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High-resolution absorption measurements by use of two-tone frequency-modulationspectroscopy with diode lasers

Avetisov, V. G; Kauranen, P

Published in:Applied Optics

DOI:10.1364/AO.36.004043

1997

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Citation for published version (APA):Avetisov, V. G., & Kauranen, P. (1997). High-resolution absorption measurements by use of two-tone frequency-modulation spectroscopy with diode lasers. Applied Optics, 36(18), 4043-4054.https://doi.org/10.1364/AO.36.004043

Total number of authors:2

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https://doi.org/10.1364/AO.36.004043

https://portal.research.lu.se/portal/en/publications/highresolution-absorption-measurements-by-use-of-twotone-frequencymodulation-spectroscopy-with-diode-lasers(b5e9b5a8-1fed-4a48-ae87-e0cee36ab640).html

https://doi.org/10.1364/AO.36.004043

High-resolution absorption measurementsby use of two-tone frequency-modulationspectroscopy with diode lasers

Viacheslav G. Avetisov and Peter Kauranen

The capability of two-tone frequency-modulation spectroscopy ~TTFMS! in deriving spectral line-shapeinformation was investigated. Two oxygen A-band transitions at 760 nm were selected, and the Voigtprofile and two different collisionally narrowed line profiles were employed in their analysis. By meansof a least-squares fitting procedure, we obtained accurate information regarding transition strengths andpressure-induced broadening, shift, and narrowing coefficients. Both TTFMS and direct absorption lineshapes were modeled with deviations as small as 0.3% over a wide pressure range by use of thecollisionally narrowed line profiles. Line parameters measured with TTFMS showed excellent agree-ment with the parameters measured with direct absorption. The experimental technique used constant-current fast-wavelength scanning, which improved measurement accuracy. © 1997 Optical Society ofAmerica

Key words: frequency-modulation spectroscopy, high-resolution spectroscopy, diode laser, oxygen.

1. Introduction

Diode-laser absorption spectroscopy based on fre-quency modulation ~FM! is a method of choice for gasconcentration measurements in which high sensitiv-ity and fast time response are required.1–6 Al-though a variety of FM schemes exist, the commonunderlying concept is a movement of the detectionband to a high-frequency region in which the lasersource ~1yf ! noise is avoided and signal detection isperformed with frequency and phase-sensitive elec-tronics. Depending on the number of modulationtones, the choice of modulation frequency relative tothe width of the studied absorption feature, and thedetection frequency, the method is referred to aswavelength modulation spectroscopy ~WMS!,7 fre-quency modulation spectroscopy ~FMS!,8 or two-tonefrequency modulation spectroscopy ~TTFMS!.9 InTTFMS the laser is modulated by two closely spaced

When this research was performed, V. G. Avetisov and P. Kau-ranen were with the Division of Atomic Physics, Lund Institute ofTechnology, P.O. Box 118, S-221 00 Lund, Sweden. V. G. Avetisovis now with the Norsk Elektro Optikk AyS, Solheimveien 62A, P.O.Box 384, 1471 Skårer, Norway.

Received 11 December 1995; revised manuscript received 16August 1996.

0003-6935y97y184043-12$10.00y0© 1997 Optical Society of America

frequencies, which are comparable with or largerthan the width of the absorbing feature of interest,and the phase-sensitive detection of an absorption-related beat tone is at the intermediate frequency,i.e., the difference frequency, chosen to lie in the low-noise region of the laser. Both FMS and TTFMS,with high-frequency detection, offer the possibility ofquantum noise limited detection at a 1027–1028 frac-tional absorption.10–12 In practice, however, opticalinterference effects in the laser beam path often limitthe sensitivity to approximately 1026. Neverthe-less, as we show in our paper, this detection limitallows accurate analysis of line shapes from veryweak absorptions.

In quantitative concentration measurements by FMtechniques, the line-shape peak-to-peak value is cali-brated typically with a reference gas or an absorptionspectrum recorded with direct detection. However,accurate measurements under variable linewidth con-ditions, such as those over a wide range of concentra-tions, temperatures, and pressures, requireinformation about the line parameters and an ade-quate theory of the heterodyne-detected signal. Line-shape analysis by least-squares fitting of an assumedline profile to experimental data, yielding accurateline-shape information, was demonstrated only in afew papers with FMS13 and TTFMS.14,15 A least-squares fitting procedure, with an appropriate theoryof the detected signal and well-known line parameters,

20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4043

is used to increase not only the accuracy of quantita-tive concentration measurements but also the accu-racy of measurements of gas parameters such astemperature, velocity, and pressure.16,17 The abilityto perform high-sensitivity measurements with highaccuracy in a varying environment is important inareas such as combustion diagnostics and chemicalanalysis.

Of the various FM techniques, TTFMS seems to bethe most suitable for high-accuracy quantitative mea-surements based on data modeling. The reasons forthis are many. The electric-field approach adopted inthe theories of FMS and TTFMS is valid at arbitrarymodulation frequencies,18 whereas the intensity ap-proach commonly adopted in WMS theory is limited tolow modulation frequencies. In WMS the large mod-ulation index necessary to achieve an optimum signalamplitude generates a multitude of sideband compo-nents in the field spectrum, which makes the line-shape calculation in the electric-field approachcomplicated. In the simpler intensity approach, inwhich the WMS line-shape information is extractedfrom the dependence of the signal amplitude on theapplied modulation current,19,20 the entire line shapeis not considered, which makes the accuracy of theapplied theoretical model and the retrieved line pa-rameters difficult to evaluate. In FMS the absorptionand dispersion components in quadrature are of simi-lar amplitude, which requires high accuracy in theadjustment of the detection phase for unambiguousretrieval of the line-shape information. TTFMS, onthe other hand, with its low detection frequency, has anearly negligible dispersion contribution in recordedline shapes, and, in comparison with WMS, the rela-tively low number of generated sidebands allows accu-rate theoretical representation of line shapes, which inturn can result in greater retrieval accuracy.

