Towards spectral intensity interferometry -...

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).

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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.