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(C)RGBingham2005.Allrightsreserved.OpticsandOpticalDesignSession3R.G.BinghamSession3

ParaxialraytracingandrelatedissuesAtheoreticalfiniteray,threeselectedsystemsandtipson“pastingin”

(C)RGBingham2005.Allrightsreserved.StagesindoingthejobWhWavefrontPropagationdirectionW=a2.h2+a4.h4+a6.h6+a8.h8…+???Thereisnouniversaltheorythatleadstodesignsforthecompoundlensesandotheropticalsystemswiththelowestaberrations.Thereisnotheorythattellsuswheretostart.Thereareusuallymultiplefeasiblesolutions,andwemayhave,say,twoorthreeworkingdaystomakesomeprogress.Wecannotavoidthinking.Weneedknowledgeablytopieceopticstogethertogivetherequiredfunctions,tobeawareofdifferentpossiblesolutions,andtoworktorefinethesebasicideas.Amongstmanyconceivablenumericalapproaches,twolevelsofapproximationenableus(a)tosetupschematicinitialconceptsthatcanberay-tracedbeforecorrectingaberrations,and(b)possiblytoelaborateontheseideasandinanycasetocheckthattheoptimiser(thecomputingalgorithmthatcanreducetherealaberrations)isworkingcorrectly.Atypicalwavefrontprofileinalensshowsaberrations;itwillnotconvergetoapoint.Itsproblemsareconceptuallycomplex,eveninalensthatisaxiallysymmetrical.(C)RGBingham2005.Allrightsreserved.Paraxialraytracingandrelatedissues17pages(C)RGBingham2005.Allrightsreserved.UsesoftheParaxialandSeidelapproximationsWhWavefrontLensaxisandpropagationdirectionParaxialformulaeassume:W=a2h2.Mainlyforstartingadesign.Derivelenscurvesanddiametersfromtheimageposition,focallength,f-numberormagnification.Checkfeasibility.Likea“thinlens”calculation,butitappliestothicklensesandcomplexsystems.MuchcanbedoneusingthesolvesinZEMAXorwithapocketcalculator.Seidelapproximation:W=a2h2+a4

h4

(andrelatedtermsoff-axis).ExploittheSeidelaberrationsbyinspection.Solvesomeproblems.Inalatersession,weshalldiscusshowtousetheresidualSeidelaberrationstoreviewanopticaldesignthatisnominallyfinished,inparticulartoseewhethertheoptimiserwaswellsetupforit.W=a2h2+a4h4+a6h6+a8h8…+???(C)RGBingham2005.Allrightsreserved.UseZEMAX’ssolves,orcalculatebyhand?ZEMAXprovidesmanyconvenient‘solves’withinitsLensDataEditorforderivingcurvaturesandthicknesses.Iguessthattheysupersedetheformulaeformostusers.Idocalculateoccasionalresultsbyhandfromequationsdiscussedinthissession:paraxialray-tracing;Lagrange’sinvariant(thatupholdsthesecondlawofthermodynamics);andfromNewton’sconjugatedistanceequation.Idothiswithapocketcalculatorandinroughnotes.Iusetheseexpressionstoputdimensionsontoasketchofsomeparticularidea,tocheckfeasibility,ortentativelytocalculatelengthsoranglesinthecourseofadiscussion.Seethedetailedexample(laterslide).Thatsetsupapreliminarylensdesignwithpencilandpaperanditcopeswithsome‘greyareas’.Alternatives.(a)SimilarideasmightbeentereddirectlyintoZEMAXusingthe‘solves’orbytrialanderror,or(b)wemightstartwithsomeexistinglensdesignthatcanbescaled,modifiedandre-optimisedforanewapplicationwithinZEMAX.(C)RGBingham2005.Allrightsreserved.Example:ZEMAX’sLensDataEditor:‘pickups’intheballlensFromEx00-Ball_Lens.zmxtoEx20-Ball_Lens.zmx‘Pickups’areatypeof‘Solve’.Pickupscalculateentriesinthelensdatafromotherdimensionsorfromtherays.Dimensionselinked.Thiscanbuildindimensionalsymmetries,etc.,canholdthemevenwhenthedimensionsarevaried,eitherinoptimisationormanually.Thenfewernumbersaretreatedasvariable,andtherequiredlensstructurefollowsthem.Pickupsareillustratedhereforaballlens.Theyforcetheentranceandexitareasoftheballtohaveequalbutoppositecurvaturesandtolieattheseparationrequiredforasphere.Thegivenexamplealsousesasolvetopositionthefinalfocus.Onlyonedimensionnowneedstobevariedwhenchangingthediameterofthesphere.WORDfileBall_Lens_Pickups.docliststheSolvesthatprovidetheseeffectsinEx00-Ball_Lens.zmx,leadingtoEx20-Ball_Lens.zmx.Trychangingtheballdiameterto,say,6mmbychangingtheradiusto3mminsteadof2.5onsurface2.Thenupdatethelayoutandaberrationplot.AlsoseeZEMAX’sautomaticfocallengthresult,etc.(C)RGBingham2005.Allrightsreserved.Paraxialray-tracingformulaeandwhatwemightdowiththemThesurfaceofathicklens,orathinlenswithpowerK(seenextslide)uu'hnn'd'h1ThenextopticalsurfaceTheraytracedisusuallythemarginal,oredge,rayoftheaxialraypencil.Itisarayarisingfromtheaxialfieldpoint.Itpassesthroughtheedgeoftheaperturestop.Anglesuandu'aretreatedassmallanglesbetweentherayandtheopticalaxis.Asdiscussed,u'isnegative.Therefractiveindicesbeforeandafterrefractionarenandn'.(1)Ray-tracing.Therayshownintersectsasurfaceof‘power’Kataheighthandthefollowingsurfaceatdistanced'atheighth1.