We recently investigated the validity of the TTFMStheory over a wide range of modulation and line pa-rameters, and we modeled numerically generatedline shapes with a nonlinear least-squares fitting pro-cedure for the purpose of providing the optimumchoice of modulation frequencies and FM indices foraccurate line-shape analysis.15 Also, the weak dis-persion component, which appears in the heterodyne-detected signal when the detection phase is adjustedfor maximum signal amplitude, was investigated forvarious experimental situations. Moreover, nonlin-ear distortion in the modulation response of the diodelaser21 was considered in the theory, which provideda better description of data.

In the present study we extend our previous inves-tigation of TTFMS for high-accuracy quantitativemeasurements15 to experimental data recorded overa broad range of pressures. An extensive test of thereliability of the line parameters derived by the least-squares fitting procedure is performed by comparisonof the result with line parameters derived from spec-tra recorded by use of direct absorption. Further,the high resolution and high sensitivity of the exper-imental technique employed makes a qualitativecomparison of the Voigt profile and collisionally nar-

4044 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997

rowed line profiles possible with both TTFMS anddirect absorption.

Experimental techniques for accurate recording ofline shapes must exclude effects arising from a varia-tion in the laser spectral characteristics during wave-length scanning. It is known that diode-laserintrinsic properties, such as linewidth, longitudinalmode structure, amplitude and detuning of therelaxation-oscillation sidebands, vary with the laserinjection current.22,23 Also, the modulation charac-teristics, such as the FM and amplitude modulation~AM! indices and the AM–FM phase difference, dependstrongly on the injection current.23 Thus wavelengthscanning of a diode laser by superimposing a saw-toothcurrent waveform on the laser drive current can pro-duce significant measurement errors. In this studywe reduce the effect of injection current variations bysuperimposing a rectangular waveform on the diode-laser drive current, which produces constant-currentwavelength scans.

The experimental arrangement required for re-cording data with high sensitivity and low instru-mental distortion is described. This descriptionincludes the constant-current wavelength scanningtechnique, the data acquisition, and an optical dual-beam subtraction scheme for eliminating backgroundsignals and excess laser noise.

2. Experimental

Figure 1 shows the basic experimental arrangement.A detailed description of the apparatus is given else-where.3 Here, we describe those parts that are ofimportance for high-resolution measurements.

A. Experimental Setup

The diode laser, a double-heterostructure GaAlAs la-ser manufactured by Mitsubishi ~ML4405!, operates

Fig. 1. Schematic of the experimental setup for combined TTFMSand direct absorption measurements.

in a single mode near 760 nm with a maximum out-put power of 5 mW. The diverging beam from thelaser is collimated by a Newport F-LA40 diode-laserlens and is divided by three cube beam splitters.The first polarizing cube beam splitter passes most ofthe intensity ~70%! to a dual-beam subtractionscheme in which the oxygen cell measurements aremade. It passes the rest of the intensity into a sidearm with a half-wave ~ly2! plate and a second polar-izing cube beam splitter, which divides the beam anddirects it onto two photodetectors, D1 and D2. Byrotating the half-wave plate, we can optimize theintensity in either of the two beams after the polar-izing beam splitter. At photodetector D1 referencespectra of the oxygen absorption from the ambient airare recorded by use of either direct absorption orTTFMS. The reference spectra are used to deter-mine pressure-induced line shifts and to monitor thelaser intensity during the TTFMS cell measure-ments. At photodetector D2 we obtain a wave-number scale by recording the intensity transmittedthrough a solid Fabry–Perot etalon with a free spec-tral range of 0.08033 cm21 and a finesse of 4. Theetalon can be replaced by a confocal Fabry–Perot in-terferometer ~FPI! with a free-spectral range of 0.25cm21 ~7.5 GHz! and a finesse of approximately 100 fora coarse estimation of the FM index from thesideband-to-carrier intensity ratio and for monitoringthe laser output spectral characteristics. The use ofa quarter-wave plate ~ly4! reduces optical feedbackinto the diode laser from the etalon or the FPI.

The dual-beam subtraction scheme cancels the ab-sorption signal from the ambient oxygen in the beampath outside the absorption cell. The beam is di-vided by a 50y50 nonpolarizing beam splitter anddirected onto two identical photodiodes ~photodetec-tor D3!, which are mounted on a movable platform formatching the distances travelled by both beams ~sig-nal and reference! in open air. The photodiodes,which were carefully selected to have similar transfercharacteristics, are connected with opposite polari-ties to a common transimpedance amplifier. Thisphotodetector configuration maintains backgroundcancellation with high linearity over a broad band-width. The passband of the detector was dc to 15MHz ~3 dB!, and in this band the subtraction wasbetter than 40 dB. A polarizing cube beam splitter,which deflects part of the reference beam, balancesthe intensities of the two beams by rotation aroundthe beam axis.

The photodetectors D1, D2, and D3 contain re-versed biased p-i-n photodiodes ~S-1190 Hamamatsu!and a transimpedance amplifier with a low-noise op-erational amplifier ~Burr–Brown OPA-620!.

The stainless steel gas cell, equipped with wedgedwindows, accommodates an absorption path length of90 cm. The cell pressure was measured by a CCMInstruments CCM-1000 capacitance manometer,which has a specified accuracy of 0.5% of the reading.Oxygen of 99.998% purity with natural isotope abun-dance was used in the measurements.

A Melles Griot 06 DLD103 diode-laser driver was

used for current supply and temperature control.The diode laser was biased by a dc current below thethreshold. A rectangular current waveform derivedfrom a pulse generator ~Tektronix RG501!, with cur-rent pulses selected with 0.1- to 1-ms duration at rep-etition rates in the range of 1–0.1 kHz, wassuperimposed on the laser dc current. The two mod-ulation frequencies ~nm 6 1y2 V! were generated bymixing the 585-MHz output of an rf generator~Wavetek 2510A! with the 5.35-MHz output of a sine-wave generator ~Tektronix SG 503! in a mixer ~Mini-Circuit ZFM-4H!. The modulation current passing avariable attenuator was superimposed on the laserdrive current by means of a bias tee. The beat signalsat V 5 10.7 MHz from detectors D1 and D3 werefiltered and amplified at 30 dB by a low-noise amplifier~Miteq AU-2A-0110! before demodulation in a double-balanced mixer ~Mini-Circuit ZFM-3!. A phaseshifter with variable delay lines was incorporated for aprecise adjustment of the local oscillator phase. Thedemodulated signals were amplified and low-pass fil-tered at 1 MHz ~Stanford SR560! before data acquisi-tion. The detection bandwidth was carefully selectedwith respect to the rate at which the laser wavelengthis scanned across the absorption lines. When mea-suring direct transmission, we connected detectors D1and D3 directly through a low-pass filter ~1 MHz! fordata acquisition. Great effort was placed on filteringand proper grounding of the electronics.

