Interferometric data reduction with AMBER/VLTI. Principle, estimators, and illustration

14 Pages·2007·0.46 MB·English

A&A464,29–42(2007) Astronomy
DOI:10.1051/0004-6361:20064799 &
(cid:1)c ESO2007 Astrophysics
Special feature
AMBER:Instrumentdescriptionandfirstastrophysicalresults
Interferometric data reduction with AMBER/VLTI.
(cid:1)
Principle, estimators, and illustration
E.Tatulli1,2,F.Millour1,3,A.Chelli1,G.Duvert1,B.Acke1,14,O.HernandezUtrera1,K.-H.Hofmann4,S.Kraus4,
F.Malbet1,P.Mège1,R.G.Petrov3,M.Vannier3,6,13,G.Zins1,P.Antonelli5,U.Beckmann4,Y.Bresson5,M.Dugué5,
S.Gennari2,L.Glück1,P.Kern1,S.Lagarde5,E.LeCoarer1,F.Lisi2,K.Perraut1,P.Puget1,F.Rantakyrö6,
S.Robbe-Dubois3,A.Roussel5,G.Weigelt4,M.Accardo2,K.Agabi3,E.Altariba1,B.Arezki1,E.Aristidi3,C.Baffa2,
J.Behrend4,T.Blöcker4,S.Bonhomme5,S.Busoni2,F.Cassaing7,J.-M.Clausse5,J.Colin5,C.Connot4,
A.Delboulbé1,A.DomicianodeSouza3,5,T.Driebe4,P.Feautrier1,D.Ferruzzi2,T.Forveille1,E.Fossat3,R.Foy8,
D.Fraix-Burnet1,A.Gallardo1,E.Giani2,C.Gil1,15,A.Glentzlin5,M.Heiden4,M.Heininger4,D.Kamm5,
M.Kiekebusch6,D.LeContel5,J.-M.LeContel5,T.Lesourd9,B.Lopez5,M.Lopez9,Y.Magnard1,A.Marconi2,
G.Mars5,G.Martinot-Lagarde5,9,P.Mathias5,J.-L.Monin1,D.Mouillet1,16,D.Mourard5,E.Nussbaum4,
K.Ohnaka4,J.Pacheco5,C.Perrier1,Y.Rabbia5,S.Rebattu5,F.Reynaud10,A.Richichi11,A.Robini3,
M.Sacchettini1,D.Schertl4,M.Schöller6,W.Solscheid4,A.Spang5,P.Stee5,P.Stefanini2,M.Tallon8,
I.Tallon-Bosc8,D.Tasso5,L.Testi2,F.Vakili3,O.vonderLühe12,J.-C.Valtier5,andN.Ventura1
(Affiliationscanbefoundafterthereferences)
Received2January2006/Accepted1March2006
ABSTRACT
Aims.Inthispaper, wepresent aninnovative datareduction method for single-mode interferometry. Ithasbeen specificallydeveloped for the
AMBERinstrument,thethree-beamcombineroftheVeryLargeTelescopeInterferometer,butitcanbederivedforanysingle-modeinterferometer.
Methods.Thealgorithmisbasedonadirectmodellingofthefringesinthedetectorplane.Assuch,itrequiresapreliminarycalibrationofthe
instrumentinordertoobtainthecalibrationmatrixthatbuildsthelinearrelationshipbetweentheinterferogramandtheinterferometricobservable,
whichisthecomplexvisibility.Oncethecalibrationprocedurehasbeenperformed,thesignalprocessingappearstobeaclassicalleast-square
determination of a linear inverse problem. From the estimated complex visibility, we derive the squared visibility, the closure phase, and the
spectraldifferentialphase.
Results.Thedatareductionprocedureshavebeengatheredintotheso-calledamdlibsoftware,nowavailableforthecommunity,andarepresented
inthispaper.Furthermore,eachstepinthisoriginalalgorithmisillustratedanddiscussedfromvariouson-skyobservationsconductedwiththe
VLTI,withafocusonthecontrolofthedataqualityandtheeffectiveexecutionofthedatareductionprocedures.Wepointoutthepresentlimited
performancesoftheinstrumentduetoVLTIinstrumentalvibrationswhicharedifficulttocalibrate.
Keywords.technique:interferometric–methods:dataanalysis–instrumentation:interferometers
1. Introduction baseline lengths and number of recombined telescopes, the in-
terestofusingthepracticalcharacteristicsofsingle-modefibers
AMBER is the first-generation near-infrared three-way beam
to carry and recombine the light, as first proposed by Connes
combiner (Petrov et al. 2007) of the Very Large Telescope
etal.(1987)withhisconceptualFLOATinterferometer,isnow
Interferometer(VLTI).Thisinstrumentsimultaneouslyprovides
well established. Furthermore, in the light of the FLUOR ex-
spectrally dispersed visibility for three baselines and a closure
periment on the IOTA interferometer, which demonstrated the
phase at three different spectral resolutions. AMBER has been
“on-sky” feasibility of such interferometers for the first time,
designed to investigate the milli-arcsec surrounding of astro-
CoudéDuForestoetal.(1997)showedthatmakinguseofsingle
physical sources like young and evolved stars or active galac-
modewaveguidescouldalsoincreasetheperformancesofopti-
tic nuclei, and to possibly detect exoplanet signal. The main
calinterferometry,thankstotheirremarkablepropertiesofspa-
new feature of this instrumentcomparedto other interferomet-
tial filtering, which change the phase fluctuations of the atmo-
ric instrumentsisthesimultaneoususe ofmodalfilters(optical
spheric turbulent wavefront into intensity fluctuations. Indeed,
fibers)andadispersedfringecombinerusingspatialcoding.The
bymonitoringthesefluctuationsinrealtimethankstodedicated
AMBER team has therefore carefully investigated a data pro-
photometricoutputsandbyperforminginstantaneousphotomet-
cessingstrategyforthisinstrumentandisprovidinganewtype
ric calibration, he experimentally proved that single-mode in-
ofdatareductionmethod.
terferometrycould achieve visibility measurementswith preci-
Given the astonishingly quick evolution of ground based sions of 1% or lower. Achievement of such performance level
optical interferometers in only two decades, in terms of hassincebeenconfirmedwiththeIONICintegratedopticbeam
combiner set up on the same interferometer (LeBouquin et al.
(cid:1) Based on observations collected at the European Southern 2004).
Observatory,Paranal,Chile.
Article published by EDP Sciences and available at http://www.aanda.orgor http://dx.doi.org/10.1051/0004-6361:20064799
30 E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration
Spatial filtering DK P1 P2 IF P3
Recombination Spectral dispersion 2200
h
IFP3 WavelengthWavelengt 2180
P2 2160
P1
K
D 2140
50 100
Single−mode fibers Output pupils Spectrograph Raw Data #Pipxiexlesls
Fig.1.Leftpanel:SketchoftheAMBERinstrument.Thelightenterstheinstrumentfromtheleftandispropagatingfromlefttorightuntilthe
rawdataarerecordedonthedetector.Furtherdetailsaregiveninthetext.Rightpanel:AMBERreconstitutedimagefromtherawdatarecorded
duringthe3-telescopeobservationofthecalibratorHD135382inFebruary2005,inthemediumspectralresolutionmode.DKcorrespondstoa
darkregion,Pkaretheverticallydispersedspectraobtainedfromeachtelescope,andIFisthespectrallydispersedinterferogram.
