Uncertainty Analysis of Absolute Concentration Measurement with Continuous-Wave Cavity Ringdown Spectroscopy

Uncertainty analysis of absolute concentrationmeasurement with continuous-wave cavityringdown spectroscopy

Jae Wan Kim, Yong Shim Yoo, Jae Yong Lee, Jae Bong Lee, and Jae Won Hahn

To evaluate the uncertainty of concentration measurement using cavity ringdown spectroscopy, weanalytically derived expressions for uncertainty for parameters, such as temperature, laser frequency,and ringdown time deviation, from the model equation. The uncertainties that are due to systematicerrors in a practical cavity ringdown system were assessed through an experimental study of the PQ�35�transition in an A band of molecular oxygen. We found that, except for the line strength that is regardedas a reference value independent of the measurement, the laser frequency jitter is the largest uncertaintysource in the system. Some practical requirements for minimizing the uncertainty in concentrationmeasurements are discussed. We also demonstrated determination of the line strength of the PQ�35�transition line of oxygen to be 8.63�3� � 10�27 cm�1 with a relative uncertainty of less than 0.4%.© 2001 Optical Society of America

OCIS codes: 300.0300, 020.3690.

1. Introduction

The present vacuum technology demands diagnosticsthat can measure the absolute concentration of a mi-nor species in a gas mixture. In many vacuum ap-plications the partial pressure of a certain componentspecies in contained gas is of more interest than thetotal pressure.1,2 One should precisely determinethe residual components of, to name a few, water,nitrogen, hydrogen, carbon monoxide, and oxygen todiagnose the performance of the vacuum system.For the qualitative and the quantitative analysis ofthe composition of a residual gas, mass spectrometer-based instruments are commonly used. When onechooses to use these instruments, a careful calibra-tion is the critical requirement for converting theinstrument output readings into absolute pressure.

Absorption spectroscopy can be another approach

to measuring the concentration of gas-phase mole-cules, atoms, or radicals.3,4 Absorption in a linearspectroscopic technique permits both the molecularnumber density and the absorption strength. Thelinear absorption signal is more easily quantifiableand will scale more favorably over wide ranges ofmolecular absorption strength and concentration.Provided that an absorption spectroscopy is sensi-tive enough to measure the concentration of minorspecies in vacuum systems, it is attractive to em-ploy absorption spectroscopy to establish the refer-ence standard.

Among the various absorption techniques, cavityringdown spectroscopy �CRDS� can easily routinelyachieve the high sensitivity of 10�7 for single-passabsorbance.5,6 The sensitivity of this level will per-mit one to measure the concentration of residualgases. In addition to the promising sensitivity,CRDS features the advantage of the immunity tolaser intensity fluctuation and the simplicity in theimplementation. CRDS has been demonstratedsuccessfully for use in detecting free radicals in com-bustion systems,7,8 for studying molecular ions inplasmas,9 and for determining the absolute concen-tration of methane in supersonic expansions.10

To be accepted as reliable, any quantitative mea-surement needs a detailed analysis of its uncertaintybudget. When one measures, based on an absorp-tion spectroscopic technique, the concentration of a

J. W. Kim, Y. S. Yoo, J. Y. Lee, and J. W. Hahn are with the LaserMetrology Group, Korea Research Institute of Standards and Sci-ence, P.O. Box 102, Yusong, Taejon 305-600, South Korea. J. B.Lee is with the School of Electronic and Computer Engineering,Chungbuk National University, 48 Gaesindong, Cheongju Chung-buk 361-763, Korea. J. W. Kim is now with the Length Group atKRISS. The e-mail address for J. W. Hahn is [email protected].

Received 27 July 2000; revised manuscript received 23 April2001.