B. Data-Acquisition System

The spectra were collected in 1024 channels of databy an 8-bit Tektronix 2431L digital oscilloscope andaveraged 1024 times in a personal computer. Toenhance the vertical digital resolution of the oscillo-scope, we found it necessary to employ different off-sets to the incoming waveforms to ensure that thesignal was not always digitized by the same portion ofthe analog-to-digital converter ~ADC!. During anaveraging cycle ~for as many as 1024 spectra! thesignal offset was moved gradually by the computersoftware through a total of 16 digitizing levels on theADC. This way the differential nonlinearity causedby a nonuniform spacing of the digitizing levels in theADC was effectively reduced.24 A resolution of bet-ter than 12 bits was achieved, which is more than 2bits better than that obtained by conventional signalaveraging. The total nonlinearity, i.e., the generaltransfer function of the digital oscilloscope, was cor-rected after a full averaging cycle with a previouslymeasured table of the deviation from an ideal lineartransfer function. This procedure improved the lin-earity of the digital oscilloscope from 0.5% to 0.1% ofthe full scale of the ADC. The linearity of the com-plete data-acquisition system, which includes a pho-todiode, amplifiers, and mixers, was estimated to bebetter than 0.5% of the maximum scale used.

C. Wavelength Scanning

A rectangular current pulse with a rise time that wasshort compared with the growth rate of the junctiontemperature was used to produce a heat-induced


constant-current wavelength scan under the dura-tion of the pulse.25,26 There are several advantagesto keeping the wavelength-shift-inducing currentconstant during the wavelength scan. First, varia-tions of the laser intensity and spectral characteris-tics during the scan are minimized. Second, whendirect absorption is used, the zero intensity level canbe recorded directly. Third, depending on the pulseduration and repetition rate, the laser can be oper-ated at higher optical powers and temperatures. Atthe beginning of the current pulse a very high scan-ning rate in the range of 0.1–1 cm21yms can beachieved. This has the additional advantage that asingle spectrum can be recorded on a short time scale,which reduces the influence of environmental fluctu-ations. The drawback to a current pulse in compar-ison with a current ramp is that the inducedwavelength shift is more nonlinear. In high-resolution measurement, however, this is not a seri-ous disadvantage because calibration of the relativewave-number scale of the recorded absorption spec-tra is always required. In our measurements we userecordings of the spectral throughput from a solidFabry–Perot etalon to linearize the wavelength scaleof our spectra computationally.

Figure 2 presents a comparison of the spectralthroughput from the FPI, in which an FM laser iswavelength scanned with a current ramp and a rect-angular current pulse. A significant change in thesideband-to-carrier ratio ~i.e., the FM index! occursduring the current ramp, whereas a nearly constantFM index is maintained during the current pulse.Note that whereas the laser carrier intensity increaseslinearly with the current ramp, the sideband intensi-ties that depend on the FM current remain constant.

Fig. 2. Comparison of the power spectra of a two-tone frequencymodulated laser, which is wavelength scanned with ~a! a currentramp and ~b! a current pulse. The FM index changes with thelaser intensity during the current ramp, whereas a constant FMindex is maintained over the wavelength scan induced by thecurrent pulse. The free-spectral range of the Fabry–Perot inter-ferometer was 7.5 GHz.


Note that heating of the laser-active region duringa current pulse slightly raises the threshold current,which in turn causes a small change in the FM index.The relative FM index change during the wavelengthscan can be estimated from the change in the laserpower. By measuring the FM index from thesideband-to-carrier ratio at different drive currentswith an FPI, we observed that the FM index scaleswith the laser power P0 approximately as 1y=P0.Thus a relative FM index change db is related to therelative laser power change dP0 as udbu ' 0.5udP0u.In the measurements reported here the thresholdcurrent change over a 2-cm21 single-mode scan pro-duced a 6% change in the laser power, which mayresult in an FM index change of 3%. For analyzingline shapes with half-widths of approximately 0.05cm21, we note that the corresponding change of theFM index is negligible. If necessary, a highly uni-form FM index can be obtained by stabilization of thelaser power by electrical feedback or by use of currentpulses of a specially designed shape.

D. Signal Subtraction by a Dual-beam Configuration

To eliminate the ambient oxygen absorption, we im-plemented a dual-beam subtraction scheme. Figure3 demonstrates the performance of the dual-beamsubtraction scheme over a 2-cm21 laser scan with theabsorption cell evacuated. The upper trace is thetransmitted intensity with one beam blocked, and thelower trace is the recorded signal after subtraction.Both waveforms were accumulated, which yielded aneffective detection bandwidth of 2 kHz. The insertshows two magnified atmospheric oxygen lines withthe peak absorbance of 1.3%. In the lower trace thedetected laser intensity is subtracted to better than5 3 1024 over the whole tuning range, and the ab-sorption lines are subtracted to the noise level. Theresidual slope after subtraction originates from a

Fig. 3. Demonstration of the efficiency of the dual-beam subtrac-tion scheme. ~a! Intensity transmitted through the referencebeam showing two atmospheric oxygen lines. The peak absor-bance of the stronger line is 1.3%. ~b! Signal after subtractionmultiplied by 103.

wavelength-dependent reflectance of the nonpolariz-ing cube beam splitter.

