《Zemax光学与光学设计课件》由会员分享，可在线阅读，更多相关《Zemax光学与光学设计课件（132页珍藏版）》请在金锄头文库上搜索。
1、光学与光学设计RichardG.BinghamSession 1IntroductionCoordinatesRays that may appear oddImage rotation and parity and an image slicerSpherical Trigonometry advanced option(C)RGBingham2005.Allrightsreserved.ThebeginningSix pages of introductory notes(C)RGBingham2005.Allrightsreserved.AimsRay-tracingsoftwareis
2、amazinglyaccurateandcomprehensive,butpractisingwiththebuttonsonthetoolbarisnotenough.Ouraimsare:tounderstandopticalsystems;tounderstandwhattheprogramisdoing;togainafeelingforwhatproblemsaretractableandwhatopticalcapabilitiesareprovidedbyexistingdesigns;toassesstheperformanceofopticaldesignsconstruct
3、ively;toseehowcomplexopticsmaybecomprisedofsub-systemsexploitinglocalaberrationcorrectionorspecialcases;toexpressourideaswiththeprogram;tohandlethetoolsthatareavailablefordesigningnewsystems;toseewhatfactorsmaystopopticalaberrationsfrombeingmadesmaller;tobeawareofcostandweight;toavoidredundantcomple
4、xityinadesignoritsmanufacturingspecification;andtobepracticalandeconomicasregardstheopticsthatwedesignandtheuseofourtime.Wehopetoachieveallthisbyproperlyunderstandingthedesignersmethods,sothatwecanlearnandkeepuptodateinourownnewdesignsorothersthatwemayevaluate.(C)RGBingham2005.Allrightsreserved.Form
5、atofthistextThismaterial,althoughcreatedinPowerPoint,wasnotintendedfor,andisnotsuitablefor,projectionasslides.PowerPointwasusedtoencapsulatedsuccessiveideasandtopicsinaformatthatultimatelywillbeneatlyseparatescreens.Aslideisreferredtohereasa“page”.Thematerialisarrangedseparate“sessions”ofabout25page
6、seach.Afewsessionsmayincludeoneortwo.docfilesaswell.Asmentionedabove,thematerialcouldbeviewedassuccessivescreensbyanindividualuser,butinitially,itwillbeprinted(quitelarge),againtobereadindividually.Thesuggestedprocedureistofirsttoreadasessionindividuallyandtofollowthatbydiscussingitwithothersinagrou
7、p,orifpossible,toarrangesomekindoftutorialfollowingtheindividualreading.(C)RGBingham2005.Allrightsreserved.HowtohandlethisIthinkthatitisimpossiblefullytopractiseeachtopicbeforemovingon.Perhapsmaterialcanbeskimmedoverifitdoesnotseemhelpful.Orperhapstheconceptsandterminologyfromcomplexareasmaybeacquir
8、edinaneffortlessosmosis,later.Mostequationsaregivenherewithoutderivations;theaimsofthisparticularcoursearesuchthatweneedtopressontotheworkofthedesigner.Thefundamentals,however,maybeusefulforaphysicistorengineertounderstand.TheyarewrittenupbyWelfordin“AberrationsofOpticalSystems”,Hilger1986.Itmaybewo
9、rthconsideringwhethertoworksystematicallythroughthewholebook,asindeedIdidwiththepreviousedition,foracomprehensivefoundation.IrefertothebookasWelfordthroughout.ExamplesaregiveninZemax.Ifboththe.zmxandthe.sesfilesareinthesamefolder,graphicswindowsshouldopenwhentheLensDataEditoropens.Alsoexploreanyofth
10、eotherZemaxfeatures,theirlocalhelpfilesandthemainmanual(.pdf).WithintheZEMAXexamples,thereisawindowcalledTitle/NotesthatappearsonatabaftertheGenbuttonispressed.Asyouviewoneoftheexamples,itisvitaltoreadthosenotesthatarewithintheZemaxdata,asIhaveoftenputcriticalinformationthere.Irecommendthatyoudonotp
11、ersonalisethedefaulttoolbarbuttonsinyourowncopyofZemax.Ifyoudo,youmayfindthatadifferentcomputerisslowertouse,anditwillbecomemarkedlymoredifficulttodemonstratethingstootherusers.(C)RGBingham2005.Allrightsreserved.HowcanIgettopractiseonsomerealdesigns?Ifyouarealreadyworkingwithanyoptics,awaytopractice
12、onrealopticswithoutspendingmoneyistocreatearay-tracingmodelofyourexperimentassoonasthereisevenaminormodificationtoexplore.Soforexample,ifsomeexistinginstrumentneedsmerelytobedifferentlyfocusedortohaveathickercolourfilter,orsomeas-madedatabecomesavailable,usethatasareasontosetupaZEMAXlensdatafile,how
13、eversimple,andthenuse your ZEMAX file to check the modification. OnceyouhaveafewZEMAXmodels,youwillalsobebetterplacedtosolveanyfurtherproblemsarisingwithyourexperiment,andwillbeabletocreaterealisticgraphicsforpublication.However,Iwouldsaythatwhilstsetting up and usingZEMAXyourselfforevensimplepurpos
14、esisveryinstructive,collectingcomplexdatafilesfromotherpeoplewouldbeawasteoftime.(C)RGBingham2005.Allrightsreserved.ZEMAXZEMAXwascreatedbyKenMoorefrom1990todate.ItisforPCsonly.WeshalluseZemaxinthiscourse.Thehardkeyallowsonlyonepersontoaccessitatonce.Itisworththewholevalueofthesoftware.YoucanopenZema
15、xtwiceonthesamemachine(exceptwithRemoteDesktop).ThesecondinstanceofZemaxisusefulforrunninglongcomputationswhilstgettingonwithsomethingelse,orforcuttingandpastingbetweensystems.Youneedindividual,unrestricted,continuousaccesstotheprogram,alongwithtechnicalsupportandupdates,todomuchproductiveworkinopti
16、caldesign,ifthatisyourtask.Sharingasystemisfrustratingandhindersthedesignofreallenses.(C)RGBingham2005.Allrightsreserved.ExamplesettingupaballlensTheexampleisaglassball,asavailablefromMellesGriot.Itisusedasalens(e.g.forfibre-opticswork).ItcanbesetupbyfollowingthedetailsinBall_lens_data.doc.Thisisthe
17、mainexamplewithsuchinstructionsforcreatingitfromabsolutescratch,whichwillprovideausefulexerciseifyouhavenotpreviouslysetuplensesinZEMAX.Inanycase,pleasefindthefurthernotesthatIwrotewithinthat.zmxfile.TheycanbefoundontheGeneral/Title/Notestab(ontheGenbutton).SuchfurthernotesareheldwithinallmyZEMAXexa
18、mplesintheselectures,soitwillbeusefultobeabletofindthem.Tocheckyourresults,theintendeddataisinfile:Ex00-Ball_Lens.zmx.IftheEx00-Ball_Lens.SESfileisalsopresent,itwillbringuprelevantgraphicsasbelow.(C)RGBingham2005.Allrightsreserved.Coordinatesystems13 pages(C)RGBingham2005.Allrightsreserved.+zInZemax
19、,lightmustleavetheobjectsurfaceinthe+z direction.Thethicknessoftheobjectsurfacemustbepositive.x, y, z+y+z+x inwardsRight-handedaxesPositivesagzPositiveradiusofcurvatureSurfaceofalens+yNegativesag,negativeradiusofcurvature(C)RGBingham2005.Allrightsreserved.Lens.Sequentialraytracing1.Thethicknessoraxi
20、althicknessismeasuredpositiveinthe+zdirection2.ThecurvatureofthefrontsurfaceispositivehereExamplefile:Ex01-Lens.ZMX+z+y5.Thisisanopticalsystem4.Lensdrawingsinmostdiagramsarecross-sectionsthatarenotshaded3.Thefrontofalensisthefacethatthelighthitsfirst(C)RGBingham2005.Allrightsreserved.Localcoordinate
21、s.AsurfaceS1hasanorigin(O1here)thatservestolocatethissurfacewithintheopticalsystem.ThefigureofsurfaceS1isdefinedbyitssagz(x,y) thatisthusmeasuredorthogonaltothex,y plane. Eachsurfacealsohasafollowingthickness.Forsequentialraytracing,theoriginO2 ofthenextopticalsurfaceS2ispositionedbythisthicknessofS
22、1alongthez axis.Eachsurfacecanbepositionedinthiswaywithreferencetotheprevioussurface.Definingsurfacesforraytracing+y+z+x isinwardsz(x,y)Some weird lensO1O2S1S2thickness(C)RGBingham2005.Allrightsreserved.ThicknessafteramirrorThethicknessofglassoranairspaceenteredintoZEMAXsLensDataEditorchangessignata
23、mirror.(C)RGBingham2005.Allrightsreserved.Sequentialraytracing-aManginmirrorThisonepieceofglasshastoappearinthedatatwiceforsequentialraytracing.Ithaspositivethicknessbeforethemirrorbutnegativethicknessafterthemirror;inthesecondpass,thethicknessismeasuredintheminuszdirectionfromtheprecedingsurface,th
24、emirror.+y+zExamplefile:Ex02-Mangin.ZMX1234Thethreeraysinthis2Dlayoutaredrawnoptionallyfromaflatsurfacenumbered1thatisinthedatafile.Thissurface1isadummysurfacehere.Adummysurfaceisonethathasthesamerefractiveindexonbothsides.ThesignsofthesecurvaturesareallnegativeGlassA back-surface MirrorThethreerays
25、startedfromanobjectatinfinityatsurfacenumber0.5(C)RGBingham2005.Allrightsreserved.ThemanufacturingspecificationisdifferentrzzGlassremovalmightcorrespondtopositivez intheray-tracingdatabutnegativez onamachine.Lens on machineLens in the ray-tracing dataWronglymadeconvexbecauseapositivesignwasquoted?Ac
26、urvaturewithapositivesignisnotnecessarilyconvexinZemax!Thereisanecessaryintermediatestagewherewesupplyanengineeringdrawingoratleastasketch.Zemaxwilloutputagoodstartingpointforaproperdrawing.LensAsphericlens(C)RGBingham2005.Allrightsreserved.LocalTiltsandDecentresTheprogramprovidesforsurfacesthataret
