Optical system

Idealopticalsystem

Theidealopticalsystemisanimagingsystemthatcanproduceaclear,completelysimilarimagetotheobject.Thebeamsinwhicheachlightrayoritsextension

linesallintersectatthesamepointarecalledconcentricbeams.Aftertheincidentconcentricbeampassesthroughtheidealopticalsystem,theoutgoingbeammustalsobeaconcentricbeam.Theintersectionoftheincidentandoutgoingconcentricbeamsiscalledtheobjectpointandtheimagepoint,respectively.Theidealopticalsystemhasthefollowingproperties:①Afterallthelightrayscrossingtheobjectpointpassthroughtheopticalsystem,theoutgoinglightraysareallcrossingtheimagepoint.viceversa.Thepointatwhichthispairofobjectimagescanbeinterchangediscalledtheconjugatepoint.②Eachstraightlineontheobjectsidecorrespondstoastraightlineontheimagesidecalledaconjugateline;thecorrespondingsurfaceiscalledaconjugatesurface.③Anyplaneperpendiculartotheopticalaxis,itsconjugatesurfaceisstillperpendiculartotheopticalaxis.④Forapairofconjugateplanesperpendiculartotheopticalaxis,thelateralmagnificationisconstant.Thetheoryofstudyingtheone-to-onecorrespondencebetweenthetwoobjectsinanidealopticalsystemiscalledGaussianoptics.ItwasfirstclarifiedbytheGermanscientistC.Gaussinhisworkin1841.Infact,thereisnotrulyidealopticalsystem.Thecoaxialsphericalsystemcanapproximatelymeettherequirementsofanidealopticalsystemunderparaxialconditions.

Basepointandbasesurface

Severalpairsofspecialpointsandsurfacesthatdeterminetheconjugationrelationshipbetweentheobjectandimageofanidealopticalsystem.

Focusandfocalplane

Thepointontheopticalaxisthatisconjugatetotheinfinityimagepointiscalledtheobjectfocus(orthefirstfocus).AsF;thepointontheopticalaxisthatisconjugatetotheinfinityobjectpointiscalledtheimage-sidefocalpoint(orsecondfocalpoint),andisdenotedasF'.TheplanespassingthroughtheFandF′pointsandperpendiculartotheopticalaxisarecalledtheobjectfocalplane(firstfocalplane)andimagefocalplane(secondfocalplane).

Principalpointandprincipalsurface

Apairofconjugatesurfaceswithalateralmagnificationequalto1iscalledtheprincipalsurface,andtheintersectionofthetwoprincipalsurfacesandtheopticalaxisiscalledMainpoint.AnylightrayemittedfromthefocusoftheobjectF,afterpassingthroughtheopticalsystem,becomesarayparalleltotheopticalaxis.ExtendthepairofconjugatelightraystogettheirintersectionpointM,thisintersectionpointThesetof,constitutesthemainsurfaceoftheobject(thefirstmainsurface),andtheintersectionofthemainsurfaceandtheopticalaxisHiscalledthemainpointoftheobject(thefirstmainpoint).Afterthelightparalleltotheopticalaxisenters,theoutgoinglightintersectsattheimagefocalpointF'.ExtendthepairofconjugatelightraystoobtaintheintersectionpointM'.ThesetofintersectionsconstitutesTheprincipalsurfaceoftheimageside(thesecondprincipalsurface),theintersectionpointH'betweenitandtheopticalaxisiscalledtheprincipalpointoftheimageside(thesecondprincipalpoint).Thetwoprincipalsurfacesareapairofconjugatesurfaces,andthetwoprincipalpointsareapairofconjugatepoints.Theheightofanypairofconjugatepointsonthetwomainsurfacesfromtheopticalaxisisequal,andthelateralmagnificationis1.

Nodesandnodalplanes

Apairofconjugatepointswithanangularmagnificationof1ontheopticalaxisiscalledanode,whichpassesthroughthenodeandisperpendiculartotheopticalaxis.Thefaceiscalledthesectionface.

