Design of a far-infrared spectrometer for atmospheric thermal emission measurements

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Designofafar-infraredspectrometerforatmospheric

thermalemissionmeasurements

DavidG.Johnson

NASALangleyResearchCenter,Hampton,Virginia,USA

ABSTRACT

Globalmeasurementsoffarinfraredemissionfromtheuppertropospherearerequiredtotestmodelsofcloudradiativeforcing,

watervaporcontinuumemission,andcoolingrates.Spectrawithadequateresolutioncanalsobeusedforretrievingatmo-

spherictemperatureandhumidityprofiles,andyettherearefewspectrallyresolvedmeasurementsofoutgoinglongwavefluxat

wavelengthslongerthan16µm.Ithasbeendifficulttomakemeasurementsinthefarinfraredduetotheneedforliquid-helium

cooleddetectorsandlargeopticstoachieveadequatesensitivityandbandwidth.Wereviewdesignconsiderationsforinfrared

Fouriertransformspectrometers,includingthedependenceofsystemperformanceonbasicsystemparameters,anddiscussthe

prospectsforachievingusefulsensitivityfromasatelliteplatformwithalightweightspectrometerusinguncooleddetectors.

Keywords:Fouriertransformspectrometers;infraredspectroscopy;atmosphericremotesensing;atmospherictemperature;

atmospherichumidity;radiativetransfer

1.INTRODUCTION

Fouriertransformspectrometers(FTS)operatinginthefarinfrared(FIR;definedhereaswavelengthslongerthan16µm)have

generallyusedcooleddetectors,eitherphotoconductorsoperatingat4.2Korbolometersoperatingatevencoldertemperatures.

Thelowtemperaturesarerequiredbothtoachieveusefulsensitivityand,inthecaseofascanningFTS,toprovidetheneces-

saryelectronicbandwidth.Recentimprovementsinthesensitivityandbandwidthofuncooleddetectorshavemotivatedusto

reconsiderthepossibilityofdemonstratingusefulsensitivitywithanFTSoperatingintheFIRwithuncooleddetectors.

InthefollowingsectionswederiveageneralexpressionforthesensitivityofaFouriertransformspectrometerasafunction

ofbasicsystemparameters.Wethenestimatethesensitivitythatcanbeachievedwiththepresentgenerationofuncooled

detectors,andtheimprovementinperformancethatcouldberealizedwithdetectorsoperatingatthethermalnoiselimit.

2.INTERFEROGRAM

WeconsiderfirstaconventionalscanningFTS,shownschematicallyinFigure1.Foramonochromaticsourcetheinterferometer

outputisgivenby

IxI005ηkSdcos2πkx(1)

whereI(x)istheoutputpower,xistheopticalpathdifference(OPD)betweenthetwoarmsoftheinterferometer,I0S205ηkSd,SdS1S2,ηk4RkTk,S1andS2arethepowerateachinput,kisthewavenumber(1/wavelength),and

RkandTkarethebeamsplitterreflectanceandtransmittance,respectively.

ForabroadbandsourcetheinterferogramisgivenbytheintegralofEquation(1)asshownbelow:

IxI005∞

∞dkηkSdkcos2πkx(2)

I0∞

∞dkS2k05ηkSdk;(3)

whereSdk,thenetinputspectralpowerperunitwavenumberinterval,isgivenbyS1kS2k,andS1kandS2karethe

spectralpowerperunitwavenumberintervalatinputs1and2,respectively.

+TS2RS1

12TS+RSImage of

fixed mirror

S1

2SOPD ( ) = 2xy

I(x)BeamsplitterFixed mirror

Moving mirror

y

Figure1.GenericFouriertransformspectrometerlayout.

WeestimatethenetinputspectrumbytakingtheFouriertransformofEquation(2)aftersubtractingtheconstanttermI0:

Sdk∞

∞dxexp2πikx∞

∞dk05ηkSdkcos2πkx(4)

05ηkSdk;(5)

whereSkistheestimatednetinputspectrum.

InpracticetheinputsS1andS2(ortheoutputIx)areopticallyfilteredtoproduceaband-limitedinterferogram,andwe

furtherreducethenoisebypassingthedetectoroutputthroughanelectronicfiltermatchedtotheopticalbandpassandmirror

scanvelocity.WethenestimateSdkfromthediscreteFouriertransform(DFT)oftheband-limitedinterferogram1:

Sdkn∆Hn;(6)

HnN21∑

jN2exp2πknxjIjI0(7)

whereHnistheDFTofIjI0,IjIxj,xjj∆,knnN∆,Nisthetotalnumberofsamples,and∆isthedifferencein

opticalpathbetweeninterferogramsamples.Notethatwehaveassumedthattheinterferogramisfullytwo-sided,

WecombineEqs.(5)and(6)tosolveforHnasafunctionofthenetinputpowerperunitwavenumber:

Hn05ηkSdkn∆(8)

Toestimatetheresponseofaparticularinterferometertoaninputofknownnetspectralintensityweneedtoconsider

opticalthroughputandspectralresponseofthedetector.Tosimplifythepresentdiscussionweassumethatthespectrometer

fullyilluminatesadetectorofareaAwithabeamhavingthefocalratiof,andthatthespectralresponseofthedetector,filters,

andotheroptics(exceptthebeamsplitter)isrepresentedbyasingleefficiencyεk.Theeffective(detected)netspectralpower

perunitwavenumber(Sd)isgivenby:

SdkεkFdkπA4f2(9)

whereFdisthenetinputspectralintensity.3.NOISE

Wenowconsidertheeffectofnoiseontheinterferogram.Weassumethatnoiseisgivenbythefunctiongx,andthatthe

observedinterferogramisgivenbyIxgx.Wedefinegjgxj,wherexjisdefinedabove.ThevarianceintheDFTof

gjisrelatedtothevarianceofgjbythediscreteversionofParseval’sTheorem1:

VarGnNVargj(10)

whereGnistheDFTofgj,Varysignifiesthevarianceofy,andNisdefinedabove.Thevarianceofgjisrelatedtothespecific

detectivityandelectronicbandwidthofthedetectorandpreamplifierasfollows:

VargjAνBD2(11)

whereνBisthepreamplifierbandwidthandDisthespecificdetectivityofthedetector.AftersubstitutingEquation(11)in