Surprisingly,theeffectofsingle-modewaveguidesonthein- reductionmethodshavealreadybeenproposedforsingle-mode
terferometric signal has only been studied recently from a the- interferometersusingtemporalcoding(Colavita1999b;Kervella
oreticalpointofview.Ruilieretal.(1997)usednumericalsim- etal.2004),thispaperisthefirsttopresentasignal-processing
ulationsinthe presenceofpartialcorrectionbyadaptiveoptics algorithm dedicated to single-mode interferometrywith spatial
to show that spatial filtering provided a gain on the visibility beamrecombination.Moreover,inthecaseofAMBER,thecon-
signaltonoiseratio.Howeverhisstudywaslimitedtothecase figurationoftheoutputpupils,i.e.thespatialcodingfrequency,
ofapointsource.Thecaseofsourceswithagivenspatialextent imposes a partial overlap of the in the three telescopes case
wasfirsttheoreticallyaddressedbyDyer&Christensen(1999) interferometric peaks in the Fourier plane. As a consequence,
fromageometricalpointofview.Theyprovedthatthevisibility data reduction based on the classical estimators in the Fourier
obtained from single-mode interferometry was biased, the ob- plane (Roddier & Lena 1984; Mourard et al. 1994) cannot be
jectbeingmultipliedbytheantennalobe(thepointspreadfunc- performed. The AMBER data reduction procedureis based on
tion of one single telescope) exactly as it happens in radio in- a direct analysis in the detector plane, a principle that is an
terferometry(Guilloteau2001).Anequivalentgeometricalbias optimization of the “ABCD” estimator as derived in Colavita
was also characterizedfor the closurephase (Longueteauet al. (1999b). The specificity of the AMBER coding and its sub-
2002).Then Guyon(2002)noticedin his simulationsthattook sequent estimation of the observables arises from the desire
the presenceofatmosphericturbulenceintoaccount,thatinter- to characterize and to make use of the linear relationship be-
ferometric observations of extended objects (resolved by one tweenthepixels(i.e.theinterferogramsonthedetector)andthe
single telescope) could not be completely corrected for atmo- observables (i.e. the complex visibilities). In other words, the
spheric perturbations, therefore lowering the performances of AMBERdatareductionalgorithmisbasedonmodellingthein-
single-mode interferometry. Finally, by thoroughly describing terferograminthedetectorplane.
thepropagationoftheelectricfieldthroughsingle-modewaveg- InSect.2,wepresenttheAMBERexperimentfromasignal-
uidesinthegeneralcaseofpartialcorrectionbyadaptiveoptics processing point of view and we introduce the interferometric
andforasourcewitha givenspatialextent,Mègeetal. (2003) equationgoverningthisinstrument.Wedevelopthespecificdata
unified previous studies and introduces the concept of modal reduction processes of AMBER in Sect. 3, and then derive the
visibility, which in the general case does not equal the source estimators of the interferometric observables. Successive steps
visibilityV andexhibitsajointlygeometricalandatmospheric inthedatareductionmethodaregiveninSect.4,asperformed
o
bias. Neverthelesstheyalsoshowthatforcompactsources,i.e. by the software provided to the community. Finally, the data-
smallerthanoneAirydisk,themutualcoherencefactorµcould reductionalgorithmisvalidatedinSect. 5throughseveral“on-
bewrittenintheformofasimpleproductµ = TT V whereT sky” observations with the VLTI (commissioning and science
i a o i
and T are, respectively, the instrumental and the atmospheric demonstrationtime(SDT)).Presentandfutureperformancesof
a
transferfunctionsthatcanbe calibrated.Recently,Tatullietal. thisinstrumentarediscussed.
(2004)deducedfromananalyticalapproachthatinthespecific
caseofcompactobjects,thebenefitofsingle-modewaveguides
issubstantial,notonlyintermsofthesignal-to-noiseratioofthe 2. Presentationoftheinstrument
visibilitybutalsooftherobustnessoftheestimator.
2.1.Imageformation
Hence, following the path opened by the FLUOR experi-
ment, the AMBER instrument – the three-beam combiner of TheprocessofimageformationofAMBERissketchedinFig.1
theVLTI(Petrovetal.2007)–makesuseofthefilteringprop- (left)fromasignal-processingpointofview.Itconsistsofthree
erties of single-mode fibers. However, in contrast to FLUOR, major steps. First, the beams from the three telescopes are fil-
PTI (Colavita 1999a) or VINCI on the VLTI (Kervella et al. teredbysingle-modefiberstoconvertphasefluctuationsofthe
2003), where the fringes are coded temporally with a movable corrugatedwavefrontsintointensityfluctuationsthatare moni-
piezzo-electric mirror, the interference pattern is scanned spa- tored. The fraction of light entering the fiber is called the cou-
tially thanks to separated output pupils, the separation fixing pling coefficient (Shaklan & Roddier 1988) and it depends on
the spatial coding frequency of the fringes, as in the case of the Strehl ratio (CoudéduForesto et al. 2000).At this point, a
the GI2T interferometer (Mourard et al. 2000). Thus, if data pair of conjugated cylindrical mirrors compresses, by a factor
E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration 31
Table1. Detectorproperties. The individualimage that is recorded during the detector inte-
grationtime(DIT)iscalledaframe.Acubeofframesobtained
Detectorspecifications duringtheexposuretimeiscalledanexposure.
Society/Name Rockwell/HAWAII
Composition HgCdTe
Numberofpixels 512×512 2.2.AMBERinterferometricequation
Pixelsize 18.5µm×18.5µm
The following demonstration is given considering a generic
SRpeeacdtoraultwnoidisteh 0.8µm9−e−2.5µm Ntel ≥ 2 telescope interferometer. In the specific case of
e−/ADU 4.18 AMBER, however, Ntel = 2 or Ntel = 3. Each line of the
Cooling Liquidnitrogen detector being independent of each other, we can focus our
Temperature 78K attention on one single spectral channel1, which is assumed to
Autonomyofcryostat 24h be monochromatichere. The effect of a spectral bandwidth on
theinterferometricequationistreatedinSect.3.6.1.
Interferometric output: when only the ith beam is illu-
minated, the signal recorded in the interferometric channel is
of about 12, the individual beams exiting from fibers into one the photometricflux Fi spreadon theAiry patternai, whichis
k
dimensionalelongatedbeamsto be injectedin the entranceslit the diffraction pattern of the ith output pupil weighted by the
ofthespectrograph.Foreachofthethreebeams,beam-splitters single-modeof the fiber, k is the pixelnumberon the detector,
placedinsidethespectrographselectpartofthelightandinduce andαistheassociatedangularvariable.Then, Fi resultsinthe
threedifferenttiltanglessothateachbeamisimagedatdifferent totalsourcephotonfluxNattenuatedbythetotaltransmissionof
locationsofthedetector.Thesearecalledphotometricchannels the ithopticaltrainti,i.e. theproductofthe opticalthroughput
andareeachonerelativetoacorrespondingincomingbeam.The (including atmosphere and optical train of the VLTI and the
remaining parts of the light of the three beams are overlapped instrument) and the coupling coefficient of the single-mode
onthedetectorimageplanetoformfringes.Thespatialcoding fiber:
frequenciesofthefringes f arefixedbytheseparationofthein-
dividualoutputpupils.Theyare f = [1,2,3]d/λ,wheredisthe Fi =Nti. (1)
outputpupildiameter.Sincethebeamshitaspectraldispersing
Whenbeamsiand jareilluminatedsimultaneously,thecoherent
element(a prism gluedon a mirroror one of the two gratings)
additionofbothbeamsresultsinaninterferometriccomponent
in the pupil plane, the interferogram and the photometries are
superimposedonthephotometriccontinuum.Theinterferomet-
spectrally dispersed perpendicularly to the spatial coding. The
ric part, i.e. the fringes, arises from the amplitude modulation
dispersed interferogramarisingfromthe beamcombination,as
of the coherent flux Fij at the coding frequency fij. The co-
wellasthephotometricoutputsarerecordedontheinfraredde- c
herentfluxisthegeometricalproductofthephotometricfluxes,
tector,whichcharacteristicsaregiveninTable1.