The dual-beam subtraction not only cancels theambient oxygen absorption and the laser intensitybut also cancels etalon fringes that originate frommultiple reflections in the beam path before the non-polarizing beam splitter and reduces excess lasernoise to below the shot-noise level. By attenuatingthe laser intensity with neutral-density filters, weverified shot-noise limited detection for both directabsorption and TTFMS. Because shot-noise variesproportionally to the square root of the incident laserpower, it can be distinguished from excess laser noise.The measured shot-noise rms value was approxi-mately 2 3 1025 of the incident laser power in asingle scan with a 1-MHz bandwidth. This valuecoincides with the theoretical value calculated for 2mW of optical power, accounting for a factor of =2that is due to the two independent photodiodes.

When the signal-to-noise ratio was being enhancedby signal accumulation, the diode laser was mechan-ically dithered by a loudspeaker attached to thediode-laser mounting to wash out unwanted etalonfringes. The loudspeaker was driven by a sine waveasynchronously with the wavelength-shifting wave-form superimposed on the diode-laser drive current.Vibrating the diode laser is more efficient than vi-brating single optical elements or photodetectors27

because it causes all optical interference in the beampath to be destabilized. Because a spectrum wasscanned over a time scale that was short in compar-ison with the mechanical vibrations, the recorded lineshapes were not affected. After signal averaging,etalon fringes limited the detection level to approxi-mately 2 3 1026 for both TTFMS and direct absorp-tion. In direct absorption, however, the prominentsloping background did not allow this detectivitylimit to be practically achieved.

3. Measurement Technique

Two oxygen A-band lines, R15Q16 and R17R17 cen-tered at 13156.272 and 13156.616 cm21, respectively,were selected for use in this investigation. Theselines appeared within the continuous tuning range of alaser mode and were recorded simultaneously in onedata set. Direct absorption and TTFMS spectra wererecorded in two separate measurement series with ox-ygen pressures ranging from 5 to 400 Torr. As thetwo oxygen lines are closely spaced, overlapping occursat high pressures. For better measurement selectiv-ity, the FM index b was chosen to be approximately0.7, although b ' 1 provides optimum signal ampli-tude.15 At three different pressures ~30, 200, and 400Torr! additional spectra were recorded with b changedto approximately 0.5 and 1, respectively. These ad-ditional spectra were used to determine the FM indexmore accurately, as we describe in Subsection 4.C.

Before and after the measurement of each series ofabsorption spectra, we recorded the spectral through-put of the solid etalon and background spectra withthe sample cell evacuated. Because the backgroundwas stable throughout the measurements, the back-

ground spectra recorded by use of direct absorptionand TTFMS were subtracted numerically from therecorded absorption spectra. The recordings of theintensity transmitted through the etalon were usedto obtain a relative wave-number scale in the absorp-tion spectra. The reproducibility of the wave-number scale was 0.01% of the 2-cm21 single-modescans, which provided sufficient accuracy for the lineprofile measurements. The accuracy in the relativewave-number scale calibration was determined bythe uncertainty of the etalon free-spectral range cal-ibration to approximately 0.05%.

Figures 4~a! and 4~b! show typical spectra of oxy-gen at 200 Torr recorded with the dual-beam subtrac-tion scheme by use of direct absorption and TTFMS,respectively. Figure 4~c! shows the correspondingetalon spectral throughput. Note the fast wave-length sweep in the beginning of the current pulse.The nonvarying envelopes of the etalon fringes indi-cate constant laser spectral properties during thewavelength scan.

The pressure dependence of the line-center positionswas investigated in a separate measurement sequencewith TTFMS. The pressure-induced line shift wasmeasured relative to the line-center position of theambient oxygen line in a simultaneously recordedTTFMS reference spectrum. Because variations inthe ambient pressure controlled by a mercury manom-eter were less than 0.1% during the measurement se-quence, the atmospheric oxygen lines provided a goodreference. The peak absorption in the reference beamwas 3%. After linearization of the wavelength scalein the recorded spectra, the line-shape peaks were fit-ted to parabolas and the line shift determined. At themodulation frequencies employed in this study, use ofTTFMS for determining line shifts was advantageousbecause the peak of a recorded line shape was nar-rower than the peak of the corresponding direct ab-sorption line shape.15

Fig. 4. ~a! Direct absorption and ~b! TTFMS spectra of oxygen at200 torr recorded with constant-current wavelength scanning anddual-beam subtraction. ~c! Corresponding throughput from thesolid etalon.


4. Data Analysis

A. Processing of Spectra

Although a dual-beam subtraction is performed, theambient oxygen absorption in the air path cannot bedisregarded in an accurate line-shape analysis. Ac-cording to the Beer–Lambert law, the direct absorp-tion signal after subtraction IS~n! is

IS~n! 5 I0~n!exp@2ac~n! 2 aa~n!# 2 I0~n!exp@2aa~n!#,(1)

where I0~n! is the nonabsorbed transmitted intensity,which is assumed to be equal in both beams, and ac~n!and aa~n! are dimensionless absorbances caused byoxygen absorption in the cell and in the air path,respectively. From Eq. ~1! the absorbance of the ox-ygen in the cell is

ac~n! 5 2ln$1 1 @IS~n!yIair~n!#%, (2)

where

Iair~n! 5 I0~n!exp@2aa~n!# (3)

is the intensity transmitted through the open air.The intensity Iair~n! was recorded with the signalbeam blocked, and in the subsequent analysis all thedirect absorption spectra were processed according toEq. ~2!.

For a detailed analysis of TTFMS spectra, appro-priate expressions for the heterodyne-detected signalare essential. In previous research15 we developedthe TTFMS theory for accurate line-shape modeling;here, we give only a summary relevant to this study.

When a laser at frequency nc is modulated by twoclosely spaced radio frequencies of n1 5 nm 1 1y2Vand n2 5 nm 2 1y2V, the electric field produced isboth phase and amplitude modulated. Theabsorption-related signal, neglecting the dispersioncontribution, after phase-sensitive detection of theintensity component at V is given by

IV 5 2I0 cos~u! (n,m

Re~Rn,m!exp@21y2~an,m 1 an11,m21!#,

(4)

where

Rn,m 5 rnrmr*n11r*m21 1 Jn11~b!Jm21~b!An,m

1 Jn~b!Jm~b!A*n11,m21, (5)

rn~b, M, C! ; Jn~b! 1 ~My2i!@Jn21~b!exp~iC!

