Measurement of transparent plates with wavelength-tuned phase-shifting interferometry

wavelength-tunedphase-shiftinginterferometry

PeterdeGroot

Awavelength-tunedFizeauinterferometerisappliedtotheproblemofflatnesstestingoftransparentplates.Whentheplateispositionedataspecificdistancefromthereferencesurfaceandaninteger-math13-framephase-shiftingalgorithmisapplied,thesystemdirectlyfiltersoutunwantedinterferencearisingfrombacksurfacereflections.Theresultingfront-surfaceprofileexhibitslessthan2nmofresidualerrorattributabletospuriousreflectionsfromwithintheplate.©2000OpticalSocietyofAmerica

OCIScodes:120.3180,120.3940,120.5050,120.5060,120.6650,220.4840.

1.Introduction

Acommonprobleminopticalsurfaceprofilingisspu-riousreflections,suchasareencounteredwhentheflatnessofatransparentplateistestedinaFizeauinterferometer.1UnwantedinterferenceeffectsarisingfrombacksurfacereflectionsresultinthecomplexmixtureoffringesshowninFig.1.Theoverlappingpatternsrenderineffectivestandardtechniquesinphase-shiftinginterferometry͑PSI͒,whichrelyontheanalysisofasequenceofinterfer-enceimagesshiftedinphasebymechanicalmodula-tionofthereferenceflat.Commonpracticeistodefeatbacksurfacereflectionsbyoneapplyinganindex-matchingcoatingandthenproceedingwithstandardPSI.Itwouldbedesirabletodispensewithsuchsurfacetreatmentsforroutineinspectionoftransparentplates,particularlyforprocesscontrolinaproductionenvironment.

Severalalternativesarenowavailable,includingwhitelight,2grazingincidence,3desensitizedgratinginterferometry,4useofmultimodelaserdiodes,5andcombinationsofmeasurementsindifferentorienta-tions.6Animportantclassofsolutionsinvolvestunable-wavelengthsources.Okadaetal.,forexam-ple,havemeasuredtransparentplatesinaTwyman–Greeninterferometerbyacquiring60interference

imagesatasequenceofwavelengthsandleast-squaresfittingofthefirst-ordertermstocalculatesurfaceandopticalthicknessprofiles.7,8HereIproposetomeasuretransparentplatesbyusingawavelength-tunedlaserFizeauinterferome-terandaPSIalgorithmthatdirectlysuppressesin-terferencemodulationsthatareattributabletospuriousreflections͑patentpending͒.Theresultingphasemaprepresentsonlythedesiredfront-surfacereflectionandisrelativelyfreeofdistortions,inspiteofthecomplexnatureoftheinterferencepattern.Apartfromuseofawavelength-tunablesourceandtheneedtopositiontheobjectrathercloselytothereferenceflat,thetechniqueinvolvesfewchangestoastandardlaser-basedFizeau.ThePSIalgorithmexecutesquicklyforhigh-throughputproductiontest-ing.Theapproachshouldproveusefultoqualifyopticalcomponents,flatpaneldisplays,andtranspar-entsubstratesforrigiddiskdrives.

2.InterferometryofTransparentPlates

Figure2showstheopticalgeometryleadingtotheinterferencepatternshowninFig.1.Thephasede-layfromthereferencesurfacetothefrontobjectsur-faceis

␪ϭ2k͑h1Ϫh0͒ϩ2kL,

(1)

P.deGroot͑peterd@zygo.com͒iswiththeDepartmentofRe-searchandDevelopment,ZygoCorporation,21LaurelBrookRoad,Middlefield,Connecticut055-0448.

Received28October1999;revisedmanuscriptreceived16Feb-ruary2000.

