光学检测CH01课件

1.0BasicWavefrontAberrationTheoryForOpticalMetrology

ChangchunInstituteofOpticsandFineMechanicsandPhysicsDr.ZhangXuejunThePrincipalpurposeofopticalmetrologyistodeterminetheaberrationspresentinanopticalcomponentoranopticalsystem.Tostudyopticalmetrologytheformsofaberrationsthatmightbepresentneedtobeunderstood.Formostopticaltestinginstruments,thetestresultisthedifferencebetweenareference(unaberrated)wavefrontandatest(aberrated)wavefront.WeusuallycallthisdifferencetheOpticalPathDifference(OPD).OPDTestwavefrontReferencewavefrontRayNotethattheOPDisthedifferencebetweenthereferencewavefrontandthetestwavefrontmeasured

alongtheray.Thedistanceispositiveifmeasuredfromlefttoright.TheangleispositiveifitisincounterclockwisedirectionrelativetoZaxis.(+)(-)(+angle)(-angle)Sincemostopticalsystemsarerotationallysymmetric,usingpolarcoordinateismoreconvenient.XYx=cosy=sin1.2AberrationFreeSystemIftheopticalsystemisunaberratedordiffraction-limited,forapointobjectatinfinitytheimagewillnotbea“point”,butanAiryDisk.ThedistributionoftheirradianceontheimageplaneofAiryDiskiscalledPointSpreadFunctionorPSF.SincePSFisverysensitivetoaberrationsitisoftenusedasanindicatoroftheopticalperformance.FirstmaximumSecondmaximumDiametertothefirstzeroringiscalledthediameterofAiryDisk:workingwavelengthF#:fnumberofthesystemFiniteconjugateNA:numericalApertureNA=nsinuunF#W:WorkingFnumberRuleofthumb:forvisiblelight,0.5m,DAiryF#inmicronsForaberrationfreesystem,thePSFwillbethesquareoftheabsoluteoftheFouriertransformofacircularapertureanditisgivenintheformof1storderBesselfunction.rAngularResolution-RayleighCriterion

,isinlp/mmTheCut-Offfrequencyofanopticalsystemis:Features：MirrorsalignedonaxisAdvantages：SimpleandachromaticDisadvantages：CentralobscurationandlowerMTFSmallerFOVwithlongfocallength

ObscuredSystem

UnobscuredSystemFeatures：MirrorsalignedoffaxisAdvantages：NoobscurationandhigherMTF；LargerFOVwithlongfocallengthAchromaticDisadvantages：Difficulttomanufactureandassembly1.3SphericalWavefront,DefocusandLateralShiftAperfectlenswillproduceinitsexitpupilasphericalwavefrontconvergingtoapointadistanceRfromtheexitpupil.Thesphericalwavefrontequationis:SagequationDefocusOriginalwavefront:Newwavefront:DefocustermIncreasingtheOPDmovesthefocustowardtheexitpupilinthenegativeZdirection.Inotherword,iftheimageplaneisshiftedalongtheopticalaxistowardthelensanamountz(zisnegative),achangeinthewavefrontrelativetotheoriginalsphericalwavefrontis:Lateral(Transverse)ShiftInsteadofshiftingthecenterofcurvaturealongZaxis,wemoveitalongXaxis,then:Forthesamereason,ifmovealongYaxis,then:1.4TransverseandLongitudinalAberrationIngeneral,thewavefrontintheexitpupilisnotaperfectspherebutanaberratedsphere,sodifferentpartsofthewavefrontcometothefocusindifferentplaces.Itisoftendesirabletoknowwherethesefocuspointsarelocated,i.e.,find(x,y,z)asafunctionof(x,y).WavefrontaberrationisthedepartureofactualwavefrontfromreferencewavefrontalongtheRAY.Iflooktheopticalsystemfromtherearend,weseeexitpupilplaneandimageplane.WavefrontAberrationExpansionW000W020W040W060W111W131W151W222W242Whatdoaberrationslooklike?W000W020W040W060W111W131W151W222W242W333FieldCurvatureWheredoaberrationscomefrom?DistortionAstigmatismW222ComaW131WarrenSmith,ModernOpticalEngineering,P65SphericalAberrationW=W0404+W=W0404W=W0202W=-1W0202+W0404SphericalAberration+DefocusThrough-focusDiffractionImage(WithSphericalAberration)Wavefrontmeasurementusinganinterferometeronlyprovidesdataatasinglefieldpoint(oftenonaxis).Thiscausesfieldcurvaturetolooklikefocusanddistortiontolookliketilt.Therefore,anumberoffieldpointsmustbemeasuredtodeterminetheSeidelaberration.Whenperformingthetestonaxis,comashouldnotbepresent.Ifcomaispresentonaxis,itmightresultfromtiltor/anddecenteredopticalcomponentsinthesystemduetomisalignment.Acommonerrorinmanufacturingopticalsurfacesisforasurfacetobeslightlycylindricalinsteadofperfectlyspherical.Astigmatismmightbeseenonaxisduetomanufacturingerrorsorimpropersupportingstructure.ImportanttoknowCausticSpecifiesthesizeofaberrationBasicformofaberrationTheaberrationsofagivenopticalsystemdependonthesystemparameterssuchasaperturediameter,focallength,andfieldangle,aswellassomespecificconfigurationsofthesystem.1.6AberrationCoefficientsTheLagrangeInvariantжTheLagrangeInvariantholdsatanyplanebetweenobjectandimage.ж=Atobjectplane:ж=Atimageplane:ж=Forobjectatinfinity：ParaxialRayTracingSnell’sLawL=SeidelCoefficientTableSeidelCoefficientCalculationforaSingleletCalculationbyZemaxCalculationbySeidelCoefficientFormulaTheThinLensFormTheaberrationsofagivenopticalsystemdependonthesystemparameterssuchasaperturediameter,focallength,andfieldangle,aswellassomespecificconfigurationsofthesystem.Thesystemparameterscanbefactoredoutoftheaberrationcoefficients,leavingremainingfactorswhichdependonlyupontheconfigurationofthesystem.Theseremainingfactorswewillcallthestructuralaberrationcoefficients.TheStructureAberrationCoefficientRolandV.ShackTheThinLensBendingItispossibletohaveasetoflenseswiththesamepowerandthesamethicknessbutwithdifferentshapes.X：MinimumsphericalaberrationIfYisconstant,thenIfobjectatinfinity,Y=1,n=1.5,thenMinimumcomaIfobjectatinfinity,Y=1,n=1.5,thenX=-2X=-1X=+1X=+2Forobjectatinfinity,stopatthinlens,whenlenspowerisfixed:ZemaxResultCalculationUsingThinLensFormForobjectatinfinity：ж=Forthinlensisinair,n=1，rearrangethethinlensformula:1.7ZernikePolynomialsOfteninopticaltesting,tobetterinterpretthetestresultsitisconvenienttoexpresswavefrontdatainpolynomialform.Zernikepolynomialsareoftenusedforthispurposesincetheycontaintermshavingthesameformsastheobservedaberrations(Zernike,1934).NearlyallcommercialdigitalinterferometersandopticaldesignsoftwaresuseZernikepolynomialstorepresentthewavefrontaberrations.Zernikepolynomialshavesomeinterestingproperties,IfisZernikepolynomialtermsofnthdegreeandwediscusswithinaunitcircle:Thesepolynomialsareorthogonaloverthecontinuousinterioroftheunitcircle:

