Cepstral Analysis for Baseline-Insensitive Absorption Spectroscopy Using Light Sources with Pronounced Intensity Variations
Christopher S. Goldenstein, Garrett C. Mathews, Ryan K. Cole, Amanda S. Makowiecki, Gregory B. Rieker
CCepstral Analysis for Baseline-Insensitive Absorption Spectroscopy Using LightSources with Pronounced Intensity Variations
Christopher S. Goldenstein ∗ and Garrett C. Mathews School of Mechanical Engineering, Purdue University,585 Purdue Mall, West Lafayette, IN 47907, USA
Ryan K. Cole, Amanda S. Makowiecki, and Gregory B. Rieker
Precision Laser Diagnostics Laboratory, University of Colorado Boulder,1111 Engineering Drive, Boulder, CO 80309, USA (Dated: June 5, 2020)This manuscript presents a data-processing technique which improves the accuracy and precisionof absorption-spectroscopy measurements by isolating the molecular absorbance signal from errors inthe baseline light intensity ( I o ) using cepstral analysis. Recently, cepstral analysis has been used withtraditional absorption spectrometers to create a modified form of the time-domain molecular free-induction decay (m-FID) signal which can be analyzed independently from I o . However, independentanalysis of the molecular signature is not possible when the baseline intensity and molecular responsedo not separate well in the time domain, which is typical when using injection-current-tuned lasers(e.g., tunable diode and quantum cascade lasers) and other light sources with pronounced intensitytuning. In contrast, the method presented here is applicable to virtually all light sources since itdetermines gas properties by least-squares fitting a simulated m-FID signal (comprising an estimated I o and simulated absorbance spectrum) to the measured m-FID signal in the time domain. Thismethod is insensitive to errors in the estimated I o which vary slowly with optical frequency and,therefore, decay rapidly in the time domain. The benefits provided by this method are demonstratedvia scanned-wavelength direct-absorption-spectroscopy measurements acquired with a distributed-feedback (DFB) quantum-cascade laser (QCL). The wavelength of a DFB QCL was scanned acrossCO’s P(0,20) and P(1,14) absorption transitions at 1 kHz to measure the gas temperature andconcentration of CO. Measurements were acquired in a gas cell and in a laminar ethylene-air diffusionflame at 1 atm. The measured spectra were processed using the new m-FID-based method and twotraditional methods which rely on inferring (instead of rejecting) the baseline error within thespectral-fitting routine. The m-FID-based method demonstrated superior accuracy in all cases anda measurement precision that was ≈ I. INTRODUCTION
Laser-absorption spectroscopy (LAS) is a powerfuland broadly applicable technique for providing quanti-tative measurements of gas conditions and species con-centrations [1–5]. While there are many variants ofLAS, all methods ultimately rely on discerning howmuch of the incident light was absorbed by the test gas,typically as a function of wavelength. This is most ob-vious in the context of direct-absorption-spectroscopy(DAS) techniques which rely on converting a measure-ment of the transmitted light intensity ( I t ) to the spec-tral absorbance ( α ) of the test gas using Beer’s Law (Eq.1). The incident (i.e., baseline) light intensity ( I o ) mustbe well known to execute this conversion accurately and,ultimately, to provide an accurate measurement of gasproperties.While rarely discussed in the literature, accurate de-termination of I o frequently limits the accuracy and ∗ [email protected] precision of LAS diagnostics. Time- or wavelength-dependent variations in I o due to etalon interference ef-fects (e.g., produced by windows or other planar opticalcomponents) and/or variations in the laser’s intensitycan be difficult to account for with high accuracy, evenin tame laboratory environments. For general context,many LAS applications such as characterization of ther-mochemical flame structure and combustion kinetics,atmospheric sensing, and high-fidelity characterizationof absorption lineshapes often demand measurementswith <2% error. In the case of measuring a spectral ab-sorbance of 0.05 (typical of near-infrared LAS applica-tions), achieving a 2% error in the spectral absorbanceof the target species requires achieving an effective er-ror in I o of only 0.1%. This is especially challenging toachieve when characterizing harsh combustion environ-ments where beamsteering, window fouling, mechani-cal vibration, and scattering off particulates frequentlycause the transmitted light intensity to vary on the or-der of 1 to 10% [6] and, in extreme applications (e.g.,coal gasifiers, explosive fireballs), by several orders ofmagnitude [7–9].To overcome this challenge, researchers have devel- a r X i v : . [ phy s i c s . i n s - d e t ] J un oped a variety of strategies to avoid or mitigate the im-pact of measurement errors induced by uncertainty orerror in I o . For example, wavelength-modulation spec-troscopy (WMS) with various harmonic-normalizationtechniques (e.g., WMS-2 f /1 f , RAM-normalization) [10–14] can actively account for variations in I o inducedby broadband transmission losses. These methods canbe especially advantageous in circumstances where theabsorbance spectra are spectrally broad compared tothe wavelength-scan amplitude of the laser and non-resonant wavelengths cannot be reached (e.g., in gasesat high-pressures) [5]. For example, researchers havedemonstrated that WMS-2 f /1 f is capable of providinghigh-fidelity measurements of gas properties in a varietyof high-pressure combustion environments where nar-rowband lasers such as distributed-feedback (DFB) tun-able diode lasers (TDLs) and quantum-cascade lasers(QCLs) cannot access a non-absorbing baseline or in-terrogate a sufficiently large portion of the spectrumto reliably infer the baseline via post-processing [7, 15–17]. That being said, WMS techniques remain suscep-tible to error induced by background signals originatingfrom, for example, a non-linear laser-intensity responseor intensity modulation induced by etalon effects, whichcan be time-varying and non-trivial to account for (e.g.,using background subtraction or accounting for back-ground signals in the WMS model). Such errors ulti-mately stem from a lack of understanding of I o or thesignal components comprising it [12].In the context of DAS, numerous strategies have beendeveloped to mitigate errors induced by I o [18–31].Perhaps the most widely used method utilizes a poly-nomial or spline to account for wavelength-dependentvariations in I o . This method has been widely uti-lized in both narrowband techniques (e.g., TDLAS)[18–21] and broadband techniques (e.g., using