2 Jn11~b!exp~2iC!#. (6)

Here, 1y2an,m [ 1y2a~nc 1 nn1 1 mn2! is the atten-uation experienced by the frequency component at nc1 nn1 1 mn2, u is the detection phase angle, b is theFM index, M is the AM index, C is the phase differ-ence between AM and FM, and An,m is a correctionterm for a second-order nonlinear distortion of thediode-laser frequency modulation response. In Ref.15 it was shown that accounting for the nonlineardistortion at the second-harmonic only provides suf-


ficient accuracy for modeling TTFMS line shapes.In this case An,m is

An,m ; Jm~b!dn 1 Jn~b!dm, (7)

where

dn~b, z, q! ; J1~z!@Jn22~b!exp~iq! 2 Jn12~b!exp~2iq!#.

(8)

Here z and q are the amplitude and the phase shift,respectively, of the second-harmonic distortion com-ponent.

After dual-beam subtraction, the heterodyne-detected signal can be expressed as

IV 5 2I0 cos~u! (n,m

Re~Rn,m!exp$2aa@nc 1 ~n 1 m!nm#%

3 $exp@21y2~acn,m 1 ac

n11,m21!# 2 1%, (9)

where it is assumed that two closely spaced sidebandsexperience the same attenuation from the atmosphericbroadened oxygen 1y2aa, i.e., approximating n1 > n2 >nm. The recorded TTFMS spectra were normalized tothe transmitted nonabsorbed intensity I0~n!, whichwas numerically derived from a recording of the inten-sity Iair~n!. Furthermore, the spectral dependence ofexp@aa~n!# was derived from the same recording ofIair~n! and used in Eq. ~9! to calculate the atmosphericattenuation of aa@nc 1 ~n 1 m!nm# for the differentfrequency components.

The absorbance a~n! can be expressed in terms ofthe integrated intensity S and the line-shape func-tion g~n 2 n0! as

a~n! ; Sg~n 2 n0!. (10)

The integrated intensity S is defined as S [ S0PL,where S0 is the line strength, P is the partial pressureof the absorbing gas, and L is the absorption pathlength. In this analysis the line-shape function is de-scribed by the Voigt ~V!, Galatry ~G! soft-collision,28 orRautian–Sobelman ~R! hard-collision29 profiles. Theline profiles are standardized according to Herbert,30

and the dimensionless line-shape parameters are x [~n 2 n0!ys, y [ Gys, and z [ jys, where n 2 n0 is thefrequency deviation from the possibly shifted line cen-ter, s is the Doppler half-width at 1ye intensity, G is thecollision-induced half-width at half maximum, and j isthe narrowing, i.e., the effective velocity-changing col-lision rate. In this notation the dimensionless absor-bance is expressed as

a~n! ; ~SysÎp!K~x, y, z!, (11)

where K~x, y, z! is a general standardized line profilefunction.

A nonlinear least-squares fitting procedure was de-veloped for analyzing direct absorption and TTFMSline shapes with the Voigt, Galatry, or Rautian–Sobelman profiles. The analytical expressionsadopted for the line profile functions can be found inRef. 15 and in references cited therein.

The adjustable line parameters in the least-

Fig. 5. Typical results of least-squares fits to data re-corded by use of direct absorption at three different pres-sures. The spectra are normalized to their peakamplitudes. The fitted line profiles are the Voigt, Gala-try, and Rautian–Sobelman, labeled V, G, and R, respec-tively.

squares fitting procedure were S, n0, y, and z ~denotedequally for G and R!. The Doppler half-width wasfixed to the theoretical value of 0.01443 cm21, i.e., s5 0.01733 cm21. In the fits to the TTFMS lineshapes, the AM index M was an adjustable parame-ter, the FM index was either adjustable or fixed, andthe AM–FM phase difference was fixed to C 5 py2,which is a good approximation as demonstrated nu-merically in Ref. 15. Before the nonlinear least-squares fits of the two oxygen line shapes, the weakabsorption features in the recorded spectra from thediode-laser relaxation-oscillation sidebands and oxy-gen isotopes were removed with a linear fit of a lineprofile to each of these features. The line profileused in this procedure was obtained from a nonlinearleast-squares fit to the R15Q16 line. The removal ofthe weak spectral features did not influence the lineparameter retrieval noticeably; however, it facili-tated the evaluation of the residuals of the fits.

Figure 5 shows three direct absorption spectra atdifferent pressures and plots of the residuals fromleast-squares fits with the Voigt, Galatry, andRautian–Sobelman profiles. The total oxygen pres-sures in the cell were 70, 200, and 350 Torr, whichyielded the peak absorbances of 2.4%, 4.7%, and5.9%, respectively. The characteristic deviation ofthe fitted Voigt profile from the data indicates theeffect of collisional ~Dicke! narrowing.31,15 We veri-fied that the nonlinearity of the detection system~,0.5%! and the errors in determining the relativewave-number scale ~,0.05%! could not produce a sig-nal distortion with similar appearance and magni-tude. Instead, use of the Galatry and Rautian–

Sobelman profiles in the least-squares fittingprocedure provided excellent fits to the data.

Figure 6 shows typical results of the least-squaresfits of the TTFMS line shapes. The upper traces arethe TTFMS spectra at the same pressures used in thedirect absorption recordings in Fig. 5, and below arethe residuals from fits with the Voigt, Galatry, andRautian–Sobelman profiles. The FM index b wasderived separately, as discussed in Subsection 4.C.,and fixed to 0.73. The phase shift of the second-harmonic distortion was set to q 5 p,32 and the dis-tortion amplitude z was set to z 5 0.02b, whichprovided the best fit in a fitting sequence in which thedistortion amplitude was manually adjusted. Ac-counting for the second-harmonic distortion in themodeling reduced the residuals by a factor of 3. Thecharacteristic signature in the residuals from theleast-squares fits with the Voigt profile are consistentwith the residuals obtained from fits to synthetic col-lisionally narrowed line shapes.15 Note that theseresiduals are of the same magnitude as the residualsobtained from the fits to the direct absorption lineshapes in Fig. 5. Because the results of the least-squares fits with the collisionally narrowed line pro-files are nearly identical, the Rautian–Sobelmanprofile is used for computational simplicity in thesubsequent analysis unless otherwise stated.