27、iltedormoved(decentred).Thelocalframeofreferencecanbetiltedormoved,leavingthefunctionz(x,y) unchanged.+z+x isinwardsO2+yForexample,theweirdlenscanhavearight-handsurfacethatistiltedarounditslocalxaxis.Thetiltshownispositivearoundthelocal+xaxis.Combinationsoftwoorthreetiltscanbeused.TheoriginO2canalso
28、belocallyshiftedinx andy, butnotz wehavealreadychosenaz position, basedontheprevioussurface.(C)RGBingham2005.Allrightsreserved.Rotationsthatareconsistentwiththeabove:1.Inthex, yArganddiagramapositiveangleisdrawnanticlockwise,butitisclockwiselookingalonga+zaxisemergingfromthepaper.AsignlikethatoftheA
29、rganddiagramisalsousedfortheangleofarayinsomeopticaldiagrams,andforposition angleonthesky,e.g.fortheplaneofskypolarisationmeasuredanticlockwise(fromnorth).2.Internationalstandardsformachinetoolssuchasmillingmachinesalsousethissign.x, y, z rotationssigns+y+zWerotatetheaxesinwhichsubsequentsurfacesare
30、thendefined,sothesubsequentsurfacesmovewiththerotation.ApositiverotationinZEMAXisclockwiseasviewedinapositiveaxisdirection.Apositiverotationaroundthex axis.TherelevantmatrixexpressionisintheZEMAXmanual.(C)RGBingham2005.Allrightsreserved.x, y, z rotationsthataredifferent1.OPTICA(Mathematica)istheoppo
31、site.Itappliesapositiverotationlookingtowardstheorigin,thatis,inthenegativeaxisdirection.Thedifferencearisesfromrotatinganobjecttheotherwayinglobalaxes.TheglobalaxesinZEMAXcanalsobeunaffected,andoftenoneneednotbeawareofthemanyway.Checkhowtheyaredefined.InZEMAX,withacoordinatebreak,freshlocalaxesareo
32、btainedforfollowingpartsofthesystem.2.IntheCodeVopticaldesigncode(current),arotationaroundeitherthelocalxorthelocalyaxisisnegativeascomparedtotheright-handconventionusedinZEMAX,whilstrotationaroundzisnotaffected.3.IntheGRTopticaldesigncode(obsolescentuniversityFORTRAN),thexrotationisreversedasinCode
33、V.However,theyrotationisnotreversed.ThezrotationappearedtobereversedwhenIrotatedthelinesonadiffractiongrating.4.Othercodesmaybedifferentagain.5.Doubtscanberesolvedbyinspecting3Dgraphicsthattheprogramwilldraw!(C)RGBingham2005.Allrightsreserved.SequentialraytracingaprismTheLensDataEditorcontainsaseque
34、nceofopticalsurfaces.Thepathofanyrayiscalculatedtointerceptthenextopticalsurface,intheorderinwhichtheyarelistedintheLensDataEditor.SeenextslideforanexampleinZEMAX.(C)RGBingham2005.Allrightsreserved.-35+40Coordinatebreaks-aprism1.Theprismanglesaredefinedbycoordinatebreaks.Acoordinatebreakisplacedatad
35、ummyopticalsurface,takinguponelineofdataintheLensDataEditor.Thatlinehasboxesforx&yshiftsandforx, y & ztiltsindegrees.Otherboxesonthesamelineholdthethicknesstothenextsurfaceandaflagforreversingtheorderofthetilts.Axes-252.Thecoordinatebreaksare:2thetiltofthefrontsurface;4thetiltofthezaxisleadingtother
36、equiredcentreoftheexitsurface;5thetiltoftheexitsurface;and7thetiltofthezaxisleadingtotherequiredcentreoftheimagesurface.+60Examplefile:Ex03-Seq_Prism.ZMX3.TheraysrefractpassivelyaccordingtoSnellslaw.Theraysdonotaffectthetiltsoftheaxesinthesedata.RaysThickness 42457(C)RGBingham2005.Allrightsreserved.
37、Coordinatebreaksagain1.Ifweplaceacoordinatebreakimmediatelybeforeanopticalsurface,itissometimesusefultoputinanotheronewiththeoppositesignimmediatelyafterthatsurface,undoingthetiltetc.Thenwecancontinuewiththepreviousaxisdirection.2.Reversingmultipletiltanglesneedsbedoneinthereversesequence.Readtheman
38、ual!(C)RGBingham2005.Allrightsreserved.Tiltsandshiftsagain1.SEQUENTIALRAYTRACINGuseslocalcoordinates:CoordinatebreaksareinsertedintheLens Data Editoraheadofopticalsurfacesthatneedmoving.Thevaluesoftheanglesandshiftscanbevariedinoptimisation.Againforsequentialraytracing,individualopticalsurfacescanbe
39、givenanintrinsictiltorshiftonaSurfacePropertiestab.Theeffectisthesameaswithwithcoordinatebreaksbutthedataarenotcurrentlyoptimisable.2.ForNON-SEQUENTIALRAYTRACING,awholeobjectcanhaveitsowntiltandpositionsetupinalineofdataforthatobjectwithintheNon-Sequential Components Editor.Thisisessentiallyaglobalr
40、eferencesystem,althoughalternatively,onecomponentcanbereferredtoanother.(C)RGBingham2005.Allrightsreserved.RaysthatmayappearoddTwo pages about geometrical rays in a ray-tracing program(C)RGBingham2005.Allrightsreserved.Ageometricalrayisalinetothenextsurface,butAEFADBAAisanormalraypathInsequentialray
41、tracing,rayBisavalidextrapolationtoavirtualimageinanegativespace.Ithastoendtheraytraceortobereversed.But in a sequential ray trace, rays can fail or“crash”:RayCmissesadefinedsurfaceapertureRayD,ifreflectedbyTIR,terminatesunlessthesurfaceisdefinedasamirrorRayEmissesthespherealtogetherRayFmayintersect
42、thesphereatthewrongpoint.CLens(C)RGBingham2005.Allrightsreserved.Non-sequentialraytracingaprismExample:Ex04-NSC_Prism.ZMXThisRectangularVolumeobjectisdefinedonasinglelineoftheNon-SequentialComponentsEditor.Thefacesofthisobjectwouldformacuboidbydefault,buttheycanbeangledastheyarehere,andthewholething
43、istilted30aroundx.Raysshownhereintercept0,2or3surfaces.TherearemanytypesofNSCobjects.Theycanbepositionedwithrespecttoasingleorigin,orindividuallywithrespecttootherdefinedobjects.NSCrayscansplit,scatter,etc.,butdontcrash.Twosortsofraysappearthatgothroughthisprismbutarenotdispersedinangle.(C)RGBingham
44、2005.Allrightsreserved.ImageRotationandParityandanimageslicerTen Pages(C)RGBingham2005.Allrightsreserved.Invertingtheimagemeansrotatingit180degreesaroundtheaxis.NoticethatZEMAXviewstheimageinthe+zdirection.Ifthereisanintermediaterealimage,thefinalimageiserect.Thesameappliestomirrorsastolenses.ThusaC
45、assegraintelescopegivesaninvertedimagebutaGregoriantelescope,whichhasoneintermediateimage,givesanerectfinalimage.InvertedrealimageinanaxiallysymmetricalsystemFFz(C)RGBingham2005.Allrightsreserved.ImageParityReversalofparityisathree-dimensionaleffect.Ifthetotalnumberofmirrorsisodd,theimageparityisrev
46、ersed.Withanoddnumberofmirrorsinanaxiallysymmetricalsystem,thethree-dimensionalimagemustbereversedindepth,becauseneitherlateraldirectionisspecial.Thusina3-mirrorcamera,iftheobjectmovesoneway,theimagemovesintheoppositedirectiontotheobject,justasinasingleplanemirror.Usingtheseideas,inaCassegraintelesc
47、ope,whichwaydoesthefocusmovebetweeninfinityandalaserguidestar?(C)RGBingham2005.Allrightsreserved.ImagerotatorsFieldofviewWehaveafewobjectsaroundafieldofviewthatisfixedinspace.Image1TheimagerotatorflipstheirimagesacrossadiameterA.Thatiswhatitdoes.AImage2Theniftheflippingdiameterrotatesbyanangletoline
48、B,thefieldofviewrotatesby2ascomparedwithImage1.AB(C)RGBingham2005.Allrightsreserved.Three-mirrorimagerotator(K-mirror)+Directionsofthezaxisandthesignsofthethicknesses.Theunusual,reversedfinalthicknessaffectsthesignofthefinalzrotationinthedatafile.Ex08-Rotator.ZMXAnimatewithTools,Slider(C)RGBingham20
49、05.Allrightsreserved.TwootherdeviceswithflatmirrorsThreepages(C)RGBingham2005.Allrightsreserved.DihedralmirrorsEx06-Dihedral.ZMXThedirectionoftheemergentraysisunchangedwhentiltingapairofplanemirrorstogether.Furthermore,ifthetiltisaroundthelineofintersectionofthemirrors,thedirections,positionsandtota
50、lpathlengthsoftheemergentraysareallunchanged.Soforexample,mountingapairofflatmirrorsonthesamebasecanprovidestability.TheexampleusesbothcoordinatebreaksandatiltonaSurfacePropertiestab.(C)RGBingham2005.Allrightsreserved.Imageslicer/assemblerSideviewViewintoemergingbeamEnlargedViewintobeamTheslitformed
51、ThismodifiedWalraven-typeimageslicercutsapatchoflightinto5slicestofittheslitofaspectrometer.Ex07-5-Slicer.ZMXTwomirrorssimulatingTIR(C)RGBingham2005.Allrightsreserved.Imageslicers(illustration)Silicabaseplate55mmdiameterThebHROS5-slicers.(C)RGBingham2005.Allrightsreserved.SphericalTrigonometry-Advan