FTIR,frequency-combs, supercontinuum lasers) [22–26], al-though the latter typically employs multiple polynomi-als in a piecewise-fitting approach. The polynomial(s)can be determined prior to least-squares fitting a simu-lated absorbance spectrum to the measured absorbancespectrum (e.g., by fitting to the non-absorbing regionsof I t ) or they can be determined along with the best-fitabsorbance spectrum simultaneously using the spectral-fitting routine. The latter approach is less susceptibleto user bias and Simms et al. [25] demonstrated it canreduce measurement error. In any case, these meth-ods are susceptible to errors induced by coupling be-tween the polynomial(s) and the simulated absorbancespectrum. This is particularly problematic when thespectroscopic model used for calculating the best-fit ab-sorbance spectrum is heavily flawed, in which case thefitting routine may erroneously attribute errors in theabsorbance model to errors in I o (accounted for withthe polynomial). Alternatively, the fitting routine maysimply converge on an inaccurate solution that simply leads to the smallest sum-of-squared error. Ultimatelythe modeled spectrum may match the measured spec-trum very well, however the gas properties inferred fromthe best-fit spectrum could have large errors.Other methods for correcting errors in the baselineleverage differences in the spectral "shape" between theabsorbance spectrum and I o . For example, Kranen-donk et al. [29] analyzed the first derivative of the ab-sorbance spectrum to desensitize the measurement toerrors in I o that vary slowly with frequency comparedto the absorbance spectrum. As a result, this method isbest suited for cases where the absorbance spectra con-sist of discrete, spectrally narrow lines (e.g., from smallmolecules at low pressures). One disadvantage of thistechnique is that the signal must be smoothed after dif-ferentiation as this process can amplify noise. Anothermethod, utilizes Fourier transforms and bandpass fil-tering to effectively separate the absorbance spectrumfrom the baseline intensity [22, 27, 28]. In this approach,first the Fourier transform of the measured transmis-sion spectrum is calculated. If the spectrum consists ofdiscrete absorption features, the absorption lines occurat specific frequencies (in the signal’s power spectrum)which are then isolated using bandpass filters. Afterbandpass filtering the signal from the absorption lines,the inverse Fourier transform of this signal is calculatedto yield a corrected spectrum which is less prone tobaseline errors. That being said, the corrected spectrummust still be normalized to account for the baseline in-tensity prior to comparing with simulated absorbancespectra for determination of gas properties.Most recently, Cole et al. [30] developed a techniquewhich eliminates the need to account for the baselinein post-processing. This method works by convertingthe measured transmitted intensity spectrum ( I t ) to amodified form of the molecular free-induction decay us-ing cepstral analysis. The modified free-induction decay(m-FID) signal consists of two distinct components withan additive relationship: (1) the laser-intensity response(from I o ) and (2) the molecular-absorption response(from α ). In the time domain, these signals can sep-arate from each other since the laser-intensity responsecan decay to zero more rapidly. The authors showedthat this enables gas properties to be determined, with-out knowledge of I o , by least-squares fitting a simulatedmolecular-absorption response signal (obtained from asimulated absorbance spectrum only) to the molecular-absorption response within the measured m-FID sig-nal. This approach was demonstrated with broad-band absorption measurements (synthetic and real) ofspecies (e.g., CH , C H ) with discrete and/or quasi-continuous absorbance spectra using a dual-frequency-comb spectrometer with complex frequency-dependentvariations in I o . That said, achieving baseline-free mea-surements with this technique is limited to cases wherethe laser-intensity response decays to zero in the timedomain faster than the molecular-absorption response.The work presented here builds upon the m-FID-based approach developed by Cole et al. [30] in or-der to accommodate scenarios where the laser-intensityresponse and the molecular-absorption response decayon similar timescales and, therefore, do not fully sep-arate in the time domain. In contrast to the methodof Cole et al. [30], this method relies on modeling theentire m-FID signal using an estimated I o (e.g., frombaseline fitting) and a simulated absorbance spectrum,and least-squares fitting the simulated m-FID signal tothe measured m-FID signal. As such, this method isnot "baseline-free," however we demonstrate that thisapproach reduces measurement errors significantly byseparating the molecular-absorption response from er-ror in the estimated I o . Most importantly, this ap-proach is applicable to scenarios with large and rapid(with optical frequency) variations in I o such as are en-countered in scanned-wavelength direct-absorption ex-periments conducted with injection-current-tuned semi-conductor lasers (e.g., TDLs, QCLs). As such, thismethod enables the error-reducing benefits of m-FID-based analysis to be attained in a wider variety ofLAS experiments. The remainder of this manuscriptis devoted to describing the fundamentals and operat-ing principles of this method, as well as to presentingthe experimental validation of this technique and com-parison with established data-processing methods. II. FUNDAMENTALS OF ABSORPTIONSPECTROSCOPY AND M-FID
This section describes the pertinent fundamentals ofabsorption spectroscopy and how the m-FID signal isrelated to I o and absorbance spectra. A. Absorption Spectroscopy
In LAS, a monochromatic laser beam with incidentintensity I o and frequency ν is directed through a gassample and the transmitted light intensity I t is mea-sured on a photodetector. Beer’s Law, given by Eq. 1,can be used to determine the spectral absorbance, α ,and for a uniform line-of-sight it is related to spectro-scopic parameters and thermodynamic properties usingEq. 2. I t ( ν ) = I o ( ν ) exp [ − α ( ν )] (1) α ( ν ) = (cid:88) j S j ( T ) P χ
Abs φ j ( ν ) L (2) Here, S j (cm − /atm) is the linestrength of transition j at temperature T , P (atm) is the pressure of the gas, χ Abs is the mole fraction of the absorbing species, φ j (cm) is the lineshape of transition j , and L (cm) isthe path length through the gas sample. Thermody-namic properties of the test gas (e.g., T, χ Abs ) can bedetermined by comparing measured absorbance spectrato modeled absorbance spectra, for example, using aspectral-fitting routine such as those described in Sec-tion IV.
