[工学]theopticsofspectroscopy

[工学]theopticsofspectroscopy

TheOpticsofSpectroscopy-ATUTORIALByJ.M.LernerandA.ThevenonSection1:DIFFRACTIONGRATINGSRULED&HOLOGRAPHIC·1.1BasicEquations·1.2AngularDispersion·1.3LinearDispersion·1.4WavelengthandOrder·1.5Resolving"Power"·1.6BlazedGratingso1.6.1LittrowConditiono1.6.2EfficiencyProfileso1.6.3EfficiencyandOrder·1.7DiffractionGratingStrayLighto1.7.1ScatteredLighto1.7.2Ghosts·1.8ChoiceofGratingso1.8.1WhentoChooseaHolographicGratingo1.8.2WhentoChooseaRuledGratingSection2:MONOCHROMATORS&SPECTROGRAPHS·2.1BasicDesigns·2.2FastieEbertConfiguration·2.3CzernyTurnerConfiguration·2.4CzernyTurner/FastieEbertPGSAberrationso2.4.1AberrationCorrectingPlaneGratings·2.5ConcaveAberrationCorrectedHolographicGratings·2.6CalculatingalphaandbetainaMonochromatorConfiguration·2.7MonochromatorSystemOptics·2.8ApertureStopsandEntranceandExitPupils·2.9ApertureRatio(f/value,f/Number),andNumericalAperture(NA)o2.9.1f/valueofaLensSystemo2.9.2f/valueofaSpectrometero2.9.3MagnificationandFluxDensity·2.10ExitSlitWidthandAnamorphism·2.11SlitHeightMagnification·2.12BandpassandResolutiono2.12.1InfluenceoftheSlits(P1(l))o2.12.2InfluenceofDiffractionP2(l))o2.12.3InfluenceofAberrations(P3(l))o2.12.4DeterminationoftheFWHMoftheInstrumentalProfileo2.12.5ImageWidthandArrayDetectors\no2.12.6Discussion·2.13OrderandResolution·2.14DispersionandMaximumWavelength·2.15OrderandDispersion·2.16ChoosingaMonochromator/SpectrographSection3:SPECTROMETERTHROUGHPUT&ETENDUE·3.1Definitionso3.1.1IntroductiontoEtendue·3.2RelativeSystemThroughputo3.2.1CalculationoftheEtendue·3.3FluxEnteringtheSpectrometer·3.4ExampleofCompleteSystemOptimizationwithaSmallDiameterFiberOpticLightSource·3.5ExampleofCompleteSystemOptimizationwithanExtendedLightSource·3.6VariationofThroughputandBandpasswithSlitWidthso3.6.1ContinuousSpectralSourceo3.6.2DiscreteSpectralSourceSection4:OPTICALSIGNALTONOISERATIOANDSTRAYLIGHT·4.1RandomStrayLighto4.1.1OpticalSignaltoNoiseRatioinaSpectrometero4.1.2TheQuantificationofSignalo4.1.3TheQuantificationofStrayLighto4.1.4OptimizationofSignaltoNoiseRatioo4.1.5ExampleofS/NOptimization·4.2DirectionalStrayLighto4.2.1IncorrectIlluminationoftheSpectrometero4.2.2ReentrySpectrao4.2.3GratingGhosts·4.3S/NRatioandSlitDimensionso4.3.1TheCaseforaSINGLEMonochromatorandaCONTINUUMLightSourceo4.3.2TheCaseforaSINGLEMonochromatorandMONOCHROMATICLighto4.3.3TheCaseforaDOUBLEMonochromatorandaCONTINUUMLightSourceo4.3.4TheCaseforaDOUBLEMonochromatorandaMONOCHROMATICLightSourceSection5:THERELATIONSHIPBETWEENWAVELENGTHANDPIXELPOSITIONONANARRAY·5.1TheDeterminationofWavelengthataGivenLocationonaFocalPlaneo5.1.1DiscussionofResults\no5.1.2DeterminationofthePositionofaKnownWavelengthintheFocalPlaneSection6:ENTRANCEOPTICS·6.1ChoiceofEntranceOpticso6.1.1ReviewofBasicEquations·6.2EstablishingtheOpticalAxisoftheMonochromatorSystemo6.2.1Materialso6.2.2Procedure·6.3IlluminatingaSpectrometer·6.4EntranceOpticsExampleso6.4.1ApertureMatchingaSmallSourceo6.4.2ApertureMatchinganExtendedSourceo6.4.3DemagnifyingaSource·6.5UseofFieldLenses·6.6PinholeCameraEffect·6.7SpatialFiltersReferences·Acknowledgments·Bibliography\nSection1:DiffractionGratingsRuled&HolographicDiffractiongratingsaremanufacturedeitherclassicallywiththeuseofarulingenginebyburnishinggrooveswithadiamondstylusorholographicallywiththeuseofinterferencefringesgeneratedattheintersectionoftwolaserbeams(formoredetailsseeDiffractionGratingsRuled&HolographicHandbook).Classicallyruledgratingsmaybeplaneorconcaveandpossessgrooveseachparallelwiththenext.Holographicgratinggroovesmaybeeitherparallelorofunequaldistributioninordertooptimizesystemperformance.Holographicgratingsaregeneratedonplane,spherical,toroidal,andmanyothersurfaces.Regardlessoftheshapeofthesurfaceorwhetherclassicallyruledorholographic,thetextthatfollowsisequallyapplicabletoeach;explanationsareprovidedwheretherearedifferences.1.1BasicEquationsBeforeintroducingthebasicequations,abriefnoteonmonochromaticlightandcontinuousspectramustfirstbeconsidered.Note:Monochromaticlighthasinfinitelynarrowspectralwidth.Goodsourceswhichapproximatesuchlightincludesinglemodelasersandverylowpressure,cooledspectralcalibrationlamps.Thesearealsovariouslyknownas"line"or"discreteline"sources.Note:Acontinuousspectrumhasfinitespectralwidth,e.g."whitelight".Inprincipleallwavelengthsarepresent,butinpracticea"continuum"isalmostalwaysasegmentofaspectrum.Sometimesacontinuousspectralsegmentmaybeonlyafewpartsofananometerwideandresemblealinespectrum.Theequationsthatfollowareforsystemsinairwherem0=1.Therefore,l=l0=wavelengthinair.DefinitionsUnitsa-(alpha)angleofincidencedegreesb-(beta)angleofdiffractiondegreesk-diffractionorderintegern-groovedensitygrooves/mmDV-theincludedangleordeviationangledegreesm0-refractiveindexdimensionlessl-wavelengthinvacuumnanometersl0-wavelengthinmediumofrefractiveindex,m0,wherel0=l/m0 1nm=10-6mm;1mm=10-3mm;1A=10-7mm Themostfundamentalgratingequationisgivenby:(1-1)Inmostmonochromatorsthelocationoftheentranceandexitslitsarefixedandthegrating\nrotatesaroundaplanethroughthecentreofitsface.Theangle,DV,is,therefore,aconstantdeterminedby:(1-2)Ifthevalueofaandbistobedeterminedforagivenwavelength(lambda),thegratingequationmaybeexpressedas:(1-3)AssumingthevalueofDV,isknown,aandbmaybedeterminedthroughEquations1-2and1-3(seeFigures1and2andSection2.6).Figure1.MonochromatorConfigurationFigure2.SpectrographConfigurationLA=EntrancearmlengthLB=ExitarmlengthatlnbH=AnglebetweentheperpendiculartothespectralplaneandthegratingnormalLH=PerpendiculardistancefromthespectralplanetogratingTable1showshowaandbvarydependingonthedeviationanglefora1200g/mmgratingsettodiffract500nminamonochromatorgeometrybasedonFigure1.\nTable1:Variationofa(angleofincidence)andb(angleofdiffraction),withDV(DeviationAngle),at500nminFirstOrderwith1200g/mmGrating.DVab017.45817.458(Littrow)1012.52622.526207.73627.736245.86129.861303.09433.09440-1.38238.61850-5.67044.3301.2AngularDispersionrad/nm(1-4)db=angularseparationbetweentwowavelengths(radians)dl=differentialseparationbetweentwowavelengths(nm)1.3LinearDispersionLineardispersiondefinestheextenttowhichaspectralintervalisspreadoutacrossthefocalfieldofaspectrometerandisexpressedinnm/mm,Å/mm,cm-1/mm,etc.Forexample,considertwospectrometers:oneinstrumentdispersesa0.1nmspectralsegmentover1mmwhiletheothertakesa10nmspectralsegmentandspreadsitover1mm.Itiseasytoimaginethatfinespectraldetailwouldbemoreeasilyidentifiedinthefirstinstrumentthanthesecond.Thesecondinstrumentdemonstrates"low"dispersioncomparedtothe"higher"dispersionofthefirst.Lineardispersionisassociatedwithaninstrument'sabilitytoresolvefinespectraldetail.Lineardispersionperpendiculartothediffractedbeamatacentralwavelength,l,isgivenby:nm/mm(1-5)whereLBistheeffectiveexitfocallengthinmmanddxistheunitintervalinmm.SeeFigure1.Inamonochromator,LBisthearmlengthfromthefocusingmirrortotheexitslitorifthegratingisconcave,fromthegratingtotheexitslit.Lineardispersion,therefore,varies\ndirectlywithcosb,andinverselywiththeexitpathlength,LB,diffractionorder(k),andgroovedensityn.Inaspectrograph,thelineardispersionforanywavelengthotherthanthatwavelengthwhichisnormaltothespectralplanewillbemodifiedbythecosineoftheangleofinclinationortiltangle(gamma)atwavelengthln.Figure2showsa"flatfield"spectrographasusedwithalineardiodearray.LinearDispersion(1-6)(1-7)(1-8)1.4WavelengthandOrderFigure3showsafirstorderspectrumfrom200to1000nmspreadoverafocalfieldinspectrographconfiguration.FromEquation1-1withagratingofgivengroovedensityandforagivenvalueofaandb:(1-9)sothatifthediffractionorderkisdoubled,lambdaishalved,etc.Figure3.DispersionandOrderIf,forexample,alightsourceemitsacontinuumofwavelengthsfrom20nmto1000nm,thenatthephysicallocationof800nminfirstorder(Figure3)wavelengthsof400,266.6,and200nmwillalsobepresentandavailabletothesamedetector.Inordertomonitoronlylightat800nm,filtersmustbeusedtoeliminatethehigherorders.Firstorderwavelengthsbetween200and380nmmaybemonitoredwithoutfiltersbecausewavelengthsbelow190nmareabsorbedbyair.If,however,theinstrumentisevacuatedorN2purged,higherorderfilterswouldagainberequired.1.5Resolving"Power"Resolving"power"isatheoreticalconceptandisgivenby\n(dimensionless)(1-10)where,dlisthedifferenceinwavelengthbetweentwospectrallinesofequalintensity.Resolutionisthentheabilityoftheinstrumenttoseparateadjacentspectrallines.Twopeaksareconsideredresolvedifthedistancebetweenthemissuchthatthemaximumofonefallsonthefirstminimumoftheother.ThisiscalledtheRayleighcriterion.Itmaybeshownthat:(1-11)l=thecentralwavelengthofthespectrallinetoberesolvedWg=theilluminatedwidthofthegratingN=thetotalnumberofgroovesonthegratingThenumericalresolvingpower"R"shouldnotbeconfusedwiththeresolutionorbandpassofaninstrumentsystem(SeeSection2).Theoretically,a1200g/mmgratingwithawidthof110mmthatisusedinfirstorderhasanumericalresolvingpowerR=1200x110=132,000.Therefore,at500nm,thebandpassisequalto:Inarealinstrument,however,thegeometryofuseisfixedbyEquation1-1.Solvingfork:(1-12)Buttheruledwidth,Wg,ofthegrating:(1-13)(1-14)Aftersubstitutionof(1-12)and(1-13)in(1-11),resolvingpowermayalsobeexpressedas:(1-15)Consequently,theresolvingpowerofagratingisdependenton:·Thewidthofthegrating·Thecentrewavelengthtoberesolved·ThegeometryoftheuseconditionsBecausebandpassisalsodeterminedbytheslitwidthofthespectrometerandresidualsystemaberrations,anachievedbandpassatthislevelisonlypossibleindiffractionlimitedinstrumentsassuminganunlikely100%oftheoretical(seeSection2forfurtherdiscussion).