0003-6935�01�305509-08$15.00�0© 2001 Optical Society of America

20 October 2001 � Vol. 40, No. 30 � APPLIED OPTICS 5509

species, the concentration is quantified from a spec-trum that depends on the strength of the transition,the temperature, and the pressure. One can esti-mate the probable uncertainty in the concentrationdetermined from the fit that is propagated from thestatistical uncertainty in the measured spectrum; inthis case fluctuations in the ringdown time and thelaser frequency contribute statistical noise. In caseof treating a measurement associated with manyother quantities, the resultant uncertainty is ob-tained by combining the individual uncertainties.Therefore knowledge of each individual uncertaintycontribution will suggest a guideline for minimizingthe combined uncertainty of a measurement withCRDS. In this study we derive the analytic formu-las relating uncertainties between the concentrationand some experimental parameters including ring-down time, laser frequency, line strength, and tem-perature. Finding the relationship between theuncertainty sources, we can assess the contributionsof uncertainties in terms of error propagation analy-sis. The uncertainty sources of the concentrationmeasurement are discussed for a practical CRDS sys-tem concerning the measurement with the PQ�35�transition of molecular oxygen.

2. Uncertainty in the Concentration Measurement withCavity Ringdown Spectroscopy

CRDS measures the absorption coefficient of a me-dium by monitoring the intensity evolution of a lightreflected between two highly reflective mirrors.When enough photons are put into a Fabry–Perotcavity they are reflected repeatedly by the mirrors.When no more photons are inserted into the cavity,the light intensity inside the cavity will decay expo-nentially because of the transmission of the mirrors,the absorption of a medium, and scattering. Thetime constant of such intensity decay is called ring-down time �. Since the intensity of the transmittedlight is proportional to the intensity of light inside thecavity, the transmission signal also exhibits an expo-nentially decaying function from which � can be ob-tained. The relationship between the absorptioncoefficient � and the ringdown time � is given by6

� �tr

2l �1�

�1�0� , (1)

where tr is the round-trip time for a cavity withlength l and �0 is the ringdown time of an emptycavity, which is expressed as �0 � tr�2�1 � R� formirror reflectivity R.

An absorption spectrum can be recorded by mea-suring � consecutively with a scan of laser frequencyv. With �i denoting the absorption coefficient of asample at frequency vi �i � 1, 2, . . . , N�, the absorp-tion spectrum can be expressed as11

�i � �S�T�1

�g�vi�, (2)

where � is the concentration of the sample molecule,S�T� is the line strength at temperature T, and g�vi�is the spectral line-shape function, which is normal-ized to . The integration of Eq. �2� over the fre-quency domain and the following rearrangementyield the concentration of the medium in form

� �M

S�T�, (3)

where

M � ��

i�i

�i

g�vi�. (4)

On this account the relative uncertainty of � is de-duced from

u��2

�2 �u�M�2

M2 �u�S�T� 2

S�T�2 , (5)

where u�x� is the uncertainty of a parameter x�x � �,M, and S�T� .

A. Uncertainties Caused by Absorption Coefficientand Frequency

The term u�M�2�M2 is the relative uncertainty thatcomes from fitting a spectrum to a line-shape functionsuch as Galatry or Gaussian. If �i and M are mea-sured independently, u�M�2�M2 becomes

u�M�2

M2 �

�i

u��i�2

��i

�i�2 � � �

�i

g�vi��2

� �i��g�vi�

�vi�2

u�vi�2. (6)

The first term on the right-hand side of Eq. �6� isassociated with uncertainties in the absorption coef-ficients, and the following term is related to uncer-tainties to prove laser frequencies. If u��i�

2 andu�vi�

2 for a measured data set are both known, anappropriate line-shape function is chosen, and N, thenumber of sampled data within the entire frequencyrange F, is sufficiently large, Eq. �6� can be easilycalculated in a convenient form by replacing the sum-mation with an integral,

�i�0

N

3NF �

0

F

dv. (7)

The uncertainty of the absorption coefficient u��i�2

that is due to the shot noise of the photodetector hasbeen reported by several researchers.12–14 Referringto these reports, the component of the relative uncer-tainty of the absorption coefficient arising from the

