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ASTR633AstrophysicalTechniques
SPECTROSCOPY(originalnotesbyPatHenry,editedbyMikeLiuandJonathanWilliams)
"Ifyou'renotdoingspectroscopy,you'renotdoingscience."Spectrometer/spectrograph=instrumentwithsomekindofdispersingelementthatbreakslightintoitscomponentwavelengths.Spectralanalysisofcelestialobjectsisprobablythemostimportantwayoflearningaboutthem.Alargefractionoftelescope(andyour)timeisdevotedtothisanalysis.
1. Definitions Angulardispersion(“A”)
𝑨 = 𝒅𝜷 𝒅𝝀⁄ [radians/Angstrom]Lineardispersion:linearcounterpartof“A”.measureoflinearseparationofwavelengths,usuallyinafocalplane(e.g.asseenbytheCCD).
lineardispersion:𝑑𝑙 𝑑𝜆 = 𝑓 𝑑𝛽 𝑑𝜆 = 𝑓𝐴⁄⁄ [mm/Angstrom] reciprocallineardispersion(a.k.a.theplatefactor):𝑃 = 1 𝑓𝐴⁄ [Angstrom/mm]
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Typicalslitspectrometer
d1=sizeofbeamatthecollimator(notthesizeofthecollimatoritself!)d2=sizeofbeamatthecamera.Maydifferentthand1duetodisperser.DisperserhasangulardispersionA,withdirectionofdispersionparalleltotheslitwidth(i.e.intheplaneofthediagram).Theanglesubtendedbytheslitasseenfrom3ofthesystemelements:- Asseenfromobjective:φ=w/f=angleofslitprojectedonthesky)
where“w”isthephysicalwidthoftheslit.
- Asseenfromthecollimator:da=w/f1
- Asseenfromcamera:db=w’/f2 wherew’isthewidthoftheslitimageatthebackend(e.g.onthedetector).
AnamorphicMagnification=magnificationinthedispersiondirectionIfadispersingelementisplacedbetweenthelenses,rotationalsymmetryabouttheopticalaxisislost,andtheequalityofthesubtendedanglesisnotnecessarilythesameforobjectsseenatdifferentorientations.
Inthedirectionperpendiculartothedispersion,thebeampassingthoughthedisperserisunchanged.Inthedirectionalongthedispersionaxis,therearepossiblemagnificationeffectsduetothedisperser.Theanamorphicmagnificationis:
MeganAnsdell
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BytheLagrangeInvariant(H=htanuisconserved):
Sosubstituteintoouraboveexpressionfor“r”
i.e.ratioofthebeamatthecollimatortobeamatthecamera PhysicalSlitWidthRelatedtoSystemParameters
fromourpreviousequation,theslitsizeasseenfromtheobjectiveis
(whereF1isf/number,a.k.a.focalratio)ImageofSlitWidthRelatedtoSystemParametersfromourequationfortheslitsizeasseenfromthecamera:
r � ⇤⇥⇤�
=w�
w
f1f2
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Example:Keck10-metertelescope,φ=1”Wanttobeabletomachine“w”easily,somakeit1mm. F1=w/D/φ=1e-3m/(10m)/(1”/206265”/rad)=20.6àveryslow,soeasyWant“w’“tocover2CCDpixels=44microns F2=w’/D/φ=44e-6m/(10m)/(1”/206265”/rad)=0.91àhard,butdo-ableàspectrographsonlargetelescopesneedfastcamerasi.e.forfixed(w’/φ),needfastercameraforlargertelescopes(e.g.,DEIMOSonKeck2)SpectralPurity=wavelengthchangeacrosstheimageoftheslitwidth.Bydefinitionoflineardispersion
or
δ𝜆 ≡ 𝑤3
𝑓4𝐴= 𝐷𝜙𝑓4/𝑑4𝑓4𝐴
= 𝐷𝜙𝐴𝑑4
SpectralResolution=dimensionlessmeasureofspectralpurityThisisthemorecommonquantityusedtodescribespectrographs
Thushighspectralresolutionrequires
1. Smallφ(i.e.slitwidth)–butthiscanleadtolossoflight2. Largecamera–expensive(morespecifically,cameralargeenoughto
accommodatelargebeamcomingfromthedisperser)3. Largeangulardispersion–dispersersthatdothishavelowerefficiencies