ThenfollowingWelford,n'u'-nu

=-hK…(1)h1

=h+d'u'…(2)Equation1givestheangleu'

oftheray,followingthesurface.Equation2transferstheraytointersectthenextopticalsurface.Anynumberofsurfacescanberay-traced.

(2)Seidelaberrationscouldbecalculatedfromtheseresultsoftheparaxialray-trace.(3)Moreinterestingly,startingfromrequiredrayheightsandangles,wecanworkbackwardstofindKforeachopticalsurface.Thismaywellbehelpfulasastartingpointfordesigningthelens.SowhatdoesKtellus?Seenextslide.

(C)RGBingham2005.Allrightsreserved.WaysofusingthepowerKasfoundfromtheparaxialformulaeForathinlensoffocallengthf:f=1/K.Theuseofthisformulawouldbetoselectafewthinlensesfromacatalogueorfromaspectacletrialcase(maybeallowingforimmersion)tosetupacrudeopticalsystemmatchingtheray-trace.Forathinlensbetweendifferentmedia(lessusual):f'

=n'/K(seeWelford).Forasingleopticalsurfacewithradiusofcurvaturer:r

=(n'–n)/K.Weusefullyfindrforasinglesurfaceasapreliminarytoray-tracing,dependingasitdoesonthedifference(n'–n)ofrefractiveindexacrossalenssurface.Note:Intheseparaxialformulae,refractiveindexandthicknesschangesignafteramirror,so(n'–n)=2atamirror.Forathinlensaddingopticalpathatheighth:=h2K/2.

Thisappliesbothtoordinarylensesandtothingradient-indexlenses.isthedistancebywhichtheopticalpathmustbeadvancedatheighthinordertogivetherequiredlenspowerK.hisoftentheradiusofthelensaperture.Note:alensofpositivepoweradvancesphaseatitsedgerelativetoitscentre.(C)RGBingham2005.Allrightsreserved.Noteonparaxialrays

Paraxialrayscanbetracedevenwhentherealrayswillcrash.Rayscrashduetooneoftheeffectsdiscussedinaprevioussession,suchastotalinternalreflectioninalens.Thiscanbemisleading,soitisbesttouseparaxialray-tracingwithinrealisticcontextsandinconjunctionwitharealisticsketch.Crashingrayscanalsobeavoidedbydoingthefirstrealray-tracewithareducedapertureorfieldsize,orperhapsacurvaturecanbechanged.Therequireddimensionscansometimesberestoredinstages.Aparaxialray-tracecanalsobeusedtotracechiefrays,whenlookingatquestionsofpupilimaging.Apupilexistswheretheheightofthechiefrayiszero.(C)RGBingham2005.Allrightsreserved.Example:fillinginlensdimensionsonasketchRayangles