The detector consists in a 512 × 512 pixel array with the weightedbythevisibility:
√ (cid:1) (cid:2)
vertical dimension aligned with the wavelength direction. The Fij =2N titjVijeiΦij+φipj (2)
first 20 pixels of each scanline of the detector are masked and c
neverreceiveanylight,allowingustoestimatethereadoutnoise whereVijeiΦij isthecomplexmodalvisibility(Mègeetal.2003)
and bias during an exposure. The light from the two (resp. 3)
andφij takesapotentialdifferentialatmosphericpistonintoac-
telescopes comes in three (resp. 4) beams, one “interferomet- p
count.Notethat,strictlyspeaking,themodalvisibilityisnotthe
ric” beam where the interference fringes are located, and two
sourcevisibility.However,thestudyoftherelationbetweenthe
(resp.three)“photometric”beams.These3 (resp.4)beamsare
modalvisibilityandthesourcevisibilityisbeyondthescopeof
dispersed and spread over three (resp. 4) vertical areas on the
this paper, but further informationcan be foundin Mège et al.
detector.Thedetectorisreadinsubwindows.Horizontally,these
(2003) and Tatulli et al. (2004). Here we consider our observ-
subwindowsarecenteredontheregionswherethebeamsaredis-
abletobethecomplexmodalvisibility.
persed,withatypicalwidthof32to40pixels.Vertically,thede-
Suchananalysiscanbedoneforeachpairofbeamsarising
tectorcanbesetuptoreaduptothreesubwindows(coveringup
tothreedifferentwavelengthranges).Therawdataformatused fromtheinterferometer.Asaresult,theinterferogramrecorded
onthedetectorcanbewritteninthegeneralform:
by AMBER recordsindividuallythese subframes.However,as
sketchedintherightpanelofFig.1,theAMBERrawdatacan (cid:3)Ntel (cid:3)Ntel (cid:4) (cid:5) (cid:1) (cid:2)(cid:6)
beconceivedasthegroupingtogetherofthesesubwindows: ik = aikFi+ aikakjCBijRe Fcijei2παkfij+φisj+ΦiBj . (3)
i i<j
– Theleftcolumn(noted“DK”)containsthemaskedpixels.
– The two following columns(noted“P1” and “P2”) and the Here,φijistheinstrumentalphasetakingpossiblemisalignment
s
right one in the three telescope mode (noted “P3”), usu- and/or differential phase between the beams ai and aj into
k k
ally of 32 pixels wide, are the photometric outputs. They account, and Cij and Φij are, respectively, the loss of contrast
record the photometric signal coming from the three tele- B B
and the phase shift due to polarization mismatch between the
scopes. When dealing with 2-telescope observations, only
two beams (after the polarizers), such as the rotation of the
channelsP1andP2arelit.
single-mode fibers might induce. This equation is governing
– The fourth column is the interferometric output (de-
the AMBER fringe pattern, that is the interferometric channel
noted“IF”).itexhibitstheinterferencefringesarisingfrom
therecombinationofthebeams(thatis,twoorthreebeams, 1 In practice, there is a previous image-centering step, where each
accordingtothenumberoftelescopesused).WecallNpixthe channel isre-centeredwithrespecttotheothersalongthewavelength
numberofpixelsinthiscolumn,whichisusuallyN =32. dimension,asexplainedinSect.4.
pix
32 E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration
ofthefourthcolumn.ThefirstsuminEq.(3),whichrepresents Table2.Acquisitionsequenceofcalibrationfiles.
the continuum part of the interference pattern, is called the
DC component from now on, and the second sum, which Step Sh1 Sh2 Sh3 Phaseγ DPRkey
0
describes the high frequencypart(that is the coded fringes),is 1 O X X NO 2P2V,3P2V
calledtheACcomponentoftheinterferometricoutput. 2 X O X NO 2P2V,3P2V
3 O O X NO 2P2V,3P2V
Photometric outputs: thanks to the photometric channels, 4 O O X YES 2P2V,3P2V
the number of photoevents pi(α) coming from each telescope 5 X X O NO 3P2V
6 O X O NO 3P2V
canbeestimatedindependentlywith
7 O X O YES 3P2V
pi = Fibi (4) 8 X O O NO 3P2V
k k 9 X O O YES 3P2V
wherebi isthebeamprofileintheithphotometricchannel.The Sh=Shutter;O=Open;X=Closed.
k
previousequationrulesthephotometricchannels.
3. Datareductionalgorithm We then can compute the DC continuum corrected interfero-
gramm ,
TheAMBERdata-reductionalgorithmisbasedonthemodelling k
of the interferogram in the detector plane. Such a method (cid:3)Ntel
requiresanaccuratecalibrationoftheinstrument. m =i − Pivi, (11)
k k k
i=1
3.1.Modellingtheinterferogram whichcanberewritten:
Inordertomodeltheinterferogram,wedistinguishbetweenthe m =cijRij−dijIij. (12)
astrophysical and instrumental parts in the interferometric k k k
equation.Itbecomes:
This equation defines a system of N linear equations with
pix
2N = N (N −1) unknowns(i.e. twice the number of base-
i =(cid:3)Ntel aiFi+(cid:3)Ntel (cid:7)cijRij+dijIij(cid:8) (5) linebs). Itctehlarateclterizesthe linear linkbetweenthe pixelsonthe
k k k k detectorandthecomplexvisibility:
i i<j
wcikijth=CBij (cid:4)(cid:4)(cid:9)akikaaikkjakj cos(cid:1)2παkfij+φisj+ΦiBj(cid:2), (6) ⎛⎜⎜⎜⎜⎜⎜⎜⎝mmN|1pix ⎞⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝(cid:17)(cid:1)..(cid:1)|(cid:1)..(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)c(cid:18)NciN:(cid:19)jbi1pji(cid:1)x(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)..(cid:1)|..(cid:20)(cid:17)..|(cid:1)..(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)d(cid:1)(cid:1)d(cid:18)NiN:1jip(cid:19)bjix(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)..(cid:1)|(cid:1)..(cid:1)(cid:20)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝RI:::iijj⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=V2PM⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝RI:::iijj⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (13)
(cid:4)
TheV2PMmatrix(namelyvisibilitytopixelmatrix),whichcon-
dij =Cij (cid:4) aikakj sin(cid:1)2πα fij+φij+Φij(cid:2), (7) tainsthe carryingw(cid:4)aves, holdstheinformationaboutthe inter-
k B (cid:9)kaikakj k s B ferometric beams aikakj, the coding frequencies fij, and the
instrumental differential phases φij. Together with the vi, they
s k
and entirely describe the instrumentfrom a signal processing point
(cid:10)(cid:3) (cid:7) (cid:8) (cid:10)(cid:3) (cid:7) (cid:8) ofview.Thesequantities,namelycij,dij,andvi,have,however,
k k k
Rij = aiajRe Fij , Iij = aiajIm Fij . (8) tobecalibrated.
k k c k k c
k k
3.2.Calibrationprocedure
Asananalogywithtelecomdataprocessing,cijanddijarecalled
k k
the carrying waves of the signal at the coding frequency fij, The calibration procedure is performed thanks to an internal
sincetheycarry(intermsofamplitudemodulation)Rij andIij, sourcelocatedintheCalibrationandAlignmentUnit(CAU)of
AMBER(Petrovetal.2007).Itconsistsofacquiringasequence
whicharedirectlylinkedtothecomplexcoherentflux(asshown
of high signal-to-noiseratio calibration files, whose successive
byEq.(8)).
Theestimatedphotometricfluxes Pi arecomputedfromthe configurationsaresummarizedinTable2andexplainedbelow.