Notethatthefundamentallimitofthisissetbydiffraction:φ=1.22λ/D
RD=Ad2/1.22
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2. Dispersers 2.1 Prisms
Prism=wedge-shapedpiece,composedofmaterialwithindexofrefractionn(λ).Becauseofwavelength-dependenceofdispersion,thereissomewavelengthwheretheraysintheprismgoparalleltothebase(“minimumdeviationcondition”,i.e.anglein=angleout).Fortheserays,thediagramissymmetricabouttheverticalbisectoroftheprism,asshownabove.Thissetupisaspecificpedagogicalchoicetoshowhowtheprismworks.Ifthewavelengthoflightisdifferent,youhavetochangeθtogetthissetup.Getconstructiveinterferencebetweenthe2lightpathsabovewhen Takederivativew.r.t.lambda
Fromthefigure,weseethat
Sothisgives
n(⇥)t = 2L cos �
L sin� = a⇥ + 2� + ⇤ = ⌅, or � =
12(⌅ � ⇤ � ⇥) ⇥ d�
d⇥= �1
2
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where(t/a)istheratiooftheprismbaselengthtothebeamwidth.Indexofrefractionofmostopticalglassescanbeexpressedas Sothen
àndecreaseswithλ,i.e.,dn/dλA<0i.e.,angulardispersiondecreaseswithlongerwavelengths.Bluelightisdeviatedmorethanredlight.àSinceA=dβ/dλisnumericallylargeratbluerwavelengths,aprismdispersesbluelightmorestronglythanredlight. Notethatthereisno(anamorphic)magnification(i.e.,r=1)becausetheincoming&outgoingbeamarethesamesize.Spectralresolutioninminimumdeviationconditionisgivenby(pluggingin“A”aboveintoourpreviousequationfor“R”):
𝑅 = 𝜆𝛿𝜆 =
𝜆𝐴𝑑4𝜙𝐷 =
𝜆𝐴𝑎𝜙𝐷 =
𝜆𝜙𝑡𝐷𝑑𝑛𝑑𝜆
Example:ObjectiveprismonaSchmidttelescope a=D=1meter t=150mm φ=1”(typicalseeing,sincedataarecollectedslitless) [email protected]:dn/dλ=0.066µm-1 àA=0.57degs/µm àR=103i.e.notveryhighbutstilluseful(e.g.high-zQSOs,low-Zstars) Weneedmuchhigherresolution,R~104,toresolvelinesinstars.
tdn
d⇥= (�2a)
��1
2
⇥d�
d⇥⇤ A ⇥ d�
d⇥=
t
a
dn
d⇥
7
2.2 Diffraction Gratings Grating=planar(orcurved)opticalsurfacewithaseriesoffinelyspacedgrooves(orslits).Primarydisperserinmostastronomicalapplications. Pros:SignificantlylargerRthanprismofcomparablesize. Cons:Notasefficientasprism.Limitedλcoverage(typically<2x)Gratingsworkbothintransmissionandreflection.TransmissiongratingMulti-slitversionofYoung’stwo-slitexperiment(RiekeFigure6.2):
TheincomingbeamemergesfromtheslitsasapatternofHuygenswavefronts,spreadoverarangeofanglesduetodiffraction.Lightraysthatemergeperpendiculartotheslitsaddtogethertoproducethewhitelightpeakatm=0.Atdifferentangles,interferenceprovidesawavelengthdependenceandpeaksatm=1(firstorder),m=2(secondorder),etc.Thepathdifference,p,forlightpassingthroughneighboringslitsis
𝑝 = 𝑑 sin 𝑖 + 𝑑 sin 𝜃Fori=0shownabove,thephasedifferenceis
𝛿 = 2𝜋𝑝𝜆 =
2𝜋𝑑 sin 𝜃𝜆
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Themagnitudeoftheconstructivewavefrontdecreaseswithincreasingorderbecausetheintensityofthelightdecreaseswithdiffractionangle.Thisworkstodisperselightbutisinefficientbecausemostofthelightisinthewhitelightpeak.Bettertoreplaceslitswithmirrors.ReflectiongratingNotjustmoreefficientbuteasiertocutlinesinsolidthantocuttinyslitsthroughit(thediagramhereisfromen.wikipedia.org/wiki/Blazed_gratingbutalsoseeRiekeFigure6.3):