0003-6935͞00͞162658-06$15.00͞0©2000OpticalSocietyofAmerica2658

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whereListhenominaldistancetothefrontobjectsurfaceandtheangularwavenumberkis2␲dividedbythewavelength␭.Thephasedelayfromthefronttothebackobjectsurfacesis

␾ϭ2kn͑h2Ϫh1͒ϩ2knT,

(2)

Fig.1.FringepatterninaFizeauinterferometerwhenviewingatransparentplate.Thecomplicatedinterferenceeffects,includ-ingbacksurfacereflections,defeatconventionalPSIwithmechan-icalphaseshifting.

wherenTisthenominalopticalthicknessofthetransparentplateobject.InEqs.͑1͒and͑2͒,haretheheightprofilesforthereferencefrontobject0,1,2andbackobjectsurfaces,respectively,andr0,1,2arethecorrespondingamplitudereflectivities.

Intheabsenceofthebacksurfacereflection,PSIwouldprovidethephasedifference␪givenbyEq.͑1͒toanarbitraryintegermultipleof2␲,whichtogetherwithknowledgeofthereferencesurfaceh0andthewavenumberkprovidesarelativesurfaceprofileproportionaltoh1.Inthemoregeneralsituation,theinterferenceintensityis

gϭ͉u͉2,

(3)

where

uϭ

r0ϩrЈexp͑i␪͒

1ϩrЈr␪͒,

(4)0exp͑irЈϭr1ϩr2exp͑i␾͒

1ϩr.

(5)

1r2exp͑i␾͒

Fig.2.Laser-basedFizeauinterferometerviewingatransparentplate.LD,laserdiode.

Thiscalculationincludesallpossiblemultiplereflec-tionsbetweenthereferencesurfaceandthetwosur-facesofthetransparentplate.Asimplifiedexamplecalculationassumesherethattheintensityreflectiv-ityRisidenticalforallthreesurfaces:

r0ϭϪͱR,

r1ϭͱR,r2ϭϪͱR.

(6)

UsingthissimplifyingassumptionandexpandingEqs.͑3͒–͑5͒,Ifindthatforthisexample

gϭ2R

ͫ3

2

Ϫcos͑␪͒Ϫcos͑␾͒ϩcos͑␪ϩ␾͒ͬϩO͑R2͒.(7)

Thefirst-ordertermsinRalreadyrevealthediffi-cultyofperformingPSIontransparentplates.Termsinvolvingthephase␾introduceanunwanteddependenceoftheinterferencedataontheheightprofileh2ofthebacksurfaceofthetransparentplate.StandardPSI,whichreliesonamechanicalmodula-tionof␪,isunabletomakesenseoftheresultinginterferencebehavior.

3.Wavelength-TunedPhase-ShiftingInterferometry

Controlledwavelengthtuning—forexample,byinjec-tioncurrentmodulationofalaserdiodeinanunequal-pathinterferometer—isaconvenientnon-mechanicalmeansofgeneratingphaseshiftsforPSI.9,10Wavelength-tunedPSIhasafurtheradvan-tageindealingwithoverlappinginterferencepat-ternsfrommultiplereflections.Asthewavenumberrampswithtimeataratedk͞dt,theinter-ferenceintensityvariesaccordingtog͑t͒ϭ2R

ͫ3

2

Ϫcos͑␪ϩ␯1t͒Ϫcos͑␾ϩ␯2t͒ϩcos͑␪ϩ␾ϩ␯3t͒ͬϩO͑R2͒,

(8)

where

␯1ϭ2Ldk͞dt,(9)␯2ϭ⌫␯1,(10)␯3ϭ͑⌫ϩ1͒␯1,(11)⌫ϭnT͞L.

(12)

WavelengthtuningmodulatesthevarioustermsinEq.͑8͒atdifferentrates,thereforeprovidingameansinprincipleofonedistinguishingbetweentheircon-tributionstothefinalinterferenceimage.