52、cedoptionNote.Ijoinedaprojectthathadbeenrunningsixmonths,onlytofindthatnoonehadbeenusingthecorrect3DgeometryinZEMAX.Asaresult,nothingwasquiterightandnocorrect3Dlayoutscouldbedrawn.SphericalTrigonometrymayseemaforbiddingsubjectareatostepinto,butpersonally,Idontexpecttosolvetheseproblemsinanhouroramor
53、ning.EvenifittakesafewdaystopuzzleouttheSphericalTrigonometry,itmaybewellworththeeffort.Tenpages.(C)RGBingham2005.Allrightsreserved.GreatandsmallcirclesandsphericaltrianglesSeveralgreatcirclesOnesmallcircleSphericalTriangles.Theiranglesaddtomorethan180degrees.Thesidesofthetrianglesaregreatcircles.Th
54、eyarealsoangles.Noticethesphericaltrianglesinwhichatleastoneoftheanglesisarightangle.ThesetrianglesareoftenneededandcanbesolvedbyNapiers rules.(C)RGBingham2005.Allrightsreserved.Sketchingforcalculationsin3DNorth PoleSouth PoleStereographicprojectionontotheequatorialplanefromtheSouthPole.PP as projec
55、tedMappingasphereontoaplaneSideviewthroughasphere,showingaplaneanditspolePNorth Pole90-9090-(C)RGBingham2005.Allrightsreserved.GoodNewsabouttheStereographicProjection1.Weneverneedtocalculatetheactualprojection.Itservesasawayofsketchingdiagramsof3Dobjects.2.Anglesmeasuredlocallyonthesurfaceofthespher
56、earethesameangleintheprojection.3.Acircledrawnonthesphereappearsasacircleintheprojection.(C)RGBingham2005.Allrightsreserved.Napiersrulesforaright-angledtriangle(C=90)(C)90-c90-B90-A baC=90bAcBaArrangethesymbolsinfive“parts”asshownontheright.Thensin(middlepart)=prodtanadjacentparts=prodcosoppositepar
57、tse.g.sin(90-c)=cosacosb=tan(90-A)tan(90-B)Thisistheonethatweshallusefortheexample(C)RGBingham2005.Allrightsreserved.Napiersrulesforaquadrantaltriangle(c=90)CbAc=90Ba(c)C-9090-b90-aBAsin(middlepart)=prodtanadjacentparts=prodcosoppositeparts(C)RGBingham2005.Allrightsreserved.Otherusefulformulaeforari
58、ght-angledsphericaltriangle:sina/sinA=sinb/sinB=sinc/sinCcosa =cosb cosc +sinbsinccosAcosA =cosBcosC +sinBsinCcosaThereareothersbutIhaveneverneededthemforoptics.(C)RGBingham2005.Allrightsreserved.3Dexample-PositioningaRetroreflectorUse NSC object Triangular Corner but we want the hollow side to face
59、 the light.(C)RGBingham2005.Allrightsreserved.z 3Dexample-PositioningaRetroreflectorYXZTurning the Triangular Corner so that the hollow side faces the light. The poles of all the surfaces need to be equidistant from Z.1. Plot the poles of the three starting surfaces at the black dots x, y and z. 2.
60、First, rotate the device through a negative angle around X. The Y pole goes to Y. Z goes to z. Draw the equator of Y through z.Y 3. Second, rotate the device 135 degrees around Y. The -z pole shifts to Z. The -X pole shifts to X. Z X4. The poles of the surfaces are now X, Y and Z. They can be all at
61、 an angle 90- from the original Z axis.(90- )5. Now solve for in the triangle X Z z.(90- )45(C)RGBingham2005.Allrightsreserved.Solvingtheright-angledtrianglez ZX90-45a ( )c(90- )b (45)CABOneofNapiers rulesforaright-angledtrianglesays:cos c = cos a cosb (ifC =90)Sointhiscase:sin = cos cos45 , therefo
62、re:tan = 1/2 = 35.2644degrees90- = 54.7356degreesRe-drawit:(C)RGBingham2005.Allrightsreserved.ResultofpositioningtheretroreflectorusingtheTriangularCornerEx05-RR1.ZMXThis example is developed for the advanced topic on Spherical Trigonometry.(C)RGBingham2005.Allrightsreserved.OpticsandOpticalDesignSe
63、ssion2Session 2 Spheres AspheresAberrationsDefocusSpherical aberration Coma ExamplesRichardG.Bingham(C)RGBingham2005.Allrightsreserved.11pagesonspheresandaspheres(C)RGBingham2005.Allrightsreserved.EquationofasphereasusedforraytracingZzcisthecurvature.Itisthereciprocaloftheradiusofcurvature.Thisrecip
64、rocalisconvenientinexpressionsthatavoidnumericalproblemsataflatopticalsurfaceorwhenraysarenormaltoasurface.z(x,y)YThespherepassingthroughtheoriginhastobeexpressedasz = f(r,c) wherer2=x2+y2r1/c(C)RGBingham2005.Allrightsreserved.1+(1c2r2)Equationofasphere(2)Zzrc r2z (r) =1/cThisexactequationisthatused
65、forraytracing.Ithasasimpleextensionforconicsections.Itisoftenalsousefulforcalculatingthetotaldepthofthesurfaceofasphericallensormirror.(C)RGBingham2005.Allrightsreserved.Equationofasphere(3):Taylorseriesz (r)=cr2/2+c3r4/8+c5r6/16zrTheosculatingparabolaDifferenceofthesphereandtheparabolatoorderr4.Inc
66、identally,thisleadstotheSchmidtcorrectorplate.Differenceofthesphereandtheparabolatoorderr6.ThisisnotusefulfortheSchmidtplate.Parabola-paraboloidofrevolution(C)RGBingham2005.Allrightsreserved.ConicsectionsAreflectingtelescopewithaparabolicmirrorEx11-Paraboloid.zmxTherefractingsurfacediscoveredbyDesca
67、rtesin1637Ex12-Ellipsoid_lens.zmxAsphericsurfacesareantiquebutwestillstruggletomakethem!(C)RGBingham2005.Allrightsreserved.TheStandardsurfaceandtheconicconstant1+(1(1+k)c2r2)c r2z (r) =zrConicConstantkShapeofthesurfaceEccentricitye ofaplanecurve0oblateellipsoid(Arotatedellipse)0sphere0-1k0prolateell
68、ipsoid0e 1-1paraboloid11eisusefulforfindingtheconjugatesEx10-Conics.zmx(C)RGBingham2005.Allrightsreserved.Osculatingsphereandparaboloidz (sphere)= z (paraboloid)=cr2/2rThedifferencebetweenthesphereandtheparaboloidisr4.Thereisnodifferenceterminr2.Nearthecentre,thedifferencebetweensphereandparaboloidt
69、endstozero.cr 2/2isusefulforaquickestimateofthedepthofacurveevenasphere.zcr2/2+c3r4/81/c(C)RGBingham2005.Allrightsreserved.AsphereandalltheosculatingconicoidssphereAlltheprolateellipsoids(rugbyballs)paraboloidAllthehyperboloidsTheseconicoidsofrevolutionfitthespherenearthecentre.Theyhavethesamevertex
70、curvature.Thedifferencesareallr4toafirstapproximation.Thereisnodifferencer2.Mirrorofalargetelescope?Alltheoblateellipsoids(Smarties)rz(C)RGBingham2005.Allrightsreserved.ConjugatefociofellipseandhyperbolaaaEccentricity e=(-k)=(-8 a4/c3)VFFVFFForanellipse:a=1/c (1-e2)VF=a (1-e)VF=a (1+e)Foranhyperbola
71、:a=1/c (e2-1)VF=a (1-e)VF=-a (1+e)VisthevertexForsettinguparequiredcurveorfordesigninganopticaltest(C)RGBingham2005.Allrightsreserved.+a2r2+a4r4+a6r6+a8r81+(1(1+k)c2r2)c r2z (r) =TheEvenAspheresurface(1)zra2hasdimensionsL-1,a4hasdimensionsL-3etc.Affectsscaling.Odd-powertermsarenotnecessaryinasymetri
72、calopticalsurfaceexpressedwithsuchapolynomial,becausetheyandtheirdifferentialswouldnotbesymmetricalorwouldhaveacuspattheoriginifwrittenwithyand-y.Thesameappliesbothtoanopticalsurfaceandtoaxialaberrationsinasymmetricallens,soalthoughwemightdescribeanopticalsurfacewithoddterms,thosetermsarenotoftenhel
73、pful.(C)RGBingham2005.Allrightsreserved.1.Vertex curvature. a20impliesavertexcurvature2a2evenifc=0.Ihaveneverusednon-zeroa2withnon-zeroc(althoughpeopledo).2.Exact paraboloids. Eitherk = 1withc0,ora20.Tousebothwouldberedundantatbest.kismorevisiblethana2inthedata.3.Conic sections.Eitherexactlywithk0,o
74、rasfarasther4termusinga4.z(conic)z(sphere)=,sothereisanapproximationthata4=.TheEvenAsphere(2)+a2r2+a4r4+a6r61+(1(1+k)c2r2)c r2z (r) =8c3kr 4+Therearetwowaysofprescribing:8c3kWhy use that? Is c the same?(C)RGBingham2005.Allrightsreserved.Isitbettertousepolynomialsorconicsections?Theconicsectionisspec
75、ialforitsgeometricalfoci.Ifthereisnoreasontoconsiderthegeometricalfoci,theformsavailablewiththeconicsectionmaybetoolimited.Theconicsectioncannotdescribeasurfacelikethis:Intheconicsection,theasphericitydependsonthevertexcurvature.Awkwardwhenflat!Theconicsectionisexcellentforopticaltesting.(C)RGBingha
76、m2005.Allrightsreserved.Sixslidesonwavefrontaberrationanddefocus(C)RGBingham2005.Allrightsreserved.WavefrontaberrationOPDplotwhatitisTheOPDmapsandsectionsarewellexplainedintheZEMAXmanual.CalculatingOPDinvolvesray-tracingthatisintheprogrambutwhichisprobablyunknowntotheuser,suchastracingbacktotheexitp
77、upil(seeWelford).Anothergeneralmethodisgiveninoneofthepagesondefocusinthiscourse.(Idonotknowwhythatmethodislessused.)Thereisawarninginsession4regardingusingthetwoOPDcross-sectionsinasymetricalsystems.PositiveOPDisadvanced.(C)RGBingham2005.Allrightsreserved.WavefrontWavefrontaberration-movingthefocus