Object-imagerelationship

InGaussoptics,thespecificopticalsystemisabstractedasasystemconsistingofabasepointandabasesurface.Theobjectdistance,imagedistanceandfocallengthareallbasedontwomainpointsCalculatedasabenchmark.ObjectpointQandobjectfocusFtothemainobjectpointHdistancesandfrespectivelytheobjectdistanceandtheobjectfocallength;theprincipalpointoftheimagesideH'totheimagepointQ'andthefocuspointoftheimagesideF'Thedistancesaretheimagedistances'andtheimagesidefocallengthf'.Thepositionalrelationshipbetweenobjectsandimagesisexpressedbythefollowingformula:

f'/s'+f/s=1

ThisformulaiscalledGaussianformula.Thepositionofobjectsandimagescanalsoberepresentedbyxandx'.Therelationshipbetweenthetwois:

xx'=ff'

ThisformulaiscalledNewton'sformula.

Magnification

Theratiooftheconjugatequantityrelatedtotheobjectandtheimage.Itcanbedividedintothreetypes:horizontalmagnification,verticalmagnificationandangularmagnification.

Horizontalmagnification

Theratiooftheimageheighty'totheobjectheighty,alsoknownastheverticalaxismagnification.βmeans:

β=y'/y=-ns'/n's

wherenandn'aretherefractiveindexoftheobjectspaceandtheimagespace.

Longitudinalmagnification

Theratioofthelongitudinaldepthoftheimagealongtheopticalaxistothelongitudinaldepthoftheobjectontheopticalaxis,expressedbyα.Therelationshipbetweenαandβis:

α=β2n'/n

Angularmagnification

Theanglebetweentheemittedlightandtheopticalaxisu'andtheincidentTheratiooftheangleubetweenthelightandtheopticalaxis,expressedbyγ,namely:

γ=u'/u=tanu'/tanu=ny/n'y'

Therefore:

n'y'u'=nyu

ThisistheLagrange-Helmholtztheorem.

Thethreemagnificationshavethefollowingrelationship:

αγ=β

Aperture

Anopticalelementthatrestrictsthelightbeampassingthroughtheopticalsystem.Itcanbetheframeoftheopticalelement(lens,mirror,etc.)itself,oritcanbeanadditionalopaquescreenwithholes.Thecenterofthediaphragmisusuallyontheopticalaxisandperpendiculartotheopticalaxis.

Eachopticalpartoftheopticalsystemisdefinedbyitsownlensframetodefineitslight-passinghole.Inmostcases,itisaroundhole.Sometimesfixedorvariablededicatedlightholesareaddedtothesystem.Amongalltheselightholes,onelightholemustplayaroleinlimitingtheapertureangleoftheon-axispointimagingbeam;anotherlightholeplaysaroleinlimitingtheimagingrange.Suchanapertureiscalledadiaphragm:theformeriscalledanaperturediaphragmoraneffectivediaphragm;thelatteriscalledafielddiaphragm.Anyopticalsystemmusthavesuchtwodiaphragms.

Aperturediaphragm

Amongthemultiplediaphragms,thelimitingeffectonthebeamisthegreatest,thatis,thediaphragmthatdeterminesthesizeoftheimagingbeam,alsoknownastheeffectivediaphragm.Theaperturediaphragmcanblockthelightthatdeviatesfromtheparaxiallightinthebeam,whichhasadirectimpactonthesharpness,accuracy,brightnessanddepthoftheimage.

Becausetheimagingbeamoftheon-axispointislimitedbytheaperturediaphragm,itiseasytoimaginethatwhenallthelightholesofthesystemareimagedintheobjectspacethroughtheopticalpartsinfrontofit,theon-axisobjectpointisopened.Theimagewiththesmallestangle,ortheimagewiththesmallestaperturewhentheobjectisatinfinity,mustbetheaperturestop.Theimageoftheaperturediaphragmintheobjectspaceiscalledtheentrancepupil,anditsopeningangletotheobjectpointiscalledthebeamapertureangleoftheobjectside.Similarly,theaperturediaphragmisformedintoanimageintheimagespacebytheopticalpartsbehindit,calledtheexitpupil.Itmustalsobeanapertureimagewiththesmallestopeningangleoftheimagepointontheaxis,andthisopeningangleisthebeamoftheimageside.Angleofaperture.Theentrancepupil,aperturestopandexitpupilareconjugated.Ifthediaphragmaberrationisneglected,theentrancepupilisthecommonentranceoftheimagingbeamatallpointsontheobjectplane;theexitpupilisthecommonexitoftheimagingbeam.Thelightpassingthroughthecenteroftheaperturediaphragmiscalledthechiefray,andbecauseoftheconjugaterelationship,italsopassesthroughthecenteroftheentrancepupilandthecenteroftheexitpupil.Therefore,itisgenerallysaidthatthechiefrayisthecenterlineoftheimagingbeam.