Table1summarizesthemodulationfrequencies␯1...3and␯4...9forfirst-andsecond-ordertermsofEq.͑8͒,respectively.Forexample,⌫ϭ3mightin-volveaTϭ12-mm-thicktransparentglassplateofindexnϭ1.5placedatadistanceLϭ6mmfromthereferencesurface.Theresultinginterferencemod-1June2000͞Vol.39,No.16͞APPLIEDOPTICS

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Table1.ModulationFrequenciesResultingfromWavelengthTuninga

FizeauInterferometerwithaTransparentPlateObjectaModulationFrequencyAmplitudeFrequency

2Ldk͞dt⌫ϭ3⌫ϭ0.52R͑1Ϫ4R͒

␯1111Ϫ1␯2⌫30.5Ϫ1␯3⌫ϩ141.51␯4222ϪR␯5⌫Ϫ12Ϫ0.5ϪR␯6⌫ϩ252.52R␯72⌫61ϪR␯82⌫ϩ1722R␯92⌫ϩ2

8

3

ϪR

a⌫ϭnT͞L.

ulationsincludethefundamentalfrequency␯front-surfacereflection,plusthespuriousharmonics1forthe␯2...9listedinTable1andshowngraphicallyinFigs.3and4.AlthoughtheseunwantedmodulationsconfuseconventionalPSI,itshouldbepossibleinprincipletoidentifyandsuppressthesetermsaccord-ingtotheirmodulationfrequency.

4.FourierFilteringwithPhase-ShiftingAlgorithms

Phase-shiftingalgorithmsareessentiallysingle-frequencyFouriertransformstunedtotheexpectedinterferencemodulationfrequency␯1.Conse-quently,allPSIalgorithmshave,tosomedegree,theabilitytosuppresssignaldistortionsatotherfre-quencies.11Ithereforeproposetoconstructaneffi-cientinteger-mathPSIagorithmthatisspecificallydesignedtoextractthedesiredfront-surfaceinterfer-enceinformationwhiledirectlyfilteringouttheun-wantedmodulationfrequencies␯allPSIalgorithmsareappropriate2...9.

Nottothistask,sothefirststepistoidentifyanalgorithmhavingthe

Fig.3.Single-pixelintensitymodulationinaFizeauinterferom-eterduringacontinuousshiftinsourcewavelength.Thesolidcurverepresentstheexpectedsignalfromasingleobjectsurface.Thedottedcurverepresentsthesignalwhenviewingatransparentplateobject,includingunwanteddistortionsresultingfromspuri-ousbacksurfacereflections.2660

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Fig.4.Frequencycontentofthedotted-curvesignalshowninFig.3.Thedesiredfundamentalfrequency␯1,herenormalizedto1,isaccompaniedbyseveralparasiticmodulationsasnotedinTable1.

desiredfilteringcharacteristics.MostPSIalgo-rithmscanbewrittenasfollows:

MϪ1␪ϭtan

Ϫ1

ͩM͚Ϫ1

smgm

m

gm

(13)

mϭ0

Ͳm͚cϭ0

ͪϩconst,

wheretheindexmcorrespondstosuccessivecameraframesandsmandcmarecoefficientsspecifictothePSIalgorithm.ThetotalnumberofframesMistypicallybetween4and7.Theinterferencephaseshiftscontinuouslybetweencameraframesbyanamount

⌬␣ϭ␯1⌬t,

(14)

where⌬tisthetimelapsebetweenframes.PSIalgorithmsaredesignedtobesensitivetothisphaseshiftandtoberesistanttodistortionsinthephaseshiftaswellasotherdisturbances.

OnewaytoevaluatethesensitivityofPSIalgo-rithmstounwantedinterferencetermsistoperformadiscreteFouriertransformofthecoefficientssmandcm:

M͚Ϫ1

S͑␯͒ϭ

s

m

exp͑Ϫim⌬␣␯͞␯1͒,(15)mϭ0

M͚Ϫ1

C͑␯͒ϭ

c

m

exp͑Ϫim⌬␣␯͞␯1͒.

(16)

mϭ0

TheresultingfilterfunctionsS͑␯͒andC͑␯͒aretrans-ferfunctionsinthefrequencydomain.12Ameasureofthecharacteristicsensitivityofthealgorithmtointensitynoiseistherefore

I͑␯͒ϭ͓͉S͑␯͉͒2ϩ͉C͑␯͉͒2͔1͞2.