78、z Wz axisuSupposewemovetheCCDtotherightbyz.ThatintroducesaphaseerrorW intheconvergingsphericalwavefront.Wisawavefrontaberration.Itisadistance.Itispositiveherebecausethephaseisnowtooadvanced.HowbigisW,andwhatshapeisitinxandy?Itisthedifferencebetweentwospheres.CCDThefocus(C)RGBingham2005.Allrightsrese
79、rved. zuCCDWduetodefocusonlyThefocusTheaberrationistakenaszeroforthechiefray.Thepathlengthsoftheraysareallequaltoeachotherasfarasthefocus.Thentheaxialraytravelsafurtherdistancez,butthewavefrontcorrespondingtotheperipheralraytravelsonlyz cos u z (1 u2/2).Sothesectionofthewavefrontcorrespondingtothepe
80、ripheralrayarrivesearlier:itsphaseisadvancedbyz u2/2. W=z u2/2orz =2W/u2Whenthesignalisdetected,thewavefrontshavearrivedhere.Huygenswavelets(Usefultoknow)(C)RGBingham2005.Allrightsreserved.(cosineterm)Theshapeofthedefocusedwavefront(c2 c1)r2/2z (r)=cr2/2+ Wrzc1c2Thecosinetermaffectsonlyr4andabove. W
81、Thewavefrontaberrationthatcorrespondstoashiftoffocusshifttakestheformofaparabolalocally,because:SeeEx09-Defoc_Lens.zmxEquationofasphereseelater(C)RGBingham2005.Allrightsreserved.Theshapeofthedefocusedwavefront,andtheaberrationfansOut-of-focusimage(C)RGBingham2005.Allrightsreserved.Exercise.Depthoffo
82、cuswithrays.Effectwithlongfocallength,e.g,withatelephotolens.Thisessentiallyusesaperfectcamerathatisnotfocusedtoinfinity.UseNewtonsConjugateDistanceEquation(seeWelford).Assumealenshasfocallengthf, F-numberN(f/diameter)andsetalimitptotheimagediameter.Approximatingtan=,findthedepthoffocus(linearrangeo
83、facceptableimagediameter)ataseriesofobjectdistancesandnotice its dependence onf.Nowdefineawavelengthasifpistwicethediffraction-limiteddiameter.SeewhetherW/ isneartheRayleighdiffractionlimit for the imagediameterp.(C)RGBingham2005.Allrightsreserved.Fourslidesonsphericalaberrationandcoma.(C)RGBingham2
84、005.Allrightsreserved.BalancingsphericalaberrationagainstfocusThesetwoaberrationsarenotdescribedbyorthogonalbasisfunctions.Fortunately.SphericalaberrationDefocusBalancedMany forms of aberration can and must be balanced against others. A difference from the paraxial focus is commonplace.(C)RGBingham2
85、005.Allrightsreserved.Aberrationsare:thedepartures(a)ofwavefrontsfromspherescentredonsomerequiredimagepoint,and(b)ofraysfromcrossingthefocalsurfaceattherequiredpoint.Aberrationsatasingleopticalsurfacecanhaveeithersign.Wetrytocancelthemoutamongstthedifferentsurfacesinafinisheddesign.DefocusdonethatSp
86、hericalaberrationEx13-Sphere.zmx(dropthecentralobscurations)ComaEx11-Paraboloid.zmx(C)RGBingham2005.Allrightsreserved.SphericalaberrationComa W 4 0 W y 3 1OPD=Wavefrontaberrationory=relativeradiusinaperture=distancefromtheaxialfieldpointCross-sectionsofwavefrontInthisgroupofslides,anyminussignsareta
87、kenupintheconstantofproportionalityOnaxisHalfwayoutEdgeofthefieldofviewyx(C)RGBingham2005.Allrightsreserved.Sphericalaberrationandcomawavefronts W 4 0 W 3 1cosWavefrontaberration=relativeradiusinaperture=distancefromcentreoffieldEdgeoffieldMeantiltremovedMoreonaberrationplotsinsession4.(C)RGBingham2
88、005.Allrightsreserved.Eightslidesofexamples(C)RGBingham2005.Allrightsreserved.ACassegraintelescopeThe4.2-metreWHT(WilliamHerschelTelescope)Ex14-WHTelescope.zmxOPD+/-10wavesTransverserays+/-500microns(C)RGBingham2005.Allrightsreserved.ARitchey-ChrtientelescopeThe3.9-metreAAT(Anglo-AustralianTelescope
89、)Ex15-AATelescope.zmxOPD+/-10wavesTransverserays+/-500microns(Accordingtotheorythemirrorsarenotanalyticallyparaboloids.)(C)RGBingham2005.Allrightsreserved.Atelescopeas-madeWiththeas-madedimensionsofthemirrors,thefocusisstillformedatthespecified2500mmbehindthevertexoftheprimarymirror.Refocusingbymovi
90、ngM2wouldhaveintroducedsphericalaberration.Mosttelescopeshaveanoticeableerrorhere(butnotthistelescope).IntheproductionoftheWHT,thespecificationofM2wasre-computedaftertheprimarymirrorwasfinishedtokeepthepositionoftheCassegrainfocuscorrect(zerosphericalaberration).Theas-madef-numberis10.95,not11.Theas
91、-madeaperturediameteris4180mm,not4200.Do these last two points matter?Example:the4.2-metreWHT(WilliamHerschelTelescope)Ex14-WHTelescope.zmx(C)RGBingham2005.Allrightsreserved.Off-axisparaboloid(cameraorcollimator)Intheray-tracingdata,wepositionthevertexofthemirrorratherthanthepointwheretheaxialrayhit
92、sit.Also,thenormalfocallengthofthemirrorwilldifferfromtherequiredfocallengthoftheoff-axissystem.This is typical of many geometrical problems in optics. Tosetuptheraytrace,wecaneither:(1)Dothealgebra,or(2)Setupanapproximation,computevariousdimensionswithintheprogram,thenrefinethemtoarbitrarilyhighpre
93、cisionastargetedvariablesinnumericaloptimisation(laterlecture).Foranexactparabola,thealgebramayseempreferable.Therearestillafewstepstosetupintheray-tracingdata.See:OFF_AX_P.docEx16-Trivial_parab.zmxEx17-bHROS_parab.zmxandinthenextslide,adescriptionformanufacturing.(C)RGBingham2005.Allrightsreserved.
94、Off-axisparaboloidanactualspecification(C)RGBingham2005.Allrightsreserved.AGoodPairofOff-AxisParaboloidsEx18-Good_parabs.zmx“OSCA”323mm(C)RGBingham2005.Allrightsreserved.ABadPairofOff-AxisParaboloidsArelativelymassiveaberration,butwhyarewestillcallingitcoma?Doesitmatter?Ex19-Bad_parabs.zmx(C)RGBingh
95、am2005.Allrightsreserved.WhythegoodparaboloidsworkEnd-to-endsymmetryofanopticalsystemresultsinzeroaberrationforthosetermsthatriseasanoddpowerofthefieldangle.Suchtermsarecoma( 1)anddistortion( 3).(C)RGBingham2005.Allrightsreserved.OpticsandOpticalDesignSession3R.G.BinghamSession 3Paraxial ray tracing
96、 and related issuesA theoretical finite ray, three selected systems and tips on “pasting in” (C)RGBingham2005.Allrightsreserved.Stagesindoingthejob WhWavefrontPropagationdirectionW = a2.h2 + a4.h4 + a6.h6 + a8.h8 + ?Thereisnouniversaltheorythatleadstodesignsforthecompoundlensesandotheropticalsystems
97、withthelowestaberrations.Thereisnotheorythattellsuswheretostart.Thereareusuallymultiplefeasiblesolutions,andwemayhave,say,twoorthreeworkingdaystomakesomeprogress.Wecannotavoidthinking.Weneedknowledgeablytopieceopticstogethertogivetherequiredfunctions,tobeawareofdifferentpossiblesolutions,andtoworkto
98、refinethesebasicideas.Amongstmanyconceivablenumericalapproaches,twolevelsofapproximationenableus(a)tosetupschematicinitialconceptsthatcanberay-tracedbeforecorrectingaberrations,and(b)possiblytoelaborateontheseideasandinanycasetocheckthattheoptimiser(thecomputingalgorithmthatcanreducetherealaberratio
99、ns)isworkingcorrectly.Atypicalwavefrontprofileinalensshowsaberrations;itwillnotconvergetoapoint.Itsproblemsareconceptuallycomplex,eveninalensthatisaxiallysymmetrical.(C)RGBingham2005.Allrightsreserved.Paraxialraytracingandrelatedissues17pages(C)RGBingham2005.Allrightsreserved.UsesoftheParaxialandSei
100、delapproximations WhWavefrontLensaxisandpropagationdirectionParaxialformulaeassume:W = a2 h2.Mainlyforstartingadesign. Derive lens curves and diameters from the image position, focal length, f-number or magnification. Check feasibility. Like a “thin lens” calculation, but it applies to thick lenses
101、and complex systems. Much can be done using the solves in ZEMAX or with a pocket calculator.Seidelapproximation:W = a2 h2 + a4 h4 (and related terms off-axis). Exploit the Seidel aberrations by inspection. Solve some problems. In a later session, we shall discuss how to use the residual Seidel aberr
102、ations to review an optical design that is nominally finished, in particular to see whether the optimiser was well set up for it.W = a2 h2 + a4 h4 + a6 h6 + a8 h8 + ?(C)RGBingham2005.Allrightsreserved.UseZEMAXssolves,orcalculatebyhand?ZEMAXprovidesmanyconvenientsolveswithinitsLensDataEditorforderivi
103、ngcurvaturesandthicknesses.Iguessthattheysupersedetheformulaeformostusers.Idocalculateoccasionalresultsbyhandfromequationsdiscussedinthissession:paraxialray-tracing;Lagrangesinvariant(thatupholdsthesecondlawofthermodynamics);andfromNewtonsconjugatedistanceequation.Idothiswithapocketcalculatorandinro
