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Automatedphase-measuringprofilometryof3-DdiffuseobjectsV.Srinivasan,H.C.Liu,andM.HaliouaThehighresolutionsurfaceprofileofa3-Ddiffuseobjectisobtainedbymeasurementofthephasedistribu-tionacrosstheimageofaprojectedsinusoidalgratingdeformedbythesurface.Ashearingpolarizationin-terferometerisusedforprojection.Deformedgratingimagesaredetectedbyanarraycameraandpro-cessedbyamicrocomputer.Surfaceheightresolutionsofbetterthan10tmhavebeenattained,limitedes-sentiallybyelectronicquantizationnoise.Incontrasttodirectdeformedgratinganalysis,thismethod,basedonphase-shiftinginterferometrictechniques,iscapableofaccuratemeasurementevenwithcoarseprojectedgratingsandlowdensityimagesensingarrays.Areasofapplicationincludeindustrialqualitycontrol,biostereometrics,robotics,andsolidmodelingforcomputergraphics.1.IntroductionSurfaceprofilemeasurementbynoncontactopticalmethodshasbeenextensivelystudiedbecauseofitsimportanceinautomatedmanufacturing,componentqualitycontrol,medicine,robotics,andsolidmodelingapplications.Inmostofthesemethodsaknownperi-odicpattern,suchasagrating,isprojectedonthesur-facetobemeasured,andtheimageofthegrating,de-formedbythesurface,isanalyzedtodeterminetheprofile.Demodulationofthedeformedgratingbymeansofamatchedreferencegratingresultsinthewell-knownmoirefringepatterns,1-3whichareeasilyinterpretableassurfacecontoursbyahumanobserverbutaresomewhatmorecomplicatedforcomputeranalysis.4Improvementstothemoiremethodaimedatincreasingtheaccuracyandatautomatingthemea-surementarebasedonphasemodulation.56Analternativeapproachtomoireisbyananalysisofthedeformedgratingitselfwithouttheuseofaphysi-cal1-3orvirtual4referencegrating.Directmethodsbasedongeometricalanalysisofthedeformedgratingrequiringfringepeakdeterminationarecomputation-allycomplex,slow,andresultinlowaccuracy.Anotherdirectmethod7basedonuseofafastFouriertransform(FFT)analysisofthedeformedgratinghasbeendem-onstratedtobemoresuitableforautomatedprofilo-TheauthorsarewithNewYorkInstituteofTechnology,CenterforOptics,LasersandHolography,andNYCOM,OldWestbury,NewYork11568.Received13April1984.0003-6935/84/183105-04$02.00/0.©01984OpticalSocietyofAmerica.metry.Measurementofsteepobjectslopes,stepdiscontinuities,theneedforhighresolutionimagingsystems,andcomputernumbercrunchingcapabilityaresomeofthelimitationsoftheFFTmethod.Themethoddescribedhereisbasedonuseofphase-measuringtechniquesdevelopedforclassicalinterferometry.Whenasinusoidalintensitydistri-butionisprojectedona3-Ddiffusesurface,themath-ematicalrepresentationofthedeformedgratingimageintensitydistributionissimilartothatencounteredinconventionalopticalinterferometry.Itisshownthatthesurfaceheightdistributionistranslatedtoaphasedistribution,andthusthemethodsofphasemodulationinterferometry,89wellknownfortheiraccuracy,canbeusedfortheanalysis.Furthermore,byusingseveralphase-modulatedframesofdeformedgratingimagedata,ahighdegreeofprecisioninthephasemeasure-mentcanbeachieved.Byanalogywithphase-mea-suringinterferometry,wherephasevaluescanbemeasuredwitharesolutionof1/1000ofafringeperiod(vs1/30forconventionalsingle-frameinterferometry),surfaceprofilemeasurementswith10-gmresolutionarepossiblebytheuseofanopticalsystemwithapro-jectedgratingpitchinthemillimeterrange.Theworkingsystemconsistsofasinusoidalgratingprojectorandanimagesensingarraycamera,interfacedtoamicrocomputersystem(Fig.1).Thephasemea-surementusingadiscretephase-shiftingtechniquesismoresuitableforparallelprocessingwithmatrixde-tectorarraysthanzero-crossing5andphase-lock6de-tection.II.ProjectionOpticsAsinusoidalintensitydistributioncanbeprojectedonthesurfacetobeprofiledintwoways:generatinganinterferencepatternbetweentwocoherentplane15September1984/Vol.23,No.18/APPLIEDOPTICS3105PROJECTIONSYSTEMWITHPHASE-SHIFTEROBJECTJL1/2J-[AYFig.1.Schematicdiagramofthephase-measuringsurfaceprofilometer.wavefrontsorprojectinganimageofagratingwithsi-nusoidaltransmissionfunctiondistributionilluminatedbyanincoherentlightsource.Asitisdifficulttofab-ricateasinusoidalgratingbyphotographicmethods,wehaveusedalaserilluminatedshearingpolarizationin-terferometer(Fig.2)inthiswork.Thelinearlypolar-izedoutputbeamfromthelaserisspatiallyfilteredandthenshearedbyaWollastonprismW.Thephasemodulator'consistsofacombinationofaquarter-waveplateQandarotatablepolarizerP.Byrotatingthepolarizer,thesinusoidalintensitydistributionoftheinterferencepatterncanbemodulated.Incontrasttootherinterferometers,thepolarizationinterferometeriscompact,stable,andinsensitivetoexternaldistur-bances.A180°rotationofthepolarizercorrespondstoa2rphasemodulation,andthispermitsveryprecisephaseshifts.Otherpolarization-dependentphaseshifters,suchaselectroopticmodulators,mayalsobeusedinthissystem.ThefringeperiodcanalsobeeasilychangedbyanaxialtranslationoftheWollastonW.AcollimatinglensLisusedtoconservelightandsimplifytheopticalgeometry.Ill.AnalysisoftheImageoftheDeformedGratingAsimplegeometricalapproachcanbeusedfortheanalysisasshowninFig.3.Theobjectheighth(x,y)ismeasuredrelativetothereferenceplaneindicatedonthefigure.Theprojectedsinuosidalinterferometric