u1u2u3u4u5=0Rayheightsh1=0h2h3h4h5Thicknesses

d1d2d3d4TryflatsurfacesK1,c1K2,c2Roughsketchoftelephotocollimator,f/6input,50mmbeamoutputWestartfromthisroughsketch.Fortherationaleofthesketchandhandlingit,seeParax_example.docandEx21-Parax_example.zmx.Mostofthesymbolsareinitiallyunknowns,butwecanassignvaluestothembyhandcalculation.Lenselements(C)RGBingham2005.Allrightsreserved.TheDefinitionofFocalLengthFocallengthisdefinedasf=-h/u',whereu'istheimage-sideparaxialrayangleoftheaxialraypencilwhentheobjectisatinfinity.Itdiffersfrom‘backfocaldistance’=b.f.d.Noteminussign.Ifthecollimatedinputbeamisdeviatedtoafieldangle,theimagepointmovesbyadistance'andf='/.Thisparaxial‘eter’conceptforfisequivalenttotheabovedefinition.SeeWelford.Forrealrays,aberrationsmatter.ForrealaxialraysatanangleU',f=-h/sinU'iftheimagehasnocoma.WemightusethattoestimateU'givenhandf.Conversely,iffdoesequal-h/sinU',thecomaiszero!ThatisknownastheAbbesineconditionforzerocoma.Focallengthisausefulconceptwhenthelens,etc.,canbefocusedtoinfinity.Adaptingthefirstexamplefromlecture1:hu'or

U''(C)RGBingham2005.Allrightsreserved.ParabasalRays.WhyisFocalLengthwelldefinedbyaparaxialformula?Aswehavediscussed,paraxialraysareapproximate.Forrealraysthathappentolieclosetotheopticalaxisofalens,theexpression“parabasal”raysisused.Paraxialraystendtoexactcoincidencewiththeserealparabasalraysashtendstozero.Thatiswhyfocallengthiswelldefinedbyaparaxialformula.Zemax,possiblyunlikemostothersoftware,tendstouseparabasalraysratherthanparaxialrays.Parabasalrayheights,oncescaleduptothelensaperture,arethesameasparaxialrays.ThusZemaxusesnumericalmethodsapplicabletoanyrayeventocomputequantitiessuchasfocallength,whichisdefinedparaxially.Itdoesthisratherthanusingparaxialformulaeinordertobroadentheapplicabilityoftheprogram’salgorithmsanduserfeatures.Its“paraxial”methodswouldalsobeapplicabletosomefurthercases,suchastounusualopticalprofilesforopticalsurfaces.(C)RGBingham2005.Allrightsreserved.Lagrange'sInvariantTheLagrangeinvariantHiscalculatedfromtherays.Thusitinvolvestheobjectorimageheightor',rayheightshinthelens,rayanglesandu'

andrefractiveindices.Forraysthatpassthroughthelens,Hhasacertainvalue,whethertheobjectandimagearerealorvirtual.ThevalueofHisthesameineachsuccessivespacefromobjecttoimage,whetherinairorglass.Theinvariancecannotbedefeatedinanysystem.Henablesaspecificationforalenstobecheckedforconsequencessuchasemergentrayanglesandindeedforphysicalfeasibility.ThevalueofH.nandn'aretherefractiveindicesintwoexamplespaces.Inanyspacewheretheraypenciliscollimatedinagivenopticalsystem,thevalueof-nhisthesame.Inanyspacewheretheraypencilisnotcollimated,thevalueofn'u''isthesame.H=-nh=n'u''hu''nn'Thinkofsomeexamples?(C)RGBingham2005.Allrightsreserved.Example:LenticularBeamExpanderEx22-Lens_Expander.zmxAcollimatedbeamof5mmdiameterisexpandedto8mm.ThisangularspreadfallsandsotheexampleillustratesLagrange’sInvariant.SeethenotesandreferencewithintheZEMAXfile.Thepupilpositionisnotshiftedbyinsertingthisparticularlens.ItsdesignaroseinastudyofShack-Hartmannwavefrontsensors.RealentrancepupilandvirtualexitpupilZEMAX’s‘paraxial’lenstotestcollimation(C)RGBingham2005.Allrightsreserved.Example:MirrorBeamReducer–confocalparaboloidsEx23-Mirror_Reducer.zmxusesconfocalconcaveparaboloidson-axistodeliveracollimatedbeamofreduceddiameter,withzerocoma.Alternatively,off-axispartsofsuchparaboloidscanbemade,andcanbecombinedasanunobstructedHerschelliantelescope.CommonfocusofbothmirrorsPrimarymirrorSecondarymirrorZEMAX’s‘paraxial’lenstotestcollimation(C)RGBingham2005.Allrightsreserved.PrismBeamExpander/TelescopeEx24-Prism_Expander.zmxAbeamcanbeexpandedwiththeuseofflatsurfacesonly.SeenoteswithintheZEMAXfile.(C)RGBingham2005.Allrightsreserved.PrincipalplanesP,P'andprincipalfociF,F'FocallengthfP'AnopticalsystemF'FocallengthFPThesameopticalsystemWelford’snotationParaxialfocus(C)RGBingham2005.Allrightsreserved.Newton’sconjugatedistanceequationandthemagnificationsO'OFF'z'(negativeinthisdiagram)zTheboxrepresentsanyopticalsystemoffocallengthf.Therefractiveindexistakenasthesameeachside(seeWelfordformoregenerality).O'isatanimageof

Oandviceversa;thusOand

O'are‘conjugates’or‘conjugatefocalpositions’or‘conjugatepoints’.Oand

O'andtheprincipalfociFandF'aresometimesimmersedwithinanopticalsystem.Newton’sconjugatedistanceequationstatesthatzz'=-f2.