Themultiplereflectionsfromthesetoftiltedmirrorsproducesaninterferencepatternthatisafunctionofλ.Thepathlengthrequirement(similartoabove)forwavestoaddinphasegivesthegratingequation:
𝑚𝜆 = 𝑑(sin 𝛼 + sin 𝛽)Theblazeangle,θB,setsthetiltofthemirrorandischosentomaximizetheefficiencyofthegratingataparticularblazewavelength,λB.Practicallythismeansthatthediffractionandreflectionangleslineup.ThiscanbeachievedfordifferentcombinationsofanglesandwavelengthsbuttheLittrowconfigurationiswhentheincidenceangleisthesameasthereflection(i.e.,theincidentbeamisperpendiculartothetilt).Thatisα=β=θBandtherefore
𝜃J = sinKL𝑚𝜆J2𝑑
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AngulardispersionDifferentiatethegratingequationw.r.t.lambda,atfixedα(angleofincidence)
Thisequationshowsthattheangulardispersioninagivenordermisafunctionofdandβ.i.e.choosedifferentgroovespacingorusegratingatdifferentangleofdiffractiontogetdifferentA.
Canalsore-writethisas:
SoavalueofAatagivenwavelengthissetentirelybyanglesαandβ,independentofmandd.ThusagivenAcanbeobtainedwithmanypossiblecombinationsofmandd,providedtheanglesatthegratingsareunchangedandm/d=constant.
Thereforeabletogethighangulardispersionbymakingαandβlarge(~60degs)whenusingcoarselyruledgratings(“echelles”=Frenchfor“ladder”,lowgroovedensitybutoptimizedforuseinhighorder).
§ Ordinarygratingspectrometer:m=1or2,d=1/300–1/1200mm§ Echellegratingspectrometer:m=10-100,d=1/30mm
Notelargeαàhighlytiltedgratingàneedalargegratingtofillthebeam.
m d⇥ = d cos � d�
A =d�
d⇥=
m
d cos �
A =d⇥
d⇤=
m
d cos ⇥=
sin⇥ + sin�⇤ cos ⇥
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SpectralresolutionStartingfromourgeneralequationabove
whereW=d2/cos(β)=physicalsizeofthegrating
Astelescopegetslarger,somustgratingsinordertopreservespectralresolution,e.g.Keck/HIRESgratingis1.2m(actuallymosaicof3echellegratingsegments).
Thelimitingresolutionoccursinthediffraction-limitedcase,φ=1.22λ/D:
(middlesubstitutionbasedonouraboveequationfor“A”)
whereN=W/D=total#ofgroovesinthegrating:moregroovesmeanmoreinterferenceandfinerwavelengthdiscrimination.Example:considersamesizegratingasourobjectiveprismexample,W=1meter. SamesizetelescopeD=1m Gratinggroovespacing:d=1/1200mm φ=1” λ=0.5um α=β=17.5deg
àmuchlargerthanprismFreeSpectralRangeForagivenpairof{α,β},thegratingequationissatisfiedforallλforwhich“m”isaninteger.Overlapwhen𝑚𝜆 = (𝑚 + 1)𝜆′
ð Freespectralrangeis∆𝜆 = 𝜆3 − 𝜆 = 𝜆/𝑚
e.g.m=1,λ’=0.8µm,thenλ=0.4μmandΔλ=0.4μmThereare2waystosortoutoverlappingorders:
(1) Ordinarygratingm=1andΔλislargeàuseafilterEchellegratingm=10andΔλissmallàaddacross-disperser.i.e.aprismorgratingtoseparatethepiecesintheperpendiculardirection.e.g.IRTF/Spexcross-disperseddata
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3. Practical Considerations for Actually Obtaining Useful Data Slitsize§ Thisisoneoftwothingstheastronomerhascontrolover,theotherbeingthegrating.§ Narrowslitsyieldimprovedspectralresolution.Thathelpssolongasthespectral
featuresareresolved.§ However,don’tmaketheslitwidth~2”)sowantslitsoforderthatsizetogetallthelightinthegalaxy.Besides,thelinesingalaxiesarenotnarrowanyways,duetostellarmotions,sonoadvantageinfinalspectralresolutionbyusinganarrowslit.