Thepositionoftheaperturediaphragmintheopticalsystemisrelatedtomanyfactors.Insomesystems,therearespecificrequirements.Forexample,theexitpupilofthevisualopticalsystemmustbelocatedoutsidetheeyepiecesothatthepupiloftheeyecancoincidewithit;inthetelecentricsystem,theaperturestopshouldbelocatedatthefocalpoint.Inaddition,thepositionoftheaperturestopisalsorelatedtotheaberrationcorrectionandthelateralsizeoftheopticalpartsofthesystem,andshouldbereasonablydeterminedduringdesign.

Fielddiaphragm

Thediaphragmthatdeterminestherangeofvision.Thefielddiaphragmcandeterminethesizeofthefieldofview.Thefielddiaphragmformedbythefrontopticalsystemiscalledtheentrancewindow,andtheimageformedbytherearsystemiscalledtheexitwindow.

Thefielddiaphragmisalightholeintheopticalsystemthatdeterminesitsimagingrange.Insystemswithintermediaterealimageplanes(suchasKeplertelescopesandmicroscopes)andsystemswithrealimageplanes(suchasphotographicsystems),thefielddiaphragmissetonthisimageplane.Theimageofthefielddiaphragmintheobjectspaceformedbytheopticalpartsinfrontofitiscalledtheentrancewindow.Theangleitopenstothecenteroftheentrancepupilisthesmallestofallapertureimages,andthisangleiscalledthefieldangle.Similarly,theimageformedbytheopticalpartsbehindthefielddiaphragmintheimagespaceiscalledtheexitwindow.Theentrancewindow,fielddiaphragmandexitwindowarealsoconjugate.Whenthefielddiaphragmissetontherealimageplaneortheintermediaterealimageplane,theentrancewindowandexitwindowarecoincidentwiththeobjectplaneandtheimageplanerespectively,andthefieldofviewhasaclearboundaryatthistime.Insituationswherethereisnorealimageorintermediaterealimageplane,suchaswhentheeyeisobservingthroughamagnifyingglassorGalileotelescope,thereisalwaysapartinthesystem.Itsclearapertureplaysaroleinlimitingthefieldofview.Theapertureofthetelescopeobjectivelensisthefielddiaphragmthatdeterminestherangeofthevisiblefieldofview.Obviously,theincidentwindowdoesnotcoincidewiththeobjectplaneatthistime,andthereisnoclearboundaryofthefieldofview.

Relativeaperture

TheratiooftheobjectivelensdiameterDtothefocallengthfintheimaginginstrument.Thephysicalquantityusedtodescribethelight-gatheringabilityoftheobjectivelens,becausetheluminousfluxdensityontheimagesurfaceisproportionalto(D/f)2.Thereciprocaloftherelativeapertureiscalledtheaperturefactor,orFnumber.Thephotographiclensisequippedwithanadjustablediaphragm(commonlyknownastheaperture),whichisusedtoadjusttherelativeaperturesize,therebyadjustingtheluminousfluxdensityonthephotosensitivefilm.AseriesofFnumbersareengravedonthelens.WhentheFnumberisreducedby2-1/2timesoftheoriginalvalue,theluminousfluxdensitywillincreaseby2times.ThegeneralFnumberseriesvalues​​are

1,1.4,2,2.8,4,5.6,8,11,16,22,...

Thevalues​​oftheabovefilesarecalculatedbythefollowingformula(Roundedup):11

Aperturefactor(Fnumber)=(21/2)x

xisapositiveinteger,calledtheindexoftheaperturecoefficient,alsocalledtheAVvalue.IntheaboveF-numberseries,thevalues​​oftheadjacenttwogearsdifferby2times,thecorrespondingluminousfluxdensitydiffersby2times,andtheAVvaluediffersbyonelevel.