(17)

Perhapsthebestwaytoillustrateuseofthefilterfunctionanalysisistogiveanexampleofwhatwillnotworkinthepresentapplication.Themostcom-monlyusedPSIalgorithmtodayistheSchwider–

Fig.5.Theoreticalfrequencyresponseofthefive-frame,⌬␣ϭ␲͞2PSIalgorithm,includingtheaveragingorintegratingbucketeffectofthephasechangeduringthedata-acquisitiontimeinterval⌬t.Unfortunatelythisalgorithmissensitivetotheunwantedharmon-icsshowninFig.4.

Hariharanfive-framealgorithm,1,13forwhichthecoefficientsare

sϭ͑020Ϫ20͒,

(18)cϭ͑Ϫ1

02

0Ϫ1͒.

(19)

Thefilterfunctionsare

S͑␯͒ϭisin͑␲␯͞2␯1͒,(20)C͑␯͒ϭsin2͑␲␯͞2␯1͒.

(21)

Thealgorithmhasperiodicextremaat␯͞␯5...,meaningthatitissensitivetomodulations1ϭ1,at3,thesefrequencies.Figure5showsthissensitivitygraphicallybyuseofEqs.͑17͒,͑20͒,and͑21͒.Forthe⌫ϭ3exampleinTable1,the␯overlapsthe␯͞␯2modulationunfilteredbythe1five-frameϭ3sensitivityPSIalgorithm.andisthereforeTheam-plitudeofthisunwanted␯thefundamentalandisfatal2modulationisthesameastoPSIforthisvalueof⌫Fig.,as6.isclearTherefromdoesthenotϮ20-nmappearcyclictobeerrorsavalueplottedof⌫thatinwouldmakeitpossibletoplacethefirst-order␯␯2andand3frequenciessimultaneouslyatthezerosofEqs.͑20͒͑21͒.

APSIalgorithmthatisbettersuitedtothetaskoffilteringunwantedinterferencemodulationsintransparentplatesisthefollowing13-frame,⌬␣ϭ␲͞4algorithm14,15:sϭ͑Ϫ3

Ϫ4012

21

16

0Ϫ16Ϫ21

Ϫ1204

3͒,

(22)

cϭ͑0

Ϫ4Ϫ12Ϫ12

0

16

24

16

0

Ϫ12Ϫ12Ϫ40͒.

(23)

ThefilterfunctionsinFig.7showthatthe13-framealgorithmsuppressesboththefirst-orderharmonics␯2,3forthe⌫ϭ3exampleinTable1,aswellasmostofthesecond-orderterms.Theresidualerrorthatis

Fig.6.Theoreticalcyclicerrorasafunctionofthefundamentalphase␪forafive-framePSIalgorithminthepresenceofreflec-tions.

attributabletothe␯Thusthe13-framePSI8͞␯algorithm1ϭ7modulationcombinediswithϮ2nm.thetransparentplatebeingplacedclosetothereferencesurfacenearlyeliminatestheeffectofbacksurfacereflections.

Thereareseveralothersystemconfigurationsandalgorithmsthatusethesameprinciple.Aninter-estingalternativeisthecaseinwhich⌫ϭ0.5.Thetwofirst-orderunwantedmodulationsaccordingtoTable1arenowat␯sensitivitycurvein2Fig.͞␯1ϭ80.5showsandhow␯3͞␯1theϭ1.5.followingThe15-frame,⌬␣ϭ␲͞2algorithmsuppressestheseun-wantedmodulations:

sϭ͑Ϫ1

0

90Ϫ21

029

0

Ϫ290210Ϫ901͒,(24)

cϭ͑0

Ϫ4015

0Ϫ26

0

300

Ϫ260150Ϫ40͒.