Gratings§ Makesuretopickappropriateblazewavelengthforobjects.§ Trytogetefficiencycurve,sincesomegratingsarebetterthanothers.§ Caneasilyloseafactorof2inlightinthegrating.Multi-slitspectroscopy§ Canget~10-100’sofobjectsinasingleexposure,bymillinglittleslitsalloverthefocal
plane.(~1000sifusingfiberspectrograph)§ Makesureslitlengthis>~10”,inordertogetenoughblankskydatatomakeagood
determinationofitslevel.WavelengthCalibrationLamps§ Takearcspectratogetwavelengthcalibration,i.e.matchobservedarcemissionlines
withlistofknownwavelengths.§ Frequencyoftakingarcimagesdependsonflexurepropertiesofspectrograph(e.g.at
CassorNasymthfocus).Forbestpractices,ReadtheManual(RTM).Flats§ Notasimportantaswithimages,sinceonlytoflattentheregionnearthespectrum,not
overthewholeimage§ Noneedtomatchthespectrumofthetargetandtheflat,becausethespectrographputs
thesameλonthesamepixelinbothcases.Justneedsomefluxatallλ.§ Makesurenottosaturatethedetector.§ Formulti-slits,needaflatforeveryindividualmask,b/cmappingofλtopixelsis
differentforeachmask.§ Forsingleslits,frequencyofflats(e.g.everyimage,everytarget,oreverynight)
dependsonphysicalstabilityoftheinstrument.RTFM.Spectrophotometricstandards§ Thesearestarsofknownspectraltype,socanusethisknowledgetocorrectforallthe
wavelength-dependenteffects(atmosphere,telescope,optics,grating,detector)andrecoverthetruespectrumofthesciencetarget.
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§ Trytousestandardswithoutstrongabsorptionlinesinthem.e.g.A0stars.Bestspectraltypesdependonwavelength.
§ Intheoptical,makesureyoucantakelong(>~15sec)exposures.Someshuttersarenon-linear.(Notrelevantforthenear-IR,sincenoshuttersareinvolved–arraysarecontinuallyreadout.)
4. Reduction of Data Reducingspectroscopydataisbasicallythesameasphotometry,justdoingitinonespatialdimensionandhundredsoftimesinthedispersiondirection.The“images”areofteninthebackground-limitedregime,soaccurateskysubtractionisveryimportant.
1.Dobiasremovalandflat-fieldingintheusualway.2a.Tomeasurethesciencetarget,foreverycolumn:
§ Collapsetheobjectspectrumtoonenumberbyaveraging,sincestellarprofileisonly3-4pixelswide
§ Measuretheskyspectrumbymedian-averagingovertheremainingpixels.Want(#ofskypixels)>>(#ofobjectpixels)sothatPoissonnoiseinthemeasurementoftheaverageskyleveldonotdominate.
§ Subtractthe2numberstoyieldthenetDNperpixel=DNnet(x)2b.Dothesameforstandardstardata:DNstd(x)3.Todeterminethewavelengthcalibration:
§ Identifypairsof(pixel,λ)inthearcspectrum.
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§ Usethesetoestablishthemappingfrompixel->λusingasimplefunction(e.g.linearorlow-orderpolynomialfit,b/cspectrographsarereasonablylinear).
4.Applywavelengthcalibrationtogetnetfluxa.f.o.λ:DNnet(λ),DNstd(λ)5.Thenthefinalfullycalibratedmeasurementforthesciencetargetis
Sinceweknowthetruephysicalspectrumofthestandardstar,thisallowsustoremoveallthewavelengthdependenteffects(e.g.atmosphere,telescope,grating/blaze,etc.)Bewaryofdetectedspectral“features”nearstrongnightskylines.