光学检测CH01课件

1.0BasicWavefrontAberrationTheoryForOpticalMetrology

ChangchunInstituteofOpticsandFineMechanicsandPhysicsDr.ZhangXuejun11.0BasicWavefrontAberrationThePrincipalpurposeofopticalmetrologyistodeterminetheaberrationspresentinanopticalcomponentoranopticalsystem.Tostudyopticalmetrologytheformsofaberrationsthatmightbepresentneedtobeunderstood.2ThePrincipalpurposeofopticFormostopticaltestinginstruments,thetestresultisthedifferencebetweenareference(unaberrated)wavefrontandatest(aberrated)wavefront.WeusuallycallthisdifferencetheOpticalPathDifference(OPD).OPDTestwavefrontReferencewavefrontRayNotethattheOPDisthedifferencebetweenthereferencewavefrontandthetestwavefrontmeasured

alongtheray.3Formostopticaltestinginstr1.1SignConventionTheOPDispositiveiftheaberratedwavefrontleadstheidealwavefront.Inotherword,apositiveaberrationwillfocusinfrontoftheparaxial(Gaussian)imageplane.RightHandedCoordinates:ZaxisisthelightpropagationdirectionXaxisisthemeridionalortangentialdirectionYaxisisthesagittaldirection41.1SignConventionTheOPDisThedistanceispositiveifmeasuredfromlefttoright.TheangleispositiveifitisincounterclockwisedirectionrelativetoZaxis.(+)(-)(+angle)(-angle)Sincemostopticalsystemsarerotationallysymmetric,usingpolarcoordinateismoreconvenient.XY

x=cosy=sin

5Thedistanceispositiveifme1.2AberrationFreeSystemIftheopticalsystemisunaberratedordiffraction-limited,forapointobjectatinfinitytheimagewillnotbea“point”,butanAiryDisk.ThedistributionoftheirradianceontheimageplaneofAiryDiskiscalledPointSpreadFunctionorPSF.SincePSFisverysensitivetoaberrationsitisoftenusedasanindicatoroftheopticalperformance.61.2AberrationFreeSystemIftFirstmaximumSecondmaximumDiametertothefirstzeroringiscalledthediameterofAiryDisk

:workingwavelengthF#:fnumberofthesystem7FirstmaximumSecondmaximumDiaFiniteconjugateNA:numericalApertureNA=nsinuunF#W:WorkingFnumberRuleofthumb:forvisiblelight,0.5m,DAiryF#inmicrons8FiniteconjugateunRuleofthumx,y:coordinatesmeasuredintheexitpupilx0,y0:coordinatesmeasuredinthefocalplaneI0:intensityofincidentwavefront(constant):wavelengthofincidentwavefrontf:focallengthoftheopticalsystemA:amplitudeintheexitpupil(x,y):thephasetransmissionfunctionintheexitpupilOPDPupilfunction9x,y:coordinatesmeasuredinForaberrationfreesystem,thePSFwillbethesquareoftheabsoluteoftheFouriertransformofacircularapertureanditisgivenintheformof1storderBesselfunction.10Foraberrationfreesystem,thThefractionofthetotalenergycontainedinacircleofradiusraboutthediffractionpatterncenterisgivenby：11ThefractionofthetotalenerrAngularResolution-RayleighCriterion12rAngularResolution-RayleighCGenerallyamirrorsystemwillhaveacentralobscuration.Ifeistheratioofthediameterofthecentralobscurationtothemirrordiameterd,andiftheentirecircularmirrorofdiameterdisuniformlyilluminated,thepowerperunitsolidangleisgivenby13Generallyamirrorsystemwill1414

,isinlp/mmTheCut-Offfrequencyofanopticalsystemis:15,isinlFeatures：MirrorsalignedonaxisAdvantages：SimpleandachromaticDisadvantages：CentralobscurationandlowerMTFSmallerFOVwithlongfocallength

ObscuredSystem

UnobscuredSystemFeatures：MirrorsalignedoffaxisAdvantages：NoobscurationandhigherMTF；LargerFOVwithlongfocallengthAchromaticDisadvantages：Difficulttomanufactureandassembly16Features：ObscuredSystem1.3SphericalWavefront,DefocusandLateralShiftAperfectlenswillproduceinitsexitpupilasphericalwavefrontconvergingtoapointadistanceRfromtheexitpupil.Thesphericalwavefrontequationis:Sagequation171.3SphericalWavefront,DefocDefocusOriginalwavefront:Newwavefront:DefocustermIncreasingtheOPDmovesthefocustowardtheexitpupilinthenegativeZdirection.Inotherword,iftheimageplaneisshiftedalongtheopticalaxistowardthelensanamount

z(zisnegative),achangeinthewavefrontrelativetotheoriginalsphericalwavefrontis:18DefocusOriginalwavefront:NewunDepthofFocusRuleofthumb:forvisiblelight,0.5m,Z(F#)2inmicronsByuseofRayleighCriterion:ThesmallertheF#,orthelargertherelativeaperture,thesmallertheDepthofFocus,sotheharderthealignment.19unDepthofFocusRuleofthumb:2020Lateral(Transverse)ShiftInsteadofshiftingthecenterofcurvaturealongZaxis,wemoveitalongXaxis,then:Forthesamereason,ifmovealongYaxis,then:21Lateral(Transverse)ShiftInstAgeneralsphericalwavefront:Thisequationrepresentsasphericalwavefrontwhosecenterofcurvatureislocatedatthepoint(

