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.,1.0BasicWavefrontAberrationTheoryForOpticalMetrology,ChangchunInstituteofOpticsandFineMechanicsandPhysics,Dr.ZhangXuejun,.,ThePrincipalpurposeofopticalmetrologyistodeterminetheaberrationspresentinanopticalcomponentoranopticalsystem.Tostudyopticalmetrologytheformsofaberrationsthatmightbepresentneedtobeunderstood.,.,Formostopticaltestinginstruments,thetestresultisthedifferencebetweenareference(unaberrated)wavefrontandatest(aberrated)wavefront.WeusuallycallthisdifferencetheOpticalPathDifference(OPD).,NotethattheOPDisthedifferencebetweenthereferencewavefrontandthetestwavefrontmeasuredalongtheray.,.,1.1SignConvention,TheOPDispositiveiftheaberratedwavefrontleadstheidealwavefront.Inotherword,apositiveaberrationwillfocusinfrontoftheparaxial(Gaussian)imageplane.,RightHandedCoordinates:ZaxisisthelightpropagationdirectionXaxisisthemeridionalortangentialdirectionYaxisisthesagittaldirection,.,Thedistanceispositiveifmeasuredfromlefttoright.TheangleispositiveifitisincounterclockwisedirectionrelativetoZaxis.,Sincemostopticalsystemsarerotationallysymmetric,usingpolarcoordinateismoreconvenient.,x=cosy=sin,.,1.2AberrationFreeSystem,Iftheopticalsystemisunaberratedordiffraction-limited,forapointobjectatinfinitytheimagewillnotbea“point”,butanAiryDisk.ThedistributionoftheirradianceontheimageplaneofAiryDiskiscalledPointSpreadFunctionorPSF.SincePSFisverysensitivetoaberrationsitisoftenusedasanindicatoroftheopticalperformance.,.,DiametertothefirstzeroringiscalledthediameterofAiryDisk,:workingwavelengthF#:fnumberofthesystem,.,Finiteconjugate,NA:numericalApertureNA=nsinu,F#W:WorkingFnumber,Ruleofthumb:forvisiblelight,0.5m,DAiryF#inmicrons,.,x,y:coordinatesmeasuredintheexitpupilx0,y0:coordinatesmeasuredinthefocalplaneI0:intensityofincidentwavefront(constant):wavelengthofincidentwavefrontf:focallengthoftheopticalsystemA:amplitudeintheexitpupil(x,y):thephasetransmissionfunctionintheexitpupil,.,Foraberrationfreesystem,thePSFwillbethesquareoftheabsoluteoftheFouriertransformofacircularapertureanditisgivenintheformof1storderBesselfunction.,.,Thefractionofthetotalenergycontainedinacircleofradiusraboutthediffractionpatterncenterisgivenby：,.,r,AngularResolution-RayleighCriterion,.,Generallyamirrorsystemwillhaveacentralobscuration.Ifeistheratioofthediameterofthecentralobscurationtothemirrordiameterd,andiftheentirecircularmirrorofdiameterdisuniformlyilluminated,thepowerperunitsolidangleisgivenby,.,.,isinlp/mm,TheCut-Offfrequencyofanopticalsystemis:,.,Features：MirrorsalignedonaxisAdvantages：SimpleandachromaticDisadvantages：CentralobscurationandlowerMTFSmallerFOVwithlongfocallength,ObscuredSystem,UnobscuredSystem,Features：MirrorsalignedoffaxisAdvantages：NoobscurationandhigherMTF；LargerFOVwithlongfocallengthAchromaticDisadvantages：Difficulttomanufactureandassembly,.,1.3SphericalWavefront,DefocusandLateralShift,AperfectlenswillproduceinitsexitpupilasphericalwavefrontconvergingtoapointadistanceRfromtheexitpupil.Thesphericalwavefrontequationis:,Sagequation,.,Defocus,Originalwavefront:,Newwavefront:,Defocusterm,IncreasingtheOPDmovesthefocustowardtheexitpupilinthenegativeZdirection.Inotherword,iftheimageplaneisshiftedalongtheopticalaxistowardthelensanamountz(zisnegative),achangeinthewavefrontrelativetotheoriginalsphericalwavefrontis:,.,DepthofFocus,Ruleofthumb:forvisiblelight,0.5m,Z(F#)2inmicrons,ByuseofRayleighCriterion:,ThesmallertheF#,orthelargertherelativeaperture,thesmallertheDepthofFocus,sotheharderthealignment.,.,.,Lateral(Transverse)Shift,InsteadofshiftingthecenterofcurvaturealongZaxis,wemoveitalongXaxis,then:,Forthesamereason,ifmovealongYaxis,then:,.,Ageneralsphericalwavefront:,Thisequationrepresentsasphericalwavefrontwhosecenterofcurvatureislocatedatthepoint(X,Y,Z).,TheOPDis:,Thisthreetermsareadditiveforthemisalignment,someorallofthemshouldberemovedfromthetestresultfordifferenttestconfigurations.,.,1.4TransverseandLongitudinalAberration,Ingeneral,thewavefrontintheexitpupilisnotaperfectspherebutanaberratedsphere,sodifferentpartsofthewavefrontcometothefocusindifferentplaces.Itisoftendesirabletoknowwherethesefocuspointsarelocated,i.e.,find(x,y,z)asafunctionof(x,y).,.,WavefrontaberrationisthedepartureofactualwavefrontfromreferencewavefrontalongtheRAY.,.,1.5SeidelAberrations,Inarealopticalsystem,theformofthewavefrontaberrationscanbeextremlycomplexduetotherandomerrorsindesign,fabricationandalignment.AccordingtoWelford,thiswavefrontaberrationcanbeexpressedasapowerseriesof(h,x,y):,a3termgivesrisetothephaseshiftoverthatisconstantacrosstheexitpupil.Itdoesntchangetheshapeofthewavefrontandhasnoeffectontheimage,usuallycalledPiston.b1tob5termshavefourthdegreeforh,x,ywhenexpressedaswavefrontaberrationorthirddegreeastransverseaberration,usuallycalledfourth-orderorthirdorderaberrations.,h:fieldcoordinatesx,y:coordinatesatexitpupil,.,.,Iflooktheopticalsystemfromtherearend,weseeexitpupilplaneandimageplane.,.,WavefrontAberrationExpansion,.,ClassicalSeidelAberrations,.,Whatdoaberrationslooklike?,.,.,FieldCurvature,Wheredoaberrationscomefrom?,.,Distortion,.,Astigmatism,W222,.,.,Coma,W131,.,WarrenSmith,ModernOpticalEngineering,P65,SphericalAberration,W=W0404,.,+,W=W0404,W=W0202,W=-1W0202+W0404,SphericalAberration+Defocus,.,Through-focusDiffractionImage(WithSphericalAberration),.,Wavefrontmeasurementusinganinterferometeronlyprovidesdataatasinglefieldpoint(oftenonaxis).Thiscausesfieldcurvaturetolooklikefocusanddistortiontolookliketilt.Therefore,anumberoffieldpointsmustbemeasuredtodeterminetheSeidelaberration.Whenperformingthetestonaxis,comashouldnotbepresent.Ifcomaispresentonaxis,itmightresultfromtiltor/anddecenteredopticalcomponentsinthesystemduetomisalignment.Acommonerrorinmanufacturingopticalsurfacesisforasurfacetobeslightlycylindricalinsteadofperfectlyspherical.Astigmatismmightbeseenonaxisduetomanufacturingerrorsorimpropersupportingstructure.,Importanttoknow,.,Caustic,.,Specifiesthesizeofaberration,Basicformofaberration,Theaberrationsofagivenopticalsystemdependonthesystemparameterssuchasaperturediameter,focallength,andfieldangle,aswellassomespecificconfigurationsofthesystem.,1.6AberrationCoefficients,.,.,TheLagrangeInvariant,TheLagrangeInvariantholdsatanyplanebetweenobjectandimage.,=,Forobjectatinfinity：,.,ParaxialRayTracing,SnellsLaw,.,L=,SeidelCoefficientTable,.,SeidelCoefficientCalculationforaSinglelet,.,CalculationbyZemax,.,CalculationbySeidelCoefficientFormula,.,.,TheThinLensForm,Theaberrationsofagivenopticalsystemdependonthesystemparameterssuchasaperturediameter,focallength,andfieldangle,aswellassomespecificconfigurationsofthesystem