B. m-FID Signal
1. Calculating the M-FID Signal
Recent work by Cole et al. [30] introduced the m-FIDsignal (which derives from the traditional time-domainfree-induction-decay signal through cepstral analysis), A ( t ) , which is related to I t , I o , and α according to Eq.3 and 4. A ( ν ) = − ln ( I t ) = α ( ν ) − ln ( I o ( ν )) (3) A ( t ) = F − [ A ( ν )] = F − [ α ( ν )] + F − [ − ln ( I o ( ν ))] (4)Here, F − represents the inverse Fourier Transform ofa given quantity, A ( t ) is the m-FID signal (also knownas the Cepstrum of I t ( t ) ) [32, 33], F − [ − ln ( I o ( ν ))] is the laser-intensity response, and F − [ α ( ν )] is themolecular-absorption response. Eq. 3 comes from tak-ing − ln of Eq. 1 (i.e., Beers Law). Eq. 4 illustratesthat the m-FID signal can be found from taking the in-verse Fourier Transform of A ( ν ) and that, in the timedomain, there is an additive relationship between themolecular-absorption response and the laser-intensityresponse due to the logarithmic operation in the fre-quency domain.
2. M-FID Using an Ultrafast Pulse
The physical meaning and behavior of the m-FID sig-nal and its components are best understood by consid-ering an experiment where a single ultrafast transform-limited pulse is used to measure the absorbance spec-trum of a molecule (e.g., similar to as described inTancin et al. [34]). That being said, it is importantto note that the m-FID signal can be calculated fromany measurement of a transmitted light intensity spec-trum. To elucidate the principles governing the m-FIDsignal, this section will discuss a simulated experimentwhere a transform-limited pulse with a full-width at
Figure 1. (a) Intensity ( I o , I t ) and absorbance spectra for a simulated ultrafast laser-absorption measurement of CO spectranear 2150 cm − using a transform-limited 55 fs pulse. (b) Zoom view of the beginning of the m-FID signal, laser-intensityresponse, and molecular-absorption response in the time domain which correspond to the spectra shown in the opticalfrequency domain in (a). half-maximum of 55 fs in the time domain and 267 cm − in the frequency domain is used to measure the ab-sorbance spectrum of CO’s fundamental vibration bandnear 2150 cm − at a temperature and pressure of 300K and 1 atm, respectively. Figure 1a illustrates sim-ulated intensity and absorbance spectra correspondingto this simulated experiment and Figure 1b illustratesthe corresponding m-FID signal in the time domain,which is composed of the laser-intensity response andthe molecular absorption response. In this case, thelaser-intensity response (shown in green) is largest attime zero, decays rapidly on the timescale of the pulseFWHM, and reaches near zero (1% of its initial in-tensity) within ≈ ≈ τ c,decay ≈ /π ∆ ¯ ν c assuming an instanta- Figure 2. Raw detector signal acquired in a scanned-DAexperiment with a DFB QCL scanning across CO’s P(0,20)and P(1,14) transitions at 500 Hz. The spectra were ac-quired in an ethylene-air flame at 1 atm.
Figure 3. (a) Measured I t for a single-scan across CO’s P(0,20) and P(1,14) transitions, I o determined from baseline fitting,and corresponding absorbance spectrum. (b) Zoom view of the beginning of the m-FID signal, laser-intensity response, andmolecular-absorption response in the time domain which correspond to the spectra shown in (a). The m-FID signal agreeswell with the molecular-absorption response signal at times > 1 ns where the contribution from the laser-intensity responsehas decayed to near zero. The measurements were acquired in an ethylene-air flame at 1619 K, 1 atm, and with 11.1% COby mole. neous excitation pulse of light where ∆ ¯ ν c [ s − ] is theaverage (across transitions) collisional-broadening (i.e.,Lorentzian) full-width at half-maximum. This followsfrom relations put forth to model the free-induction de-cay signal [30]. In this simulated experiment, τ c,decay =0.108 ns which agrees reasonably well with the decay ofthe m-FID signal envelope shown in Figure 1b.
