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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) UBKglass@0.55um: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⇥

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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.