Since the calibration is done in laboratory, the desired level of
photometricchannels(seeEq.(4)):
(cid:3) accuracyforthemeasurementsisinsuredbychoosingtheappro-
Pi =Fi bi. (9) priateintegrationtime.Asanexample,typicalintegrationtimes
k in“averageaccuracy”modeare(forthefullcalibrationprocess)
k τ = 17s, 30 s, 800s for,respectively,low, medium,andhigh
Ifweknowtheratiovi –whichonlydependsoftheinstrumental spectralresolutionmodesintheKbandand100timeshigherfor
k
configuration–betweenthemeasuredphotometricfluxesPiand the“highaccuracy”calibrationmode.
thecorrespondingDCcomponentsoftheinterferogram,wecan The sequence of calibration files has been chosen to ac-
haveanestimationofthelatterthankstothefollowingformula: commodatebothtwoandthree-telescopeoperations.Foratwo-
telescopeoperation,onlythe4firststepsareneeded.Raw data
aiFi = Pivi. (10) FITSfilesproducedbytheESOinstrumentsbearnoidentifiable
k k
E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration 33
Calibration matrix − 2200nm
vk functions − 2200nm ck & dk functions − 2200nm
1−2)0.015 30 1−2) 00..12
ne (0.010 ne ( 0.0
eli eli−0.1
Bas0.005 Bas−0.2
10 20 30 10 20 30
−3)0.015 20 −3)
aseline (200..000150 # pixels aseline (2−000...101
B B
−3) 10 20 30 10 −3) 10 20 30
10.015 1 0.1
ne (0.010 ne ( 0.0
seli0.005 seli−0.1
a a
B 10 20 30 B 10 20 30
#pixels 0 2 4 6 #pixels
Fig.2.Outputsofthecalibrationprocedures.Exampleshavebeenchosenforonegivenwavelength:λ = 2.2µm.Left:thevi functions.Middle:
k
the matrix containing the carrying waves; the first three columns are the cij functions for each baseline, and the three last columns are the
k
respectivedijfunctions.Onecanseethatforeachbaselinecijanddijareinquadrature.Right:anotherrepresentationofthecarryingwaves.From
toptobottomk,bothsinusoidalfunctionscorrespondtocolumkns1−4k,2−5,and3−6ofthecalibrationmatrix.
nameandcanonlybeidentifiedas,e.g.,filesrelevanttothecal- −2) 1.0
ibrationoftheV2PMmatrix,bythepresenceofdedicatedFITS ne (1 00..68 Vc.CB = 0.88
keywords (ESO’s pipeline Data PRoduct keys or “DPR keys”) Baseli 00..24
intheirheader.TheDPRkeysusedarelistedinTable2. 2140 2160 2180 2200
3) 1.0
−
etearchoFpitreesnltees(dsc.toeTppheseb1feraaanmcdt,io2ann–oiamfnfladgu5exiwmshereeancsouirrndede3d-btewelteiwtsheceoonpneltyhmethoinidstee)sr,hffueort-r- Baseline (2 0000....2468 Vc.CB = 0.79
ometric channeland the illuminatedphotometricchannelleads −3) 1.0 2140 2160 2180 2200
to an accurate estimation of the vi functions. Then, in order to ne (1 00..68
compute the carryingwaves cikj ankd dkij, one needsto have two Baseli 00..24 2140 21V60c.CB = 0. 720180 2200
independent(intermsofalgebra)measurementsoftheinterfer- Wavelength
ogramsincetherearetwounknowns(perbaseline)tocompute.
Fig.3. Contrast loss due to polarization effects and partial resolution
The principle is the following: two shutters are opened simul-
oftheinternalsourceasafunctionofthewavelength.The3-telescope
taneously(steps3/4,6/7,and8/9)andforeachpairofbeams,
P2VMusedisthesameastheonepresentedinFig.2.Theerrorsbars
then the interferogramis recorded on the detector. Such an in- areroughlyatthelevelofthecontrastlossrmsalongthewavelength.
terferogram corrected for its DC component and calibrated by Inotherwords,thecontrastlossisconstantoverthewavelengthrange.
the photometry yields the knowledge of the cij carrying wave.
k
To obtain its quadratic counterpart, the previous procedure is
repeated by introducing a known phase shift close to 90 de- However, since cij and dij are shifted by π/2, they insure the
k k
gree γ using piezoelectric mirrors at the entrance of beams 2 followingrelation:
0
and 3. Computing the dij function from the knowledge of cij
andγ isstraightforwardk.Notethatbyconstruction:(i)thecark- (cid:3)Npix
rying0waves are computed with the unknown system phase Φc c2k +dk2 =CB2Vc2. (16)
(possible phase of the internal source, differential optical path k
difference introducedat the CAU level, etc.), and (ii) since the
Hence,theconjugatedlossofvisibilityduetotheinternalsource
internal source in the CAU is slightly resolved by the largest and the polarization effects can be known and calibrated1 by
baseline (1−3)ofthe outputpupils,the carryingwaves forthis
computing the previous formula. Unfortunately, since it is not
specificbaselineareweightedbythevisibilityV oftheinternal
c possibletodisentanglebothcontrastlosses,andsincetheV fac-
c
source.Hence,atthispoint,thecarryingwavesarefollowingex- tor only affects the interferograms arising from the calibration
pressionsthatareslightlydifferentfromtheiroriginaldefinition
procedure,andnotfromtheobservation,thevisibilityestimated
givenbyEqs.(6)and(7): on a star will be affected from this factor as well, as shown in
(cid:4) Sect.3.5.1.
aiaj (cid:1) (cid:2) Figure 3 illustrates Eq. (16). For the baselines (1,2)
cij = (cid:4) k k CijVijcos 2πα fij+φij+Φij+Φij (14) and (2,3), the contrast loss arises from polarization effects,
k (cid:9) B c k s B c since the internalsource is unresolved.We findC12 (cid:4) 0.9 and
aiaj B
k k k C23 (cid:4)0.8,respectively.Forthethirdbaseline(1,3),theinternal
B
source is partially resolved, which explains an higher contrast
(cid:4)
loss,C13V13 (cid:4)0.7.
aiaj (cid:1) (cid:2) B c
dkij = (cid:4)(cid:9) k k CBijVcijsin 2παkfij+φisj+ΦiBj+Φicj . (15) 1 This step is not yet provided in the amdlib software described
aiaj inSect.4.
k k k
34 E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration
3.3.Fringefitting Itisthenstraightforwardthat
(cid:3) (cid:3)
ToestimatethecoherentfluxesRij and Iij,whichareattheba- σ2 = (ζb)2σ2(m ); σ2 = (ξb)2σ2(m ). (22)
sis of the computation of the whole AMBER observables, one Rb k k Ib k k
k k
hastosolvetheinverseproblemdescribedbyEq.(13),i.e.one
has to performan χ2 linear fit of the fringes,with the coherent
fluxesbeing the unknownparameters.The solution is givenby 3.5.Estimationoftheobservables
thefollowingequation
For each spectral channel, the squared visibility and closure
[R(cid:21)ij,(cid:21)Iij]=P2VM[m ], (17) phase(inthethreetelescopecase)canbeestimatedfromthein-
k terferogram.Takingadvantageofthespectraldispersion,thedif-
where ferentialphasecanbecomputedaswell.Inthefollowingpara-
P2VM=(cid:7)V2PMTC−1V2PM(cid:8)−1V2PMTC−1 (18) gqruaapnhtist,iewse. Tdehnisotaevweriathge(cid:5).c..a(cid:6)nthbeeenpseerfmobrmleeadveeritahgeeroofnthtehedifffraemreenst
M M
withinanexposureand/oronthewavelengths.