3. M-FID Using an Injection-Current-Tuned Laser
In practice, LAS experiments are often performedusing injection-current-tuned lasers, for example, DFBTDLs and QCLs [2, 5]. In this case, injection-currentscanning is performed to scan the frequency of the laserlight and this also leads to pronounced intensity tuning.For example, Figure 2 shows the raw detector signalfor a scanned-wavelength direct-absorption (scanned-DA) experiment performed using a DFB QCL whichwas scanned across CO’s P(0,20) and P(1,14) absorp-tion transitions near 2059.9 cm − . In this case, it isclear that the laser intensity varies rapidly with opti-cal frequency on a scale that is comparable to that ofthe absorption lineshapes. In the context of the m-FIDsignal and its components, this translates into the laser-intensity response and molecular-absorption responsedecaying on a similar timescale.Figure 3a shows an example of a single-scan mea-surement of I t , I o (inferred from fitting a polynomial tothe non-absorbing regions of I t ), and the corresponding absorbance spectrum of CO near 2059.9 cm − whichwere extracted from the data shown in Figure 2. Forcomparison, Figure 3b shows the beginning of the m-FID signal, laser-intensity response, and the molecular-absorption response in the time domain which corre-spond to the measured spectra shown in Figure 3a. Inthis case, the laser-intensity response and molecular-absorption response decay on a similar timescale andare not well separated until t ≈ t =1 ns and t =2 ns; however far too much ofthe molecular-absorption response has decayed to zeroby t = 1 ns for this method to yield an accurate mea-surement. In fact, this was attempted and the fittingroutine failed to converge on a solution which motivatedthe development of the new technique described in Sec-tion IV. III. EXPERIMENTAL SETUP
Figure 4 shows a schematic of the experimental setupused for gas-cell measurements. The wavelength ofa distributed-feedback quantum-cascade laser (AlpesLasers) was scanned across CO’s P(0,20) and P(1,14)absorption transitions near 2060 cm − to determine thegas temperature and concentration of CO. Several re- High-SpeedDAQQCL Controller QCL Tube FurnaceSignal Generator To Baratron& Gas ManifoldMirror LensBP FilterDetectorTrigger @ 500 Hz
Figure 4. Schematic of the experimental setup used to ac-quire scanned-DA measurements of gas temperature and COmole fraction at 1 kHz in a heated static-gas cell. searchers have recently used these absorption transi-tions to provide high-quality measurements in a varietyof combustion applications [16, 40–43] and additionaldetails regarding their suitability for high-temperaturecombustion gases are provided by Spearrin et al. [40].The QCL produced a collimated laser beam ( ≈ − .The laser beam was directed through a static-gas celllocated within a high-uniformity tube furnace. The gascell is thoroughly described in [44], it employs sapphireor CaF (used here) rods to ensure that the laser lightpropagates through a thermally uniform ( ± at 1 atm and temperatures of 827 and1034 K. The laser light exiting the gas cell was focusedonto a photovoltaic MCT (mercury cadmium telluride)detector (Vigo Systems, PVI-5-1x1-TO8-BaF2) usingan anti-reflection coated, plano-convex, CaF lens (25.4mm diameter, 30 mm focal length). The photodetectorhas a 3dB bandwidth of 10 MHz and it is sensitive towavelengths from approximately 3 to 6 µ m. The pho-todetector’s voltage signal was recorded using a 12-bitdata-acquisition (DAQ) card (GaGe CSE123G2) with abandwidth of 500 MHz and a sampling rate of 3 GS/s.Onboard averaging of the detector signal was performedto reach a final sampling rate of 1.875 MS/s and aneffective bit depth of 16 bits for improved signal-to-noise ratio. A bandpass filter (Spectrogon) centerednear 2060 cm − with a FWHM of 40 cm − was usedto attenuate emission from the furnace and a thin ( ≈ H ) was passed throughthe core of the burner (0.5” wide cross section) and anair curtain (1” outer cross section) was used to stabi-lize the flame. The flow rates of air and fuel were ma-nipulated to achieve a stable laminar flame. The laserbeam was directed through the flame approximately 1cm above the burner surface where the flame thickness(estimated from images of visible flame emission) wasapproximately 1.25 cm. IV. LEAST-SQUARES FITTING TO THEM-FID SIGNALA. Procedure