5510 APPLIED OPTICS � Vol. 40, No. 30 � 20 October 2001

shot noise and the constant-level fluctuation aregiven, respectively, as

u��

��

shot

� �2eg�V0

1��

1�2

, (8a)

u��

��

const

�Vnoise

V0�2

fs

1��

1�2

, (8b)

where V0 is the initial ringdown signal level, Vnoise isthe constant noise level in the ringdown signal, e isthe electron charge, g is the conversion gain of aphotodetector defined by V�t� � gP�t�, where P�t� isthe light power incident on the detector, � is theresponsivity of a photodetector defined by � � I�t��P�t�, where I�t� is the current induced in the photo-detector, and fs is the sampling rate of a data-acquisition system. In general, either a Galatry or aVoigt function can be a good model for expressing theline shape of an absorption spectrum. However,they are based on complex functions, and the analyticcalculation is often impossible indeed. Insteadthese functions are approximated either to a Lorent-zian function at very high pressure or to a Gaussianat low pressure, allowing uncertainties in the Galatryor Voigt function estimated by those of Lorentzianand Gaussian functions. With Eqs. �1�, �7�, and �8�the first term on the right-hand side of Eq. �6� be-comes

where �p is the peak absorption coefficient, �V is theabsorption coefficient in vacuum, and NW representsthe number of data attained within the linewidth�NW � vDN�F for a Gaussian function and NW �vLN�F for a Lorentzian function�. By comparing theratios of the coefficients of u��p�2��p

2 and u��V�2��V2

in Eq. �9� for the two profiles, one can determine thatu��p�2��p

2 is more sensitive for the Gaussian, whichis easily reasonable if we compare the shapes of eachprofile; because a Gaussian is a more localized func-tion than a Lorentzian, the peak value plays a moredominant role in the integration than the value of thetail �or vacuum�. Moreover the coefficients of �ln2��1�2 and 1� result from peak values of normal-ized Gaussian and Lorentzian functions, respec-tively. Given the linewidth, one can thereforepredict the amounts of uncertainty for a Galatryfunction and a Voigt function.

The second term on the right-hand side of Eq. �6� isrelated to the uncertainty in the probe frequency.Assuming that the amount of frequency fluctuation is

considered to be constant so that u(vi)vi can be re-placed by u(v)�v, we can derive

��i

g�vi��2 �i��g�vi�

�vi�2

u�vi�2 �

�� ln 2�3 1�2

NW

u�v�2

vD2 (10a)

16NW

u�v�2

vD2 (10b)

for a Gaussian function �Eq. 10�a� and for a Lorent-zian function �Eq. 10�b� .

B. Uncertainties Caused by Temperature and TransitionStrength

With the strength known at a certain temperature T0and the stimulated emission ignored, the linestrength of a molecule is expressed as11,15

S�T� � S�T0�Q�T0�

Q�T�exp��

hcEJ

k �1T

�1T0�� , (11)

where S�T0� is the strength at the temperature T0,Q�T� is the rotational partition function, EJ denotesthe ground-state energy of rotational quantum num-ber J, h is the Planck constant, k is the Boltzmann

constant, and c is the speed of light. The partitionfunction Q�T� for rotational transitions is given byQ�T0��Q�T� � T0�T for linear molecules and byQ�T0��Q�T� � �T0�T�3�2 for nonlinear molecules.Because S�T� is calculated from S�T0� and T, theuncertainty of S�T� takes the form

u�S�T� 2 � ��S�T��S�T0� 2u�S�T0�

2

� ��S�T��T 2u�T�2. (12)

With Eqs. �11� and �12� the relative uncertaintyu�S�T� 2�S�T�2 in Eq. �5� becomes

u�S�T� 2

S�T�2 �u�S�T0�

2

S�T0�2 � C�T�2 u�T�2

T 2 , (13)

where C�T�2 � �1 � hcEJ�kT�2 for linear moleculesand C�T�2 � �3�2 � hcEJ�kT�2 for nonlinear mole-cules. At a room temperature of 296 K with hc�kTkept as small as 0.005 cm�1, C�T� can be approxi-mated as 1 for linear molecules and as 3�2 for a

�i

u��i�2

��i

�i� 2 � �2�ln 2 �1�2 1

NW��1

3�1�2 u��p�2

�p2 � � 1

�2�

1

�3� u��V�2

�V2 � for a Gaussian, (9a)

21

1NW

�38

u��p�2

�p2 �

18

u��V�2

�V2 � for a Lorentzian, (9b)


nonlinear molecule with a low rotational quantumnumber.