VignettingPhenomenon

Underidealcircumstances,thebeamsattheon-axisandoff-axispointsarelimitedbytheaperturediaphragm,andhavebasicallythesamebeamapertureangle.IfthefieldofviewNottoobig,theimagesurfaceilluminanceoftheentirefieldofviewisbasicallyuniform.However,inactualopticalsystems,theoff-axispointimagingbeamisoftenlimitedbythelight-passingholesofotheropticalparts.Asaresult,thebeamangleoftheoff-axispointismuchsmallerthanthatoftheon-axispoint.Thisisbecausewhentheoff-axispointisalsoimagedwithabeamoflightthatfillstheentrancepupil,thoselensesthatarefarfromtheaperturestopneedtohavearelativelylargediameter,anditisverydifficulttocorrectforthefull-apertureoff-axisbeam.Therefore,inordertoimprovetheimagingqualityoftheoff-axispointandthelateralsizeoftheopticalpartsisnotparticularlylarge,themethodofappropriatelyreducingcertainlensdiametersisoftenusedtolimittheoff-axislightbeam.Thisphenomenoninwhichthelightbeamfromtheoff-axispointthatfillstheentrancepupilispartiallyinterceptedbysomeopticalpartsandcannotpassthroughtheopticalsystemiscalledbeamvignetting.Thefarthertheoff-axispointisfromtheopticalaxis,themoreserioustheinterceptionphenomenon(thatis,vignetting),andtheresultisthattheimagesurfaceilluminanceattheperipheryofthefieldofviewisgreatlyreduced.Ofcourse,mostopticalsystemsallowacertaindegreeofvignetting.

Imagingbeam

Theimagingbeamofanobjectpointisaspatiallightconewiththeobjectpointasthevertexandtheentrancepupilasthebase.Afterthebeampassesthroughtheopticalsystem,itsstructurewillchange.Foraxisymmetricopticalsystems(mostsystemsbelongtothiscategory),theon-axispointbeamalwayshassymmetricproperties,buttheoff-axispointbeamlosessymmetryafterpassingthroughthesystem.Inordertofacilitatetheunderstandingofthestructureofthisbeam,theplanebeamonthetwocharacteristicsurfacesisusuallyusedfordescription.

Theplanecontainingtheoff-axisobjectpointandtheopticalaxisiscalledthemeridianplane.Duetotheaxisymmetricnatureoftheopticalsystem,off-axisobjectpointscanalwaysbetakenonthedrawingplane,thatis,thepaperplaneisthemeridianplane.Thebeamlyingonthemeridianplaneiscalledthemeridianbeam.Obviously,thechiefraymustbearayinthemeridianbeam.

Theplanecontainingthechiefrayandperpendiculartothemeridianplaneiscalledthesagittalplane.Thebeamlyingonthesagittalplaneiscalledthesagittalbeam.Obviously,thechiefrayistheintersectionofthemeridianplaneandthesagittalplane.Sincethechiefraychangesitsdirectionbytherefractionandreflectionofeachsurfaceofthesystem,thesagittalplanealsochangesfacebyfaceinsteadofaunifiedplane.

Duetotheaxialsymmetryoftheopticalsystem,theon-axisspotbeamdoesnotneedtobeseparatedfromthemolecularmeridianbeamandthesagittalbeam,andtheoff-axisspotbeammustbesymmetricaltothemeridianplane.

Aberration

Theimageformedbythelens(orlensgroup)isnotexactlysimilartotheoriginalappearance.Becausetheangleofthelightemittedbytheobjectpointandthemainaxisofthelensistoolarge,itisfarawayfromtheaxis,ortherefractiveindexofthelensmaterialchangeswiththewavelengthofthelight.Thesizeofaberrationreflectstheprosandconsofimagingquality.Therearemainly7kindsofaberrations;formonochromaticlight,thereare5kinds,namely,sphericalaberration,coma,astigmatism,curvatureoffieldanddistortion.Forpolychromaticlight,therearetwokindsofchromaticaberrations,namelyaxialchromaticaberrationandverticalchromaticaberration.Eliminatingorreducingtheseaberrationsasmuchaspossibleisanimportanttaskinthedesignofopticalsystems.

Symmetricalcoaxialdrawing

Thepropertiesofsymmetricalcoaxial

①Theobjectpointontheopticalaxis,theimagepointisalsoOntheopticalaxis;②Theobjectpointinthesectionpassingtheopticalaxisiscoplanarwiththeimage;③Thepropertiesofanysectionpassingtheopticalaxisarethesame;④Aplaneperpendiculartotheaxishasthesamemagnificationinthesameplane;⑤Knowingthepositionandmagnificationoftwopairsofconjugatesurfaces,orknowingthepositionandmagnificationofapairofconjugatesurfaces,plusthetwopairsofconjugatepointsontheopticalaxis,candeterminetheimagingoftheidealopticalsystem.

Proofbydrawingmethod

①Thepositionandmagnificationoftwopairsofconjugatesurfacesareknown.Thepositionandmagnificationoftheconjugatesurface,aswellasthepositionsofthetwopairsofconjugatepointsontheaxis,areshownasfollows:

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