This section describes our approach to determin-ing gas properties from measured m-FID signals incircumstances where the laser-intensity response andmolecular-absorption response do not separate quicklyin the time domain. This is especially relevant to sce-narios where the laser’s intensity varies with optical fre-quency with a similar magnitude and spectral shapecompared to the absorbance spectrum. This methodbuilds on the fitting routine put forth by Cole et al. [30]by introducing one critical modification, specifically, anestimate for I o ( ν ) . In this method a simulated m-FIDsignal is generated using (1) an estimated I o ( ν ) and (2)a simulated absorbance spectrum and this simulatedm-FID signal is least-squares fit to the measured m-FID signal. The introduction of an estimated I o ( ν ) al-lows the fitting routine to access more of the molecular-absorption response, which is particularly important ifthe laser-intensity and molecular-absorption responsesare similar. We will show that this approach is immuneto baseline errors that vary slowly with frequency, andthus does not require a perfect estimate for the baseline.The remainder of this section is devoted to describingand demonstrating the fitting routine in detail.Figure 5 illustrates a flow chart for the fitting rou-tine used to determine gas properties from measuredm-FID signals. Prior to calculating the measured m-FID signal, any background emission (e.g., from flamegas) must be subtracted from the measured detector sig-nal to properly determine I t ( ν ) as is traditionally thecase. In addition, the measured spectrum of I t ( ν ) mustbe re-sampled onto a frequency axis with uniform spac-ing (e.g., using interpolation). This is required in ex-periments performed with, for example, DFB QCLs orTDLs since their optical frequency does not vary exactlylinearly with injection current, particularly at high scanrates or when using a large scan amplitude. Next, them-FID signal corresponding to a measured spectrum of I t ( ν ) must be calculated according to F − [ − ln ( I t ( ν ))] (see Eq. 4). The inverse Fourier Transform should be Converge? YesMeasured I t (v) Estimated I o (v)A(v) -ln()m-FID Signal: A(t)
Clipped m-FID Signal:
A(t :t ) F -1 Levenberg Marquardt Algorithm Semi-Empirical I t,SE (v) Simulated Absorbance: a (v,T,P,X,L) Beer’s LawSemi-Empirical m-FID: A SE (t) Clipped m-FID Signal: A SE (t :t )A(v) F -1 -ln()Converge? No, update T, P, X Figure 5. Flowchart illustrating principles of the least-squares fitting routine used to determine gas properties frommeasured m-FID signals. calculated such that the m-FID signal is a purely realsignal. This can be done using Python’s function irfftor equivalent.The simulated m-FID signal should be calculated asfollows. First, an estimate for the baseline light in-tensity I o ( ν ) must be obtained. Here this was doneby least-squares fitting a 3rd-order polynomial to thenon-absorbing regions of the measured I t (see Figure2). Alternatively, a background measurement of I o ( ν ) (e.g., in the absence of absorbing gas) could be used todetermine an estimate for I o ( ν ) . Next, the absorbancespectrum must be calculated at gas conditions set bythe free parameters. Here, the HITEMP2010 database[45] and a spectroscopic model similar to that describedin [46] were used to simulate the absorbance spectrumof CO at the wavelengths of interest. Next, a simu-lated, semi-empirical spectrum of the transmitted lightintensity I t,SE ( ν ) was calculated using Eq. 1 with theestimated I o ( ν ) and simulated α ( ν ) . A simulated, semi-empirical m-FID signal ( A SE ( t ) ) was then calculatedfrom F − [ − ln ( I t,SE ( ν ))] .A non-linear least-squares fitting routine employingthe Levenberg-Marquardt algorithm (Matlab’s nlinfit)was used to determine the best-fit m-FID signal andcorresponding gas conditions. The algorithm seeks tominimize the sum-of-squared error between the mea-sured and simulated m-FID signals at times between t and t which must be chosen appropriately (discussedlater) to isolate the best-fit m-FID signal from error in-troduced by uncertainty in the estimated I o . In thiswork, 5 free parameters were employed to adjust thesemi-empirical m-FID signal and, more specifically, theunderlying absorbance spectrum it corresponds to. The following model inputs were treated as free parameters:(1) the gas temperature, (2) the mole fraction of theabsorbing species, (3) a scaling parameter for ∆ ν c of allabsorption lines, and (4-5) two frequency shift param-eters to accurately position the linecenter frequency ofeach of the dominant absorption lines. Given the largedisparity in magnitude between the various free parame-ters, the temperature was scaled by 10 − and the scalingfactor on ∆ ν c was scaled by 10 − prior to feeding theseinputs to the absorption spectroscopy model called bynlinfit. This scaling is reversed within the absorptionspectroscopy model. The gas pressure and optical pathlength were held constant at the known values. Utiliz-ing a single scaling factor on the collisional widths (e.g.,to account for unknown collisional broadening in com-bustion gas) of all absorption lines is justified here sincecollisional broadening coefficients for CO’s P(0,20) andP(1,14) transitions are similar (e.g., differing by only ≈
2% percent for air broadening at the temperatures ofinterest here). This approach may not be well suited, forexample, for measurements of H O absorbance spectrawhere collisional-broadening coefficients can vary dra-matically between states and collision partners [47, 48].