104、ughnotes.Iusetheseexpressionstoputdimensionsontoasketchofsomeparticularidea,tocheckfeasibility,ortentativelytocalculatelengthsoranglesinthecourseofadiscussion.Seethedetailedexample(laterslide).Thatsetsupapreliminarylensdesignwithpencilandpaperanditcopeswithsomegreyareas.Alternatives.(a)Similarideasm
105、ightbeentereddirectlyintoZEMAXusingthesolvesorbytrialanderror,or(b)wemightstartwithsomeexistinglensdesignthatcanbescaled,modifiedandre-optimisedforanewapplicationwithinZEMAX.(C)RGBingham2005.Allrightsreserved.Example:ZEMAXsLensDataEditor:pickupsintheballlensFromEx00-Ball_Lens.zmxtoEx20-Ball_Lens.zmx
106、PickupsareatypeofSolve.Pickupscalculateentriesinthelensdatafromotherdimensionsorfromtherays.Dimensionsbecomelinked.Thiscanbuildindimensionalsymmetries,etc.,canholdthemevenwhenthedimensionsarevaried,eitherinoptimisationormanually.Thenfewernumbersaretreatedasvariable,andtherequiredlensstructurefollows
107、them.Pickupsareillustratedhereforaballlens.Theyforcetheentranceandexitareasoftheballtohaveequalbutoppositecurvaturesandtolieattheseparationrequiredforasphere.Thegivenexamplealsousesasolvetopositionthefinalfocus.Onlyonedimensionnowneedstobevariedwhenchangingthediameterofthesphere.WORDfileBall_Lens_Pi
108、ckups.docliststheSolvesthatprovidetheseeffectsinEx00-Ball_Lens.zmx,leadingtoEx20-Ball_Lens.zmx.Trychangingtheballdiameterto,say,6mmbychangingtheradiusto3mminsteadof2.5onsurface2.Thenupdatethelayoutandaberrationplot.AlsoseeZEMAXsautomaticfocallengthresult,etc.(C)RGBingham2005.Allrightsreserved.Paraxi
109、alray-tracingformulaeandwhatwemightdowiththemThesurfaceofathicklens,orathinlenswithpowerK (see next slide)uuhnndh1ThenextopticalsurfaceThe ray traced is usually the marginal, or edge, ray of the axial ray pencil. It is a ray arising from the axial field point. It passes through the edge of the apert
110、ure stop.Anglesuanduaretreatedassmallanglesbetweentherayandtheopticalaxis.Asdiscussed,uisnegative.Therefractiveindicesbeforeandafterrefractionarenandn.(1)Ray-tracing.TherayshownintersectsasurfaceofpowerKataheighthandthefollowingsurfaceatdistancedatheighth1. ThenfollowingWelford,nu - nu = - hK (1)h1
111、= h + du (2) Equation1givestheangleuoftheray,followingthesurface.Equation2transferstheraytointersectthenextopticalsurface.Anynumberofsurfacescanberay-traced.(2)Seidelaberrationscouldbecalculatedfromtheseresultsoftheparaxialray-trace.(3)Moreinterestingly,startingfromrequiredrayheightsandangles,wecanw
112、orkbackwardstofindKforeachopticalsurface.Thismaywellbehelpfulasastartingpointfordesigningthelens.SowhatdoesKtellus?Seenextslide.(C)RGBingham2005.Allrightsreserved.WaysofusingthepowerK asfoundfromtheparaxialformulae1.Forathinlensoffocallengthf :f = 1/K. Theuseofthisformulawouldbetoselectafewthinlense
113、sfromacatalogueorfromaspectacletrialcase(maybeallowingforimmersion)tosetupacrudeopticalsystemmatchingtheray-trace.2.Forathinlensbetweendifferentmedia(lessusual):f = n /K (seeWelford).3.Forasingleopticalsurfacewithradiusofcurvaturer : r= (n n)/K. Weusefullyfindrforasinglesurfaceasapreliminarytoray-tr
114、acing,dependingasitdoesonthedifference(n n) ofrefractiveindexacrossalenssurface.Note:Intheseparaxialformulae,refractiveindexandthicknesschangesignafteramirror,so(n n) = 2atamirror.4.Forathinlensaddingopticalpath atheighth : = h2K /2. Thisappliesbothtoordinarylensesandtothingradient-indexlenses. isth
115、edistancebywhichtheopticalpathmustbeadvanced atheighthinordertogivetherequiredlenspowerK. h isoftentheradiusofthelensaperture.Note:alensofpositivepoweradvancesphaseatitsedgerelativetoitscentre.(C)RGBingham2005.Allrightsreserved.NoteonparaxialraysParaxialrayscanbetracedevenwhentherealrayswillcrash.Ra
116、yscrashduetooneoftheeffectsdiscussedinaprevioussession,suchastotalinternalreflectioninalens.Thiscanbemisleading,soitisbesttouseparaxialray-tracingwithinrealisticcontextsandinconjunctionwitharealisticsketch.Crashingrayscanalsobeavoidedbydoingthefirstrealray-tracewithareducedapertureorfieldsize,orperh
117、apsacurvaturecanbechanged.Therequireddimensionscansometimesberestoredinstages.Aparaxialray-tracecanalsobeusedtotracechiefrays,whenlookingatquestionsofpupilimaging.Apupilexistswheretheheightofthechiefrayiszero.(C)RGBingham2005.Allrightsreserved.Example:fillinginlensdimensionsonasketchRayanglesu1u2u3u
118、4u5=0Rayheightsh1=0h2h3h4h5Thicknessesd1d2d3d4TryflatsurfacesK1,c1K2,c2Roughsketchoftelephoto collimator,f/6input,50mmbeamoutputWestartfromthisroughsketch.Fortherationaleofthesketchandhandlingit,seeParax_example.docandEx21-Parax_example.zmx.Mostofthesymbolsareinitiallyunknowns,butwecanassignvaluesto
119、thembyhandcalculation.Lenselements(C)RGBingham2005.Allrightsreserved.TheDefinitionofFocalLengthFocallengthisdefinedasf=-h/u,whereuistheimage-sideparaxialrayangleoftheaxialraypencilwhentheobjectisatinfinity.Itdiffersfrombackfocaldistance=b.f.d.Noteminussign.Ifthecollimatedinputbeamisdeviatedtoafielda
120、ngle,theimagepointmovesbyadistanceandf= / .Thisparaxialfocometerconceptforfisequivalenttotheabovedefinition.SeeWelford.Forrealrays,aberrationsmatter.ForrealaxialraysatanangleU,f=-h/sin U iftheimagehasnocoma.WemightusethattoestimateU given h and f. Conversely,iffdoesequal-h/sin U ,thecomaiszero!Thati
121、sknownastheAbbe sine condition forzerocoma.Focallengthisausefulconceptwhenthelens,etc.,canbefocusedtoinfinity.Adaptingthefirstexamplefromlecture1:hu or U(C)RGBingham2005.Allrightsreserved.ParabasalRays.WhyisFocalLengthwelldefinedbyaparaxialformula?Aswehavediscussed,paraxialraysareapproximate.Forreal
122、raysthathappentolieclosetotheopticalaxisofalens,theexpression“parabasal”raysisused.Paraxialraystendtoexactcoincidencewiththeserealparabasalraysashtendstozero.Thatiswhyfocallengthiswelldefinedbyaparaxialformula.Zemax,possiblyunlikemostothersoftware,tendstouseparabasalraysratherthanparaxialrays.Paraba
123、salrayheights,oncescaleduptothelensaperture,arethesameasparaxialrays.ThusZemaxusesnumericalmethodsapplicabletoanyrayeventocomputequantitiessuchasfocallength,whichisdefinedparaxially.Itdoesthisratherthanusingparaxialformulaeinordertobroadentheapplicabilityoftheprogramsalgorithmsanduserfeatures.Its“pa
124、raxial”methodswouldalsobeapplicabletosomefurthercases,suchastounusualopticalprofilesforopticalsurfaces.(C)RGBingham2005.Allrightsreserved.LagrangesInvariantTheLagrangeinvariantHiscalculatedfromtherays.Thusitinvolvestheobjectorimageheightor,rayheightshinthelens,rayanglesandu andrefractiveindices.Forr
125、aysthatpassthroughthelens,Hhasacertainvalue,whethertheobjectandimagearerealorvirtual.ThevalueofHisthesameineachsuccessivespacefromobjecttoimage,whetherinairorglass.Theinvariancecannotbedefeatedinanysystem.Henablesaspecificationforalenstobecheckedforconsequencessuchasemergentrayanglesandindeedforphys
126、icalfeasibility.The value of H. nandn aretherefractiveindicesintwoexamplespaces.Inanyspacewheretheraypenciliscollimatedinagivenopticalsystem,thevalueof-n h isthesame.Inanyspacewheretheraypencilisnotcollimated,thevalueofn u isthesame.H =-n h = n u hunnThink of some examples?(C)RGBingham2005.Allrights
127、reserved.Example:LenticularBeamExpanderEx22-Lens_Expander.zmxAcollimatedbeamof5mmdiameterisexpandedto8mm.ThisangularspreadfallsandsotheexampleillustratesLagrangesInvariant.SeethenotesandreferencewithintheZEMAXfile.Thepupilpositionisnotshiftedbyinsertingthisparticularlens.ItsdesignaroseinastudyofShac
128、k-Hartmannwavefrontsensors.RealentrancepupilandvirtualexitpupilZEMAXsparaxiallenstotestcollimation(C)RGBingham2005.Allrightsreserved.Example:MirrorBeamReducerconfocalparaboloidsEx23-Mirror_Reducer.zmxusesconfocalconcaveparaboloidson-axistodeliveracollimatedbeamofreduceddiameter,withzerocoma.Alternat
