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