Zernike-Polynomials

ZernikePolynomials1IntroductionOften,toaidintheinterpretationofopticaltestresultsitisconvenienttoexpresswavefrontdatainpolynomialform.Zernikepolynomialsareoftenusedforthispurposesincetheyaremadeupoftermsthatareofthesameformasthetypesofaberrationsoftenobservedinopticaltests(Zernike,1934).ThisisnottosaythatZernikepolynomialsarethebestpolynomialsforfittingtestdata.SometimesZernikepolynomialsgiveapoorrepresentationofthewavefrontdata.Forexample,Zernikeshavelittlevaluewhenairturbulenceispresent.Likewise,fabricationerrorsinthesinglepointdiamondturningprocesscannotberepresentedusingareasonablenumberoftermsintheZernikepolynomial.Inthetestingofconicalopticalelements,additionaltermsmustbeaddedtoZernikepolynomialstoaccuratelyrepresentalignmenterrors.TheblinduseofZernikepolynomialstorepresenttestresultscanleadtodisastrousresults.Zernikepolynomialsareoneofaninfinitenumberofcompletesetsofpolynomialsintwovariables,randq,thatareorthogonalinacontinuousfashionovertheinteriorofaunitcircle.ItisimportanttonotethattheZernikesareorthogonalonlyinacontinuousfashionovertheinteriorofaunitcircle,andingeneraltheywillnotbeorthogonaloveradiscretesetofdatapointswithinaunitcircle.Zernikepolynomialshavethreepropertiesthatdistinguishthemfromothersetsoforthogonalpolynomials.First,theyhavesimplerotationalsymmetrypropertiesthatleadtoapolynomialproductoftheformr@ρDg@θD,whereg[q]isacontinuousfunctionthatrepeatsselfevery2pradiansandsatisfiestherequirementthatrotatingthecoordinatesystembyanangleadoesnotchangetheformofthepolynomial.Thatisg@θ+αD=g@θDg@αD.Thesetoftrigonometricfunctionsg@θD=±mθ,wheremisanypositiveintegerorzero,meetstheserequirements.ThesecondpropertyofZernikepolynomialsisthattheradialfunctionmustbeapolynomialinrofdegree2nandcontainnopowerofrlessthanm.Thethirdpropertyisthatr[r]mustbeevenifmiseven,andoddifmisodd.TheradialpolynomialscanbederivedasaspecialcaseofJacobipolynomials,andtabulatedasr@n,m,rD.Theirorthogonalityandnormalizationpropertiesaregivenby‡01r@n,m,ρDr@n',m,ρDρρ=12Hn+1LKroneckerDelta@n−n'DandZernikePolynomialsForTheWeb.nbJamesC.Wyant,20031r@n,m,1D=1.Asstatedabove,r[n,m,r]isapolynomialoforder2nanditcanbewrittenasr@n_,m_,ρ_D:=‚s=0n−mH−1LsH2n−m−sL!s!Hn−sL!Hn−m−sL!ρ2Hn−sL−mInpractice,theradialpolynomialsarecombinedwithsinesandcosinesratherthanwithacomplexexponential.Itisconvenienttowritercos@n_,m_,ρ_D:=r@n,m,ρDCos@mθDandrsin@n_,m_,ρ_D:=r@n,m,ρDSin@mθDThefinalZernikepolynomialseriesforthewavefrontopdDwcanbewrittenas∆w@ρ_,θ_D:=∆w¯¯¯¯¯+„n=1nmaxikjjjjja@nDr@n,0,ρD+‚m=1nHb@n,mDrcos@n,m,ρD+c@n,mDrsin@n,m,ρDLy{zzzzzwhereDw[r,q]isthemeanwavefrontopd,anda[n],b[n,m],andc[n,m]areindividualpolynomialcoefficients.Forasymmetricalopticalsystem,thewaveaberrationsaresymmetricalaboutthetangentialplaneandonlyevenfunctionsofqareallowed.Ingeneral,however,thewavefrontisnotsymmetric,andbothsetsoftrigonometrictermsareincluded.2CalculatingZernikesFortheexamplebelowthedegreeoftheZernikepolynomialsisselectedtobe6.ThevalueofnDegreecanbechangedifadifferentdegreeisdesired.ThearrayzernikePolarcontainsZernikepolynomialsinpolarcoordinates(r,q),whilethearrayzernikeXycontainstheZernikepolynomialsinCartesian,(x,y),coordinates.zernikePolarListandzernikeXyListcontainstheZernikenumberincolumn1,thenandmvaluesincolumns2and3,andtheZernikepolynomialincolumn4.nDegree=6;i=0;Do@If@m==0,8i=i+1,temp@iD=8i−1,n,m,r@n,m,ρD,8i=i+1,temp@iD=8i−1,n,m,Factor@rcos@n,m,ρDD,i=i+1,temp@iD=8i−1,n,m,Factor@rsin@n,m,ρDDD,8n,0,nDegree,8m,n,0,−1D;ZernikePolynomialsForTheWeb.nbJamesC.Wyant,20032zernikePolarList=Array@temp,iD;Clear@tempD;Do@zernikePolar@i−1D=zernikePolarList@@i,4DD,8i,1,Length@zernikePolarListDD;zernikeXyList=Map@TrigExpand,zernikePolarListDê.9ρ→è!!!!!!!!!!!!!!!x2+y2,Cos@θD→xè!!!!!!!!!!!!!!!x2+y2,Sin@θD→yè!!!!!!!!!!!!!!!x2+y2=;Do@zernikeXy@i−1D=zernikeXyList@@i,4DD,8i,1,Length@zernikeXyListDD2.1TablesofZernikesInthetablesterm#1isaconstantorpistonterm,whileterms#2and#3aretiltterms.Term#4representsfocus.Thus,terms#2through#4representtheGaussianorparaxialpropertiesofthewavefront.Terms#5and#6areastigmatismplusdefocus.Terms#7and#8representcomaandtilt,whileterm#9representsthird-ordersphericalandfocus.Likewiseterms#10through#16representfifth-orderaberration,terms#17through#25representseventh-orderaberrations,terms#26through#36representninth-orderaberrations,andterms#37through#49representeleventh-orderaberrations.Eachtermcontainstheappropriateamountofeachlowerordertermtomakeitorthogonaltoeachlowerorderterm.Also,eachtermoftheZernikesminimizesthermswavefronterrortotheorderofthatterm.Addingotheraberrationsoflowerordercanonlyincreasethermserror.Furthermore,theaveragevalueofeachtermovertheunitcircleiszero.2.1.1ZernikesinpolarcoordinatesTableForm@zernikePolarList,TableHeadings−88,8#,n,m,PolynomialD#nmPolynomial0001111ρCos@θD211ρSin@θD310−1+2ρ2422ρ2Cos@2θD522ρ2Sin@2θD621ρH−2+3ρ2LCos@θD721ρH−2+3ρ2LSin@θD8201−6ρ2+6ρ4933ρ3Cos@3θD1033ρ3Sin@3θDZernikePolynomialsForTheWeb.nbJamesC.Wyant,200331132ρ2H−3+4ρ2LCos@2θD1232ρ2H−3+4ρ2LSin@2θD1331ρH3−12ρ2+10ρ4LCos@θD1431ρH3−12ρ2+10ρ4LSin@θD1530−1