129、ively,off-axispartsofsuchparaboloidscanbemade,andcanbecombinedasanunobstructedHerschelliantelescope.CommonfocusofbothmirrorsPrimarymirrorSecondarymirrorZEMAXsparaxiallenstotestcollimation(C)RGBingham2005.Allrightsreserved.PrismBeamExpander/TelescopeEx24-Prism_Expander.zmxAbeamcanbeexpandedwiththeuse
130、offlatsurfacesonly.SeenoteswithintheZEMAXfile.(C)RGBingham2005.Allrightsreserved.PrincipalplanesP,PandprincipalfociF,FFocallengthfPAnopticalsystemFFocallengthFPThesameopticalsystemWelfordsnotationParaxialfocus(C)RGBingham2005.Allrightsreserved.NewtonsconjugatedistanceequationandthemagnificationsOOFF
131、z (negative in this diagram)zTheboxrepresentsanyopticalsystemoffocallengthf.Therefractiveindexistakenasthesameeachside(seeWelfordformoregenerality).OisatanimageofOandvice versa ;thusOandOareconjugatesorconjugatefocalpositionsorconjugatepoints.OandOandtheprincipalfociFandFaresometimesimmersedwithinan
132、opticalsystem.Newtonsconjugatedistanceequationstatesthatz z = - f 2. Also,thelateralmagnificationmisgivenbym=-f/z =z/f. Longitudinalmagnificationism2. Thesignsofthemagnificationsrelatetoimageinversionandimageparity,asdiscussedinanotherlecture.f(C)RGBingham2005.Allrightsreserved.SpreadsheetsIwouldsay
133、thatanEXCELspreadsheetforparaxialcalculationshasbeenrenderedunnecessarybysoftwaresuchasZEMAX.Thespreadsheetwouldneedtobeintuitivetolearn,andtoworkeitherforwardsorbackwardsbetweenthesurfacedimensionsandtheraypaths,startingfromanypointinthesystem.Itisnotuniversallyapplicable,becauseparaxialcalculation
134、oftenleadstophysicallyimpossibledimensions,suchaswhenrayscrashorlensesoverlapattheiredges.C.G.Wynnecompiledatableforeachproblem.Ithadparaxialray-tracingdatainleft-handcolumnsandextendedfarenoughtotherighttofillintheSeidelaberrations.Hecompleteditbyhand,exploitingitsversatility.Itsusesincludedparaxia
135、lray-tracingandderivingcurvaturesandseparationsfromparaxialrays.However,WynnealsousedittocomputecurvaturesandseparationstochangetheSeidelaberrations,theaimsometimesbeingtobalancethefiniteaberrationsshownbytracingrealrays.Hesometimesseparatelysolvedtwoorthreesimultaneousequationsindoingthat,orusedthe
136、tableiteratively.Hewouldstartanoptimisationwithagoodsolutioninthatsense.Wynnestablesandhisray-tracingprogramalsoshowedtheanglesofincidenceofmarginalraysandofchiefraysateachopticalsurface.Largeanglesofincidencecanbesourcesofaberrationproblemsonsomesurfaces,andsmallanglesofincidencecanleadinghostimage
137、sfromstrayreflections.(C)RGBingham2005.Allrightsreserved.MatrixmethodsMatrixmethodsexistforparaxialraytracingandfortracingGaussianBeams.Asdiscussed,Iprefertousetheparaxialequationsseparately,orsolveswithinZEMAX.IfindthattheGaussianBeamfeaturesinZEMAXareconvenientandIdonotusethematrixmethodseparately
138、.Occasionally,IuseoneoftheGaussianbeamformulaeexplicitlytofindanangleorbeamsizebyhand,ineitherroughnotesorshortreports.AlltheformulaeIneedforthatseemtobeintheZEMAXmanual.(C)RGBingham2005.Allrightsreserved.Atheoreticalfiniteray,threeexamplesofselectedsystemsandtipson“pastingin”5pages(C)RGBingham2005.
139、Allrightsreserved.FiniteRayTracinganoteonSnellslawSnellslawin3Drrnnnnandn aretherefractiveindicesbeforeandafteranopticalsurfacen istheunitvectoralongthelocalsurfacenormalr and r areunitvectorsalongtheraybeforeandafterrefractionSnellslawstates:n (rn)=n (rn)Theray-tracingprogramcomputesr.Welford,Chapt
140、er4,explainshowthatishandled.Thenatureoftheequationforrmayillustratewhylow-aberrationsolutionsforthedesignsofcompoundlensesarecommonlyfoundbynumericalmethodsinvolvingraytracing,ratherthanbyanytheoreticalmethodinvolvingrefractedraysandhencer.(C)RGBingham2005.Allrightsreserved.Solves:Landscapelensexam
141、pleEx25-Landscape.zmxTheexampleshowshowacriticalfeatureofthislenscanbedesignedusingsolveswithinZEMAX.Thelensdesignissignificantinthatiteliminatessomeaberrationterms;itisaprecursortothedioptricanalogueoftheSchmidtcamera.(C)RGBingham2005.Allrightsreserved.Nosolves:fastasphericlensexampleSeeEx26-Fast_s
142、inglet.zmxGaussianlaserbeamThisexampleshowsamuchfasterlensforcomparison.Itismonochromaticandhasaverysmallfieldofview.(C)RGBingham2005.Allrightsreserved.Wemayhaveaworkingfileoflensdataintowhichwewanttoinsertorsubstitutesomesurfacesfromanotherfile.Itispossibletolosetrackofpickupsandsolvesifthesystemis
143、complex.Whensurfacesareinsertedordeleted,theprogramcannotalwaysupdatesolvessuchaspickupsandreferencestointermediatesurfacesintheMF.Theprogramcannotautomaticallyinsertpickupsbetweenexistinglinesandpasted-inlines.Tokeeptrackofthem,theeditsneedtobedoneintherightorder.Basically,thesolvesandMFsurfacerefe
144、renceswillmoreoftenbecorrectifwepastesurfacesintoaworkingfilebeforedeletinganyredundantsurfacesfromthatfile.Theresultaftersuchdeletionsbecomesthenewfile.So,(A)pasteinbeforedeletingand(B)pasteaheadofanylinesthatwillbedeleted.TheproceduredetailedinPasting-in.docminimiseserrors.ZEMAXsLensDataEditor:pas
145、ting-infromanotherfileToavoiderrors,seeWORDfilePasting-in.doc(C)RGBingham2005.Allrightsreserved.Ray-tracing the whole system aShack-HartmanwavefrontsensorEx27-S-H_WFS.zmxRaytracingthewholesystemisthemostreliableapproach.Incasessuchasthis,theinputraysaresimplyfrominfinity.Thisexampleistheperipheralwa
146、vefrontsensorintheAcquisitionandGuidersystemoftheGeminitelescopes.SeetheZEMAXfileforthenoteswithinit,andforthescaleofthisdiagram.Thediagramshowsonlypartofthefullray-tracethattakesineverythingfromtheGemini8-metretelescopetothewindowofthesensor,includingtheopticalprescriptionofthemicrolenses.Notes:The
147、doubletlensontherightmakesthespotarraysmaller.ThedesignofthatlensalsominimisesaberrationsintheShack-Hartmannspots.It takes account of the aberrations due to various microlenses and the window of the sensor,sofaraspossible,asIincludedthemintheraytrace.Inthiscase,Ididnotmodifyanycomponenttocompensatef
148、orproblemsdownstream,becausethatwouldcreateproblemsiflaterchangeswererequired.filterscollimatorpupilandmicrolenseswindowsensorLensallowingthecollimatorandmicrolensarraytobelargerthanthesensorStarimagein8-metretelescope(C)RGBingham2005.Allrightsreserved.OpticsandOpticalDesignR.G.BinghamSession4Sessio
149、n 4Looking at Seidel aberrations, and why we take the trouble.Image qualityGaussian Beams and Fibre Coupling(C)RGBingham2005.Allrightsreserved.SixpagesonrecognisingsomeSeidelaberrations(C)RGBingham2005.Allrightsreserved.WhyrecogniseSeidelaberrations?Eveninanimagingsystemthatisverycomplexindesign,tot
150、alaberrationsarenotzero.Seideltermswillbeincludedatsomelevel,balancinghigher-orderaberrations.Wecanlookforallsucheffectsinordertoseewhetherthedesignprocesswasused,orisbeingused,tobestadvantage,asdiscussedelsewhereinthiscourse.Apointthatweconsiderinthissessionisthattheprogramoffersanumberoftypesofplo
151、tinanalysiswindows,showingthesameaberrationsindifferentways.Wecanbecomefamiliarwiththesedifferentformsofoutput,anditisoftenhelpfultolookatmorethanoneoftheminunderstandingthebalanceofaberrations,perhapsinrelationtotheaimsofthedesign.Thusfollowingpagesrevisittheanalysiswindowsforthemostimportantcases,
152、sphericalaberrationandcoma.OtherSeidelswillbecomefamiliaraswecontinue.Afifthscreenwarnsusofapotentiallytime-wastingpitfall.Itarisesiftheopticalsystemdepartsfromaxisymmetryandwehypnoticallykeeplookingatitsaberrationfansinthesameway.ABCA,Seidelsphericalaberration,(x4);B,thenexttermup(x6);C,thesumofAan
153、dB-thebalance.IfitlookslikeAorB,askwhy!ItisveryusefultobeabletoseethisbothinOPDplotsandrayfans.(C)RGBingham2005.Allrightsreserved.Whyaresphericalaberrationandcomaimportantinaxisymmetricalsystems?Thisrelatestotheimportanceoftheaxialfieldpoint.Everyonelooksattheimagequalityachievedon-axis.Sphericalabe
154、rration(Seidelpluswhateverhigherorders)istheonlyaberrationatthatpointinanaxisymmetricalsystem.Next,comais“linear”.Itsamplitudeiszeroattheaxialfieldpointandproportionaltothefieldradius ;nothingcomesupquickerthanthat.Sothereisatleastasmallfieldofviewaroundtheaxialfieldpointwheretheaberrationsarepredom