B. Collisional Narrowing

The narrowing parameter j retrieved from a directabsorption spectrum is influenced by the baseline de-termination. Especially if the line wings extendover a considerable part of the recorded spectrum, a

Fig. 6. Typical results of least-squares fits to experimen-tal data recorded by use of TTFMS with measurementconditions corresponding to those in Fig. 5. The modu-lation indices were b 5 0.73 and M 5 0.004, and themodulation frequencies were 585 6 5.4 MHz. Each spec-trum is normalized to its peak-to-peak amplitude.


correct baseline determination is difficult. For thisreason we performed a separate least-squares fittingsequence with the line parameters adjustable and themodulation parameters fixed to obtain the pressuredependence of the narrowing. In the subsequentdata analysis we use the determined narrowing coef-ficient jyP to calculate the narrowing parameter j atan arbitrary pressure. This procedure improves theretrieval accuracy of the remaining adjustable pa-rameters.

The narrowing coefficient jyP was determined to be0.010 6 0.001 cm21yatm for the Rautian–Sobelmanprofile and 0.013 6 0.002 cm21yatm for the Galatryprofile. These narrowing coefficients are averagevalues of data obtained from both oxygen lines. Ourresults are in agreement with the values presentedby Ritter and Wilkerson33 of 0.01 ~R! and 0.0145cm21yatm ~G!. ~The uncertainties were not given.!

Also in TTFMS, the retrieved narrowing parameterj is influenced by the baseline determination. How-ever, because the TTFMS signal appears from a nearlyzero baseline and the spectral extents15 of the TTFMSline shapes in these measurements were smaller thanthose for the corresponding direct absorption lineshapes, the baseline can be determined more accu-rately. Accordingly, the narrowing parameters re-trieved from TTFMS line shapes showed a more linearand less scattered pressure dependence. The narrow-ing coefficient jyP was determined to be 0.0097 60.0007 cm21yatm ~R!, which agrees with the measure-ments obtained by use of direct absorption. In thefollowing analysis of both direct absorption andTTFMS line shapes, the value of 0.01 cm21yatm for thenarrowing coefficient jyP is adopted.

C. FM index Determination

Since the retrieval of the broadening parameter y andthe integrated intensity parameter S correlatesstrongly with the FM index b,15 an accurate assign-ment of the FM index parameter b is important. Byusing a method described in Ref. 14 where at leasttwo spectra are recorded at the same pressure butwith different FM indices, we can determine the ex-perimental FM index more accurately. For eachspectrum a sequence of least-squares fits is per-formed with the FM index parameter b differentlyfixed around the value estimated with an FPI. As aresult for each recorded spectrum the line parame-ters are retrieved as a function of the FM index pa-rameter b. A plot of the derived broadeningparameter y versus the integrated intensity param-eter S shows different slopes for spectra recordedwith different experimental FM indices. The plottedtraces intersect where the FM indices used in thefittings coincide with their true values. With thismethod not only is the FM index determined withhigher accuracy than it would be from a single fit withan adjustable b or by use of an FPI, but the collisionhalfwidth G and the integrated intensity S are aswell. Figure 7 shows plots of G versus S obtained forthe R15Q16 line at three different pressures for threedifferent experimental FM indices. For comparison,


data obtained from the direct absorption spectra ~cir-cles! with a fitted straight line are also shown. Fora particular pressure the traces are steeper for alarger experimental FM index b, and with increasingpressure the traces become flatter. The agreementbetween the intersection points and the direct ab-sorption measurements demonstrates the efficiencyof the intersection method in deriving accurate line-shape information with TTFMS. The accuracy inthe determined collision half-width G and the inte-grated intensity S can be estimated from the inter-section areas. In Fig. 7 the determination errors areindicated by the rectangles. The FM index param-eter b increases from the right to the left in the figure:from 0.4 to 0.65 in the flattest traces, from 0.6 to 0.85in the middle traces, and from 0.8 to 1.05 in thesteepest traces. The three experimental FM indicesestimated from the intersection points are 0.52, 0.73,and 0.92 with 1% accuracy. Note that the accuracyof the intersection point determination increaseswith the difference in the experimental FM indices.When the FM index b was adjusted in the least-squares fitting sequence, the average value of b 50.72 6 2.5% was derived. In the subsequent analy-sis, the FM index value of 0.73 is adopted.

5. Results and Discussion

A. Line Broadening

The upper part of Fig. 8 shows the collision halfwidthG retrieved with direct absorption versus pressure forboth oxygen lines. The data for the R17R17 line wasoffset by 0.006 cm21 for clarity. The broadening coef-ficients GyP were determined from straight line fits to

Fig. 7. Collision half-width G and standardized broadening y ver-sus integrated intensity S for the R15Q16 line. The traces, la-beled 1, 2, and 3, are results of multiple least-squares fits of threeTTFMS spectra recorded with different unknown FM indices.The intersection points yield accurate estimations of the collisionhalf-width G, the integrated intensity S, and the FM indices. Therectangles pointed out by arrows indicate the accuracy in deter-mining the line parameters. For comparison, data ~open circles!retrieved from direct absorption spectra are shown together with afitted straight line.

the retrieved collision half-widths. In the lower partof Fig. 8 collision half-widths obtained with TTFMSare also shown. To make a quantitative comparisonpossible, we subtracted the straight line fitted to thedirect absorption data of the R15Q16 line from theretrieved collision half-widths. This way the differ-ence in the broadening coefficients for the two linesbecomes clearly visible. The agreement between thebroadening coefficients obtained with TTFMS and di-rect absorption is better than 0.5%. The broadeningcoefficients are presented in Table 1 together with thevalues obtained by Ritter and Wilkerson,33 who used aGalatry profile in their data modeling. Our valuesobtained with a Rautian–Sobelman profile are approx-imately 3% lower. Use of a Galatry profile in theleast-squares fits of the direct absorption line shapesresulted in a 1% increase of the broadening coefficients.The collision half-widths retrieved with the Voigt pro-file exhibited a nonlinear pressure dependence, owingto the effect of collisional narrowing, with an almost10% change of the slope in the measured pressurerange. A straight line fit to these data provided 3%

Fig. 8. Collision half-width versus pressure, retrieved from directabsorption and TTFMS line shapes. For clarity the upper plotshows only the direct absorption data with fitted straight lines, andthe values for the R17R17 line are offset. The lower plot showsthe difference between the retrieved collision half-widths and thestraight line fitted to the retrieved collision half-widths of theR15Q16 line ~open squares!.

lower values of the broadening coefficients than ob-tained with the Rautian–Sobelman profile for both theTTFMS and the direct absorption spectra. As a re-sult we conclude that the prominent collisional nar-rowing of the oxygen A-band lines should be consideredin an accurate determination of the broadening coeffi-cients.