(25)

Thesecond-order␯nm.The7and␯⌫ϭ9termsresultinaresidualerrorofϮ20.5casepositionsthepart

Fig.7.Theoreticalfrequencyresponseofthe13-frame,⌬␣ϭ␲͞4PSIalgorithm.Thisalgorithmsuppressesthehigh-frequencymodulationsshowninFigs.3and5.

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Fig.8.Theoreticalfrequencyresponseofthe15-frame,⌬␣ϭ␲͞2PSIalgorithm.Thisalgorithmiseffectiveatsuppressingspuri-ousmodulationsforthecaseinwhich⌫ϭ0.5͑seeTable1͒.NotethechangeinhorizontalscalewithrespecttoFigs.6and7.

atamorecomfortabledistancethanthe⌫ϭ3exam-ple,i.e.,sixtimesfurtheraway.Therelativeadvan-tageofthe⌫ϭ3solutionisthatittoleratesalargevariationinobjectposition,anditactuallyworkswellwithvaluesof⌫rangingfrom2to5.The⌫ϭ0.5solutionrequirestheobjecttobepositionedattheprecisedistanceLtowithinafewpercent,andlike-wiseitrequiresamoreprecisecalibrationofthewavelengthexcursion.

5.Experiment

Thetechniquewasevaluatedfortheinspectionof10-cm-thick,0.6-m-diametertransparentflatsgeneratedbylargerotationalpolishingmachines.TheobjectplacementdistanceisLϭ5cmfor⌫ϭ3.Figure1showsasingleframeoftheinterferenceintensitydataforthisobject.Thelarge-apertureinterferometerusesawavelength-tunablelaserdiodesource.16Figure9showstheresultsofafirstexperiment

Fig.9.Experimentaldataprocessedwiththefive-frame,⌬␣ϭ␲͞2PSIalgorithmshowingthepredicted40-nmpeak-to-valleyprofiledistortionresultingfrombacksurfacereflectionsfromatransparentplate.2662

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Fig.10.Experimentaldatashowingthesuppressionofprofiledistortionswhenwavelength-tunedinterferometryisusedwiththe13-frame,⌬␣ϭ␲͞4PSIalgorithmdescribedinthetext.ThetransparentplateobjectisidenticaltotheoneprofiledinFig.9.

withaconventionalfive-framePSIanalysis,forwhichthehigh-frequencyprofiledistortionsareat-tributabletothebacksurfacereflection.Figure10showstheresultsofasecondexperiment,inwhichthe13-framePSIalgorithmtogetherwithwave-lengthtuningeliminatesthedistortion.Forthislat-terexperiment,theangularwave-numberexcursion⌬kpercameraframeaisnominal0.079cmϪ1.Thetotalwave-lengthexcursionforwavelength␭of0.68␮able,misconsistent0.0075nm.withThetheresidualpredictederrorimprovement.

isundetect-6.Conclusions

TheforegoingexperimentaldemonstrationverifiesonepossiblecombinationofpartplacementwithaPSIalgorithmthatsuppressesunwantedmodula-tionsintheinterferenceintensity.Therearemanyotherpossiblecombinationsthatrelyonthesameprinciple.Thekeyistomatchthemodulationfre-quenciesdeterminedbyexpansionofEqs.͑3͒–͑5͒withthePSIalgorithmsensitivitydeterminedbyEq.͑a17new͒.InPSImanyalgorithmcases,bythisusewillofrequireanyoneconstructionoftheknownoftechniques.17,18Oncethisisaccomplished,itispos-sibletoprofileaccuratelytransparentplateswithlaser-basedFizeauinterferometry.

ThisresearchwasinitiatedbyT.ConnollyandwascompletedwiththeableassistanceofJ.Soobitsky,whoperformedtheexperimentswiththeZygo24-in.phase-shiftinginterferometer.

ReferencesandNotes

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15.

Thehigh-resolutionphase-shiftingmodeintheZygoCorpora-tionMetroProPSIsoftwareemploysthe13-framealgorithmgivenbyEqs.͑22͒and͑23͒.

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