\n1.6BlazedGratingsBlazeisdefinedastheconcentrationofalimitedregionofthespectrumintoanyorderotherthanthezeroorder.Blazedgratingsaremanufacturedtoproducemaximumefficiencyatdesignatedwavelengths.Agratingmay,therefore,bedescribedas"blazedat250nm"or"blazedat1micron"etc.byappropriateselectionofgroovegeometry.Ablazedgratingisoneinwhichthegroovesofthediffractiongratingarecontrolledtoformrighttriangleswitha"blazeangle,w,"asshowninFigure4.However,apexanglesupto110°maybepresentespeciallyinblazedholographicgratings.Theselectionofthepeakangleofthetriangulargrooveoffersopportunitytooptimisetheoverallefficiencyprofileofthegrating.1.6.1LittrowConditionBlazedgratinggrooveprofilesarecalculatedfortheLittrowconditionwheretheincidentanddiffractedraysareinautocollimation(i.e.,a=b).Theinputandoutputrays,therefore,propagatealongthesameaxis.Inthiscaseatthe"blaze"wavelengthlB.(1-16)Forexample,theblazeangle(w)fora1200g/mmgratingblazedat250nmis8.63°infirstorder(k=1).Figure4.LittrowConditionforaSingleGrooveofaBlazedGrating1.6.2EfficiencyProfilesUnlessotherwiseindicated,theefficiencyofadiffractiongratingismeasuredintheLittrowconfigurationatagivenwavelength.(1-17)\n(1-18)Relativeefficiencymeasurementsrequirethemirrortobecoatedwiththesamematerialandusedinthesameangularconfigurationasthegrating.SeeFigures5aand5bfortypicalefficiencycurvesofablazed,ruledgrating,andanonblazed,holographicgrating,respectively.Asageneralapproximation,forblazedgratingsthestrengthofasignalisreducedby50%attwo-thirdstheblazewavelength,and1.8timestheblazewavelength.Figure5a.EfficiencyCurveofaBlazed,RuledGratingFigure5b.EfficiencyCurveofaNon-blazed,HolographicGrating1.6.3EfficiencyandOrderAgratingblazedinfirstorderisequallyblazedinthehigherordersTherefore,agratingblazedat600nminfirstorderisalsoblazedat300nminsecondorderandsoon.Efficiencyinhigherordersusuallyfollowsthefirstorderefficiencycurve.Foragratingblazedinfirstorder,themaximumefficiencyforeachofthesubsequenthigherordersdecreasesastheorderkincreases.TheefficiencyalsodecreasesthefurtheroffLittrowconditionsinwhichthegratingisused(e.g.a¹b).Holographicgratingsmaybedesignedwithgrooveprofilesthatdiscriminateagainsthighorders.ThismaybeparticularlyeffectiveintheUVandVISusinglaminargrooveprofiles\ncreatedbyionetching.Note:Justbecauseagratingis"nonblazed"doesnotnecessarilymeanthatitislessefficient!SeeFigure5bshowingtheefficiencycurveforan1800g/mmsinusoidalgroovedholographicgrating.1.7DiffractionGratingStrayLightLightotherthanthewavelengthofinterestreachingadetector(oftenincludingoneormoreelementsof"scatteredlight")isreferredtoasstraylight.1.7.1ScatteredLightScatteredlightmaybeproducedbyeitherofthefollowing:·Randomlyscatteredlightduetosurfaceimperfectionsonanyopticalsurface.·Focusedstraylightduetononperiodicerrorsintherulingofgratinggrooves.1.7.2GhostsIfthediffractiongratinghasperiodicrulingerrors,aghost,whichisnotscatteredlight,willbefocusedinthedispersionplane.Ghostintensityisgivenby:(1-19)where,IG=ghostintensityIP=parentintensityn=groovedensityk=ordere=errorinthepositionofthegroovesGhostsarefocusedandimagedinthedispersionplaneofthemonochromator.Straylightofaholographicgratingisusuallyuptoafactoroftentimeslessthanthatofaclassicallyruledgrating,typicallynon-focused,andwhenpresent,radiatesthrough2pisteradians.Holographicgratingsshownoghostsbecausetherearenoperiodicrulingerrorsand,therefore,oftenrepresentthebestsolutiontoghostproblems.1.8ChoiceofGratings1.8.1WhentoChooseaHolographicGrating1.Whengratingisconcave.2.Whenlaserlightispresent,e.g.,Raman,laserfluorescence,etc.3.Anytimegroovedensityshouldbe1200g/mmormore(upto6000g/mmand120mmx140mminsize)foruseinnearUV,VIS,andNIRspectralregions.4.WhenworkingintheUVbelow200nmdownto3nm.5.Forhighresolutionwhenhighgroovedensitywillbesuperiortoalowgroovedensitygratingusedinhighorder(k>1).6.Wheneveranionetchedholographicgratingisavailable.1.8.2WhentoChooseaRuledGrating\n1.WhenworkinginIRabove1.2μm,ifanionetchedholographicgratingisunavailable.2.Whenworkingwithverylowgroovedensity,e.g.,lessthan600g/mm.Remember,ghostsandsubsequentstraylightintensityareproportionaltothesquareoforderandgroovedensity(n2andk2fromEquation1-18).Bewareofusingruledgratingsinhighorderorwithhighgroovedensity.Section2:Monochromators&Spectrographs2.1BasicDesignsMonochromatorandspectrographsystemsformanimageoftheentranceslitintheexitplaneatthewavelengthspresentinthelightsource.Therearenumerousconfigurationsbywhichthismaybeachieved-onlythemostcommonarediscussedinthisdocumentandincludesPlaneGratingSystems(PGS)andAberrationCorrectedHolographicGrating(ACHG)systems.DefinitionsLA-entrancearmlengthLB-exitarmlengthh-heightofentranceslith'-heightofimageoftheentranceslita-angleofincidenceb-angleofdiffractionw-widthofentranceslitw'-widthofentranceslitimageDg-diameterofacirculargratingWg-widthofarectangulargratingHg-heightofarectangulargrating2.2Fastie-EbertConfigurationAFastie-Ebertinstrumentconsistsofonelargesphericalmirrorandoneplanediffractiongrating(seeFigure6).Aportionofthemirrorfirstcollimatesthelightwhichwillfallupontheplanegrating.Aseparateportionofthemirrorthenfocusesthedispersedlightfromthegratingintoimagesoftheentranceslitintheexitplane.Itisaninexpensiveandcommonlyuseddesign,butexhibitslimitedabilitytomaintainimagequalityoffaxisduetosystemaberrationssuchassphericalaberration,coma,astigmatism,andacurvedfocalfield.\nFigure6.PlaneGratingConfiguration2.3Czerny-TurnerConfigurationTheCzerny-Turner(CZ)monochromatorconsistsoftwoconcavemirrorsandoneplanodiffractiongrating(seeFigure7).AlthoughthetwomirrorsfunctioninthesameseparatecapacitiesasthesinglesphericalmirroroftheFastie-Ebertconfiguration,i.e.,firstcollimatingthelightsource(mirror1),andsecond,focusingthedispersedlightfromthegrating(mirror2),thegeometryofthemirrorsintheCzerny-Turnerconfigurationisflexible.Byusinganasymmetricalgeometry,aCzerny-Turnerconfigurationmaybedesignedtoproduceaflattenedspectralfieldandgoodcomacorrectionatonewavelength.Sphericalaberrationandastigmatismwillremainatallwavelengths.Itisalsopossibletodesignasystemthatmayaccommodateverylargeoptics.Figure7.Czerny-TurnerConfiguration\n2.4Czerny-Turner/Fastie-EbertPGSAberrationsPGSspectrometersexhibitcertainaberrationsthatdegradespectralresolution,spatialresolution,orsignaltonoiseratio.Themostsignificantareastigmatism,coma,sphericalaberrationanddefocusing.PGSsystemsareusedoffaxis,sotheaberrationswillbedifferentineachplane.Itisnotwithinthescopeofthisdocumenttoreviewtheconceptsanddetailsoftheseaberrations,(reference4)however,itisusefultounderstandtheconceptofOpticalPathDifference(OPD)whenconsideringtheeffectsofaberrations.Basically,anOPDisthedifferencebetweenanactualwavefrontproducedanda"referencewavefront"thatwouldbeobtainediftherewerenoaberrations.Thisreferencewavefrontisaspherecenteredattheimageoraplaneiftheimageisatinfinity.Forexample:Defocusingresultsinraysfindingafocusoutsidethedetectorsurfaceproducingablurredimagethatwilldegradebandpass,spatialresolution,andopticalsignal-to-noiseratio.AgoodexamplecouldbethesphericalwavefrontilluminatingmirrorM1inFigure7.DefocusingshouldnotbeaprobleminaPGSmonochromatorusedwithasingleexitslitandaPMTdetector.However,inanuncorrectedPGSthereisfieldcurvaturethatwoulddisplaydefocusingtowardstheendsofaplanarlineardiodearray.GeometricallycorrectedCZconfigurationssuchasthatshowninFigure7nearlyeliminatetheproblem.TheOPDduetodefocusingvariesasthesquareofthenumericalaperture.Comaistheresultoftheoff-axisgeometryofaPGSandisseenasaskewingofraysinthedispersionplaneenlargingthebaseononesideofaspectrallineasshowninFigure8.Comamayberesponsibleforbothdegradedbandpassandopticalsignal-to-noiseratio.TheOPDduetocomavariesasthecubeofthenumericalaperture.ComamaybecorrectedatonewavelengthinaCZbycalculatinganappropriateoperatinggeometryasshowninFigure7.Figure8.TheEffectofComaSphericalaberrationistheresultofraysemanatingawayfromthecentreofanopticalsurfacefailingtofindthesamefocalpointasthosefromthecentre(SeeFigure9).TheOPDduetosphericalaberrationvarieswiththefourthpowerofthenumericalapertureandcannotbecorrectedwithouttheuseofasphericoptics.