To get the spectrum of a transition line, the tem-perature and the frequency are measured along withthe absorption coefficient. But the line strengthshould be known before the measurement. In thissense the line strength S�T0� differs from the otherthree parameters. The line-strength uncertaintygiven in the literature15 is approximately 1%, and theexpressions of the relative uncertainties of parame-ters are listed in Table 1.

The effect of temperature drift is obvious for a highquantum number J. For example, when the ring-down cavity is stabilized within 30 mK, u�� atroom temperature is �10�4 for the PQ�1� transitionof the oxygen A band, whereas it amounts to 8 � 10�4

for the PQ�35� transition. Figure 1 shows an exam-ple that demonstrates the temperature dependence ofS�T� in the PQ�35� transition of the oxygen A band.We took 50 spectra with a ringdown cavity whose

temperature was not stabilized and that drifted �2 Kfor 12 h. The temperature variation during the ac-quisition of a single spectral line was less than 40mK. With this temperature variation, Eq. �13� givesthe relative uncertainty of u�� 0.1%. The sam-ple gas-leak rate of the cavity was as low as 10 mPa�hfor molecular oxygen filled to a pressure of 50,000 Pa.Thus the concentration could be regarded as constantduring the entire measurement. The temperaturewas measured for every spectral position and aver-aged for each spectrum. The standard deviation inthe concentration measurement was 1.3% when wedid not take into account the temperature drift, but itreduced to 0.4% after correcting with the tempera-ture drift term. Thus stabilization of a ringdowncavity temperature is necessary for high-precisionmeasurement of the concentration. Although thiscan barely be achieved, at least the temperatureshould be monitored simultaneously with the spec-trum and Eq. �11� should be applied.

3. Experiment

Since our CRDS apparatus has the same scheme asproposed by Hahn et al.16 �Fig. 2�, we give only a briefdescription here. The light source was an external-cavity diode laser �New Focus, 6224� with a linewidth

Table 1. Relative Uncertainties in Concentration Measurements from Several Sources

Source ofUncertainty Relative Uncertainty in Concentration

Laser frequency u��2

�2 �v

�� ln 2�3 1�2

NW

u�v�2

vD2 �Gaussian�

u��2

�2 �v

�

16NW

u�v�2

vD2 �Lorentzian�

Absorption coefficient u��2

�2 ��

� 2� ln 2 � 1�2 1

NW��1

3� 1�2 u��p�2

�p2 � � 1

�2�

1

�3� u��V�2

�V2 � �Gaussian�

u��2

�2 ��

� 21

1NW

�38

u��p�2

�p2 �

18

u��V�2

�V2 � �Lorentzian�

Temperature u��2

�2 �T

� �1 � hcEJ�kT�u�T�2

T2

Line strength u��2

�2 �S�T0�

�u�S�T0�

2

S�T0�2

Fig. 1. Temperature dependence of the line intensity �S of thePQ�35� transition in the oxygen A band. Ground energy of PQ�35�is 1803.2 cm�1: solid circles, experimental data; solid line, calcu-lation from Eq. �6�. Fig. 2. Schematic setup of a cw CRDS: D, diaphragm.