X,

Y,Z).TheOPDis:Thisthreetermsareadditiveforthemisalignment,someorallofthemshouldberemovedfromthetestresultfordifferenttestconfigurations.22Ageneralsphericalwavefront:1.4TransverseandLongitudinalAberrationIngeneral,thewavefrontintheexitpupilisnotaperfectspherebutanaberratedsphere,sodifferentpartsofthewavefrontcometothefocusindifferentplaces.Itisoftendesirabletoknowwherethesefocuspointsarelocated,i.e.,find(

x,y,z)asafunctionof(x,y).231.4TransverseandLongitudinaWavefrontaberrationisthedepartureofactualwavefrontfromreferencewavefrontalongtheRAY.24Wavefrontaberrationisthede1.5SeidelAberrationsInarealopticalsystem,theformofthewavefrontaberrationscanbeextremlycomplexduetotherandomerrorsindesign,fabricationandalignment.AccordingtoWelford,thiswavefrontaberrationcanbeexpressedasapowerseriesof(h,x,y):a3termgivesrisetothephaseshiftoverthatisconstantacrosstheexitpupil.Itdoesn'tchangetheshapeofthewavefrontandhasnoeffectontheimage,usuallycalledPiston.b1tob5termshavefourthdegreeforh,x,ywhenexpressedaswavefrontaberrationorthirddegreeastransverseaberration,usuallycalledfourth-orderorthirdorderaberrations.h:fieldcoordinatesx,y:coordinatesatexitpupil251.5SeidelAberrationsInarea2626Iflooktheopticalsystemfromtherearend,weseeexitpupilplaneandimageplane.27IflooktheopticalsystemfroWavefrontAberrationExpansion28WavefrontAberrationExpansionClassicalSeidelAberrations29ClassicalSeidelAberrations29W000W020W040W060W111W131W151W222W242Whatdoaberrationslooklike?30W000W020W040W060W111W131W151W2W000W020W040W060W111W131W151W222W242W33331W000W020W040W060W111W131W151W2FieldCurvatureWheredoaberrationscomefrom?32FieldCurvatureWheredoaberraDistortion33Distortion33AstigmatismW22234AstigmatismW222343535ComaW13136ComaW13136WarrenSmith,ModernOpticalEngineering,P65SphericalAberration

W=W040

437WarrenSmith,ModernOpticalE+

W=W040

4

W=W020

2

W=-1W020

2+W040

4SphericalAberration+Defocus38+W=W0404W=W0202W=-1W020Through-focusDiffractionImage(WithSphericalAberration)39Through-focusDiffractionImagWavefrontmeasurementusinganinterferometeronlyprovidesdataatasinglefieldpoint(oftenonaxis).Thiscausesfieldcurvaturetolooklikefocusanddistortiontolookliketilt.Therefore,anumberoffieldpointsmustbemeasuredtodeterminetheSeidelaberration.Whenperformingthetestonaxis,comashouldnotbepresent.Ifcomaispresentonaxis,itmightresultfromtiltor/anddecenteredopticalcomponentsinthesystemduetomisalignment.Acommonerrorinmanufacturingopticalsurfacesisforasurfacetobeslightlycylindricalinsteadofperfectlyspherical.Astigmatismmightbeseenonaxisduetomanufacturingerrorsorimpropersupportingstructure.Importanttoknow40WavefrontmeasurementusinganCaustic41Caustic41SpecifiesthesizeofaberrationBasicformofaberrationTheaberrationsofagivenopticalsystemdependonthesystemparameterssuchasaperturediameter,focallength,andfieldangle,aswellassomespecificconfigurationsofthesystem.1.6AberrationCoefficients42Specifiesthesizeofaberrati4343TheLagrangeInvariantж