\nFigure9.TheEffectofSphericalAberrationAstigmatismischaracteristicofanoff-axisgeometry.Inthiscaseasphericalmirrorilluminatedbyaplanewaveincidentatanangletothenormal(suchasmirrorM2inFigure7)willpresenttwofoci:thetangentialfocus,Ft,andthesagittalfocus,FS.Astigmatismhastheeffectoftakingapointattheentranceslitandimagingitasalineperpendiculartothedispersionplaneattheexit(seeFigure10),therebypreventingspatialresolutionandincreasingslitheightwithsubsequentdegradationofopticalsignaltonoiseratio.TheOPDduetoastigmatismvarieswiththesquareofnumericalapertureandthesquareoftheoffaxisangleandcannotbecorrectedwithoutemployingasphericoptics.Figure10.EffectsofAstigmatisminaConcaveMirrorused"Off-Axis"2.4.1AberrationCorrectingPlaneGratingsRecentadvancesinholographicgratingtechnologynowpermitscompletecorrectionofALLaberrationspresentinasphericalmirrorbasedCZspectrometeratonewavelength,with\nexcellentmitigationoverawidewavelengthrange(Reference12).2.5ConcaveAberrationCorrectedHolographicGratingsBoththemonochromatorsandspectrographsofthistypeuseasingleholographicgratingwithnoancillaryoptics.Inthesesystems,thegratingbothfocusesanddiffractstheincidentlight.Withonlyoneopticintheirdesign,thesedevicesareinexpensiveandcompact.Figure11aillustratesanACHGmonochromator.Figure11billustratesanACHGspectrographinwhichthelocationofthefocalplaneisestablishedby:bH-Anglebetweenperpendiculartospectralplaneandgratingnormal.LH-Perpendiculardistancefromspectralplanetograting.Figure11a.AnACHGMonochromatorFigure11b.AnACHGSpectrograph\n2.6CalculatingalphaandbetainaMonochromatorConfigurationFromEquation(1-2),(remainsconstant)TakingthisequationandEquation(1-3),(2-1)UseEquations(2-1)and(1-2)todetermineaandb,respectively.SeeTable3forworkedexamples.Note:Inpracticethehighestwavelengthattainableislimitedbythemechanicalrotationofthegrating.Thismeansthatdoublingthegroovedensityofthegratingwillhalvethespectralrange(seeSection2.14).2.7MonochromatorSystemOpticsTounderstandhowacompletemonochromatorsystemischaracterized,itisnecessarytostartatthetransferopticsthatbringslightfromthesourcetoilluminatetheentranceslit(seeFigure12). Herewehave"unrolled"thesystemanddrawnitinalinearfashion.AS-aperturestopL1-lens1M1-mirror1M2-mirror2G1-gratingp-objectdistancetolensL1q-imagedistancefromlensL1F-focallengthoflensL1(focusofanobjectatinfinity)d-theclearapertureofthelens(L1indiagram)W-half-angles-areaofthesources'-areaoftheimageofthesource\n2.8ApertureStopsandEntranceandExitPupilsAnaperturestop(AS)limitstheopeningthroughwhichaconeoflightmaypassandisusuallylocatedadjacenttoanactiveoptic.Apupiliseitheranaperturestoportheimageofanaperturestop.Theentrancepupiloftheentrance(transfer)opticsinFigure12isthevirtualimageofASasseenaxiallythroughlensL1fromthesource.Theentrancepupilofthespectrometeristheimageofthegrating(G1)seenaxiallythroughmirrorM1fromtheentranceslit.TheexitpupiloftheentranceopticsisASitselfseenaxiallyfromtheentranceslitofthespectrometer.TheexitpupilofthespectrometeristheimageofthegratingseenaxiallythroughM2fromtheexitslit.2.9ApertureRatio(f/value,f/Number),andNumericalAperture(NA)ThelightgatheringpowerofanopticisrigorouslycharacterizedbyNumericalAperture(NA).NumericalApertureisexpressedby:whereμistherefractiveindex(μ=1inair)(2-2)andf/valueby:(2-3)Table2:Relationshipbetweenf/value,half-angle,andnumericalaperturef/valuef/2f/3f/5f/7f/10f/15n(degrees)14.489.65.74.02.91.9NA0.250.160.100.070.050.032.9.1f/valueofaLensSystemf/valueisalsogivenbytheratioofeithertheimageorobjectdistancetothediameterofthepupil.When,forexample,alensisworkingwithfiniteconjugatessuchasinFig.12,thereisaneffectivef/valuefromthesourcetoL1(withdiameterAS)givenby:(2-4)andfromL1totheentranceslitby:(2-5)Inthesectionsthatfollowf/valuewillalwaysbecalculatedassumingthattheentranceorexitpupilsareequivalenttotheaperturestopforthelensorgratingandthedistancesare\nmeasuredtothecenterofthelensorgrating.Whenthef/valueiscalculatedinthiswayforf/2orgreater(e.g.f/3,f/4,etc.),thensinwis~tanwandtheapproximationisgood.However,ifanactiveopticistofunctionatanf/valuesignificantlylessthanf/2,thenthef/valueshouldbedeterminedbyfirstcalculatingNumericalAperturefromthehalf-angle.2.9.2f/valueofaSpectrometerBecausetheangleofincidencealphaisalwaysdifferentineithersignorvaluefromtheangleofdiffractionbeta(exceptinLittrow),theprojectedsizeofthegratingvarieswiththewavelengthandisdifferentdependingonwhetheritisviewedfromtheentranceorexitslits.InFigures13aand13b,thewidthsW'andW''aretheprojectionsofthegratingwidthasperceivedattheentranceandexitslits,respectively.Figure13.Projectionofthegratingwidthon(a)theEntranceand(b)theExit.Todeterminethef/valueofaspectrometerwitharectangulargrating,itisfirstnecessarytocalculatethe"equivalentdiameter",D',asseenfromtheentranceslitandD"asseenfromtheexitslit.ThisisachievedbyequatingtheprojectedareaofthegratingtothatofacirculardiscandthencalculatingthediameterD'orD".(2-6)(2-7)Inaspectrometer,therefore,thef/valueinwillnotequalthef/valueout.(2-8)(2-9)where,forarectangulargrating,D'andD"aregivenby:\n(2-10)(2-11)where,foracirculargrating,D'andD"aregivenby:(2-12)(2-13)Table3showshowthef/valuechangeswithwavelength.Table3Calculatedvaluesforf/valueinandf/valueoutforaCzerny-Turnerconfigurationwith68x68mm,1800g/mmgratingandLA=LB=F=320nm.Dv=24°.l(nm)abf/valueinf/valueout2001.4022.604.174.343205.1229.124.184.4650015.3939.394.254.7468026.7350.734.415.2480035.4059.404.625.842.9.3MagnificationandFluxDensityInanyspectrometersystemalightsourceshouldbeimagedontoanentranceslit(aperture)whichisthenimagedontotheexitslitandsoontothedetector,sample,etc.Thisprocessinevitablyresultsinthemagnificationordemagnificationofoneormoreoftheimagesofthelightsource.Magnificationmaybedeterminedbythefollowingexpansions,takingasanexamplethesourceimagedbylensL1inFigure12ontotheentranceslit:(2-14)Similarly,fluxdensityisdeterminedbytheareathatthephotonsinanimageoccupy,sochangesinmagnificationareimportantifafluxdensitysensitivedetectororsamplearepresent.Changesinthefluxdensityinanimagemaybecharacterizedbytheratiooftheareaoftheobject,S,totheareaoftheimage,S',fromwhichthefollowingexpressionsmaybederived:(2-15)Theserelationshipsshowthattheareaoccupiedbyanimageisdeterminedbytheratioofthe\nsquareofthef/values.Consequently,itistheEXITf/valuethatdeterminesthefluxdensityintheimageofanobject.Thoseusingphotographicfilmasadetectorwillrecognizetheserelationshipsindeterminingtheexposuretimenecessarytoobtainacertainsignal-to-noiseratio.2.10ExitSlitWidthandAnamorphismAnamorphicopticsarethoseopticsthatmagnify(ordemagnify)asourcebydifferentfactorsintheverticalandhorizontalplanes(seeFigure14).Figure14.(a)Verticaland(b)HorizontalMagnificationInthecaseofadiffractiongrating-basedinstrument,theimageoftheentranceslitisNOTimaged1:1intheexitplane(exceptinLittrowandperpendiculartothedispersionplaneassumingLA=LB).Thismeansthatinvirtuallyallcommercialinstrumentsthetraditionofmaintainingequalentranceandexitslitwidthsmaynotalwaysbeappropriate.Geometrichorizontalmagnificationdependsontheratioofthecosinesoftheangleofincidence,alpha,andtheangleofdiffraction,beta,andtheLB/LAratio(Equation216).Magnificationmaychangesubstantiallywithwavelength(seeTable4).(2-16)Table4illustratestherelationshipbetweenalpha,beta,dispersion,horizontalmagnificationofentranceslitimage,andbandpass.Table4RelationshipBetweenDispersion,HorizontalMagnification,andBandpassinaCzernyTurnerMonochromator.LA=320mm,LB=320mm,Dv=24°,n=1800g/mmEntranceslitwidth=1mm\nWavelength(nm)a(degrees)b(degrees)dispersion(nm/mm)horizontalmagnificationbandpass*(nm)200-1.422.601.601.081.742601.8425.841.561.111.743205.1229.121.461.141.733808.4732.471.411.171.7244011.8835.881.341.211.7050015.3939.391.271.251.6756019.0143.011.191.291.6462022.7846.781.101.351.6068026.7350.731.001.411.5574030.9154.910.881.491.4980035.4059.401.601.601.42Exitslitwidthmatchedtoimageofentranceslit.*Astheinclinationofthegratingbecomesincreasinglylarge,comainthesystemwillincrease.Consequently,inspiteofthefactthatthebandpassat800nmissuperiortothatat200nm,itisunlikelythatthefullimprovementwillbeseenbytheuserinsystemsoflessthanf/8.2.11SlitHeightMagnificationSlitheightmagnificationisdirectlyproportionaltotheratiooftheentranceandexitarmlengthsandremainsconstantwithwavelength(exclusiveoftheeffectsofaberrationsthatmaybepresent).(2-17)Note:Geometricmagnificationisnotanaberration!2.12BandpassandResolutionInthemostfundamentalsensebothbandpassandresolutionareusedasmeasureofaninstrument'sabilitytoseparateadjacentspectrallines.Assumingacontinuumlightsource,thebandpass(BP)ofaninstrumentisthespectralintervalthatmaybeisolated.Thisdependsonmanyfactorsincludingthewidthofthegrating,systemaberrations,spatialresolutionofthedetector,andentranceandexitslitwidths.Ifalightsourceemitsaspectrumwhichconsistsofasinglemonochromaticwavelengthlo(Figure15)andisanalyzedbyaperfectspectrometer,theoutputshouldbeidenticaltothespectrumoftheemission(Fig.16)whichisaperfectlineatpreciselylo.\nInreality,spectrometersarenotperfectandproduceanapparentspectralbroadeningofthepurelymonochromaticwavelength.Thelineprofilenowhasfinitewidthandisknownasthe"instrumentallineprofile"(instrumentalbandpass)(seeFigure17).Theinstrumentalprofilemaybedeterminedinafixedgratingspectrographconfigurationwiththeuseofareasonablymonochromaticlightsourcesuchasasinglemodedyelaser.Foragivensetofentranceandexitslitparameters,thegratingisfixedattheproperorientationforthecentralwavelengthofinterestandthelaserlightsourceisscannedinwavelength.Theoutputofthedetectorisrecordedanddisplayed.Theresultanttracewillshowintensityversuswavelengthdistribution.Foramonochromatorthesameresultwouldbeachievedifamonochromaticlightsourceisintroducedintothesystemandthegratingrotated.ThebandpassisthendefinedastheFullWidthatHalfMaximum(FWHM)ofthetraceassumingmonochromaticlight.Anyspectralstructuremaybeconsideredtobethesumofaninfinityofsinglemonochromaticlinesatdifferentwavelengths.Thus,thereisarelationshipbetweentheinstrumentallineprofile,therealspectrumandtherecordedspectrum.LetB(l)betherealspectrumofthesourcetobeanalysed.LetF(l)betherecordedspectrumthroughthespectrometer.LetP(l)betheinstrumentallineprofile.