is the generalized inverse of the V2PM matrix, C being the
M
covariance matrix of the measurements m , and XT denoting
k
the transpose of the X matrix. P2VM means Pixel to Visibility 3.5.1. Thesquaredvisibility
Matrixsinceitallowsestimationofthecomplexvisibilityfrom
Theoretically speaking, the squared visibility is given by com-
the interferogram recorded on the detector. Assuming that the
pixels on the detector are uncorrelated, the C matrix is diag- putingtheratiobetweenthesquaredcoherentfluxandthepho-
M tometricfluxes.FollowingEqs.(1),(2),(8)and(9)itbecomes:
onal,with eachtermofthediagonaldefinedbythe varianceof
the DC-correctedinterferogramσ2(mk). Thefundamentalerror |Fij|2 Rij2+Iij2 |Vij|2
ontheDCcorrectedinterferogramarisesfromthephotonnoise c = (cid:9) = · (23)
anddetectornoise(ofvarianceσ)corruptingthemeasurements, 4FiFj 4PiPj kvikvkj Vcij2
thatis,eachpixeloftheinterferogrami andtheestimatedpho-
k
tometricfluxesPi.Itbecomes: Notethat,thankstothecalibrationprocess,thecomputedvisibil-
ityisfreeoftheinstrumentalcontrast,whichisthelossofcon-
(cid:3)Ntel (cid:7) (cid:8) trastduetotheinstrument;butasmentionedinSect.3.2,theob-
σ2(m )=i +σ2+ P +N σ2 (vi)2. (19)
k k i pix k jectvisibilityisweightedbythevisibilityoftheinternalsource.
i=1 However,thisfactor(Vij2)automaticallydisappearswhendoing
c
thenecessaryatmosphericcalibration(seeSect.3.6.2).
3.4.Fringedetection
Asaresultthevisibility–atmosphericissuesapart–hasstill
tobecalibratedbyobservingareferencesource.Inpractice,be-
Positive detection of fringes in the measurements requires, at
cause data are noisy, we perform an ensemble average on the
thesametime,enoughfluxenteringthefibersandhighenough
framesthatcomposethedatacube(seeSect.2.1)toestimatethe
fringe contrast, so that the fringesrise with the noise level. As
expectedvaluesofthesquarecoherentfluxandthephotometric
a result, the computationof the signal-to-noiseratio of the co-
fluxes,respectively.Takingtheaverageofthesquaredmodulus
herent flux, which takes both of the parameters into account,
ofthecoherentflux,i.e.doingaquadraticestimation,allowsus
appearsnaturallyastherelevantcriteriontouse.
⎡⎛ ⎞ ⎛ ⎞⎤ to handlethe problemofthe randomdifferentialpiston φij, but
SNR2(t)= 1 1 (cid:3)Nb (cid:3)Nl ⎢⎢⎢⎢⎢⎣⎜⎜⎜⎜⎜⎝Rb2(l,t) −1⎟⎟⎟⎟⎟⎠+⎜⎜⎜⎜⎜⎝Ib2(l,t) −1⎟⎟⎟⎟⎟⎠⎥⎥⎥⎥⎥⎦ (20) introducesaquadraticbiasduetothezero-meanphotonanpdde-
N N σ2 σ2 tectornoises(Perrin2003).Theexpressionofthesquaredvisi-
b l b l Rb Ib bilityestimator,unbiasedbyfundamentalnoisesistherefore:
bbeingforsakeofsimplicitythebaselinenumberthatdescribes (cid:29) (cid:30)
eachpairoftelescopes(i, j),N thenumberofbaselines,andN |V(cid:28)ij|2 Rij2+Iij2 − Bias{Rij2+Iij2}
b l = (cid:31) (cid:9) · (24)
tqhueanntuitmiebseRrbo2fasnpdeIcbtr2atlenchdatnonwealrsd. Iσn2thaendabσs2en,creesopfecftriivnegleys,,ththues Vcij2 4 PiPj kvikvkj
Rb Ib
driving the fringe criterion toward 0. In contrast, the presence Thequadraticbiasofthesquaredamplitudeofthecoherentflux
of fringes above the noise level, that is, when Rb2(l,t) > σ2 writes as the quadratic sum of the biases of R2 and I2. From
Rb Eq.(22),weget:
and/or Ib2(l,t) > σ2Ib imposes the fringe criterion to be strictly (cid:3)(cid:7) (cid:8)
superiorto0 andisdirectlylinkedtothequalityoftheframes. Bias{Rij2+Iij2}= (ζij)2+(ξij)2 σ2(m ). (25)
Itthusallowsustooperateafringeselectionpriortotheproper k k k
k
estimation of the observables,a step thatcan be usefulforsets
of data recorded in bad observational conditions, as shown in The previousequation is nothing but the mathematical expres-
Sect.5.2. sionthatdescribesthebiasasthequadraticsumoftheerrorsof
The values σ2Rb and σ2Ib, the bias part of R2 and I2, can be themeasurementsσ2(mk)(asdefinedbyEq.(19))projectedon
easily computed from the definition of the real and imaginary therealandimaginaryaxisofthecoherentflux.
partofthecoherentfluxes,whicharelinearcombinationsofthe Usingthesquaredvisibility estimatorofEq.(24),thetheo-
DCcontinuum-correctedinterferogramsm .Ifζb andξb arethe reticalerrorbarsonthesquaredvisibilitycanbecomputedfrom
k k k
coefficientsoftheP2VMmatrix,Rb andIbverifytherespective itssecond-orderTaylorexpansion(Papoulis1984;Kervellaetal.
followingequations: 2004):
⎡ ⎤
Rb =(cid:3)Npixζbm , Ib =(cid:3)Npixξbm . (21) σ2(|Vij|2)= 1 ⎢⎢⎢⎢⎢⎢⎢⎣σ2(|Cij|2) + σ2(PiPj)⎥⎥⎥⎥⎥⎥⎥⎦|Vij|22 (26)
k=1 k k k=1 k k M |Cij|22 PiPj2
E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration 35
where |Cij|2 = Rij2 + Iij2 − Bias{Rij2 + Iij2} is the unbiased 3.5.4. Thepiston
squared coherent flux. In practice, the expected value and the
Theinterferometricphaseinducedbytheachromaticpistonterm
varianceofthesquaredcoherentfluxandthephotometricfluxes
takestheform
are computedempirically from the M available measurements.
Wethengetthefollowingsemi-empiricalformula: 2πδij
φij = =2πδijσ (32)
⎡(cid:29) (cid:30) (cid:29) (cid:30) λ λ
σ2stat(|V(cid:28)ij|2) = M1 ⎢⎢⎢⎢⎢⎢⎢⎢⎣ |Cij|4(cid:31)|MCi−j|2 |2Cij|2 2M swchoepreeiδaijndisj,thλe iascthhreomwaatviceledniffgtehr,enatniadlσpiisstotnhebwetawveeennumteblee-r
M
(cid:29) (cid:30) (cid:29) (cid:30) ⎤ (i.e.σ=1/λ).
+ Pi2Pj(cid:31)2PMiP−j 2PiPj 2M⎥⎥⎥⎥⎥⎥⎥⎥⎦|V(cid:28)ij|22. (27) First order Taylor expansion: at first order, the estimated
M differential phase of Eq. (31) is a linear function that takes the
genericform∆φ =φ +2π(σ −σ )δ.Itsslopeδdependson
12 1 2 1
Notefinallythat,althoughquadraticestimationofthevisibility thesumofatmosphericpistonδ ,whichvariesframebyframe,
p
hasbeencomputed,thesquaredvisibilitywillbesystematically and of the linear componentof the objectdifferentialphaseδ .
o
decreasedbythe atmospherejitter duringtheframeintegration A good estimate of this slope in the presence of noise is the
time.WefocusonthiseffectinSect.3.6. argumentoftheaveragecrossspectrumalongthewavelengths:
(cid:29) (cid:30)
3.5.2. Theclosurephase δ!ij+!δij = ar(cid:31)g Wλi2jl,λ2l+1 l · (33)
Bydefinition,theclosurephaseisthephaseoftheso-calledbis- p o 2π σλ2l+1 −σλ2l l
pectrumB123.Thebispectrumresultsintheensembleaverageof Estimation of the piston is unbiased when the wave number
thecoherentfluxtripleproductandthenestimatedas varies linearly with the spectral pixel index (linear grating dis-
persion law). Thiscan be true with an excellentapproximation
(cid:29) (cid:30)
B(cid:21)123 = C12C23C13∗ (28) atmediumspectralresolutionandhighspectralresolutioninthe
AMBER case. However,for the low spectralresolution, biases
ashighas5%intheestimationofpistoncanoccur.