155、inantlysphericalpluscoma.Withlargerfieldsofview,designsmaywellaimtominimiseandalsoequalisetheimagespreadatvariouspointsoverthefieldofviewandovertherangeofwavelengths.Thatmaybeachievedbynumericaloptimisationifeverythingissetupwell.Itresultsinsphericalaberrationandcomathatarenotreducedpracticallytozer
156、o.Theycanbalanceagainsthigher-orderaberrationsoff-axis.(C)RGBingham2005.Allrightsreserved.Coma;itswavefrontaberrationplot(OPD).Areminderfromsession2withtwoformsofdisplay W 3 1cosWavefrontaberration=relativeradiusinaperture=distancefromcentreoffieldEdgeoffieldMeantiltremovedOPDfans“WavefrontFunction”
157、Recall how this differs from spherical aberration(C)RGBingham2005.Allrightsreserved.SphericalaberrationandcomaraysSphericallyaberratedraysinayfanAnywhere in the field?ComaticraysinayfanOff-axisinasymmetricalsystem.RaysastheyreachtheimageItissometimessaidthatperipheralraysexperienceadifferentfocallen
158、gthfromaxialraysifthereiscoma.SeediscussionontheSineRuleelsewhere.Marginalrays(atedgeofstop)FocalsurfaceChiefray(throughcentreofstop)(C)RGBingham2005.Allrightsreserved.RayFansSphericalaberrationComa y 3 0orx 3 0 y 2 1=Transverserayaberration;y=radiusinstop;=distanceofimagefromcentreoffieldyxSpherica
159、laberrationComaThreefieldpointsineach*SeeEx11-Paraboloid.zmxandEx11A-Sphere.zmx(C)RGBingham2005.Allrightsreserved.Sphericalaberrationandcomaspots r 3 0 y 2 1=Transverserayaberration;r=radiusinaperture;=distancefromcentreoffield.AlsoseeWelford.(C)RGBingham2005.Allrightsreserved.Awarningregardingtheus
160、eofaberrationfanswhentheopticalsystemisnotaxisymmetrical.SeeEx28_Fan_Warning.ZMXInthissystem,bothOPDfansandbothrayfansunderstatetheamplitudeoftheaberrationsthatareactuallypresent.Thismayariseinanynon-axisymmetricalsystem.Also,Seideltheorydoesnotapplytoanon-axisymmetricalsystem.Wavefrontmap(C)RGBingh
161、am2005.Allrightsreserved.ImageQualityEightfollowingscreensrelatetothetopicofImageQuality.(C)RGBingham2005.Allrightsreserved.ImageQualityImage qualityreferstothesizeandshapeoftheimageofapointobject.Thecontextisthatsmallandsymmetricalaregood.Weworktoacriterionforimagequality.Thatcriterionwillbedecided
162、forthepurposeinhand.Forscientificimaging,itwillrelatetothecapabilityoftheimagetoyieldinformation.Itmayaimtolimitmorethanoneparameteroftheimageprofile.Inthedesignprocess,weexplorethetheoretical image quality.Itispartofanerror budgetthatwillhavemanyothercontributingfactorsseealatersession.Criteriafori
163、magequalityusuallyaimtocontroloneof:thepointspreadfunction(PSF);itsencircledenergyplot;itsModulationTransferFunction(MTF)-seeWelford;oraparameteroftheOPDmap.TheseapproachesallrelatetothePSFitself,butonemaybeclearlyrelevantormaybemuchfastertocomputeormeasure.Whenstraylightorthebackgroundsignallevelis
164、anissue,orwhenthepointspreadfunctionhasasharpcorebutextendedwings,contrastsuffers.Forone-offscientificoptics,theseissuesaredealtwithindividually.Formass-producedcompletesystems,MTFisoftenspecifiedandmeasuredforoverallcontrol.TheMTFcriterionandthetestmustthenreflectallpossibleoperatingconditions.Foll
165、owingslidesrelatetothequalityofafocalsystems,diffraction-limitedopticsandlasers.(C)RGBingham2005.Allrightsreserved.AfocalSystems,ImageQualityandTelescopesAnafocal systemisthespecialcasewhenaraypenciliscollimatedbothonentrancetoandexitfromanopticalsystem.Thesysteminthediagramcarriestworaypencils.Inan
166、afocalsystem,imagequalitycanbedefinedintermsofangulardeviationsoftheexitrays,butitisanalysedasanimageinray-tracing.WeinsertZemaxsParaxialsurfacebeforetheimagesurfacetocreaterealimages,asmentionedinrelationtothebeamexpandersofsession3.Weevaluatethoseimagesasusual.Wecansometimeschoosethefocallengthoft
167、heParaxialsurfaceandthecurvatureoftheimagesurfacetohavesomephysicalsignificanceinthecontext.Formally,atelescopeisanotherwordforanafocalopticalsystem.So,whenwelookatadistantscenethroughatelescope,includingbinoculars,weshouldfocustheeyetoinfinity.Forcomparison,whentheopticalsystemformsanormalimage,iti
168、sacamera.IoftenusethetermCassegraincameraratherthanCassegraintelescope,howeverlarge,ifitscurrentpurposeistoplaceadirectimageonaCCD.AsinEx23-Mirror_Reducer.zmx(C)RGBingham2005.Allrightsreserved.Obscurationandaberrationsatthediffractionlimit(1)Obscuration.Indiffraction-limitedimages,theproportionofthe
169、energyfallinginthecentralpeakoftheimagefallsroughlyinproportiontothe“fillingfactor”,theproportionoftheoverallbeamareathatisunobstructed.So,ifanunaberrated,unobscuredtelescopebecomes10 percentmaskedarea-wisebyacentralobscuration,thecentralpeakintensityinthediffraction-limitedimagefallsto80percentofwh
170、atitwas.Halfthis20percentlossisthelightsimplyblockedbytheobscuration,andhalfisthediffractioneffect,sendingenergyintoouterdiffractionrings.Neglectofthisdoubledlight-lossisapotentiallyseriouspitfall.Itaffectssomeexistingastronomicalinstrumentsandtelescopes.SeetheaxialimageinEx29_Diffraction_Peak.ZMX.(
171、2)Aberrationsdistributeyetmoreenergyoutofthecentraldiffractionpeak,initiallyintotheouterAiryrings.Largeraberrationsgiverisetospeckles.Thisgraphicshowscomainanoff-axisimageinEx29_Diffraction_Peak.ZMX.UsetheFFTortheHuygensmethodhere.Weseepartsofaboutfourbrightrings,includingtwospeckles.StrehlRatioisdi
172、scussedonthenextscreen.(C)RGBingham2005.Allrightsreserved.StrehlRatio-exampleAnidealimagingsystemwithacircularaperturedeliversanAirydiskasintheleft-handcross-section(relativeirradiance).Itwillbedegradedbyaberrations.Theaberrationintroducedtogeneratethefollowingdiagramsisr4sphericalaberrationplusfocu
173、sthatkeepsthepeak-to-valleyOPDminimised.ThevalueoftheStrehlRatioisequaltob/a, 0.5inthisexample.Whatisactuallyrequiredofotherapertureshapesmayneedtobeconsideredcase-by-case.Furthernotesandcommentsappearonthetwofollowingpages.ab(C)RGBingham2005.Allrightsreserved.StrehlRatiocalculationsandconventionAbe
174、rrationssendlightoutofthecentralpeakofanidealimageintoouterdiffractionringsandultimatelyintomorecomplexfeaturesofthePSF.TheStrehl Ratioisthepeakintensitydividedbythatwhichwouldhavebeenobtainediftherewerenoaberrations.SeeWelford.Strehl Ratio = 1 (2 / )2 (mean square wavefront aberration)Ifstartingfro
175、mther.m.s.aberrationcomputedseparately,checkwhetheritwasexpressedinthesamelengthunitsasthewavelengthabove!AfinalStrehlRatioequaltoorabove0.8isconsideredtobegoodinmostapplications.Thereishardlysuchathingasperfection. Raleighs quarter-wavelength rule. Ifthewavefrontforminganimagewilljustfitbetweentwoc
176、oncentricsphericalsurfaces/4apart,theimageisdiffraction-limited.Noticethatthisdeterminesthepeak-to-valley wavefront aberration.Ithasbeenshowntheoreticallythatwith/4p-vaberration,theStrehlRatiois0.8forarangeofdifferentwavefrontprofiles.Practicalcasesalsofollowtherule,aswillbeseeninZEMAXinthenormalcou
177、rseofwork.(C)RGBingham2005.Allrightsreserved.StrehlRatiowarningsTheStrehlratioisdefinedintextbooksfromtheexpectedaberrationsalone,andthatisamajorexampleofthepitfall.Obscurationinthebeam(seeapreviousslide)affectstherelativeintensityatthecentralpeakofanimage.ItcanmakeanopticalsystemfailtomeetaStrehlsp
178、ecification,orthespecificationmaybeimpossibletomeet.InZemaxsanalysiswindows,thetotaleffectofobscurationsandaberrationsoncentralintensityneedstobeconsideredinstages.ThefulleffectscannotbeseeninanysingleanalysiswindowinZemaxuntilwegetintoPhysicalOpticsPropagationandlookatthescaleofirradiance.All-refle
179、ctingopticalsystemsareoftensaidtobetotallyachromatic,performingequallywellatanywavelength-butweshouldnotneglectdiffractionsocasually.TheStrehlRatiowillbeafunctionofwavelength.(C)RGBingham2005.Allrightsreserved.Wavelength-compensationofimagesizeSeeEx30_Wavelength_Compensation.zmx.Aspecificationforper
180、formancemaybesatisfiedbyaPSFfallingshortofthediffractionlimitbutinwhichtheAirypatternisdiscernible.Ifso,ausefuleffectappliestoall-reflectingsystems.Itstabilisestheimageprofileoveralargerangeofwavelengths.Itworksasfollows.Atlongerwavelengths,theAirypatternislarger,butaberrationsaresmallerintermsofwav