TheLagrangeInvariantholdsatanyplanebetweenobjectandimage.ж=Atobjectplane:ж=Atimageplane:ж=Forobjectatinfinity：44TheLagrangeInvariantжTheLaParaxialRayTracingSnell’sLaw45ParaxialRayTracingSnell’sLaL=SeidelCoefficientTable46L=SeidelCoefficientTable46SeidelCoefficientCalculationforaSinglelet47SeidelCoefficientCalculationCalculationbyZemax48CalculationbyZemax48CalculationbySeidelCoefficientFormula49CalculationbySeidelCoeffici5050TheThinLensFormTheaberrationsofagivenopticalsystemdependonthesystemparameterssuchasaperturediameter,focallength,andfieldangle,aswellassomespecificconfigurationsofthesystem.Thesystemparameterscanbefactoredoutoftheaberrationcoefficients,leavingremainingfactorswhichdependonlyupontheconfigurationofthesystem.Theseremainingfactorswewillcallthestructuralaberrationcoefficients.51TheThinLensFormTheaberrati5252TheStructureAberrationCoefficientRolandV.Shack53TheStructureAberrationCoeffTheThinLensBendingItispossibletohaveasetoflenseswiththesamepowerandthesamethicknessbutwithdifferentshapes.X：MinimumsphericalaberrationIfYisconstant,thenIfobjectatinfinity,Y=1,n=1.5,then54TheThinLensBendingItisposMinimumcomaIfobjectatinfinity,Y=1,n=1.5,thenX=-2X=-1X=+1X=+2Forobjectatinfinity,stopatthinlens,whenlenspowerisfixed:55MinimumcomaIfobjectatinfinZemaxResultCalculationUsingThinLensForm56ZemaxResultCalculationUsingForobjectatinfinity：ж=Forthinlensisinair,n=1，rearrangethethinlensformula:57Forobjectatinfinity：ж=Fort1.7ZernikePolynomialsOfteninopticaltesting,tobetterinterpretthetestresultsitisconvenienttoexpresswavefrontdatainpolynomialform.Zernikepolynomialsareoftenusedforthispurposesincetheycontaintermshavingthesameformsastheobservedaberrations(Zernike,1934).NearlyallcommercialdigitalinterferometersandopticaldesignsoftwaresuseZernikepolynomialstorepresentthewavefrontaberrations.581.7ZernikePolynomialsOftenZernikepolynomialshavesomeinterestingproperties,IfisZernikepolynomialtermsofnthdegreeandwediscusswithinaunitcircle:Thesepolynomialsareorthogonaloverthecontinuousinterioroftheunitcircle:

59Zernikepolynomialshavesomecanbeexpressedastheproductoftwofunctions.Onedependsonlyontheradialcoordinate

andtheotherdependsonlyontheangularcoordinate

.nandlareeitherbothevenorbothodd.Ithasrotationalsymmetryproperty.Rotatingthecoordinatesystembyanangledoesn'tchangetheformofthepolynomials:

60canbeexpressedasthepro

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

cosfunctionisusedforn-2m

061canbeexpressedas:,whereSothewavefrontaberrationcanbeexpressedasalinearcombinationofZernikecircularpolynomialsofkthdegree:WhereAnmisthecoefficientofZerniketermUnm.62Sothewavefrontaberrationca4thZernikepolynomials634thZernikepolynomials63Re-orderedZernikepolynomials(first36terms)64Re-orderedZernikepolynomials12354678PlotsofZernikepolynomials#1~#86512354678PlotsofZernikepolyn9101112131415PlotsofZernikepolynomials#9~#15669101112131415PlotsofZernikePlotsofZernikepolynomials#16~#2416171819202122232467PlotsofZernikepolynomials#33PlotsofZernikepolynomials#25~#36252628272930323135346833PlotsofZernikepolynomialsZernikepolynomialsareeasilyrelatedtoclassicalaberrations.W(,

)isusuallyfoundthebestleastsquaresfittothedatapoints.SinceZernikepolynomialsareorthogonalovertheunitcircle,anyoftheterms:alsorepresentsindividuallyabestleastsquaresfittothedata.Anmisindependentofeachother,sotoremovedefocusortiltweonlyneedtosettheappropriatecoefficientstozerowithoutneedingtofindanewleastsquaresfit.AdvantagesofusingZernikepolynomials69ZernikepolynomialsareeasilyCautionsofusingZernikepolynomialsMidorhighfrequencyerrorsmightbe“smoothedout”.ForexampletheDiamondTurnedsurfaceprofilecannotbeaccuratelyexpressedbyusingreasonablenumberofZerniketerms.Zernikepolynomialsareorthogonalonlyoverthecontinuousinteriorofanunitcircle,generallynotorthogonaloverthediscretesetofdatapointswithinaunitcircleoranyotherapertureshape.70CautionsofusingZernikepolyRelationshipBetweenZernikepolynomialsandSeidelAberrationsThefirst9Zernikepolynomialsareexpressedas:ThesameaberrationcanbeexpressedinSeidelform:71RelationshipBetweenZernikepUsingtheidentity:72Usingtheidentity:7273731.8PeaktoValleyandRMSWavefrontAberrationPeaktoValley(PV)issimplythemaximumdepartureoftheactualwavefrontfromthedesiredwavefrontinbothpositiveandnegativedirections.WhileusingPVtospecifythe