(2-18)TherecordedfunctionF(l)istheconvolutionoftherealspectrumandtheinstrumentallineprofile.Theshapeoftheinstrumentallineprofileisafunctionofvariousparameters:·thewidthoftheentranceslit·thewidthoftheexitslitorofonepixelinthecaseofamultichanneldetector·diffractionphenomena·aberrations·qualityofthesystem'scomponentsandalignmentEachofthesefactorsmaybecharacterizedbyaspecialfunctionPi(l),eachobtainedbyneglectingtheotherparameters.TheoverallinstrumentallineprofileP(l)isrelatedtotheconvolutionoftheindividualterms:\n(2-19)2.12.1InfluenceoftheSlits(P1(l))Iftheslitsareoffinitewidthandtherearenoothercontributingeffectstobroadentheline,andif:Went=widthoftheimageoftheentranceslitWex=widthoftheexitslitorofonepixelinthecaseofamultichanneldetectorDl1=lineardispersionxWentDl2=lineardispersionxWexthentheslit'scontributiontotheinstrumentallineprofileistheconvolutionofthetwoslitfunctions(seeFigure18).Figure18.ConvolutionofEntrancewithExitSlits2.12.2InfluenceofDiffraction(P2(l))Ifthetwoslitsareinfinitelynarrowandaberrationsnegligible,thentheinstrumentallineprofileisthatofaclassicdiffractionpattern.Inthiscase,theresolutionofthesystemisthewavelength,l,dividedbythetheoreticalresolvingpowerofthegrating,R(Equation1-11).2.12.3InfluenceofAberrations(P3(l))Ifthetwoslitsareinfinitelynarrowandbroadeningofthelineduetoaberrationsislargecomparedtothesizeduetodiffraction,thentheinstrumentallineprofileduetodiffractionisenlarged.2.12.4DeterminationoftheFWHMoftheInstrumentalProfileInpracticetheFWHMofF(l)isdeterminedbytheconvolutionofthevariouscausesoflinebroadeningincluding:dl(resolution):thelimitingresolutionofthespectrometerisgovernedbythelimitinginstrumentallineprofileandincludessystemaberrationsanddiffractioneffects.dl(slits):bandpassdeterminedbyfinitespectrometerslitwidths.dl(line):naturallinewidthofthespectrallineusedtomeasuretheFWHM.Assumingagaussianlineprofile(whichisnotthecase),areasonableapproximationoftheFWHMisprovidedbytherelationship:\n(2-20)Ingeneral,mostspectrometersarenotroutinelyusedatthelimitoftheirresolutionsotheinfluenceoftheslitsmaydominatethelineprofile.FromFigure18theFWHM,duetotheslits,isdeterminedbyeithertheimageoftheentranceslitortheexitslit,whicheverisgreater.Ifthetwoslitsareperfectlymatchedandaberrationsminimalcomparedtotheeffectoftheslits,thentheFWHMwillbehalfthewidthatthebaseofthepeak.(Aberrationsmay,however,stillproducebroadeningofthebase).Bandpass(BP)isthengivenby:BP=FWHM~lineardispersionx(exitslitwidthortheimageoftheentranceslit,whicheverisgreater).InSection2-10imageenlargementthroughthespectrometerwasreviewed.TheimpactonthedeterminationofthesystembandpassmaybedeterminedbytakingEquation2-16tocalculatethewidthoftheimageoftheentranceslitandmultiplyingitbythedispersion(Equation1-5).Bandpassisthengivenby:(2-21)ThemajorbenefitofoptimisingtheexitslitwidthistoobtainmaximumTHROUGHPUTwithoutlossofbandpass.ItisinterestingtonotefromEquations(2-21)and(1-5)that:·Bandpassvariesascosa·Dispersionvariesascosb2.12.5ImageWidthandArrayDetectorsBecausetheimageintheexitplanechangesinwidthasafunctionofwavelength,theuserofanarraytypedetectormustbeawareofthenumberofpixelsperbandpassthatareilluminated.Itisnormaltoallocate3-6pixelstodetermineonebandpass.Iftheimageincreasesinsizebyafactorof1.5,thenclearlyphotonscontainedwithinthatbandpasswouldhavetobecollectedover4-9pixels.ForadiscussionoftherelationbetweenwavelengthandpixelpositionseeSection5.TheFWHMthatdeterminesbandpassisequivalenttothewidthoftheimageoftheentranceslitcontainingatypicalmaximumof80%ofavailablephotonsatthewavelengthofinterest;theremainderisspreadoutinthebaseofthepeak.Anyimagemagnification,therefore,equallyenlargesthebasespreadingtheentirepeakoveradditionalpixels.2.12.6Discussiona)BandpasswithMonochromaticLightTheinfinitelynarrownaturalspectralbandwidthofmonochromaticlightis,bydefinition,lessthanthatoftheinstrumentalbandpassdeterminedbyEquation2-20.(Averynarrowbandwidthistypicallyreferredtoasa"line"becauseofitsappearanceinaspectrum).Inthiscaseallthephotonspresentwillbeatexactlythesamewavelengthirrespectiveofhowtheyarespreadoutintheexitplane.Theimageoftheentranceslit,therefore,willconsistexclusivelyofphotonsatthesamewavelengtheventhoughthereisafiniteFWHM.Consequently,bandpassinthisinstancecannotbeconsideredasawavelengthspreadaroundthecenterwavelength.If,forexample,monochromaticlightat250nmispresentand\ntheinstrumentalbandpassissettoproduceaFWHMof5nm,thisdoesNOTmean250nm±2.5nmbecausenowavelengthotherthan250nmispresent.Itdoesmean,however,thataspectrumtracedout(wavelengthvs.intensity)willproducea"peak"withanapparentFWHMof"5nm"duetoinstrumentalandNOTspectrallinebroadening.b)Bandpasswith"Line"SourcesofFiniteSpectralWidthEmissionlineswithfinitenaturalspectralbandwidthsareroutinelyfoundinalmostallformsofspectroscopyincludingemission,Raman,fluorescence,andabsorption.Inthesecasesspectramaybeobtainedthatseemtoconsistoflineemission(orabsorption)bands.If,however,oneofthese"lines"isanalysedwithaveryhighresolutionspectrometer,itwouldbedeterminedthatbeyondacertainbandpassnofurtherlinenarrowingwouldtakeplaceindicatingthatthenaturalbandwidthhadbeenreached.DependingontheinstrumentsystemthenaturalbandwidthmayormaynotbegreaterthanthebandpassdeterminedbyEquation2-20.Ifthenaturalbandwidthisgreaterthantheinstrumentalbandpass,thentheinstrumentwillperformasiftheemission"line"isaportionofacontinuum.Inthiscasethebandpassmayindeedbeviewedasaspectralspreadof±0.5BParoundacenterwavelengthatFWHM.Example1:Figure19showsasomewhatcontrivedspectrumwherethefirsttwopeaksareseparatedontherecordingby32mm.TheFWHMofthefirstpeakisthesameasthesecondbutislessthanthethird.Thisimpliesthatthenaturalbandwidthofthethirdpeakisgreaterthanthebandpassofthespectrometerandwouldnotdemonstratespectralnarrowingofitsbandwidthevenifevaluatedwithaveryhighresolutionspectrometer.Thefirstandsecondpeaks,however,maywellpossessnaturalbandwidthslessthanthatshownbythespectrometer.Inthesetwocases,thesameinstrumentoperatingunderhigherbandpassconditions(narrowerslits)maywellrevealeitheradditional"lines"thathadpreviouslybeenincorporatedintojustoneband,orasimplenarrowingofthebandwidthuntileitherthelimitofthespectrometerorthelimitingnaturalbandpasshavebeenreached.Figure19.StripChartRecordingPlottingWavelengthvs.Intensitywhere*BP=FWHM(inmm)xDispersion\nExample2:Aresearcherfindsaspectruminajournalthatwouldbeappropriatetoreproduceonaninhousespectrometer.Thefirsttaskistodeterminethebandpassdisplayedbythespectrum.Ifthisinformationisnotgiven,thenitisnecessarytostudythespectrumitself.Assumingthatthewavelengthsofthetwopeaksareknown,thenthedistancebetweenthemmustbemeasuredwitharulerasaccuratelyaspossible.Ifthewavelengthdifferenceisfoundtobe1.25nmandthisincrementisspreadover32mm(seeFigure19),therecordeddispersionofthespectrum=1.25/32=0.04nm/mm.ItisnowpossibletodeterminethebandpassbymeasuringthedistanceinmmattheFullWidthatHalfMaximumheight(FWHM).Letussaythatthisis4mm;thebandpassoftheinstrumentisthen4mmx0.04nm/mm=0.16nm.AlsoassumingthatthespectrometerdescribedinTable4istobeused,thenfromEquation2-21andthelistofmaximumwavelengthsdescribedinTable6,thefollowingoptionsareavailabletoproduceabandpassof0.16nm:Table5:VariationofDispersionandSlitWidthtoProduce0.16nmBandpassina320mmFocalLengthCzerny-TurnerGrooveDensity(g/mm)Dispersion(nm/mm)EntranceSlitWidth(microns)3009.2176004.63512002.37018001.510724001.1513936000.77208Thebestchoicewouldbethe3600g/mmoptiontoprovidethelargestslitwidthpossibletopermitthegreatestamountoflighttoenterthesystem.2.13OrderandResolutionIfagivenwavelengthisusedinhigherorders,forexample,fromfirsttosecondorder,itisconsideredthatbecausethedispersionisdoubled,soalsoisthelimitingresolution.Inamonochromatorinwhichthereareancillaryopticssuchasplaneorconcavemirrors,lenses,etc.,alinearincreaseinthelimitingresolutionmaynotoccur.Thereasonsforthisinclude:·Changesinsystemaberrationsasthegratingisrotated(e.g.,coma)·Changesinthediffractedwavefrontofthegratinginhigherorders(mostseriouswithclassicallyruledgratings)·Residualsystemaberrationssuchassphericalaberration,coma,astigmatism,andfieldcurvatureswampinggratingcapabilities(particularlylowf/value,e.g.,f/3,f/4systems)Evenifthefullwidthathalfmaximumismaintained,adegradationinlineshapewilloftenoccur-thebaseofthepeakusuallybroadenswithconsequentdegradationof\nthepercentageofavailablephotonsintheFWHM.2.14DispersionandMaximumWavelengthThelongestpossiblewavelength(lmax1)aninstrumentwillreachmechanicallywithagratingofagivengroovedensityisdeterminedbythelimitofmechanicalrotationofthatgrating.Consequently,inchangingfromanoriginalgroovedensity,n1,toanewgroovedensity,n2,thenewhighestwavelength(lmax2)willbe:(2-22)Table6:VariationinMaximumWavelengthwithGrooveDenistyinaTypicalMonochromatorLA=LB=F=320mm,DV=24°.Inthisexamplemaximumwavelengthatmaximumpossiblemechanicalrotationofa1200g/mmgrating=1300nmGrooveDensity(g/mm)Dispersion(nm/mm)MaxWavelength(nm)15018.4104003009.252006004.6260012002.3130018001.586724001.1565036000.77433FromTable6itisclearthatifa3600g/mmgratingisrequiredtodiffractlightabove433nm,thesystemwillnotpermitit.If,however,adispersionof0.77nm/mmisrequiredtoproduceappropriateresolutionat,say,600nm,asystemshouldbeacquiredwith640mmfocallength(Equation(1-5)).Thiswouldproduceadispersionof0.77nm/mmwitha2400g/mmgratingandalsopermitmechanicalrotationupto650nm.2.15OrderandDispersionInExample2,Section2.12.6,thesolutiontothedispersionproblemcouldbesolvedbyusinga2400g/mmgratingina640mmfocallengthsystem.Asdispersionvarieswithfocallength(LB),groovedensity(n),andorder(k);forafixedLBatagivenwavelength,thedispersionequation(Equation1-5)simplifiesto:kn=constantTherefore,iffirstorderdispersion=1.15nm/mmwitha2400g/mmgratingthesamedispersionwouldbeobtainedwitha1200g/mmgratinginsecondorder.Keepinginmindthatkl=constantforagivengroovedensity,n,(Equation1-9),usingsecondorderwithan1800g/mmgratingtosolvethelastproblemwouldnotworkbecausetofind600nminsecondorder,itwouldbenecessarytooperateat1200nminfirstorder,whenitmaybeseen\ninTable6thatthemaximumattainablefirstorderwavelengthis867nm.However,ifadispersionof0.77nm/mmisnecessaryintheWat250nm,thiswavelengthcouldbemonitoredat500nminfirstorderwiththe1800g/mmgratingandobtainasecondorderdispersionof0.75nm/mm.Inthiscaseanyfirstorderlightat500nmwouldbesuperimposedontopofthe250nmlight(andvice-versa).Wavelengthselectivefiltersmaythenbeusedtoeliminatetheunwantedradiation.Themaindisadvantagesofthisapproacharethatthegratingefficiencywouldnotbeasgreatasanoptimizedfirstordergratingandorder-sortingfiltersaretypicallyinefficient.Ifaclassicallyruledgratingisemployed,ghostsandstraylightwillincreaseasthesquareoftheorder.2.16ChoosingaMonochromator/SpectrographSelectaninstrumentbasedon:·Asystemthatwillallowthelargestentranceslitwidthforthebandpassrequired.