.Thesystemparameterscanbefactoredoutoftheaberrationcoefficients,leavingremainingfactorswhichdependonlyupontheconfigurationofthesystem.Theseremainingfactorswewillcallthestructuralaberrationcoefficients.,.,.,TheStructureAberrationCoefficient,RolandV.Shack,.,TheThinLensBending,Itispossibletohaveasetoflenseswiththesamepowerandthesamethicknessbutwithdifferentshapes.,X：,Minimumsphericalaberration,IfYisconstant,then,Ifobjectatinfinity,Y=1,n=1.5,then,.,Minimumcoma,Ifobjectatinfinity,Y=1,n=1.5,then,Forobjectatinfinity,stopatthinlens,whenlenspowerisfixed:,.,ZemaxResult,CalculationUsingThinLensForm,.,Forobjectatinfinity：,=,Forthinlensisinair,n=1，rearrangethethinlensformula:,.,1.7ZernikePolynomials,Ofteninopticaltesting,tobetterinterpretthetestresultsitisconvenienttoexpresswavefrontdatainpolynomialform.Zernikepolynomialsareoftenusedforthispurposesincetheycontaintermshavingthesameformsastheobservedaberrations(Zernike,1934).NearlyallcommercialdigitalinterferometersandopticaldesignsoftwaresuseZernikepolynomialstorepresentthewavefrontaberrations.,.,Zernikepolynomialshavesomeinterestingproperties,IfisZernikepolynomialtermsofnthdegreeandwediscusswithinaunitcircle:Thesepolynomialsareorthogonaloverthecontinuousinterioroftheunitcircle:,.,canbeexpressedastheproductoftwofunctions.Onedependsonlyontheradialcoordinateandtheotherdependsonlyontheangularcoordinate.nandlareeitherbothevenorbothodd.Ithasrotationalsymmetryproperty.Rotatingthecoordinatesystembyanangledoesntchangetheformofthepolynomials:,.,canbeexpressedas:,wheremn,l=n-2m.SoZerniketermUnmcanbeexpressedas:,Where:sinfunctionisusedforn-2m0cosfunctionisusedforn-2m0,.,SothewavefrontaberrationcanbeexpressedasalinearcombinationofZernikecircularpolynomialsofkthdegree:,WhereAnmisthecoefficientofZerniketermUnm.,.,4thZernikepolynomials,.,Re-orderedZernikepolynomials(first36terms),.,1,2,3,5,4,6,7,8,PlotsofZernikepolynomials#1#8,.,9,10,11,12,13,14,15,PlotsofZernikepolynomials#9#15,.,PlotsofZernikepolynomials#16#24,16,17,18,19,20,21,22,23,24,.,33,PlotsofZernikepolynomials#25#36,25,26,28,27,29,30,32,31,35,34,.,Zernikepolynomialsareeasilyrelatedtoclassicalaberrations.W(,)isusuallyfoundthebestleastsquaresfittothedatapoints.SinceZernikepolynomialsareorthogonalovertheunitcircle,anyoftheterms:alsorepresentsindividuallyabestleastsquaresfittothedata.Anmisindependentofeachother,sotoremovedefocusortiltweonlyneedtosettheappropriatecoefficientstozerowithoutneedingtofindanewleastsquaresfit.,AdvantagesofusingZernikepolynomials,.,CautionsofusingZernikepolynomials,Midorhighfrequencyerrorsmightbe“smoothedout”.ForexampletheDiamondTurnedsurfaceprofilecannotbeaccuratelyexpressedbyusingreasonablenumberofZerniketerms.Zernikepolynomialsareorthogonalonlyoverthecontinuousinteriorofanunitcircle,generallynotorthogonaloverthediscretesetofdatapointswithinaunitcircleoranyotherapertureshape.,.,RelationshipBetweenZernikepolynomialsandSeidelAberrations,Thefirst9Zernikepolynomialsareexpressedas:,ThesameaberrationcanbeexpressedinSeidelform:,.,Usingtheidentity:,.,.,1.8PeaktoValleyandRMSWavefrontAberration,PeaktoValley(PV)issimplythemaximumdepartureoftheactualwavefrontfromthedesiredwavefrontinbothpositiveandnegativedirections.WhileusingPVtospecifythewavefronterrorisconvenientandsimple,butitcanbemisleading.Ittellsnothingaboutthewholearea