whereCij =Rij+iIij.Theclosurephasethenisstraightforward:
⎡ ⎤ Fitting the complex phasor: the achromatic piston can
φ!123 =atan⎢⎢⎢⎢⎣Im(B(cid:21)123)⎥⎥⎥⎥⎦· (29) alsobeestimatedfromaleast-squarefitofthecomplexcoherent
B (cid:21) flux.Ifwedefinethecomplexphasoras
Re(B123)
Theclosurephasepresentsthe advantageofbeingindependent Ψλ =Cλij×e2iπλδij, (34)
of the atmosphere(e.g. Roddier 1986).However in the case of
δij can be retrieved by minimizing the phase of such complex
AMBER,theclosurephaseoftheimagemightnotcoincidewith
phasoror,equivalently,by minimizingthetangentofthe phase.
theoneoftheobjectandmightbebiasedbecauseofthecalibra-
Theχ2isthendefinedas:
tion process. If the so-called system phase presents a non-zero
(cid:1) (cid:2)
closure phase Φ1c2 +Φ2c3 −Φ1c3, this biasmustbe calibratedby (cid:3) Im(Ψλ) 2
observingapointsourceoratleastacentro-symmetricalobject. Re(Ψλ)
So far, no theoretical computation of the error of the closure σ2 +σ2
phase has been provided for the AMBER data-reduction algo- χ2 = (cid:3)λ Rλ Iλ · (35)
1
rithm.Thus,closure-phaseinternalerrorbars(i.e.thatdoesnot
σ2 +σ2
includesystematicserrors)are computedstatistically bytaking λ Rλ Iλ
the root mean square of all the individual frames, then divid-
ing by the square root of the number of frames, as illustrated Thisχ2 ishighlynon-linearandsimpletechniquessuchasgra-
inSect.5.5. dient fitting cannot be used here. On the contrary, non-linear
fitting techniques such as genetic or simulated annealing algo-
rithms(Kirkpatricketal.1983)mustbeusedinstead.
3.5.3. Thedifferentialphase Note that, in order to distinguish between the atmospheric
piston δ andthe linearcomponentofthe differentialphaseδ ,
The differential phase is the phase of the so-called cross spec- p o
thefittingtechniquesdescribedabovecanbeperformedbyonly
trumW .Foreachbaseline,thelatterisestimatedfromthecom-
12 using spectral channels correspondingto the continuum of the
plexcoherentfluxtakenattwodifferentwavelengthsλ1andλ2: source(i.e.outsidespectralfeatures)whereitsdifferentialphase
(cid:29) (cid:30) oftheobjectisassumedtobezero.
(cid:28) ∗
Wij = CijCij . (30)
12 λ1 λ2
3.6.Biasesofthevisibility
Andthedifferentialphaseis:
3.6.1. Lossofspectralcoherence
⎡ " #⎤
∆(cid:28)φi1j2 =atan⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣IRme"WW(cid:28)(cid:28)1ii2jj#⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦· (31) Tmofhoeonnoaecbhosrvpoeemcdtaertariiclvascpthieaocnntnroaeflltcihsheannionnnteelrz.feIernroopmaranecdttriidcceee,pqethuneadtssioponencatsrthasulemwreeidssotha-
12 lution R of the spectrograph. As a consequence the coherence
36 E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration
length L of the interferogram is finite and equals L = λ R, hardly calibratable, are clearly identified and suppressed. Note
c c 0
where λ is the reference wavelength in the spectral channel. as well thatthe use of the accuratefringetrackerFINITO(Gai
0
Assuming a lineardecompositionof the phaseof theinterfero- etal.2002),soonexpectedtooperateontheVLTI,shoulddrasti-
gram andneglectinghigherorders,the interferogramis attenu- callyreducethejitterattenuation,henceallowingintegrationon
atedbyafactorρ ,whichcanbewritten muchlongertimesthanthecoherencetimeoftheatmospherein
k
$$ & ’$$ ordertoreachfainterstars.
$ δ +δ +δ $
ρ =$$F% π k p o $$, (36)
k $ L $
c
4. Theamdlibdatareductionsoftware
whereF%istheFouriertransformofthespectralfilterfunction,
δ isthespatialsamplingoftheinterferogram(thatis,thepixel A dedicatedsoftwaretoreduceAMBERobservationshasbeen
k
coordinates expressed in optical path difference (OPD) units), developed by the AMBER consortium. This consists of a li-
δ and δ the atmospheric piston and the slope of the object braryofCfunctions,calledamdlib,plushigh-levelinterfacepro-
p o
spectraldifferentialphase,respectively,asdefinedinSect.3.5.4. grams. The amdlib functionsare used at all stages of AMBER
Notethatforasquarefilter,theattenuationcoefficienttakesthe dataacquisitionandreduction:intheobservationsoftware(OS)
forwavelengthcalibrationandfringeacquisition,inthe(quasi)
well-knownformofthesincfunction:
$$ & ’$$ real time display program used during the observations, in the
$ δ +δ +δ $
ρ =$$sinc π k p o $$· (37) online data reduction pipeline customary for ESO instruments,
k $ L $ andinvariousofflinefrontendapplications,noticeablyaYorick
c
implementation(AmmYorick).Theamdliblibraryismeanttoin-
Inthelow-resolutionmodewhereR = 35,theattenuationcoef-
corporateall the expertiseonAMBER data reductionandcali-
ficient stronglydependson the pixelpositionδ , which is cali-
k brationacquiredthroughoutthelifeoftheinstrument,whichare
bratablequantity.Nonetheless,the compensationfor thiseffect
boundtoevolvewithtime.
requiresaniterativeprocessintwostepswhere(i)theestimation
The data obtainedwith AMBER (“raw data”) consistof an
of δ +δ is performedas described in Sect. 3.5.4 and (ii) the
p o exposure,i.e.,atimeseriesofframesreadontheinfraredcam-
ρ attenuationcorrectionisapplieddirectlytotheDCcorrected
k era, plus all relevant information from AMBER sensors, ob-
interferogramsm .Theloopisthenrepeateduntilconvergence.
k servedobject,VLTI setup, etc.,stored in FITSTABLE format,
Thisalgorithm,whichhasnotyetbeenimplementedinthesoft-
according to ESO interface document VLT-ICD-ESO-15000-
ware,willbedescribedingreaterdetailinaforthcomingpaper.