of �5 MHz. To prevent feedback from the ringdowncavity to the laser, we placed an optical isolator �Op-tics for Research� at the beginning of the laser beampath. We made the transverse profile of a laserbeam Gaussian by spatial filtering with two lenses� f � 5 cm� in combination with a pinhole. The laserwavelength was monitored with a wavemeter �Bur-leigh WM1500� providing a frequency resolution of 60MHz. The laser beam was well coupled to the cav-ity’s fundamental transverse mode by a mode-matching lens � f � 75 cm�. The light transmittedthrough the output cavity mirror was focused into amultimode fiber that could deliver the light intensityto a photomultiplier tube, PMT �Hamamatsu R955with a load resistance of 600 ��. The ringdown sig-nals were recorded with a 10-MHz bandwidth data-acquisition board �Compuscope, GaGe 1020� offeringa 12-bit resolution. The ringdown cavity consistedof two identical 1.27-cm-diameter concave mirrorsseparated by 30 cm. The reflectivity and the radiusof curvature of the mirrors were quoted as 99.99%and 4 m, respectively. One mirror was attached to atubular piezoelectric transducer �PZT�, which permit-ted the scan of a ringdown cavity length. The cavitychamber was evacuated to 0.1 mPa with a turbomo-lecular pump. A capacitance diaphragm gauge�Vacuum General CMO� was used to measure thechamber pressure. The pressure gauge was cali-brated with a reference of the ultrasonic interferom-eter manometer at the Korea Research Institute ofStandards and Science. The calibration uncertaintyof the pressure gauge was 0.5%. With the valve con-necting the cavity to the pump closed, the gas leakwas measured to be at a rate of �10 mPa�h. Aplatinum resistance thermometer was embedded inthe cavity to monitor the gas temperature. Oxygenwith a stated purity of 99.99% was used without fur-ther purification.

A. Measurement of Molecular Parameters of Oxygen

When we apply the cw CRDS technique to the ab-sorption measurement, the lower and the upper lim-its of the dynamic range of the measurement aredetermined by the ultimate sensitivity of the tech-nique and the bandwidth of the detection system,respectively. Thus, to measure the concentration ofa sample molecule, we need to choose a proper tran-sition line that gives the wide dynamic range of themeasurement.

The ultimate sensitivity of the absorbance mea-surement of our CRDS apparatus is approximately10�7 and the upper limit is 10�3. If we select aPP�7� transition line of the oxygen A band �linestrength, 8.51�4� � 10�24 cm�1�molecule cm�2 forconcentration measurement,17 the detectable pres-sure range is from 0.01 to 100 Pa. For an easy com-parison of the CRDS technique and a conventionalpressure gauge operating in the pressure range of 105

Pa, we chose the oxygen transition line of PQ�35�whose line strength is a thousand times smaller thanthat of PP�7�.

The line intensity �S of the cavity filled with oxy-

gen is determined by a fit of the spectrum to a Galatryprofile. Figure 3 shows the spectrum of the PQ�35�transition in the oxygen A band and the fit of a Ga-latry line shape to the data. The pressure is 3.0 �104 Pa and the vacuum ringdown time is 50.73�14��s. The best-fit parameters obtained were a centerfrequency of 12,966.8093 cm�1, a �S of 0.622 cm�1, aDoppler width of 0.0146 cm�1, and a collision-broadened width of 0.009 cm�1. Figure 4 shows theline-intensity variation as a function of oxygen pres-sure. The error bars in Fig. 4 show the standarddeviation of the four measurements. To compensatefor a temperature drift, we corrected all the datapoints with a temperature drift from 296 K. Thesolid line in Fig. 4 is the least-squares fit of the datapoints. The relative standard deviation of the data

Fig. 3. Spectrum of the PQ�35� transition and the fit of the Ga-latry profile to the data. The pressure was 3.0 � 104 Pa. Thebest-fit parameters suggest a center frequency of 12,966.8093cm�1, �S of 0.622 cm�1, a Doppler width of 0.0146 cm�1, and acollision-broadened width of 0.009 cm�1.