B. Selection of the fitting window start time, t Selecting an appropriate value for t is critical tomaximizing the accuracy of the best-fit parameters (i.e.,gas conditions) since this parameter governs which ofthe strong early-time m-FID signal components are ig-nored by the least-squares fitting routine. We will showthat errors in the simulated m-FID signal which are in-troduced by errors in the estimated I o appear at veryearly times in the m-FID signal, and thus selecting avalue of t that is too small will retain the influenceof those errors. On the other hand, using a value of t that is too large (e.g., to avoid all dependence on I o )could correspond to ignoring too much of the molecular-absorption response, thereby making it difficult or im-possible to accurately infer the underlying absorbancespectrum and gas conditions it corresponds to. Theremainder of this section describes how to use an esti-mated error in I o ( ν ) (obtained from a spectral-fittingroutine) to determine an appropriate value for t . Thisapproach follows from recognizing that I o can be de-scribed by Eq. 5: I o ( ν ) = I o,estimate ( ν ) I o,error ( ν ) (5)where I o is the true incident laser intensity, I o,estimate is an estimate for I o (e.g., from baseline fitting), and I o,error is an unknown frequency-dependent correctionfactor which accounts for the error in I o,estimate . In thiscase, the m-FID signal is given by Eq. 6: Figure 6. (a) Example single-scan measurements of CO’s absorbance spectrum in an ethylene-air flame, correspondingbest-fit absorbance spectrum, estimated error in I o , and peak-absorbance-normalized residual. (b) Time domain signalscorresponding to the spectra shown in (a). The baseline error decays rapidly in the time domain thereby enabling baseline-insensitive measurements of gas properties to be obtained via the m-FID signal despite using a small t . All spectra andsignals shown correspond to a scanned-DA measurement at 1618 K with 11.1% CO by mole at 1 atm. A ( t ) = F − [ α ( ν )] + F − [ − ln ( I o,estimate ( ν ))]+ F − [ − ln ( I o,error ( ν ))] (6)which shows that the m-FID signal consists of threedistinct components with an additive relationship. Asa result, to achieve an accurate measurement from least-squares fitting a simulated m-FID signal to a measuredm-FID signal, t must simply be chosen such that thecontribution from F − [ − ln ( I o,error ( ν ))] has decayed tozero. It should be noted that the error in I o,estimate canalso be accounted for inside the exponential of Beer’slaw via a frequency-dependent shift in the absorbance.In this case the additive relationship of the three m-FID signal components holds, but the time-domain sig-nal associated with baseline error would be given by F − [ − I o,error ( ν )] (i.e., differing only by ln ). This ap-proach was taken here for convenience.Figure 6a shows an example of a single-scan mea-surement of CO’s absorbance spectrum in the ethylene-air flame, the corresponding best-fit spectrum, the es-timated error in I o , and the residual between the mea-sured and best-fit spectrum. The measured absorbancespectrum was calculated using an I o that was obtainedusing the traditional method of fitting of a 3rd-orderpolynomial baseline to the non-absorbing regions of I t .The best-fit spectrum was calculated using a spectral-fitting routine analogous to that described previouslyfor determining the best-fit m-FID signal; however, inaddition a 3rd-order polynomial was superimposed onto the simulated absorbance spectrum in an effort to ac-count for and estimate errors in I o ( ν ) that were inducedby the imperfect nature of inferring I o from the tra-ditional method of fitting a polynomial to the "non-absorbing" regions of I t . This approach is later referredto as "Method 2." The coefficients of the polynomialwere treated as free-parameters in the model, therebyleading to a total of 9 free parameters (compared to5 needed for reliable measurements via m-FID signals).Using this method, peak-absorbance-normalized residu-als typically <2% were achieved. In this case, the base-line error (inferred from the polynomial incorporatedwithin the spectroscopic model) varied monotonicallyfrom absorbance equivalent values of -0.014 to -0.007.Figure 6b illustrates that the baseline error decays tozero rapidly in the time domain, which is expected giventhat it varies slowly and smoothly in frequency space.The time required for the estimated baseline error todecay to within 1% of its initial value was used to de-termine t . In this case, t =0.15 ns and this correspondsto the 4th data point in the time history of the m-FIDsignal. As a result, the first 3 data points in the timehistory were ignored by the least-squares fitting routineused to determine the best-fit m-FID signal.Figure 6b also shows the measured and best-fit m-FID signals at times between t and t . The measuredand simulated semi-empirical m-FID signals at timesless than t and greater than t were not used in anymanner to determine the best-fit m-FID signal and cor-responding gas conditions. The best-fit m-FID signalagrees within 2% of the measured m-FID signal at alltimes. The value of t (3.25 ns) was chosen to be suf-ficiently large such that the molecular-absorption re-sponse had decayed to within 0.1% of its initial value,thereby retaining the vast majority of information per-taining to the absorbance spectrum. Using a larger t was not found to significantly impact the gas conditionscorresponding to the best-fit signal. It should be notedthat utilizing both sides (i.e., at the beginning t to t and end t end − t to t end − t ) of the m-FID signaltime-history (as done by Cole et al. [30]) which approxi-mately mirror each other was not found to significantly(i.e., >0.2% change) impact the gas conditions corre-sponding to the best-fit m-FID signal. This may notalways be the case depending on the spectrum of thenoise in the data. V. EXPERIMENTAL RESULTS
