceitisasumofthephasesofthreebaselines(detectorsspacings)overaclosedloop.

Thereisaspecialcase,whenallthreedetectorsarelocatedonthesameline3.Ifnowdetectors2and3arefixedinplaceandseparated by a distance∆ , and detector 1 is free tomove instepsofsamedistanceinaline,wehavetheintensityateachoneofthedetectorsas ∆ , , , and ∆ , ,re‐spectively.Thetriplecorrelationisthen

⟨ ∆ , , ∆ , ⟩ ⟨ ∆ , ⟩⟨ , ⟩⟨ ∆ , ⟩

⟨ ∆ , ⟩ Γ ∆

⟨ , ⟩ Γ 1 ∆

⟨ ∆ , ⟩ Γ ∆ 2 Γ 1 ∆ Γ ∆ Γ ∆ cos 1 ∆ ∆ ∆ .

(11)

Normalizing,weobtainthethirdorderdegreeofspatialcoher‐ence

, , 1 ∆ 1 ∆

∆ 2 1 ∆ ∆ ∆ cos 1 ∆ ∆ ∆

(12)

Eq. (12)Error!Reference sourcenot found. tells us that bymeasuringthethirdordercorrelationfunctionandtheabsolutevalueofthefirstordercorrelationfunction(fromtheSiegertre‐lation)thephasecanbefoundby

Φ ∆ ≜ 1 ∆ ∆ ∆

cos, , 1 ∆ 1 ∆ ∆

2| 1 ∆ || ∆ || ∆ |(13)

Solving Error!Reference source not found. as a differenceequationwecanobtainthephaseby

∆ Φ ∆ ∆ ; 1,2,… (14)

Theterm ∆ isaconstanttermforsettingthecenterofthereconstructedimageandthuscanbesetarbitrary.ThesignoftheΦ ∆ termateachstepofthetriplecorrelationmeasure‐mentcanbefoundfromtakingdoubleintervals, Φ ∆ ≜ 2 ∆ ∆ 2∆ . (15)Fromthedifferentialphasesforthedoubleintervalsweobtainthesignofthecos function,exceptwhereΦ ∆ 0.Inthiscasetripleintervalscanbeconsidered.Sincethismethodrequiresthreedetectorstobeplacedonthesame

line,inordertofindthephasevaluesonaplane,arotationoftwodetec‐torsaroundthethirdone, locatedatorigin, isrequired.Skyrotation

againstthelightcollectorscanbeusefulathighlatitudes,butotherwiseitisnecessarytomovethe(heavy)collectorsaround.Thelocationofspaceandgrounddetectorsshouldbeknownverypreciselyinordertoimprovetheresults.Indeed,theprecisioninthiscaseplaysaveryim‐portantrole,sinceEq.(14)isarecurrencerelation,i.e.anydeviationofameasuredphasevaluefromtherealonewillnotonlyaffectthecurrentmeasurement,butallofthefollowingonesaswell.Followingtheideaabove,wecancalculatethephasesonaplaneus‐

ingfourdetectors.Threearefixedinplacewithbaselines∆ and∆ andthefourthistheonlyonewhichismovingandmappingtheplane.Then,the intensity at each one of the detectors is , y , , ∆ , , , , ∆ , and ∆ , ∆ , . Then,Eq.(13)becomesasetoftwoequationsfordetectors1,2,4and1,3,4,re‐spectively,

Φ ∆ , ∆ ≜ 1 ∆ , ∆ ∆ , ∆ ∆ , 0Φ ∆ , ∆ ≜ ∆ , 1 ∆ ∆ , ∆ 0, ∆ . (16)

Solving Error!Reference source not found. as a differenceequationwecanobtainthephaseby

∆ , ∆ Φ ∆ , ∆ ∆ , ∆ , 1,2, … (17)

where ≜ |gcd , |isagreatestcommondivisorof and .Thisway,wecanconstructmultiplepathsfromtheorigin , y tothepointofthemeasuredphase ∆ , ∆ .Forexample,thephaseat∆ , 2∆ canbecalculatedthroughthreedifferentpaths,leadingto

amoreaccuratevalue(Fig.16),∆ , 2∆ 2 0, ∆ ∆ , 0 Φ 0, ∆ Φ 0,2∆ #∆ , 2∆ 2 0, ∆ ∆ , 0 Φ 0, ∆ Φ ∆ , ∆ #∆ , 2∆ 2 0, ∆ ∆ , 0 Φ ∆ , 0 Φ ∆ , ∆ #

(18)

Fig.16.Differentpathsforevaluationof ∆ , 2∆ accordingto(18).Amongthebenefitsofthistechnique:onlyonedetectorismoving,

providingbetterprecisiononthe locationsofthedetectors;multiplepathsimprovetheaccuracyofthephasecalculation(withfurthercalcu‐latedpointsawayfromtheorigin,moredifferentpathscanbetakenintoaccount26).Afteracquiringthephaseofthecoherencefunctionthroughthispro‐

cess,itisaddedtotheabsolutevalueofthecoherencefunction(found,forexample,fromthesecondordercoherencemeasurements),leadingtothecomplexcoherencefunctionγ(1).Finally,theimageoftheobjectisreconstructedbytakinganinverseFouriertransformofγ(1)usingthevanCittert‐Zerniketheorem.