·Thehighestdispersion.·Thelargestopticsaffordable.·Longestfocallengthaffordable.·Highestgroovedensitythatwillaccommodatethespectralrange.·Opticsandcoatingsappropriateforspecificspectralrange.·Entranceopticswhichwilloptimizeetendue.·Iftheinstrumentistobeusedatasinglewavelengthinanon-scanningmode,thenitmustbepossibletoadjusttheexitslittomatchthesizeoftheentranceslitimage.Remember:f/valueisnotalwaysthecontrollingfactorofthroughput.Forexample,lightmaybecollectedfromasourceatf/1andprojectedontotheentranceslitofanf/6monochromatorsothattheentireimageiscontainedwithintheslit.Thenthesystemwilloperateonthebasisofthephotoncollectioninthef/lconeandnotthef/6coneofthemonochromator.SeeSection3.Section3:SpectrometerThroughputandEtendue3.1DefinitionsFlux:fluxisgivenbyenergy/time(photons/sec,orwatts),emittedfromalightsourceorslitofgivenarea,intoasolidangle(Q)atagivenwavelength(orbandpass).Intensity(I):Thedistributionoffluxatagivenwavelength(orbandpass)persolidangle(watts/steradian).Radiance(Luminance)(B):Theintensitywhenspreadoveragivensurface.AlsodefinedasB=Intensity/SurfaceAreaoftheSource(watts/steradian/cm2).3.1.1IntroductiontoEtendue\nFigure20.TypicalMonochromatorSystemS=areaofsourceS'=areaofentranceslitS"=areaofmirrorM1S*=areaofexitslitW=halfangleoflightcollectedbyL1W'=halfangleoflightsubmittedbyL1W"=halfangleoflightcollectedbyM1W*=halfangleoflightsubmittedbyM2L1=lensusedtocollectlightfromsourceM1=sphericalcollimatingCzernyTurnermirrorM2=sphericalfocusingCzernyTurnermirrorAS=aperturestopLS=illuminatedareaoflensL1p=distancefromobjecttolensL1q=distancefromlensL1toimageofobjectattheentranceslitG1=diffractiongratingGeometricetendue(geometricextent),G,characterizestheabilityofanopticalsystemtoacceptlight.Itisafunctionofthearea,S,oftheemittingsourceandthesolidangle,Q,intowhichitpropagates.Etenduetherefore,isalimitingfunctionofsystemthroughput.(3-1)(3-2)FollowingintegrationforaconicalbeamtheaxisofwhichisnormaltosourceofareaS(seeFigure20),(3-3)\nEtendueisaconstantofthesystemandisdeterminedbytheLEASToptimizedsegmentoftheentireopticalsystem.Geometricetenduemaybeviewedasthemaximumbeamsizetheinstrumentcanaccept,therefore,itisnecessarytostartatthelightsourceandensurethattheinstrumentincludingancillaryopticscollectsandpropagatesthemaximumnumberofphotons.FromEquation(3-3),etenduewillbeoptimizedif(3-4)Asomewhatsimplerapproximationmaybeusedifthespectrometerf/valueisslowerthanf/5(f/6,f/7,etc.).Then,(3-5)where(3-6)ThefollowingisusedinsteadofEquation3-5etc.(3-7)ThisapproximationisgoodwhentanW~sinW~(radians).Theerroratf/5~1%andatf/1~33%.Sincenumericalaperture=msinW=NA,then:(3-8)Thisformisveryusefulwhenworkingwithfiberopticsormicroscopeobjectives.3.2RelativeSystemThroughput3.2.1CalculationoftheEtendueh=heightofentranceslit(mm)w=entranceslitwidthbandpass/dispersion(mm)F=focallengthLA(mm)n=groovedensityofgrating(g/mm)GA=illuminatedgratingarea(mm2)Sg=projectedilluminatedareaofgrating=GAxcosalpha(mm2)k=\norderBP=bandpass(nm)SES=areaofentranceslit(mm2)TheareaofentranceslitSES=wxh(refertoEquation2-21)where:(mm)(3-9)Therefore,(mm2)(3-10)Tocalculateetendue,G,(3-11)and(3-12)then(3-13)Relativesystemthroughputis,therefore,proportionalto:·h/F·groovedensity(n)·order(k)·areaofgrating(GA)·bandpass(BP)Theratioh/Fimpliesthattheetenduemaybeincreasedbyenlargingtheheightoftheentranceslit.Inpracticethiswillincreasestraylightandmayalsoreduceresolutionorbandpassresultingfromanincreaseinsystemaberrations.3.3FluxEnteringtheSpectrometerFluxisgivenbyradiancetimesetendue:\n(3-14)(3-15)whereBisafunctionofthesource,S'istheareaoftheentranceslit(oremittingsource),andW',isthehalfconeangleilluminatingthespectrometerentranceslit.Becauseflux,etendue,andradiancemustbeconservedbetweenobjectandimage,assumingnootherlosses,theabovetermsareallweneedtodeterminethetheoreticalmaximumthroughput.3.4ExampleofCompleteSystemOptimizationwithaSmallDiameterFiberOpticLightSourceFirst,calculatetheetendueofthelightsourcegiventhat:Thefiberhasacorediameterof50mmandemitslightwithaNA=0.2wheretheareaofthefibercoreis:thenTherefore,etendueofthelightsource=G=2.46x10-4Nextcalculatetheetendueofthespectrometerassumingabandpassof0.5nmat500mm:n=1800g/mm(Given)k=1(Given)DV=24°(Given)LA=F=LB=320mm(Given)GAA=58x58mmruledareaofthegrating(Given)a500nm=15.39°(FromEquation2-1)b500nn=39.39°(FromEquation1-2)f/valuespectrometer=f/5(FromEquation2-10)NAspectrometer=0.1(FromEquation2-3)f/valuefiberoptic=f/2.5(FromEquation2-3)NAfiberoptic=0.2(Given)h=tobedeterminedCalculateoperatingslitdimensions:\nEntranceslitwidth,w,fromEquation(3-9)From(2-16)exitslitwidth==0.3725mmInthiscaseweshallkeeptheentranceslitheightandexitslitheightat0.2987mm.TheetendueofthespectrometerisgivenbyEquation3-13.Then,G=[(0.2987).(1800).(1).(58x58).(0.5)]/[(320).(106)]Consequently,theetendueofthelightsource(2.46x10-4)issignificantlylessthantheetendueofthespectrometer(2.83x103).Ifthefiberwassimplyinsertedbetweentheentranceslitjaws,theNA=0.2ofthefiberwoulddrasticallyoverfilltheNA=0.1ofthespectrometer(f/2.5tof/5)bothlosingphotonsandcreatingstraylight.InthiscasetheSYSTEMetenduewouldbedeterminedbytheareaofthefiber'scoreandtheNAofthespectrometer.Thepointnowistore-imagethelightemanatingfromthefiberinsuchawaythattheetendueofthefiberisbroughtuptothatofthespectrometertherebypermittingtotalcaptureandpropagationofallavailablephotons.Thisisachievedwiththeuseofancillaryopticsbetweenthefiberopticsourceandthespectrometerasfollows:(NA)in=NAofFiberOptic(NA)out=NAofSpectrometerthen(2-15)and(2-14)Thethinlensequationis(3-16)\nwhereFinthiscaseisthefocallengthforanobjectatinfinityandpandqarefiniteobjectandimagecoordinates.Takinga60mmdiameterlensasanexamplewhereF=100mm,thenmagnification=NAin/NAout=q/p=0.2/0.1=2SubstitutinginEquation(3-16)1/100=1/p+1/2pAftersolving,p=150mmandq=300mmbutf/valueout=1/2(NA)out=q/dthend=300x0.2=60mm.f/valuein=1/2(NA)in=q/dthend=150x0.4=60mm.Therefore,thelightfromthefiberiscollectedbyalenswitha150mmobjectdistance,p,andprojectsanimageofthefibercoreonthespectrometerentranceslit300mm,q,fromthelens.Thef/valuesarematchedtoboththelightpropagatingfromthefiberandtothatofthespectrometer.Theimage,however,ismagnifiedbyafactorof2.Consideringthatwerequireanentranceslitwidthof0.2987mmtoproduceabandpassof0.5nm,theresultingimageof100mm(2x50mcorediameter)underfillstheslit,therebyensuringthatallthelightcollectedwillpropagatethroughthesystem.Asamatterofinterest,becausetheimageofthefibercorehasawidthlessthantheslitjaws,thebandpasswillbedeterminedbytheimageofthecoreitself.Straylightwillbelessenedbyreducingtheslitjawstoperfectlycontainthecore'simage(seeSection4).3.5ExampleofCompleteSystemOptimizationwithanExtendedLightSourceAn"extendedlightsource"isonewherethesourceitselfisconsiderablylargerthantheslitwidthnecessarytoproduceanappropriatebandpass.Inthiscasetheetendueofthespectrometerwillbelessthanthatofthelightsource.UsingaHgspectrallampasanexampleofanextendedsource,theetendue1isasfollows:Areaofsource=50mm(height)x5mm(width)(Given)=250mm2W=90°Then,Assumingthesamespectrometerandbandpassrequirementsasinthefiberopticsourceexample(3-4)theslitwidthsandetendueofthespectrometerwillalsobethesameaswillthespectrometeretendue.Therefore,theetendueofthelightsourceisdrasticallylarger(785comparedto2.8x103)thanthatofthespectrometer.BecausetheetendueofthesystemisdeterminedbythesegmentwiththeLEASTetendue,themaximumlightcollectionfromthelightsourcewillbegovernedbythelightgatheringpowerofthespectrometer.Inthepreviousexampletheentranceslitheight(h)wastakenas0.2987mm.Withanextendedsource,however,itispossibletouseagreaterslitheight,sointhiscasewewilltakeentranceandexitslitheightsof3mm(evenhigherslitsmaybepossiblebutstraylightisdirectlyproportionaltoslitheight).Thespectrometeretendue,therefore,increasesfrom4.7x103to4.7x102\nThisthenwillbetheeffectiveetendueofthesystemandwillgovernthelightsource.ThebestwaytoaccommodatethisistosampleanareaoftheHglampequivalenttotheentranceslitareaandimageitontotheentranceslitwiththesamesolidangleasthatdeterminedbythediffractiongrating(Equation3-12).Todeterminethegeometricconfigurationoftheentranceopticstakethesame60mmdiameterlens(L1)witha100mmfocallengthasthatusedinthepreviousexample.Weknowthattheentranceslitdimensionsdeterminetheareaofthesourcetobesampled,therefore,SES=areaofthesourceS.Thesourceshouldbeimaged1:1ontotheentranceslit,therefore,magnification=1.Takingthethinlensequation1/F=1/p+1/qwhereq/p=1p=2Fandq=2FTheHglampshouldbeplaced200mmawayfromlensL1whichinturnshouldbe200mmfromtheentranceslit.Thediameterrequiredtoproducethecorrectf/valueisthendeterminedbythespectrometerwhosef/value=5.Therefored=200/5=40mmThe60mmlensshould,therefore,beaperturestoppeddownto40mmtopermitthecorrectsolidangletoenterthespectrometer.Thissystemwillnowachievemaximumlightcollection.3.6VariationofThroughputandBandpasswithSlitWidthsAssume:Theimageofthesourceoverfillstheentranceslit.wi=originalentranceslitwidth(e.g.,100mm)wo=exitslitwidth(originalwidthofentranceslitimage,e.g.,110mm)3.6.1ContinuousSpectralSourceForexample,atungstenhalogenlamporaspectrumwherelinewidthsaresignificantlygreaterthaninstrumentalbandpass(thisisoftenthecaseinfluorescenceexperiments).Throughputwillvaryasafunctionoftheproductofchangeinbandpassandchangeinetendue.Case1:Doubletheentranceslitwidth,wi,butkeepexitslitunchanged,therefore:entranceslit=2wi(200mm)exitslit=wo(110mm)Etendueremainsthesame(determinedbyexitslit).Bandpassisdoubled.Throughputisdoubled.Case2:Doubletheexitslitwidth,wo,butkeepentranceslitunchanged,therefore:entranceslit=wi(100mm)exitslit=2wo(220mm)Etendueremainsthesame(determinedbyentranceslit).Bandpassisdoubled.Throughputisdoubled.Note:Doublingtheexitslitallowsabroadersegmentofthespectrumthroughtheexitand,therefore,increasesthephotonflux.Case3:Doubleboththeentranceandexitslitwidths,therefore:\nentranceslit=2wi(200mm)exitslit=2wo(220mm)Etendueisdoubled.Bandpassisdoubled.Throughputisquadrupled.3.6.2DiscreteSpectralSourceAlightsourcethatwillproduceanumberofmonochromaticwavelengthsiscalledadiscretespectralsource.Inpracticeanapparentlymonochromaticlinesourceisoftenadiscretesegmentofacontinuum.Itisassumedthatthenaturallinewidthislessthantheminimumachievablebandpassoftheinstrument.Throughputthenvariesasafunctionofchangeinetendueandisindependentofbandpass.Case1:Doubletheentranceslitwidth,wi,butkeepexitslitunchanged,therefore:entranceslit=2wi(200mm)exitslit=wo(110mm)Etendueremainsthesame(determinedbyexitslit)Bandpassisdoubled.Throughputremainsthesame.Case2:Doubletheexitslitwidth,wo,butkeepentranceslitunchangedtherefore:entranceslit=wi(100mm)exitslit=wo(220mm)Etendueremainsthesame(determinedbyentranceslit).Bandpassisdoubled.Throughputremainsthesame.Note:Foradiscretespectralsource,doublingtheexitslitwidthwillnotcauseachangeinthethroughputbecauseitdoesnotallowanincreaseofphotonfluxfortheinstrument.Case3:Doubleboththeentranceandexitslitwidths,therefore:entranceslit=2wi(200mm)exitslit=2wo(220mm)Etendueisdoubled.Bandpassisdoubled.Throughputisdoubled.Section4:OpticalSignaltoNoiseRatioandStrayLightStraylightandtheeffectithasonOpticalSignaltoNoiseratio(S/N)fallsintooneoftwomajorcategorieseithera)randomscatterfrommirrors,gratings,etc.orb)directionalstraylightsuchasreflections,reentryspectra,gratingghostsandgratinggeneratedfocusedstraylight.4.1RandomStrayLightConsiderfirsthowmuchlightthereistobeginwithattheprimarywavelengthofinterestthencompareittootherwavelengthsthatmaybepresentasscatter.4.1.1OpticalSignaltoNoiseRatioinaSpectrometer\nTodeterminetheratioofsignaltonoiseeachofthecomponentsmustfirstbequantified.4.1.2TheQuantificationofSignal,fuFluxEnteringtheInstrument(fT):Ses=areaoftheentranceslit=(hw)BT=totalradianceoflightenteringtheinstrumentGA=totalilluminatedareaofthegratingThenfromEqns.(3-14),(3-11)and(3-12)totalfluxenteringtheinstrumentisgivenby:(4-1)Sotocalculatethefluxatagivenwavelengththatwillexittheinstrumentfu,letElbetheefficiencyofthegratingatwavelengthlandBltheradianceoflightatwavelengthlinthefocalplane.Assumenowthattheareaoftheexitslitisperfectlymatchedtotheimageoftheentranceslit.IfSex=theareaoftheexitslit=(h'w')(orifaspectrograph,thetotalareaofthepixels).However,therearemanycaseswhenthesizeoftheimageoftheentranceslitislargerthantheexitslitduetoimageaberrations.Lightlossesofthiskindare"geometriclosses"andmaybecharacterizedbythetransmissionthroughthesystemTg.Tg=1foraperfectsystem.Thefluxatagivenwavelengthcollectedbythedetectorisgivenby:(4-2)whereTgisthegeometrictransmissionatwavelengthl.4.1.3TheQuantificationofStrayLight,fd,andS/NRatio,fu/fdTheluminanceofrandomlyscatteredlightisproportionaltothefluxperunitareaonthescatteringoptic.Tocalculatestraylightduetorandomscatter:LetG=theetenduebetweenthegratingandthedetectorelement.(4-3)LetBd=theradianceofstraylightproportionaltothetotalfluxdensityfT/GAC=afactorwhichexpressesthequalityoftheoptics(includingthegrating)asafunctionofrandomscatter.\n(4-4)Totalscatteredfluxisproportionaltotheradianceofthescatteredlight,totheareaoftheentranceslit,andthesolidanglewithwhichtheexitslitperceivestheilluminatedoptic.Randomfluxisgivenby:fd=BdGthen,(4-5)andtheratiooffluxatthewavelengthofinterestfuandtherandomfluxfdis:(4-6)4.1.4OptimizationofSignaltoNoiseRatioOptimizationrequirestwothings:themaximizationof(fu/fd)andtheeliminationofstrayreflections.TakingthetermsofEquation(4-6)inturn:C:Obtainthehighestqualityopticsincludingaholographicgratingifoneisavailable.El:Ensurethatthegratingisoptimizedtobemostefficientatthewavelengthsofinterest.LA2/(hw):Unfortunately,thesemaynotbetotallyfreeparametersbecauseofdispersionandbandpassrequirements.Tgl:Thedominantcauseofimageenlargementperpendiculartodispersionisastigmatism.Ifpresent,theheightoftheexitslitmustbeenlargedtocollectallavailablelightwithsubsequentlossinopticalsignaltonoiseratio.NewaberrationcorrectingplanegratingsforuseincertainCZspectrometersenhanceS/Nratiobysignificantlyreducingastigmatism.Bl/BT:Thistermistheratioofthebrightnessatthewavelengthofinterestltothetotalbrightnessofthesource.;itisnotusuallyauseraccessiblefunction.4.1.5ExampleofS/NOptimizationThisisanexerciseincompromise.Forexample,takearesearcherwhoownsa500mmfocallengthmonochromatorandisdissatisfiedwiththesignaltonoiseratio.Equation4-6suggeststhatS/Nimprovementmaybeachievedbyutilizingalongerfocallengthinstrument;a1000mmspectrometerjusthappenstobeavailable.Assumingthebandpassrequirementisconstantforbothexperiments,thegroovedensity,wavelengthoptimization,andsizeofthegratingisthesame,thenthroughputishalved(fromEquation3-13,allotherthingsbeingequal,etenduewillbeproportionaltotheratioofthefocallengths).OpticalS/Nratiowouldbeimprovedbyafactorof2.ReferringtoEquation4-6,theratioofthesquaresofthefocallengthsgivesafactoroffourandassumingtheslitheightsremainthesametheslitwidthsinthe1000mmfocallengthsystemwouldproducedoubletheareaofthe500mmsystem,thereby,losingafactoroftwo.Thequestiontoresolveiswhether\npickingupafactorof2inS/Nratiowasworthlosinghalfthethroughput.Inthisexample,theremayalsobeareductioninthevalueofTg,astigmatismbeingproportionaltothenumericalaperture(whichinthiscasewouldbedoublethatofthe500mmsystem).Itisalsoworthcheckingtheavailabilityofamoresensitivedetector.Itissometimespossibletoobtainsmallerdetectorswithgreatersensitivitythanlargerones.Ifthisisthecasethetotalthroughputlossmaynotbeassevereasoriginallyanticipated.4.2DirectionalStrayLight4.2.1IncorrectIlluminationoftheSpectrometerIftheopticsareoverfilled,thenacombinationofstrayreflectionsoffmirrormounts,screwheads,fluorescencefromanodizedcastings,etc.maybeexpected.Thesolutionissimple:optimizesystemetenduewithwelldesignedentranceopticsandusefieldlensestoconjugateaperturestops(pupils).Thisisachievedbyprojectinganimageoftheaperturestopoftheentranceopticsviaa"field"lensattheentranceslitontotheaperturestopofthespectrometer(usuallythegrating)andthenimagethegratingontotheaperturestopoftheexitopticswithafieldlensattheexitslit.ThisisreviewedinSection6.4.2.2ReentrySpectraInsomeCZmonochromatorconfigurations(especiallywithlowgroovedensitygratingsusedinthevisibleorUV)adiffractedwavelengthotherthanthatonwhichtheinstrumentissetmayreturntothecollimatingmirrorandbereflectedbacktothegratingwhereitmayberediffractedandfinditswaytotheexitslit.Ifthisproblemisserious,agoodsolutionistoplaceamaskperpendiculartothegroovesacrossthecenterofthegrating.Themaskshouldbethesameheightastheslits.Iftheprecisewavelengthisknown,itispossibletocalculatetheexactimpactpointonthegratingthatthereflectedwavelengthhits.Inthiscasetheonlymaskingnecessaryisatthatpoint.Amorecommonexampleofthisproblemisfoundinmanyspectrometers(irrespectiveoftype)whenalinearormatrixarrayisusedasthedetector.Reflectionsbacktothegratingmaybesevere.Thesolutionistoeithertiltthearrayuptothepointthatresolutionbeginstodegradeorifthesystemisbeingdesignedforthefirsttimetoworkoutofplane.4.2.3GratingGhostsClassicallyruledgratingsexhibitghostsandstraylightthatarefocusedinthedispersionplaneand,therefore,cannotberemediedotherthanbyobtainingadifferentgratingthatdisplaysacleanerperformance.Oneofthebestsolutionsistoemployanionetchedblazedholographicgratingthatprovidesgoodefficiencyatthewavelengthofinterestandnoghostswhatsoever.Anyremainingstraylightisrandomlyscatteredandnotfocused.4.3S/NRatioantSlitDimensionsThissectionreviewstheeffectsofslitdimensionsonS/Nratioforeitheracontinuumoramonochromaticlightsourceinsingleordoublemonochromator.Itisassumedthattheentranceandexitslitdimensionsarematched.4.3.1TheCaseforaSINGLEMonochromatorandaCONTINUUMLightSource\nVariationwithSlitWidthObservation:S/NratiodoesNOTvaryasafunctionofslitwidth.Explanation:FromEquation3-13andareviewofSection3,signalthroughputincreasesasthesquareoftheslitwidth(slitwidthdeterminestheentranceetendueandthebandpass).Because,thelightsourceisacontinuumtheincreaseinsignalvariesdirectlywithbothbandpassandetendue.The"noisesignal"alsovarieswiththesquareoftheslitwidthsasshowninEquation4-5.Consequently,boththesignalandthenoisechangeinthesameratio.VariationwithSlitHeightObservation:S/Nratiovariesinverselywithslitheight.Explanation:Signalthroughputvarieslinearlywithslitheight(fromEquation3-13).Noise,however,variesasthesquareofslitheight(fromEquation4-5).Consequently,S/Nratiovariesinverselywithslitheight.4.3.2TheCaseforaSINGLEMonochromatorandMONOCHROMATICLightVariationwithSlitWidthObservation:S/Nratiovariesinverselywithslitwidth.Explanation:Signalthroughputvariesdirectlywithslitwidth.