canbeexpressedastheproductoftwofunctions.Onedependsonlyontheradialcoordinateandtheotherdependsonlyontheangularcoordinate.nandlareeitherbothevenorbothodd.Ithasrotationalsymmetryproperty.Rotatingthecoordinatesystembyanangledoesn'tchangetheformofthepolynomials:

canbeexpressedas:,wheremn,l=n-2m.SoZerniketermUnmcanbeexpressedas:Where:sinfunctionisusedforn-2m>0

cosfunctionisusedforn-2m0SothewavefrontaberrationcanbeexpressedasalinearcombinationofZernikecircularpolynomialsofkthdegree:WhereAnmisthecoefficientofZerniketermUnm.4thZernikepolynomialsRe-orderedZernikepolynomials(first36terms)12354678PlotsofZernikepolynomials#1~#89101112131415PlotsofZernikepolynomials#9~#15PlotsofZernikepolynomials#16~#2416171819202122232433PlotsofZernikepolynomials#25~#3625262827293032313534Zernikepolynomialsareeasilyrelatedtoclassicalaberrations.W(,)isusuallyfoundthebestleastsquaresfittothedatapoints.SinceZernikepolynomialsareorthogonalovertheunitcircle,anyoftheterms:alsorepresentsindividuallyabestleastsquaresfittothedata.Anmisindependentofeachother,sotoremovedefocusortiltweonlyneedtosettheappropriatecoefficientstozerowithoutneedingtofindanewleastsquaresfit.AdvantagesofusingZernikepolynomialsCautionsofusingZernikepolynomialsMidorhighfrequencyerrorsmightbe“smoothedout”.ForexampletheDiamondTurnedsurfacepronotbeaccuratelyexpressedbyusingreasonablenumberofZerniketerms.Zernikepolynomialsareorthogonalonlyoverthecontinuousinteriorofanunitcircle,generallynotorthogonaloverthediscretesetofdatapointswithinaunitcircleoranyotherapertureshape.RelationshipBetweenZernikepolynomialsandSeidelAberrationsThefirst9Zernikepolynomialsareexpressedas:ThesameaberrationcanbeexpressedinSeidelform:Usingtheidentity:1.8PeaktoValleyandRMSWavefrontAberrationPeaktoValley(PV)issimplythemaximumdepartureoftheactualwavefrontfromthedesiredwavefrontinbothpositiveandnegativedirections.WhileusingPVtospecifythewavefronterrorisconvenientandsimple,butitcanbemisleading.Ittellsnothingaboutthewholeareaoverwhichtheerrorareoccurring.AnopticalsystemhavingalargePVerrormayactuallyperformbetterthanasystemhavingasmallPV.ItismoremeaningfultospecifywavefrontqualityusingtheRMSwavefronterro