Fig. 4. Measured line intensities at several pressure readingsfrom the pressure gauge.


from the solid line is approximately 0.4%, which iscomparable with the uncertainty of the pressuregauge. In this measurement we obtain a linestrength of 8.68�3� � 10�27 cm�1�molecule�cm2,which is in good agreement with the literature value�8.71 � 10�27 cm�1�molecule�cm2� within its uncer-tainty.18

The center frequency and collision-broadened line-width determined by the fit are shown in Fig. 5. Thesolid and the dotted lines are the results of weightedlinear regressions. From the slope of the lines, wecan get line-shift and self-broadening coefficients of0.0068 and 0.029 cm�1�atm, respectively. The tran-sition frequency of 12966.8113�4� cm�1 obtained inthis study shows a slight difference from the value of12,966.8260 cm�1 listed in the HITRAN96 data.

B. Analysis of Uncertainty in Concentration Measurement

As discussed above, the statistical noise of the ring-down signals causes a fluctuation in ringdown time.In the typical operation conditions of our system, theringdown signals have an initial signal level of 100mV and a ringdown time of 50 �s. The estimatedphoton number contained in each ringdown signal is�5 � 106. At the beginning of the ringdown signal,the corresponding shot-noise level is �1.2 mV, calcu-lated with a gain coefficient ��g of 6.38 � 106 V�A.The relative uncertainty of the absorption coefficientu�� calculated with Eq. �8a� is then 0.064%. Theproper design and choice of a detection system couldpermit one to reach the shot-noise limit. However,this is true only when the ringdown signal is in aperfect exponential decay waveform and the othertechnical noise sources are effectively isolated. Tocharacterize the noise source of the ringdown signal,we measured the background-noise level of the detec-tion system. The background noise of the detectionsystem was reduced by placing an aperture in front ofthe entrance mirror of the ringdown cavity and bycovering the exit mirror to the coupling fiber with a

black curtain. The resultant background noise ofthe detection system in the bandwidth of the digitizerwas 0.42 mV. For a ringdown signal with an initialsignal level of 100 mV, Eq. �7b� gives u�� 0.075%. The relative uncertainty of the absorptioncoefficient owing to background noise is comparablewith that of shot noise.

The distribution of the fit residuals of a ringdownsignal provides direct information about inherentnoise. Figure 6�a� shows a ringdown signal at vac-uum and a fit of the exponential decay function. Thefit gives the parameters, an onset voltage of 0.11 V,and a ringdown time of 50.78�3� �s. Figure 6�b�shows the residuals of the experimental data to the fit�solid curve� and the shot-noise level calculated �dot-ted lines�. The residual is the largest at the signalonset, where the ringdown signal is at its maximumand then decreases monotonically with time. At thetail of a ringdown signal the residuals are at a con-stant level. Thus we infer that the shot-noise andthe constant-background noise are the dominantnoises in a ringdown signal and they cause a decay-time fluctuation of u�� 0.1%.

If some other systematic errors severely influencethe ringdown time itself in shot-by-shot events, the

Fig. 5. Center frequency and collision-broadened linewidth: �,center frequency; F, collision-broadened linewidth of the PQ�35�transition as a function of oxygen pressure; dotted and solid lines,least-squares fits of the measured data. From the slopes of thetwo straight lines the pressure shift and the broadening coefficientare determined as 0.0068 and 0.029 cm�1�atm, respectively.

Fig. 6. �a� Ringdown signal trace of an evacuated cavity and thefit to an exponential function. The ringdown time results in50.78�3� �s. �b� Residuals �solid line� superimposed on the shot-noise level �dotted line� of the ringdown signal for comparison.


relative deviation determined by repeated measure-ments would be generally greater than 0.1%, whichwas previously predicted with the residuals of theringdown signal. We often observed the ringdowntime fluctuating periodically; the period somewhatcoincided with the period of bias voltage variationthat was applied to the PZT to maintain the reso-nance at the center of the ramp voltage. We couldminimize the amplitude of such a fluctuation by ad-justing the PZT in the azimuth angle but could notcompletely get rid of it. The relative uncertainty ofthe absorption coefficient u�� with our spectrom-eter was 0.3% in practice.