(Eventhoughbandpassincreases,onlytheetenduegovernsthenumberofphotonsavailable).The"noise"isproportionaltothesquareoftheslitwidth.Consequently,S/Nratioisinverselyproportionaltotheslitwidth.VariationwithSlitHeightObservation:S/Nratiovariesinverselywithslitheight.Explanation:Signalthroughputvarieslinearlywithslitheight.Noisevariesasthesquareoftheslitheight.Consequently,S/Nratiovariesinverselywithslitheight.4.3.3TheCaseforaDOUBLEMonochromatorandaCONTINUUMLightSourceVariationwithSlitWidthObservation:S/Nratiovariesinverselywithslitwidth.Explanation:S/Nratioattheexitofthefirstmonochromatordoesnotvarywithslitwidth,however,thelightnowilluminatingtheopticsofthesecondmonochromatorisapproximatelymonochromaticandtheS/Nratiowillnowvaryinverselywithslitwidthinthesecondmonochromator.VariationwithSlitHeightObservation:S/Nratiovariesastheinversesquareofslitheight.Explanation:TheS/Nratiovarieslinearlywithslitheightattheexitofthefirstmonochromator.Thesecondmonochromatorviewing"monochromatic"lightwillalsochangetheS/Nratioinverselywithslitheight,therefore,thetotalvariationinS/Nratioattheexitofthesecondmonochromatorwillvaryasthesquareoftheslitheight.4.3.4TheCaseforaDOUBLEMonochromatorandaMONOCHROMATICLightSourceVariationwithSlitWidthObservation:S/Nratiovarieswiththeinversesquareoftheslitwidth.Explanation:AttheexitofthefirstmonochromatorS/Nvariesinverselywithslitwidth.ThesecondmonochromatoralsoilluminatedbymonochromaticlightagainchangestheS/Nratio\ninverselywithslitwidth.Consequently,thetotalchangeinS/Nratioisproportionaltotheinversesquareoftheslitwidth.VariationwithSlitHeightObservation:S/Nratiovarieswiththeinversesquareofslitheight.Explanation:EachofthetwomonochromatorsvariestheS/NratioinverselywithslitheightsothetotalvariationinS/Nratiovariesastheinversesquareoftheslitheight.Section5:TheRelationshipBetweenWavelengthandPixelPositiononanArrayForamonochromatorsystembeingusedinspectrographconfigurationwithasolidstatedetectorarray,theusershouldbeawareofthefollowing:·Thefocalplanemaybetiltedbyanangle,g.Therefore,thepixelpositionnormallyoccupiedbytheexitslitmayNOTmarkthenormaltothefocalplane.·Thedispersionandimagemagnificationmayvaryoverthefocalplane.·Asaconsequenceof(b),thenumberofpixelsperbandpassmayvarynotonlyacrossthefocalplanebutwillalsovarydependingonthewavelengthcoverage.Figure21(a)illustratesatiltedfocalplanethatmaybepresentinCzerny-Turnermonochromators.Inthecaseofaberrationcorrectedholographicgratings,g,bH.andLHareprovidedasstandardoperatingparameters.OperatingmanualsformanyCzernyTurner(CZ)andFastieEbert(FE)monochromatorsrarelyprovideinformationonthetiltofthefocalplane,therefore,itmaybenecessaryfortheusertodeducethevalueofgamma.Thisismosteasilyachievedbytakingawellknownspectrumanditerativelysubstitutingincrementalvaluesof±g,untilthewavelengthappearingateachpixelcorrespondstocalculatedvalues.Figure21.SpectrographwithFocalPlane(a)Inclinedand(b)NormaltotheCentralWavelength\n5.1TheDeterminationofWavelengthataGivenLocationonaFocalPlaneThetermsusedbelowareconsistentforaberrationcorrectedholographicconcavegratingsaswellasCzernyTurnerandFastieEbertspectrometers.lc-Wavelength(innm)atcenterofarray(whereexitslitwouldusuallybelocated)LA-Entrancearmlength(mm)LBln-Exitarmlengthtoeachwavelengthlocatedonthefocalplane(mm)LBlc-Exitarmlengthtolc(CzernyTurnerandFastieEbertmonochromatorsLA=LBlc=F)LH-Perpendiculardistancefromgratingorfocusingmirrortothefocalplane(mm)F-Instrumentfocallength.ForCZandFEmonochromatorsLA=F=LB.(mm)bH-AnglefromLHtothenormaltothegrating(thiswillvaryinascanninginstrument)bln-Angleofdiffractionatwavelengthnblc-AngleofdiffractionatcenterwavelengthHBln-DistancefromtheinterceptofthenormaltothefocalplanetothewavelengthlnHBlc-DistancefromtheinterceptofthenormaltothefocalplanetothewavelengthlcPmin-Pixel#atextremitycorrespondingtolmin(e.g.,#1)Pmax-Pixel#atextremitycorrespondingtolmax(e.g.,#1024)Pw-Pixelwidth(mm)Pc-Pixel#atlc(e.g.,#512)Pl-Pixel#atlng-Inclinationofthefocalplanemeasuredatthelocationnormallyoccupiedbytheexitslit,lc.(Thisisusuallythecenterofthearray.However,providedthatthepixelmarkingthislocationisknown,thearraymaybeplacedastheuserfindsmostuseful).Forthisreason,itisveryconvenienttouseaspectrometerthatpermitssimpleinterchangefromscanningtospectrographbymeansofaswingawaymirror.Theinstrumentmaythenbesetupwithastandardslitusing,forexample,amercurylamp.Switchingtospectrographmodeenablesidentificationofthepixel,Pc,illuminatedbythewavelengthpreviouslyattheexitslit.TheequationsthatfollowareforCzernyTurnertypeinstrumentswhereg=0°inonecaseandg¹0°intheother.Case1g=0°.SeeFigure21(b).LH=LB=Fatlc(mm)bH=batlcHBln=Pw(Pl-Pc)(mm)HBisnegativeforwavelengthsshorterthanlc.HBispositiveforwavelengthslongerthanlc.(5-1)Note:Thesecretofsuccess(andreasonforfailure)isfrequentlythelevelofunderstandingofthesignconvention.Beconsistent,makereasonablyaccuratesketcheswheneverpossible.Tomakeacalculation,aandbatlccanbedeterminedfromEquations1-2and2-1.Atthispointthevalueforaisusedinthecalculationofallvaluesblnforeachwavelength.Then\n(5-2)Case2:gdoesnotequal0°SeeFigure21(a).LH=Fcosg(whereF=LBlc)(5-3)bH=blc+g(5-4)HBlc=Fsing(5-5)HBln=Pw(PlPc)+HBlc(5-6)bln=bHtan-1(HBln/LH)(5-7)AgainkeepingsignificantconcernforthesignofHBln,proceedtocalculatethevalueblnafterfirstobtainingaatlcthenuseEquation(5-2)tocalculateln.INPRACTICE,THIRDANDFOURTHDECIMALPLACEACCURACYISNECESSARY.Indeedthelongertheinstrument'sfocallength,thegreaterthecontributionofroundingerrors.Toillustratetheabovediscussionaworkedexample,takenfromareadilyavailablecommercialinstrument,isprovided.Example:Thefollowingaretypicalresultsforafocalplaneinclinedby2.4°inCzernyTurnermonochromatorusedinspectrographmode.LB=320mmatlc=Fn=1800g/mmD=24°LH=319.719mmg=2.4°HBlc=13.4mmArraylength=25.4mm;lcappears12.7mmfromendofarraylmin,lmax=wavelengthatarrayextremitieslerrormin,max=wavelengththoughttobeatarrayextremityifg=0°Disp=dispersion(Equation1-5)(nm/mm)mag=magnificationindispersionplane(Equation2-16)Dl(g=0°)lminorlmax-lerror(nm)Dd=Actualdistanceoflerrorfromextremepixel(μm)Table7OperatingParametersforaCZSpectrometerwitha2.4°TiltatlcontheSpectralPlaneComparedtoa0°Tilt.nmlmin229.9463lc250lmax269.7469lmin381.4545lc400lmax418.1236lmin686.1566lc700lmax713.1999a1.298649.595028.0963bH27.698635.995054.496b23.031725.298627.573231.328033.595035.869549.829452.096354.3707Disp.1.591.571.541.481.451.411.121.071.01Mag1.091.111.131.161.181.221.371.441.51\nDl0.05100.0150.04800.0140.03700.011Dd+320-10+320-10+320-105.1.1DiscussionofResultsExaminationoftheresultsgivenintheworkedexampleindicatesthefollowingphenomena:A.Ifanarraywith25mmpixelswasusedandthefocalplanewasassumedtobenormaltolcratherthantheactual2.4°,atleastaonepixelerror(32mm)wouldbepresentatlmin(thismaynotseemlikemuch,butitisincrediblehowmuchlostsleepanddiscussiontimehasbeenspentattemptingtorationalizethisdilemma).B.A25mmentranceslitisimagedinthefocalplanewithawidthof27.25mm(1.09x25)at229.946nm(whenlc=250nm)butisimagedwithawidthof37.75mmat713.2nm(1.51x25)(whenlc=700nm),Indeedinthislastcasethedifferenceinimagewidthatlmincomparedtolmaxvariesbyover10%acrossthearray.C.Ifthearraydidnotlimittheresolution,thena25mmentranceslitwidthwouldproduceabandpassof0.04nm.Giventhat,intheaboveexamplewithg=0°ratherthan2.4°,thewavelengtherroratlminexceeds0.04nm.Therefore,aspectrallineatthisextremeendofthespectralfieldcould"disappear"thecloserlccomestothelocationoftheexitslit.D.Thespectralcoverageoverthe25.4mmarrayvariesintheexamplescalculatedasfollows:lc(nm)(lmax-lmin)(nm)25039.8040036.6770027.045.1.2DeterminationofthePositionofaknownWavelengthIntheFocalPlaneInthiscase,providedlcisknown,a,bH,andLHmaybedeterminedasabove.Iflnisknown,theblnmaybeobtainedfromtheGratingEquation1-1.ThenHBln=LHtan(bH-bln)(5-9)Thisformulaismostusefulforconstructingalignmenttargetswiththelocationofknownspectrallinesmarkedonascreenoretchedintoaribbon,etc.Section6:EntranceOptics6.1ChoiceofEntranceOpticsWhilelensesareusedintheexamplesthatfollow,frontsurfaceconcavemirrorscoatedforthespectralregionofchoicearepreferred.Acoatingsuchasaluminiumishighlyreflectivefrom170nmtothenearIRwhereascrownandflintglassesstartlosingtransmissionefficiencyrapidlybelow400nm."AchromaticDoublets"areroutinelycementedwithUV\nabsorbingresinsandtheirantireflectivecoatingsoftendiscriminateagainsttheUVbelow425nm(thisisduetothefactthatsuchlensesareoftenusedincameraswherephotographicfilmmaybeveryUVsensitive).IflensesmustbeusedinthebluetoUV,thenchooseuncoatedquartzsingletsorairspaceddoublets.Figure22.TypicalMonochromatorSystemASaperturestopL1lens1M1mirror1M2mirror2G1grating1pobjectdistancefromlensL1qimagedistancefromlensL1FfocallengthoflensL1dtheclearapertureofthelens(L1indiagram)Theabovediagramshowsatypicalmonochromatorsystemwithonefixedexitslitandonedetector,however,allthatfollowsisequallyapplicabletoaspectrograph.6.1.1ReviewofBasicEquationsThinlensequation:(3-16)Magnification(m):(2-14)Forsimplicitythediameterofanopticorthatofitsaperturestop(AS)(assumingitisveryclosetotheopticitself)isusedtodeterminethef/value.InwhichcaseEquations(2-4)and(2-5)simplifyto:f/valuein=p/d     