1826. Saving the raw, uncalibrated data, although more space-
In the medium and high resolutions (where R = 1500 and
consuming, permits us to benefit afterwards, by replaying the
R = 10000, respectively), however, the OPD δ due the spa-
k calibration sequences and the data reduction anew, from all
tialsamplingofAMBERcanbeneglected.Indeedthisapprox-
the improvementsthat could have been deposited in amdlib in
imation leads to a relative error of the coefficient below 10−3
themeantime.
and 10−5, respectively, which is within the specified error bars
Thelibrarycontainsasetof“softwarefilters”thatrefinethe
of the visibility. In such a case, the loss of spectral coherence
raw data sets to obtain calibrated “science data frames”. This
simply resultsin biasingthe visibility frameto framebya fac-
treatmentisperformedonalloftherawdataframes,irrespective
torρ(δ +δ ).Thisbiascanbecorrectedbyknowingtheshape
p o of their futureuse (calibrationor observation).A secondset of
ofthespectralfilterandbyestimatingthepistonδ +δ thanks
p o functionsperformshigh-leveldataextractiononthesecalibrated
toEq.(33).
frames,eithertocomputetheV2PM(seeSect.4.3)fromasetof
calibrationdataortoextractthevisibilitiesfromasetofscience
3.6.2. Atmosphericjitter targetobservations,theendproductinthiscasebeingareduced
setofvisibilitiesperobject,storedintheopticalinterferometry
Although a quadratic estimation of the visibility has been per-
standardOI_FITSformat(Paulsetal.2005).
formedtoavoidthedifferentialpistontocompletelycancelout
thefringes,thehighfrequencyvariationsofthelatterduringthe
integration time – so called high-pass jitter – neverthelessblur 4.1.Detectorcalibration
thefringes.Asaresult,thecoherentflux,thusthevisibility,isat-
First,allframespixelsaretaggedvalidifnotpresentinthecur-
tenuated.Onaverage,theattenuationcoefficientΓofthesquared
rentlyavailablebadpixellistoftheAMBERdetector.Thenthey
visibilityisgivenbyColavita(1999b):
are converted to photoevent counts. This step necessitates, for
Γ=exp(−σ2 ) (38) each frame,precisely modellingof the spatially andtemporar-
φp
hf ilyvariablebiasaddedbytheelectronics.Thedetectorexhibits
whereσ2 isthevarianceofthehigh-passjitterφp . a pixel-to-pixel(highfrequency)biaswhosepatternisconstant
φp hf
Forthheftimebeing,thisatmosphericeffectiscompensatedby intimebutwhichdependsonthedetectorintegrationtime(DIT)
andthesizeandlocationofthesubwindowsreadonthedetector.
calibratingthesourcevisibilitywithareferencesourceobserved
Thus, after each change in the detector setup, a new pixel bias
shortlybeforeandafterthescientifictargettoinsuresimilarat-
map(PBM)ismeasuredpriortotheobservationsbyaveraginga
mosphericconditions.Wehavealsoplannedinthenearfutureto
largenumberofframesacquiredwiththedetectorfacingacold
provideamoreaccuratecalibrationofthiseffect,basedoncom- shutter2.ThisPBMisthensimplyremovedfromallframesprior
puting the variance of the so-called “first difference phase jit-
toanyothertreatment.
ter”,whichisthedifferenceoftheaveragepistontakenbetween
Oncethisfixedpatternhasbeenremoved,thedetectormay
two successive exposures,as proposedbyColavita (1999b)for still be affected by a time-variable “line” bias, i.e., a variable
thePTIinterferometerandsuccessfullyappliedbyMalbetetal.
(1998).However,jitteranalysis(asillustratedinSect.5.2)can- 2 Duetomechanical overheads, “hot dark” observations, i.e.,using
notbetestedandvalidatedaslongastheextra-sourcesofvibra- onlyanambienttemperaturebeamshutterexternaltothedewarofthe
tionsduetoVLTIinstabilities(delaylines,adaptiveoptics,etc.), detector,arecurrentlyusedtocomputethePBM.
E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration 37
offsetforeachdetectorline.Thisbiasisestimatedforeachscan them with their “governing”calibration matrix. The amdlib li-
lineandeachframeasthemeanvalueofthecorrespondingline brary takes the opportunity of the P2VM file being pivotal to
ofmaskedpixels(“DK”columninFig.1)andthensubstracted the data reduction,and unique,to make it a placeholderfor all
from the rest of the line of pixels. The detector has an image the other calibration tables needed to reduce the science data,
persistenceof∼10%;consequently,allframesarecorrectedfor namelythespectralcalibration,badpixels,andflatfieldtables.
this effect before calibration. Pixels are then converted to pho-
toeventcountsbymultiplyingbythepixel’sgain.Currentlythe
mapofthepixelgainsusedissimplyaconstante−/ADUvalue 4.4.Fromsciencedatatovisibilities
(seeTable1)multipliedbya“flatfield”mapacquiredduringlab- The computation of visibilities is performed by the
oratorytests. Finally,thermsofthevaluesin themaskedpixel amdlibExtractVisibilities() function, using a valid
set, which were calibrated as the rest of the detector, gives the P2VM file. Then amdlibExtractVisibilities()is able to
frame’sdetectornoise. performvisibilityestimates on aframe-by-framebasis, or over
agroupofframes,calledbin.
The amdlibExtractVisibilities() function, in se-
4.2.Imagealignmentandsciencedataproduction
quenceis
Once the cosmetics on the pixels is done, amdlib corrects the
1. inverttheV2PMcalibrationmatrix;
datafromthespatialdistortionspresentintheimage.Presently,
theonlycorrectedeffectisadisplacementofthespectraacquired 2. extractrawvisibilities;
3. correctforbiases,computedebiasedV2visibilities;
inthe“photometricchannels”(labeledP1,P2,P3inFig.1)with
4. computephaseclosures;
regards to the fringed spectrum in the interferometric channel.
5. computecrossspectra;
Thisdisplacementof a few pixelsin the spectraldispersiondi-
6. fitpistonvaluesfromcrossspectra;
rectionisduetoaslightmisalignmentofthebeam-splittersde-
scribed in Sect. 2.1,and correctingfor this effect is mandatory 7. writetheOI-FITSoutputfile.
forcomputingtheDCcontinuuminterferogram(Eq.(11)).The
A typical data reduction process will first process all
calibration of this displacementis performedby amdlib during
raw data files related to the calibration procedures per-
thespectralcalibrationprocedure,oneofthefirstcalibrationse-
formed before acquiring the science data, thus perform-
quencestobeperformedpriortoobservations.
ing spectral calibration (e.g., using the command line pro-
Finally,eachframeisconvertedtothehandier“sciencedata”
gram amdlibComputeSpectralCalibration), then P2VM
structure, which contains only the calibrated image of the “in-
file computation (e.g., using the command line program
terferometric channel” and (up to) three 1D vectors, the corre-
amdlibComputeP2vm).OncetheP2VMfileiscomputed,itcon-
spondinginstantaneousphotometryofeachbeam,correctedfor
tains all the calibration quantities needed to process science
theabove-mentionedspectraldisplacement.
object observations. One then uses the amdlibExtractVis
programonasciencedatasettogetthefinalOI-FITSfilecon-
tainingthemeasuredscienceobjectvisibilities.
4.3.Calibrationmatrixcomputation
ComputationoftheV2PMmatrixisperformedbythefunction
amdlibComputeP2vm().Thisfunctionprocessesthe4or9files
5. Illustrationsanddiscussions
describedinSect.3.2applyingbyeachofthemthedetectorcal-
ibration,imagealignment,andconversionto“sciencedata”de- This section aims to present, step-by-step, the data reduction
scribedabove,thencomputingthevi (Eq.(10))andthecarrying procedures performed on real interferometric measurements
k
waves cij and dij of the V2PM matrix (Eq. (13)). The result is arisingfromVLTIobservations.Resultsarediscussed,focusing
k k
stored in a FITS file, improperly called, for historical reasons, onkeypointsintheprocess.
“theP2VM”3.
The P2VM matrix is the most important set of calibration
valuesneededto retrievevisibilities. Theshape of the carrying 5.1.Fringefitting
waves(thec sandd s)and,inlessermeasure,theassociatedv s
k k k
aretheimprintsofallthechangesinintensityandphasethatthe Assuming the calibration processhas been properlyperformed
beamssufferbetweenthe outputofeach fiberanddetectionon followingSect.3.2,thefirststepinthederivationoftheobserv-
theinfraredcamera.AnychangeintheAMBERopticssituated ablesistoestimatetherealandimaginarypartsofthecoherent
in this zone, either by moving, e.g., a grating, or just thermal flux.Thisisdonebyinvertingthecalibrationmatrixandobtain-
long-termeffects,rendertheP2VM unusable.Thus,theP2VM ing the P2VM matrix, shown by Eqs. (17) and (18). Figure 4
matrix must be recalibrated each time a new spectral setup is givesanexampleofthefringefittingprocessforanobservation
calledthatinvolveschangingtheopticalpathbehindthefibers. ofthecalibratorstarHD135382withthreetelescopes.