181、elengthsandsendlessenergyintoouterrings.Theneteffectisdramatic,givingimageswithaconstantsamplingrequirementorpixelsizeoveratleasta4:1rangeofwavelengths.Theencircledenergycurvesintheexampleremainreasonablyclosetogetheratwavelengths0.25,0.50and1.00microns.(C)RGBingham2005.Allrightsreserved.Thecorrectf
182、ocusshouldgivethecorrectaberrationsFocusing.Achievingthecorrectfocusisacriticaltestoftheassembledlens.Wemightlookattheimageprofilepassingthroughfocus,comparingitwithspotdiagrams,orwemightlookatthewavefrontaberrationswithaninterferometer.Theaimisnottoseehowsmallwecanfocusanimage.Theaimistoseewhethert
183、heaberrationsinthefinishedlensresemblethetheoreticalaberrations.Perhapssurprisingly,thisusuallyworksevenwhenmanufacturingerrorshavenotbeenincludedinthelatestray-trace.Strehlratio.Obtaininglowaberrationsundertestbyfocusingasingleimage,andsodemonstratingaStrehlRatio,maynottellthewholestory.Forexample,
184、itcanconcealanassemblyerrorwherebyresidualsphericalaberrationhasthewrongsign.Otherwavelengthsorotherfieldpointswillfail.Ontheotherhand,iftheaberrationsaremeasurableandareclosetoexpectation,wearehomeanddry.Allowabledefocus.Focusaccuracyisalwaysintheerrorbudget.Theproblemismechanicalorthermal.Itismost
185、seriousinspaceinstruments,duetolaunch-stressormechanismissues.Ourdesignworkandtolerancingrun(latersession)allowustocheckthattheerrorbudgetisreasonable,beingawareoffocussensingandcontrol,mechanicalengineeringandcost.Wecanneverriskhavingasignificantdefocusaberrationintheerrorbudget.Tomeettherequiredto
186、talaberrations,wecouldenduptryingtoremoveotheraberrationsbymakingthelensmorecomplicated.(C)RGBingham2005.Allrightsreserved.GaussianBeamsEightpagesonGaussianBeamsandrelatedissues(C)RGBingham2005.Allrightsreserved.ModesinfreespaceIftheamplitudeandphaseofsomeaberratedwavefrontcanbemodelledalgebraically
187、,itmaybepossibletodescribethepropagationofthatwavefrontbytheory.Iftheanalyticalresultcouldberepresentedbyafewparameters,itcouldperhapsbeusedasifitwereanewly-discovered“mode”oftheelectromagneticwaveinfreespace.However,therewouldbeaninfinityofdifferentcasesandfewofthemwouldberemotelyworththecandle.Wha
188、tisimportantistobeawareofsomerelativelysimplemodesthatareusedinnumericalworkandintheoreticaldevelopment.Suchsimpleformscanhavepracticalrelevance,orcanbeusedasbasisfunctionsformorecomplexcases.Forexample,planewaves:anaberratedwavefrontcanbereducedtothesumofplanewaves,sothatafterthepropagationofthepla
189、newaves,theevolved,aberratedwavefrontisfoundbyre-addingthem.ThenHuygensexpandingsphericalwaveletsunderlieusefultheoryandvariousalgorithmsfornumericalwavepropagation.TheGaussian Beamisanotherfree-spacemode.(C)RGBingham2005.Allrightsreserved.GaussianbeamItisthebeamprovidedbylasers.Itpropagatesaccordin
190、gto“paraxial”formulae(ref.ZEMAXmanualandotherreferencestherein)thatarefasttocompute,althoughasopticaldesignerswehardlyusethem,exceptperhapsinveryearlystagesofthework.TheABCDMatrixisapackagedwayofdoingaparaxialray-trace.Theparaxialresultsuffersfromsevereapproximationsasmaybecomeclearinthefollowingpag
191、es.(C)RGBingham2005.Allrightsreserved.PropagatingGaussianBeamsinZEMAX.TheStandardorParaxialformulae.1:ZeroaberrationsTheparaxialGaussianBeamformulaedonotusenormalraytracingandZEMAXhandlesthemspecially.Inanyevent,theformulaeofferonlyanapproximation,merelyaparaxialcalculationthoughtheopticalsystem;the
192、resultisinvalidwhenusedforapracticallens.Thisisazero-aberrationexamplewithpropagationinfreespaceonly.SeeEx32_Parax_Gaussian.zmxTheopticalsysteminthisexampleismerelyanemptyspacewithathickness.TheonlydimensionsthataresetupintheLensDataEditorherearethewavelengthandthe1000mmzthicknessfromthestoptotheima
193、gesurface.ThenZEMAXsParaxialGaussianBeamwindowissettolaunchabeamwithawaistof0.05mmradius.Thisisnottobeconfusedwitharadiusofcurvature.Itistheradiusinthex and y directions tothe1/eamplitudepoint(the1/e2intensitypoint).Thetheoreticalbeamtowhichtheseconventionalformulaeapplyisinfiniteinlateralextentinal
194、lspaces;thisisaseriousapproximationintheuseoftheseformulaeforalens,asthereisnotruncationofenergyattheedgesoflenses,etc.ThebeamradiusandthesurfacefromwhichthebeamislaunchedarespecifiedintheSettingswindow.Itcanbeseeninthatwindowthatthebeamdilatesto3.5mmradiusinthe1000mmpathandthereareotherresults.This
195、SettingswindowisunusualinZemax.Tryinteractivelychangingtheparametersofthebeamthatislaunched.(C)RGBingham2005.Allrightsreserved.TheStandardorParaxialformulae.2:BewareofM2.Asmentioned,theGaussianBeamformulaeofferonlyanapproximation,merelyaparaxialcalculationthoughtheopticalsystem.Thisisanexamplewithab
196、errations.SeeEx33_Parax_Gaussian+abns.zmx(C)RGBingham2005.Allrightsreserved.2.NormalRay-tracingforGaussianbeamsAtruncatedGaussianBeamisjustanotherraypencil.ItcanbelaunchedinZEMAXbymeansofGaussianapodisation,uptoapodisationfactor4.Ordinaryray-tracingandthecorrespondinganalysisfeaturesarevalidfortheGa
197、ussianBeamandcorrectlyshowthebeamwaist,forexample.SeeEx31_Gaussian_Amp_Apod.zmx.ThereareexceptionswherePhysicalOpticsPropagationhastobeusedinanycase.Forexample,thebeammaybesignificantlytruncatedatsomeintermediatesurfaceandwemaywishtoevaluatethediffractionpathsoftheremainingenergy.ThentheuseofPOPisne
198、cessary.Thelocalwavefrontprofilemayneedtobestudied,ratherthanthatintheexitpupil,andagainPOPprovidestheresultrequired.Othercasesarerelated.Wemightwishtostudydiffractionataslit,ortheeffectofdiffractiononanominallycollimatedbeamoveralongpath,whichisnotshownbyparallelrays.However,theexampleusingnormalra
199、y-tracingistypicalofmanypracticalcases.(C)RGBingham2005.Allrightsreserved.3.PhysicalOpticsPropagationinZEMAXasappliedtoGaussianBeams.Thisisrelevantforlatersessions.ItshouldalsobeclearfromtheZEMAXmanual(whenPOPitselfis).(C)RGBingham2005.Allrightsreserved.Fibre-opticcouplingintegral(1)Thecouplinginteg
200、ralisusedtofindhowmuchpowerisreceivedbyanopticalfibreinanarbitraryincidentbeam.Theresultisoftenrequiredforfibre-couplingoptics,whereabeamistransmittedfromanotherfibreatsomekindofjoint.Itisalsousefultomodelphysicsexperimentswherethefibrepicksupunfocusedillumination.Itisaquestionofhowmuchpowergetsinto
201、thelow-orderwave-guidemodeofthefibre(theonethatisnormallyused).SeetheZEMAXmanualandreferencesthereinfortheformoftheintegral.SeeourexampleEx34_Fibre_Coupling.zmxandthenotesinitsTitle/Noteswindow!Furthercommentsontheintegralappearonthefollowingpage.TestbeamatinputisaGaussianBeam.Positionoffibretip(C)R
202、GBingham2005.Allrightsreserved.Fibre-opticcouplingintegral(2)Zemaxfullyevaluatesthecouplingintegral.Itisinterestingtoseehowitusesamplitudeandphase,notintensities.Owingtothesquaredtermswithinit,thevalueoftheintegralisthefractionoftheincidentbeampowerthatiscoupledintoawaveguidemodeinthefibre.Theintegr
203、alisevaluatedoveranyareawheretheincomingwaveandthewavemodeofthereceivingfibreoverlap.(Byreciprocity,thefibrewavemodecorrespondstotheprofileofphaseanamplitudethatthefibrewouldtransmit.)Thisareaoverwhichtheintegralisperformedcanbelocatedanywhere.Theintegralscanbecomplex.Ifthefibreisreceivingroughlycol
204、limatedillumination,wecanfindausefuleffectivecollectingareaofthefibretipforincomingpower.Theintegralisperformedoverthesmallbeamwaistatthecutendofthefibre.Thephaseofthefibremodeisflatthere.Letthebeamwaistradiustothe1/eamplitudepointber0,andlettheangletothatamplitudepointinthefarbeamfromthefibrebe0.Ne
205、xt,lettheareaofthefibretipbesmallsothatweconsidertheincomingwaveasaplanewavetiltedatangle,whichisagreatsimplification.Theeffectivecollectingareaofafibretipforincomingenergyistwicetheareaofthebeamwaisttothe1/eamplitudepoint.Power in = Irradiance * 2r0r0* exp(-2(tan /tan 0)2)CollectingareaforpowerTiltfactor(C)RGBingham2005.Allrightsreserved.