objectf/value(6-1)f/valueout=q/d     imagef/value(6-2)\n6.2EstablishingtheOpticalAxisoftheMonochromatorSystem6.2.1Materials·HeNelaser·Lenses,mirrors,andotheropticalcomponentsasrequiredforoptimization(seeSection3)·Threepinholeaperturesoffixedheightabovethetable·Precisionpositioningsupportsforabove·Opticalbench,rail,orjigplate6.2.2ProcedureAssembletheabovecomponentssothatthelaserbeamactsastheopticalaxiswhichpassesfirstthroughtwopinholeapertures,followedbythemonochromator,andfinallythroughthethirdpinholeaperture.Theexternalopticsandsourcewilleventuallybeplacedontheopticalaxisdefinedbythepinholeaperturesandlaserbeam.Positionthepinholeaperturessothatthelenses,etc.maybeaddedwithoutdisturbingthem.Note:Reverseilluminationmaysometimesbepreferredwherethelaserpassesfirstthroughtheexitslitandproceedsthroughalltheopticsuntilitilluminatesthelightsourceitself.Alignmentofthecomponentsisaniterativeprocess.Thegoalisforthelaserbeamtopassthrougheachslitcenterandtostrikethecenterofeachopticalelement.Thefollowingstepsachievethis:1.Ifamonochromatorhasasinedrive,thensetthemonochromatortozeroorder.2.Aimthelaserbeamthroughthecenteroftheentranceslit.3.Centerthebeamonthefirstoptic.4.Centerthebeamonthenextoptic,andsoonuntilitpassesthroughthecenteroftheexitslit.5.Ifthelaserdoesnotstrikethecenteroftheopticfollowingthegrating,thenrotatethegratinguntilitdoes.Manyspectrometersarenotaccuratelycalibratedatzeroorder,therefore,someoffsetistobeexpected.6.3IlluminatingaSpectrometerIfalightsourcesuchasasampleoracalibrationlampistobefocusedintotheentranceslitofaspectrometer,then:Ensurethatthefirstactiveopticishomogeneouslyilluminated(planemirrorsarepassive).Placeawhitescreenbetweentheentranceslitandthefirstactiveoptic(inaCZmonochromator,thecollimatingmirrorandinanaberrationcorrectedconcavegrating,thegratingitself).Checkfor"images",ifthereisauniformhomogeneouslyilluminatedarea,alliswell.Ifnot,adjusttheentranceopticsuntilthereis.6.4EntranceOpticsExamplesThemajorityofcommercialspectrometersoperatebetweenf/3andf/15,butthediagrams\nthatfollowusedrawingsconsistentwithf/3andallthecalculationsassumef/6.Intheexampleswhichfollow,thelens(L1)usedisasinglethinlensof100mmfocallength(foranobjectatinfinity)and60mmindiameter.Thef/valueoutoftheentranceopticsmustbeequaltothef/valueinofthemonochromator.Ifnecessary,anaperturestopshouldbeusedtoadjustthediameteroftheentranceoptics.Rememberwhencalculatingthediameterofaperturestops,toslightlyunderfillthespectrometeropticstopreventstrayreflectionsinsidethespectrometerhousing.6.4.1ApertureMatchingaSmallSourceExample1(Figure23)Theemittingsourceissmallerinwidththanthewidthoftheentranceslitforarequiredbandpass.Figure23.SmallSourceCase1.Calculatetheentranceslitwidthforappropriatebandpass(Equation3-9).Forthisexample,lettheslitwidthbe0.25mm.ExampleObject:afiberof0.05mmcorediameterandNAof0.25.Objectemitslightatf/2(NA=0.25).Spectrometer=f/6.2.Projectedimagesizeoffiberthatwouldbeaccommodatedbythesystem(givenbyentranceslitwidth)=0.25mm.Calculatemagnificationtofillentranceslit.m=imagesize/objectsize=0.25/0.05=5.0.Therefore,q/p=5,q=5p.SubstitutingintothelensEquation3-16givesp=120mm,andq=600mm.Tocalculated,lightmustbecollectedatf/2andbeprojectedatf/6toperfectlyfillthegrating.Therefore,p/d=2,d=120/2=60mm.Therefore,aperturestop=fulldiameterofL1.Projectionf/value=600/60=10.Inotherwords,thegratingofthemonochromator,eventhoughreceivinglightcollectedatf/2,isunderfilledbytheprojectedconeatf/10.Allthelightthatcouldhavebeencollected\nhasbeencollectedandnofurtherimprovementispossible.Example2If,however,thefiberemittedlightatf/1,lightcollectioncouldbefurtherimprovedbyusingalensinthesameconfiguration,but120mmindiameter.Thiswould,however,produceanoutputf/valueof600/120=f/5Becausethisexceedsthef/6ofthespectrometer,maximumsystemlightcollectionwouldbeproducedbyalenswithdiameterd=q/(f/value)=600/6=100mmtherebymatchingthelightcollectionetenduetothelimitingetendueofthespectrometer.Thecollectionf/valueis,therefore,f/valuein=p/d=120/100=1.2Sinceetendueisproportionaltothesquareofthe(f/value)-1,about70%oftheavailableemittedlightwouldbecollectedatf/1.2(seeSection3).Iftheuserhadsimplyplacedthefiberattheentranceslitwithnoentranceoptics,only3%oftheavailablelightwouldhavebeencollected(lightinthiscasewascollectedatthespectrometer'sf/6ratherthanthef/1.2withetenduematchingentranceoptics).6.4.2ApertureMatchinganExtendedSourceFigure24.ExtendedSouceCaseTheobjectwidthisequaltoorgreaterthantheentranceslitwidth(seeFigure24).Thef/valueoutoftheentranceopticsmustbeequaltothef/valueinofthemonochromator.Theobjectdistanceshouldbeequaltotheimagedistance(absolutemagnification,m,equals1).Aperturestopsshouldbeusedtomatchetendueoftheentranceopticstothemonochromator.Becausetheobjectislargerthantheslitwidth,itisthemonochromatoretenduethatwilllimitlightcollection.Inthiscase,image1:1atunitmagnification.1.TakinglensL1SoforF=100mm,p=200mm,q=200mm(2F).2.f/valueofthemonochromator=q/d=p/d=6.Then\nd=q/(f/value)=200/6=33.3Therefore,aperturestop=33.33mmtofillthediffractiongratingperfectly.6.4.3DemagnifyingaSourceInthiscasethef/valueofthesourceisnumericallylargerthanthatofthespectrometer.Thisisoftenseenwithatelescopewhichmayprojectatf/30butistobemonitoredbyaspectrometeratf/6.Inthiscaseetenduematchingisachievedbythedemagnificationofthesource(seeFigure25).Figure25.DemagnifiedSourceCase1.Calculatetheentranceslitwidthfortheappropriatebandpass(Equation(2-21)).Take,forexample,1.0mm=finalimagesize=entranceslitwidth.2.Imageprojectedbytelescope=5mmandformstheobjectforthespectrometer.m=1/5=0.2,thenfromEquation(3-16).TakinglensL1withF=100mm(given),p=600mm,q=120mm.Calculatedknowingthemonochromatorf/value=6.q/d=6,d=120/6=20mm.Theaperturestopwillbe20mmdiameter.Lightisgatheredateithertheapertureoftheprojectedimageor600/20=f/30,whicheverisnumericallygreater.6.5UseofFieldLensesTheconceptsgiveninthissectionhavenotincludedtheuseoffieldlenses.Extendedsourcesoftenrequireeachpupilinthetraintobeimagedontothenextpupildownstreamtopreventlightlossduetooverfillingtheoptics,vignetting(seeSection2.8).·Usedwhenentranceslitheightislargeandthelightsourceisextended.·Afieldlensimagesonepupilontoanother.InFigure26,ASisimagedontoG1.\nFieldlensesensurethatforanextendedsourceandfiniteslitheight,alllightreachesthegratingwithoutvignetting.InFigures26and27theheightoftheslitisintheplaneofthepaper.Figure26.ExampleofFieldLensImagingOnePupilontoAnotherFigure27.ExampleofIlluminationSystemwithFieldLens6.6PinholeCameraEffectWhenentranceopticsareabsent,itispossiblefortheentranceslittoprojectanimageofjustabouteverythingbeforetheslitintothespectrometer.Thismayincludethelamp,thesample,rimsoflenses,evendistantwindows.Section3describeshowtocorrectlyilluminateaspectrometerforhighestthroughput.Followingthisprocedurewilleliminatethepinholecameraeffect.Multipleimagingmayseverelydegradeexitimagequalityandthroughput.Ontheotherhand,thepinholecameraeffectisveryusefulintheVUVwhenrefractivelensesarenotavailableandmirrorswouldbeinefficient.6.7SpatialFiltersApertureandfieldstopsmaybeusedtoreduceoreveneliminatestructureinalightsource,andblocktheunwantedportionsofthelight(e.g.,thecladdingaroundanopticalfiber).Inthiscapacity,aperturestopsarecalledspatialfilters(seeFigure28).Thelightsourceimageisfocusedontotheplaneofthespatialfilter.whichthenbecomesthelightsourceforthesystem.\nFigure28.SpatialFilterCaseReferencesACKNOWLEDGMENTSWewishtothankthemanypeoplewhohavecontributedtheirideasandhelpinthewritingofthistutorial.ThankstoJeanFlamand,JeanPierreLaude,andBrunoTouzetofHORIBAJobinYvon,andKimMarshallofHORIBAJobinYvonInc.SpecialthankstoRamonBarnesoftheUniversityofMassachusettsandRickFeinbergofWesternWashingtonUniversityfortheirpoignantandconstructivecomments.BIBLIOGRAPHY1)DiffractionGratingsRuledandHolographicHandbook,publishedbyHORIBAJobinYvonInc,Edison,NewJersey08820USAandHORIBAJobinYvonDiv.d'InstrumentsSA.,1618RueduCanal91160Longjumeau,France.2)Chi,ChangH.,Ed.,"PeriodicStructures,Gratings,MoirePatternsandDiffractionPhenomena",SPIEProc.240(1980).3)Goldstein,S.A.andWalters,J.P.,"AReviewofConsiderationsforHighFidelityImagingofLaboratorySpectroscopicSourcesParts1and2",SpectrochimicaACTA,31B,201316,(1976).4)Hecht,E.andA.Zajac,Ontics,AddisonUesley,Reading,Massachusetts,ITRA(1979).5)James,J.F.andR.S.Sternburg,TheDesignofOpticalSpectrometersChapman&HallLtd.,London,England,(1969).6)Lerner,JeremyM.,Ed.,"InternationalConferenceontheApplication,TheoryandFabricationofPeriodicStructures,DiffractionGratingsandMoirePhenomena",SPIEProc.503(1984).7)Lerner,JeremyM.,Ed.,"InternationalConferenceontheApplication,TheoryandFabricationofPeriodicStructures,DiffractionGratingsandMoirePhenomena",SPIEProc.815(1987).8)Reader,J.,"OptimizingCzernyTurnerSpectrographs:AComparisonBetweenAnalyticTheoryandRayTracing",J.Opt.Soc.Am.59,(9),11891196.(1969).9)Reader,J.,"WhatisDynamicDispersion",AppliedOptics20,(13),21732172,(1981).\n10)Shafer,A.B.,Megill,L.R.,andDroppleman,L.,"OptimizationoftheCzernyTurner",J.Opt.Soc.Am.54,(7),879887(1964).11)Stroke,G.W.,HandbuchderPhysikXXIX,SpringerBerlin,(1967),(ChapterisinEnglishandiscomprehensive).12)Thevenon,A.,Flamand,J.,Laude,J.P.,Touzet,B.,andLerner,J.M.,"AberrationCorrectedPlaneGratings",SPIEProc815,136145,(1987).