This section presents measurements of gas tempera-ture and CO concentration acquired in a static-gas celland ethylene-air diffusion flame. Results are presentedusing the new m-FID approach as well as two traditionaldata-processing techniques in order to provide propercontext for the results obtained using the new m-FID-based signal-processing technique. Specifically, resultsfrom the following methods are presented. "Method1" corresponds to least-squares fitting an absorbancespectrum provided by a two-line Voigt model [21] witha 3rd-order polynomial baseline correction, "Method2" corresponds to least-squares fitting a complete ab-sorbance spectrum (obtained using HITEMP2010 [45])with a 3rd-order polynomial baseline correction, and"Method 3" corresponds to least-squares fitting a sim-ulated, semi-empirical m-FID signal to the measuredm-FID signal as described in Section IV. All methodsrely on the same initial estimate for I o which is ob-tained from least-squares fitting a 3rd-order polynomialto the non-absorbing regions of I t . Method 1 employs10 free-parameters ( ν o , ν c , and A α (the integrated ab-sorbance) for both lines and 4 polynomial coefficientsfor correcting the baseline) with the Doppler full-widthat half-max ( ∆ ν D ) fixed according to the tempera-ture obtained from the two-color ratio of integratedabsorbance. Method 3 employed 5 free-parameters tomodel the absorbance spectrum and, hence, m-FID sig-nal (as described in Section IV), and Method 2 em-ployed an additional 4 free-parameters (for a total of 9)for modeling the baseline error with a 3rd-order poly-nomial (as done in Method 1).Using Method 1, the gas temperature was calculatedfrom the two-color ratio of integrated absorbances pro-vided by the fitting routine, and the mole fraction ofCO was calculated from the integrated absorbance ofthe P(0,20) line. Using Methods 2 and 3, the gas tem-perature and mole fraction of CO are free-parameters and, therefore, are direct outputs of the fitting rou-tine. In flame experiments, the path length through theflame was assumed to be 1.25 cm (estimated from vis-ible images). The spectroscopic parameters employedby all three methods were taken from HITEMP2010[45] which is known to be accurate for these transitions[40] and the measurements reported here support thisfurther. A. Gas-Cell Measurements
Measurements of temperature and CO mole fractionwere acquired at 1 kHz (due to using up-scan and down-scan measurements) over a 100 ms period (100 mea-surements) in a mixture of 2% CO in N at 1 atm andtemperatures of 827 and 1034 K. Figure 7a shows anexample of a single-scan measurement of CO’s P(0,20)and P(1,14) absorption transitions at 1 atm and 1034K, as well as the best-fit spectra corresponding to Meth-ods 1 through 3 and, if applicable, the error in baselineintensity inferred from the polynomial baseline correc-tion (i.e., "poly shift") which was superimposed on thesimulated absorbance spectra within the spectral-fittingroutine (for Methods 1 and 2 only). It is important tonote, that each method was applied to the same spectrawith the same initial estimate for the baseline light in-tensity. As a result, Methods 1, 2, and 3 were all appliedto a measurement with the same initial baseline error.First, the results shown in Figure 7a illustrate that thereis an error in I o of ≈ I o are coupled to the spectroscopicmodel. Methods 1 and 2 attempt to account for this er-ror via a 3rd-order polynomial (i.e., the "poly shift",best-fit is shown) and Method 3 escapes this error viause of the m-FID signal with a t > 0. The gas tem-perature and CO mole fraction inferred from Method 1,2, and 3 for this measurement are 1064 K and 2.11%,1038 K and 2.05%, and 1034 K and 2.04%, respectively.As a result, Methods 2 and 3 provided nearly identicalresults which agree with expected values within 0-0.4%for temperature and 2-2.5% for CO mole fraction. Incontrast, the gas temperature and CO mole fractioninferred from Method 1 exhibit a significantly largererror, specifically, 2.9% for temperature and 5.5% forCO mole fraction. The best-fit spectra associated withMethods 2 and 3 are virtually identical, as expectedgiven the nearly identical gas conditions associated witheach. However, the best-fit spectrum associated withMethod 1 exhibits subtle but significant differences (seezoom view within Figure 7a). In addition, the baselineerror inferred from Methods 1 and 2 differ significantly,thereby illustrating how it is difficult to reliably infer theerror in I o and, therefore, motivating the use of Method3 (i.e., the m-FID-based approach presented in Section0 Figure 7. (a) Example single-scan measurement of CO absorbance spectra with baseline error and best-fit spectra calculatedusing Methods 1-3 which address baseline error via a polynomial shift or using the m-FID signal between t and t . (b)Measured and best-fit m-FID signal (calculated using Method 3) corresponding to the measured absorbance spectra shownin (a). The measurements were acquired in a mixture of 2% CO in N at 1 atm and 1034 K.Table I. Comparison of results obtained using various data-processing techniques for 100 measurements of temperature andCO mole fraction acquired in a static-gas cell at 827 and 1034 K.Method Average Spread Error 1- σ T, K X CO T, K X CO T, % X CO ,% T, % X CO , %1. 2-Line Voigt + poly 916.3, 1077.8 0.0224, 0.0216 160.7, 75.1 3.8E-3, 1.8E-3 10.8, 4.2 12.0, 8.0 10.4, 3.9 10.4, 4.92. α from HITEMP + poly 833.1, 1035.3 0.0203, 0.0205 24.1, 19.6 3.4E-4, 3.3E-4 0.7, 0.1 1.5, 2.5 1.7, 1.1 1.0, 1.13. m-FID from HITEMP 831.6, 1034.5 0.0202, 0.0204 13.6, 11.7 1.1E-5, 2.4E-5 0.6, 0.04 1.0, 2.0 1.1, 0.8 0.5, 0.6 IV). For this measurement, it seems that the additionalflexibility provided by floating the collisional FWHMof both lines in Method 1 (the 2-line Voigt method)prevented the spectral-fitting routine from accuratelyinferring the error in the baseline, thereby introducingbiases in the integrated absorbance inferred for one orboth of the transitions and ultimately leading to errorsin the gas conditions corresponding to the best-fit spec-trum. This is supported by the fact that the best-fitcollisional FWHM for the P(0,20) and P(1,14) lines ac-cording to Method 1 were 0.0489 and 0.0537 cm − . Thiscorresponds to a difference of 9.8% where calculationsperformed using air-broadening coefficients and temper-ature exponents from HITEMP2010 [45] suggest, albeitassuming an air bath gas, that the collisional FWHMfor these lines should agree within 2.2% at 1034 K.Table I shows the average value, average spread (i.e.,difference) between up-scan and down-scan measure-ments, error in the average value, and the 1- σ precision(i.e., 1 standard deviation) for each dataset which con-sists of 100 individual measurements (50 up-scans and50 down-scans). The results illustrate two key find- ings. First, in all cases, the m-FID based approach(Method 3) was the most accurate. The temperatureand CO mole fraction inferred from the best-fit m-FIDsignal were accurate to within 0.6% and 1.0% at 827K and 0.04% and 2.0% at 1034 K. Method 2 providedmeasurements with slightly larger errors, and Method1 was considerably less accurate presumably due to itsincreased sensitivity to the baseline error encounteredin this experiment. Second, in all cases, Method 3provided a smaller measurement precision and smallerspread between up-scan measurements and down-scanmeasurements (this was also the case in flame exper-iments, see zoom view within Figure 8). The spreadis caused by differences in baseline error between up-scans and down-scans. The influence of this error uponthe temperature and CO mole fraction inferred fromthe data is reduced in Method 3 because the m-FID ap-proach is insensitive to baseline errors. For example, forthe dataset acquired at 1034 K, the spread between themean gas temperature inferred from up-scan measure-ments and down-scan measurements differed by only11.7 K (1.1%) using Method 3, but 19.6 K (1.9%) for1 Figure 8. Measured time histories of (a) temperature and (b) CO mole fraction acquired using Methods 1-3 for dataacquired in an ethylene-air diffusion flame at 1 atm. The results demonstrate that Method 3 provides the most precisemeasurements.Table II. Comparison of results obtained using various data-processing techniques for measurements of temperature andCO concentration acquired in a laminar ethylene-air diffusion flame at 1 atm.Method Average Spread 1- σ T, K X CO T, K X CO T, % X CO , %1. 2-Line Voigt + poly 1633.3 0.115 121 0.018 4.1 8.52. α from HITEMP + poly 1627.1 0.113 42.8 0.008 1.4 3.53. m-FID from HITEMP 1617.3 0.112 12.1 0.004 0.7 1.9 Method 2 and 75.1 K (7.3%) for Method 1. Method 3exhibited a spread in temperature measurements thatwas ≈ σ measurement precision for temperature and COmole fraction measurements. Collectively these resultsdemonstrate the ability of Method 3 to considerablyimprove the accuracy and precision of temperature andconcentration measurements acquired using LAS withinjection-current-tuned lasers. B. Flame Measurements
Measurements of temperature and CO mole fractionwere acquired in an ethylene-air diffusion flame at 1atm to evaluate the performance of Method 3 in a testenvironment with beamsteering, which can introducetime-varying errors in the baseline. Figure 8 illustratesmeasured time histories of temperature and CO molefraction acquired using Methods 1 through 3 and Ta- ble II shows the average temperature and CO molefraction, as well as the spread between up-scans anddown-scans and 1- σ precision for the 800 ms time his-tories shown (800 total measurements). The measuredtime histories illustrate that the flame conditions werequasi-steady during the test. The mean gas tempera-ture and CO mole fraction agree within precision forall three methods. Method 3 provided a significantlysmaller spread and precision for temperature and COmole fraction. Specifically, the spread in temperatureprovided by Method 3 was 3.5 and 10 times smaller thanMethods 2 and 3, respectively. Similarly, the spread inCO mole fraction provided by Method 3 was 2 and 4.5times smaller compared to Methods 2 and 3, respec-tively. These results further demonstrate that Method3 provides superior measurement precision compared toMethods 1 and 2, which further suggests that utilizingthe best-fit m-FID signal to determine gas conditionsis a more robust technique when errors in the baselinelight intensity are present.2 VI. CONCLUSIONS
This manuscript presented a new method which im-proves the accuracy and precision of LAS measurementsof gas properties by isolating the molecular-absorptionsignal from errors induced by unknown variations in thebaseline light intensity. This method relies on least-squares fitting a simulated m-FID signal to a mea-sured m-FID signal in the time domain, where the for-mer is obtained from an estimated I o and simulatedabsorbance spectrum. This modified approach is re-quired (in comparison to other m-FID-based techniques[30]) when the laser-intensity response and molecular-absorption response do not separate well in the timedomain (as is the case in many applications employingTDLs, QCLs, and other light sources which exhibit pro-nounced intensity tuning). While this particular imple-mentation is not "baseline-free," it was demonstratedthat error induced by the estimated I o can be avoidedby ignoring the beginning of the m-FID signal time his-tory in the fitting routine.This approach was demonstrated using scanned-wavelength direct-absorption-spectroscopy measure-ments of CO’s P(0,20) and P(1,14) absorption lines us-ing a DFB QCL. Measurements of gas temperature and CO were obtained in a static-gas cell and ethylene-airdiffusion flames. The new m-FID-based method demon-strated the ability to provide improved measurement ac-curacy and precision in all cases compared to two estab-lished methods which rely on inferring baseline errorsvia polynomial corrections.The theory and results presented here suggest thatthis m-FID-based approach can improve the mea-surement accuracy and precision of a wide range ofabsorption-spectroscopy diagnostics. VII. ACKNOWLEDGEMENTS
This work was supported by Grant FA9300-19-P-1506with Dr. John W. Bennewitz of the Air Force ResearchLaboratory (AFRL) as program monitor. G.C.M wassupported by the National Science Foundation Gradu-ate Research Fellowship Program (NSF GRFP, Grant:1842166-DGE). In addition, G.B.R. and A.S.M. ac-knowledge support from the Air Force Office of Scien-tific Research (AFOSR, Grant: FA9550-17-1-0224) withDr. Chiping Li as program monitor, and R.K.C. wassupported by the National Aeronautics and Space Ad-ministration under the Earth and Space Sciences Fel-lowship program (PLANET18R-0018). [1] M. G. Allen, “Diode laser absorption sensors for gas-dynamic and combustion flows.,”
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