All the instrument observing strategies and operations are However,beforegoingfurtherinthedatareductionprocess,
governedbytheneedtoavoidunnecessaryopticalchanges,and itmightbeworthwhilefortheuserstocheckthevalidityofthe
care is taken at the operating system level to assure a recali- fitandthentodetectanypotentialproblemsinthedata.Sucha
bration of the P2VM whenever a “critical” motor affecting the step canbeeasilydonebycomputingtheresidualχ2res between
optical path is set in action. To satisfy these needs, the P2VM themeasurementsmk andthemodelm(cid:21)k:
computationhasbeenmademandatorypriortoscienceobserva- [m(cid:21) ]=V2PM[R(cid:21)ij,(cid:21)Iij] (39)
tionsandisgivenanuniqueIDnumber.Allthesciencedatafiles k
produced after the P2VM file inherit this ID, which associates
and
3 Actually“theV2PM”shouldbethepropername. χ2res =[m(cid:21)k−mk]TC−M1[m(cid:21)k−mk]. (40)
38 E.Tatullietal.:InterferometricdatareductionwithAMBER/VLTI.Principle,estimators,andillustration
Chi2 residual = 1.21628 Then, we have added the photometry taken on the (cid:14) Sco data,
whichallowedustokeeprealisticphotometricrealizationstak-
100 ing the correcttransmissionsof the instrumentinto account.In
thatcase,selectingthebestfringeshasnoothergoalthantoim-
prove the SNR of the observables by excluding the data with
50
poorflux.
mk
In the presence of atmospheric turbulence and lack of a
0
fringetracker,thefringesaremovingduringtheintegrationtime,
leadingtolowerthevisibility.Onaverage,thesquaredvisibility
−50 isattenuatedbyafactorexp(−σ2 ),whereσ2 isthevariance
φp φp
hf hf
of the atmospheric high pass jitter, as explained in Sect. 3.6.2.
10 20 30
The frame-by-framevisibility, though,undergoesa randomat-
# pixels
tenuationaroundthisaveragelossofcontrast.Anexampleofthe
Fig.4. Example of fringe-fitting by the carrying waves in the effectoftheatmosphericjitterisgiveninFig.6(middle),where
3-telescope case. The DC-corrected interferogram is plotted (dashdot
aprevioussetofsimulateddatahasbeenused,addingaframe-
line)withtheerrorbars.Theresultofthefitisoverplotted(solidline).
by-frame randomattenuation taking the τ = 25 ms integration
time of the (cid:14) Sco observation into account. Once again, fringe
Using this checking procedure,the user can verify two critical selectiononlyenablesustoincreasetheSNRoftheobservables
pointsofthedataprocessing: here.
However,whenwelookattherealsetofdataobtainedfrom
– thecorrectsubtractionoftheDCcomponent(seeEq.(11)): theobservationof(cid:14) Sco, weobtaintheplotdisplayedinFig.6
if such a condition is not fulfilled, the computed visibility
(right).Thedispersionofthevisibility,especiallyforlowfringe
will inevitably be biased since the fringe fitting by the car-
SNRisunexpectedlylargeandcandefinitivelynotbeexplained
ryingwavessupposestheonlypresenceofspecificfrequen-
bypureatmosphericOPDvibrations.Asamatteroffact,these
cies,thatis,thespatialcodingfrequenciesoftheinstrument.
variations are due to the present strong vibrations along the
A wrong DC subtraction might occur with sudden atmo-
VLTIinstrumentation(adaptiveoptics,delaylines,etc.),asthis
sphericchangesbetweentherecordingoftheinterferometric effectwaspreviouslyrevealedbytheVINCIrecombiner.These
channelandtheassociatedphotometricones,asthesechan-
vibrationsstronglyreducethefringe contrastand subsequently
nels are not on the same line of the detector, as mentioned
thevalueoftheestimatedvisibilities,whichexplainsthebehav-
inSect.4.2.
ior of the visibilities as a function of the fringe SNR. Indeed,
– theuseofacorrectbadpixelmap:ifnot,thepresenceofbad
whenthevisibilitytendstoward0,becauseofseverejitteratten-
pixelsinduceshighfrequenciesinthefringes,whichcannot
uation, the fringe criterion tends toward 0 as well. In contrast,
betakenintoaccountbythecarryingwaves,drivingoneto
the visibility plotted as a function of the fringe SNR saturates
compute biased visibility, as well. Note that the bad pixel
forhighvaluesofthelatter.
map is computed every time a detector calibration is per-
The major issue is thatsuch an effect is hardly calibratable
formedinthemaintenanceprocedure.
because potentially non stationary.Hence, one convenientway
toovercometheproblem,besideincreasingtheerrorbarsartifi-
ciallytotakethisphenomenonintoaccount,istoonlyselectthe
5.2.Fringecriterionandfringeselection fringesthatarelessaffectedbythevibrations,thatis,thefringes
For each frame of the set of data, Eq. (20) provides an es- withthehighestfringeSNR.Onecanthenchoosethepercentage
timation of the fringe signal-to-noise ratio. As an example, of selected frames from which the visibility will be estimated.
Fig. 5 presents100 fringesrecordedon the detectorduringthe Thethresholdmustbechosenaccordingtothefollowingtrade-
2-telescopeobservationofthecalibrator(cid:14)ScoinJuly2005,first off:reducingthenumberofframesconsideredallowstogetrid
intheordertheyappearedduringtheobservationandthenafter ofmostofthejitterattenuation,but,fromacertainnumberwhen
re-orderingthemfollowingthefringecriterion. thesampleisnotlargeenoughtoperformstatistics,itincreases
Theaimofcomputingthiscriterioncanbetwofold:(i)dur- thenoiseonthevisibility.Furthermore,itleadstomis-estimate
ing the observations, as mentioned in Sect. 3.4, it allows us to thequadraticbias(seeEq.(25)),whichisbyessenceastatisti-
detect the fringes and therefore to initiate the recording of the cal quantity,and consequentlydrives to introducea bias in the
dataonlywhenitismeaningful;and(ii)calculatedaposteriori visibility.
duringthe data reductionphase, it enablesus to select the best Obviously, such a selection process must be handled with
frames(intermsofSNR)beforeestimatingtheobservables.This care,anditsrobustnesswithregardtotheselectionlevelhasto
secondpointisespeciallyimportantwhereframesarerecorded beestablishedforanygivenobservation.Inotherwords,forthis
inthepresenceofstrongandvariablefringejitter. method to be valid, the expected value for the calibrated visi-
In the ideal and unrealistic case where the fringes are not bility must remain the same, with only the error bars changing
movingduringtheintegrationtime,thefringecontrastisnotat- and eventually reaching a minimum at some specific selection
tenuated by vibrations, and the frame-by-frameestimated visi- level. In particular, this method seems well adapted, above all,
bility is constant, no matter what the photometric flux level is to cases where the calibrator exhibits a magnitude close to the
in each arm of the interferometer. As a result, the visibility as source’sone,wherethevisibilitydistributionversustheSNRis
a functionthe function of the fringe SNR is constant, with the expected to behave similarly. Going into further details af this
errorbarsincreasingasthefringeSNRdecreases.Thisisillus- point is nevertheless beyond the scope of this paper as it will
trated in Fig. 6 (left). To obtain this set of jitter-free data, we bedeeplydevelopedinMillouretal.(2007).Howevernotethat
have built interferograms using the carrying waves of the cal- we experimentallyfoundthis procedureto be generallyrobust,
ibration matrix that simulate perfectly stable AMBER fringes. andfortypicalobservationsperformeduntilnowwiththeVLTI,