B. Integrated Intensity

The integrated intensities retrieved with TTFMS anddirect absorption agree better than 1% in the mea-sured pressure range, as illustrated in Fig. 9, inwhich the obtained TTFMS data are plotted versusthe direct absorption data. The lower part of thefigure shows the difference between these data sets.The line strength ~S0! determination with TTFMSwas possible after calibrating the TTFMS spectrumof the R17R17 line at 300 torr to the correspondingdirect absorption spectrum. The integrated inten-sity ~S! values retrieved from the TTFMS and thedirect absorption line shapes coincide better than 1%in the measured pressure range, which demonstratesthe reliability of TTFMS in measuring the integratedintensity over a wide pressure range. The linestrength S0 5 SyPL of both oxygen transitions wasdetermined from straight line fits to the integratedintensity data. The obtained values, which are pre-sented in Table 1, are between the values presentedin Refs. 33 and 34.

In many applications only relative integrated in-

Fig. 9. Integrated intensities retrieved by TTFMS versus inte-grated intensities retrieved by direct absorption. The lower plotis the difference.

Table 1. Line Parameters Determined in This Study at T 5 296 K and Literature Dataa

A-bandOxygen Line

Self-Broadening Coefficient~cm21yatm! Line Strength ~10224 cm21ymol cm22!

Self-Shift Coefficient~1023 cm21yatm!

TTFMS Direct Ref. 33 TTFMS Direct Ref. 33 Ref. 34 TTFMS Ref. 35

R15Q16 0.0437 ~3! 0.0435 ~2! 0.0449 ~9! 4.51 ~6! 4.50 ~3! 4.89 ~11! 4.21 25.75 ~10! 25.5R17R17 0.0422 ~2! 0.0424 ~2! 0.0433 ~5! 2.93 ~4! 2.93 ~3! 3.14 ~4! 2.74 26.10 ~15! 25.8

aNumbers in parentheses represent statistical error ~2 std!. Systematic errors are discussed in the text.


tensities are of interest, e.g., in gas temperature de-termination from the relative intensity ratio ofdifferent vibrational-rotational transitions and inconcentration measurements calibrated to a refer-ence absorption line. The ratio of the line strengthsof the two oxygen lines was determined with the pre-viously derived integrated intensities. The obtainedaverage values were 1.538 6 0.008 and 1.536 6 0.005with TTFMS and direct absorption, respectively.The value provided by the HITRAN database34 of 1.535is within our experimental uncertainty.

C. Self-induced Pressure Shift

Pressure-induced line shifts were measured withTTFMS. Figure 10 shows a plot of the line-centershift as a function of pressure. The data for theR17R17 line were offset for clarity. From thestraight line fits to the measured line shifts of theR15Q16 and R17R17, the self-induced pressure-shiftcoefficients were determined to be 2~5.75 6 0.10! 31023 and ~6.10 6 0.15! 3 1023 cm21yatm, respec-tively. The obtained values are in close agreementwith the previous results35 presented in Table 1.

6. Error Analysis

The uncertainties presented in Table 1 are statisticalerrors. Possible systematic errors are discussed be-low.

A. Line Broadening

A very decisive parameter in the least-squares fit ofTTFMS line shapes is the FM index b because itcorrelates with both the broadening y and the inte-grated intensity S parameters. When the TTFMSdata were refitted with the FM index b adjustable, itproduced approximately six times larger scatter ofthe retrieved line parameters. The values of the de-rived broadening coefficients, however, were not af-fected. This coincides with the results of thenumerical modeling in Ref. 15, in which the empiricalexpression for the absolute broadening error dyb in-duced by an error db of the FM index was derived:

dyb > 2Qby~1 1 y!b1.5 db, (12)

Fig. 10. Self-induced line shifts measured by TTFMS versus pres-sure. The values for the R17R17 line are offset for clarity.


where the coefficient Qby depends on the ratio nmyD

and D is the absorption line half-width at half maxi-mum. Figure 11 shows plots of the absolute broad-ening error dyb, normalized to b1.5 db, versus thebroadening parameter y for different values of thenormalized modulation frequency n#m 5 nmys. Thedata are results of least-squares fits with a Voigtprofile to numerically generated TTFMS line shapes.For n#m 5 1, dyb varies little if y # 1.5, which impliesthat in this region an error in the assignment of theFM index b only offsets the retrieved broadening pa-rameters y and does not affect the determination ofthe broadening coefficient GyP. Thus with n#m 5 1.13and y ranging from 0 to 1.35 ~see Fig. 7!, an FM indexerror should induce a negligible systematic error inthe determination of the broadening coefficients.This was also verified by refitting the experimentalTTFMS line shapes with the incorrectly fixed FMindex b.

The differences in phase shifts experienced by dif-ferent frequency components give rise to a weak inten-sity component beating at the intermediate frequencyV in quadrature with the absorption component.This dispersion-induced component might contributesignificantly in a recorded line shape if the heterodyne-detection phase is not carefully adjusted for the ab-sorption component or if the phase is adjusted foroptimum signal amplitude.15 When determining thecollision halfwidth G, we find that the dispersion com-ponent can cause systematic errors. For this reasonthe detection phase should be carefully adjusted for theabsorption component. In this study, in which thedetection phase was adjusted to the absorption com-ponent with an accuracy of 65°, the dispersion com-ponent can contribute as much as 0.2% of the recordedline shape.15 We estimate the introduced uncertaintyin the derived collision half-width at 0.2%.