As shown in Table 1 the uncertainty in concentra-tion due to the laser-frequency fluctuation dependson the root mean square of the frequency jitter and isinversely proportional to the square root of the num-ber of data points. In the experiment we put thefrequency step of the data points with the minimumtuning step, 50 MHz, of a commercial cw diode laser.Since the linewidth of an absorption spectrum of ox-ygen is approximately 450 MHz at room tempera-ture, we had only nine data points in the linewidth.The diode laser used in the present study has a long-term �5-s� frequency jitter of 5 MHz. By applyingthese parameters to Eq. �10�, we can calculate therelative uncertainty in concentration that is due tothe frequency jitter. The estimated uncertaintiesthat are due to the major noise sources of our CRDSsystem are summarized in Table 2.

Note that the contribution of laser-frequency noiseto the uncertainty in concentration is approximatelyfive times greater than the uncertainty due to theringdown-time fluctuation. According to Eq. �10�,the uncertainty can be reduced further either by sta-bilizing the laser frequency with a smaller jitter or byincreasing the number of data points. This indi-cates that frequency stabilization is an importantrequisite for accurate concentration measurementwhen the cw CRDS technique is used. When weapply the CRDS technique to a low-pressure environ-ment such as the vacuum, the bandwidth of theDoppler-broadened spectral line of a test molecule isapproximately a few hundred megahertz at room

temperature. In this case we should be careful tomake the tuning-frequency step as small as possible.

Furthermore we indicate that the Doppler shiftthat is due to the moving mirror of the cw CRDSspectrometer can induce additional frequency broad-ening of the spectral line. To implement the CRDStechnique with a cw laser source, we need to accu-mulate high-intensity photon flux into the cavity andthen quickly switch off the cw laser source to obtaina photon decay signal. One method of switching offa laser source is to scan the cavity from an on-resonance position to an off-resonance position.14

The abrupt mirror movement in our CRDS spectrom-eter could introduce a sizable frequency shift to thespectral line, where the shift is given by N�vV�c� withV, the mirror speed, and N, the number of reflectionswithin the switching time. The magnitude of thespectral line shift is estimated to be 300 MHz for thevalues of N � 1000, V � 0.23 m�s, and v � 4.0 � 1014

Hz. The uncertainty induced from these effects isnot analyzed in detail in this paper.

4. Conclusions

To evaluate the uncertainty in the concentrationmeasurement with a CRDS technique, we have ana-lytically derived expressions for the relative uncer-tainty of parameters such as temperature, linestrength, laser-frequency jitter, and ringdown-timedeviation. Using a CRDS technique, we have deter-mined the absorption coefficient of an oxygen mole-cule with an uncertainty of less than 0.4%. We haveestimated the uncertainties that are due to noisesources in a practical CRDS system applied to theconcentration measurement of an oxygen molecule.To achieve an accurate concentration measurement,it is necessary to determine the accurate transitionstrength of the line before the measurement. It hasbeen found that, except for the line strength S�T0�,which is a reference value independent of the mea-surement, the laser-frequency jitter causes a majoruncertainty in the measurement with the presentCRDS system.

We are grateful to Seung Hee Nam for assistancewith the experiment.

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Table 2. Experimentally Attained Relative Uncertaintiesof Concentration

Source of Uncertainty u�� %�

Laser frequency �5 MHz� 0.66a

Absorption coefficient 0.13b

Line strength 1.0c

Temperature �40 mK� 0.1%d

Overall uncertainty 1.2

Note: The uncertainties were calculated for the Gaussian pro-file with the values.

aThe absorption linewidth was 450 MHz and Nf was 9.bu��p�2��p

2 is 0.5% and u��V�2��V2 is 0.3%.

cThe uncertainty of the line strength is estimated with the val-ues in Ref. 15.

dThe reference temperature was 296 K and calculated with thePQ�35� transition.


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18. The software of HITRAN96 was developed under contractF19628-91-C-0132 for the Phillips Laboratory Geophysics Di-rectorate under the direction of Laurence Rothman, OntarCorporation, 9 Village Way, North Andover, Mass. 01845.


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Uncertainty Analysis of Absolute Concentration Measurement with Continuous-Wave Cavity Ringdown Spectroscopy