Also,thelateralmagnification

misgivenbym=-f/z=z'/f.Longitudinalmagnificationism2.Thesignsofthemagnificationsrelatetoimageinversionandimageparity,asdiscussedinanotherlecture.f(C)RGBingham2005.Allrightsreserved.SpreadsheetsIwouldsaythatanEXCELspreadsheetforparaxialcalculationshasbeenrenderedunnecessarybysoftwaresuchasZEMAX.Thespreadsheetwouldneedtobeintuitivetolearn,andtoworkeitherforwardsorbackwardsbetweenthesurfacedimensionsandtheraypaths,startingfromanypointinthesystem.Itisnotuniversallyapplicable,becauseparaxialcalculationoftenleadstophysicallyimpossibledimensions,suchaswhenrayscrashorlensesoverlapattheiredges.C.G.Wynnecompiledatableforeachproblem.Ithadparaxialray-tracingdatainleft-handcolumnsandextendedfarenoughtotherighttofillintheSeidelaberrations.Hecompleteditbyhand,exploitingitsversatility.Itsusesincludedparaxialray-tracingandderivingcurvaturesandseparationsfromparaxialrays.However,WynnealsousedittocomputecurvaturesandseparationstochangetheSeidelaberrations,theaimsometimesbeingtobalancethefiniteaberrationsshownbytracingrealrays.Hesometimesseparatelysolvedtwoorthreesimultaneousequationsindoingthat,orusedthetableiteratively.Hewouldstartanoptimisationwithagoodsolutioninthatsense.Wynne’stablesandhisray-tracingprogramalsoshowedtheanglesofincidenceofmarginalraysandofchiefraysateachopticalsurface.Largeanglesofincidencecanbesourcesofaberrationproblemsonsomesurfaces,andsmallanglesofincidencecanleadinghostimagesfromstrayreflections.(C)RGBingham2005.Allrightsreserved.Matrixmethods

MatrixmethodsexistforparaxialraytracingandfortracingGaussianBeams.Asdiscussed,Iprefertousetheparaxialequationsseparately,orsolveswithinZEMAX.IfindthattheGaussianBeamfeaturesinZEMAXareconvenientandIdonotusethematrixmethodseparately.Occasionally,IuseoneoftheGaussianbeamformulaeexplicitlytofindanangleorbeamsizebyhand,ineitherroughnotesorshortreports.AlltheformulaeIneedforthatseemtobeintheZEMAXmanual.(C)RGBingham2005.Allrightsreserved.Atheoreticalfiniteray,threeexamplesofselectedsystemsandtipson“pastingin”5pages(C)RGBingham2005.Allrightsreserved.FiniteRayTracing–anoteonSnell’slawSnell’slawin3Drr'nnn'nandn'aretherefractiveindicesbeforeandafteranopticalsurfacenistheunitvectoralongthelocalsurfacenormalrandr'areunitvectorsalongtheraybeforeandafterrefractionSnell’slawstates:n'(r'

n)=n

(r

n)Theray-tracingprogramcomputesr'.Welford,Chapter4,explainshowthatishandled.Thenatureoftheequationforr'mayillustratewhylow-aberrationsolutionsforthedesignsofcompoundlensesarecommonlyfoundbynumericalmethodsinvolvingraytracing,ratherthanbyanytheoreticalmethodinvolvingrefractedraysandhencer'.

(C)RGBingham2005.Allrightsreserved.Solves:LandscapelensexampleEx25-Landscape.zmxTheexampleshowshowacriticalfeatureofthislenscanbedesignedusing‘solves’withinZEMAX.Thelensdesignissignificantinthatiteliminatessomeaberrationterms;itisaprecursortothedioptricanalogueoftheSchmidtcamera.(C)RGBingham2005.Allrightsreserved.Nosolves:fastasphericlensexampleSeeEx26-Fast_singlet.zmxGaussianlaserbeamThisexampleshowsamuchfasterlensforcomparison.Itismonochromaticandhasaverysmallfieldofview.(C)RGBingham2005.Allrightsreserved.Wemayhaveaworkingfileoflensdataintowhichwewanttoinsertorsubstitutesomesurfacesfromanotherfile.Itispossibletolosetrackofpickupsandsolvesifthesystemiscomplex.Whensurfacesareinsertedordeleted,theprogramcannotalwaysupdat