The main systematic errors in determining theself-broadening coefficients originate from the detec-tion nonlinearity of 0.5% and the absolute calibrationof the capacitance manometer of 0.5%. There is alsoa deviation in the results obtained by the differentcollisional model, e.g., the soft collision model ~G!provides approximately 1% larger values of thebroadening coefficients than the hard collision model~R!. The adopted value of the narrowing coefficient

Fig. 11. Error dy of the broadening parameter induced by an FMindex error db versus the broadening parameter y for differentnormalized modulation frequencies n#m. In the regions where theerror dy is nearly constant, an FM index error only offsets thepressure dependence of retrieved broadening parameters.

jyP influences the derived broadening coefficient GyPnegligibly; e.g., a 10% change in the assignment ofnarrowing coefficient j leads to a less than 0.3%change in the derived broadening coefficient. Addi-tional systematic errors that are due to the accuracyof the relative wave-number scale and determinationof the gas temperature and gas purity are negligible.

Extrapolation of the collision half-widths in Fig. 8to zero pressure reveals a residual broadening of ap-proximately 20 MHz for both lines, which we at-tribute to the laser linewidth and the instrumentalbroadening of the line shape during signal accumu-lation caused by jitter of the trigger position andcurrent and temperature instabilities. However,this residual broadening does not introduce system-atic errors in the broadening coefficient GyP when it isdetermined from the slope of the pressure depen-dence.

B. Integrated Intensity

When we refitted the TTFMS line shapes with theFM index b fixed to different values, we found that anFM index assignment error db induces an integratedintensity error dSb, which approximately follows therelation dSbyS ' 22dbyb. This is in agreementwith the numerically derived relation15

dSbyS > 2~2.2 1 QbSb1.5!dbyb, (13)

where the coefficient QbS is approximately zero for

the ratio of nmyD experimentally realized in thisstudy. Relation ~13! implies that an error db in theFM index affects the determination of the linestrength from the pressure dependence of the inte-grated intensity parameter S. Although the accu-racy of 1% in the determined FM index shouldintroduce a 2% uncertainty in the derived integratedintensity, the integrated intensities determined withTTFMS and direct absorption show a surprisinglygood agreement, better than 1% as mentioned, overthe investigated pressure range.

Because the absorption spectra and the laser in-tensity were recorded with different oscilloscopescales, an uncertainty of 2% owing to the relativecalibration of the scales is introduced in the linestrengths. The ratio of the resonant main mode tothe total laser intensity of 0.96 6 0.02 was deter-mined by estimating the fraction of intensity in thecavity side modes and the relaxation-oscillation side-bands with the high finesse FPI. The accuracy withwhich this ratio was determined introduces an ap-proximately 2% uncertainty in the line strengths.Again, the detection nonlinearity ~0.5%! and the ca-pacitance manometer measurement accuracy ~0.5%!contribute to the line strength errors.

Since the same value of the FM index b was em-ployed in the least-squares fits of both oxygen lines,the 0.1% agreement between the line strength ratiosderived with TTFMS and direct absorption confirmsthat an FM index change during the wavelength scanacross the lines was negligible.

7. Conclusion

The capability of TTFMS in performing quantitativeline-shape measurements was demonstrated by theaccurate determination of transition line parameters.The results were compared with the line parametersretrieved with direct absorption. Two closely spacedoxygen A-band lines at 760 nm were modeled over awide range of oxygen pressures. A dual-beam sub-traction scheme eliminated excess laser noise and theatmospheric oxygen absorption in the beam path out-side the sample cell. The high sensitivity achievedwith both TTFMS and direct absorption permitted anaccurate analysis of the recorded line shapes and ameaningful comparison of the results obtained by thetwo methods. By scanning the diode laser wave-length by use of a waveform with rectangular currentpulses, we brought the FM index change over thewavelength scan to a minimum, which significantlyreduced distortion of the recorded line shapes. It wasobserved that a constant laser output power over thewavelength scan indicates a constant FM index. Inaddition, a constant-current wavelength scan also re-duces the influence from variations of intrinsic diode-laser characteristics such as linewidth, amplitude anddetuning of the relaxation-oscillation sidebands.

A nonlinear least-squares fitting procedure with aVoigt, Galatry, or Rautian–Sobelman profile wasused to analyze line shapes recorded with TTFMSand direct absorption. By using a few spectra re-corded with different unknown FM indices, whichwere multiple fitted with the FM index differentlyfixed, we accurately determined the experimental FMindex. Least-squares fits with the FM index adjust-able provided nearly the same FM index values; how-ever, the scatter of the retrieved line parametersincreased by approximately six times. This impliesthat in quantitative gas analysis with high accuracyrequirements efforts should be made to determine theFM index carefully.

When employing a Voigt line profile in the least-squares fits of line shapes recorded with both TTFMSand direct absorption, we observed systematic dis-crepancies that clearly indicate the presence of colli-sional ~Dicke! narrowing. By using a Galatry and aRautian–Sobelman profile, we significantly reducedthese deviations. TTFMS line shapes were fittedover a wide range of pressures, and the obtainedresiduals were less than 0.3%.

Within the experimental uncertainties, the mea-surements with TTFMS provided the same values ofthe integrated intensities and pressure-induced self-broadening, shift, and narrowing coefficients as thedirect absorption measurements. The determinedline parameters were consistent with literature val-ues. The line strength ratio of the two oxygen lineswas measured accurately, which is advantageous, forexample, for temperature measurements.

In this study TTFMS line shapes from absorbancesof a few percent were recorded and treated with suchaccuracy that even the small effect of collisional nar-rowing can be quantified. The combination of high


sensitivity and possibility of extracting detailed line-shape information makes TTFMS a useful tool foraccurate quantitative gas measurements in a broadarea of applications, in particular where direct ab-sorption methods are not feasible.

We are pleased to acknowledge Sune Svanberg forsupport and valuable suggestions on the manuscript.This research was funded by the Swedish ResearchCouncil for Engineering Science and the SwedishBoard for Industrial and Technical Development.

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