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Laser-induced fluorescence measurements oftranslational temperature and relative cycle numberby use of optically pumped trace-sodium vapor

Chris C. Dobson

Sodium fluorescence induced by a narrow-bandwidth tunable laser has been used to measure tempera-ture, pressure, axial velocity, and species concentrations in wind tunnels, rocket engine exhausts, and theupper atmosphere. Optical pumping of the ground states of the sodium, however, can radically alter theshape of the laser-induced fluorescence excitation spectrum, complicating such measurements. Here astraightforward extension of rate equations originally proposed to account for the features of the pumpedspectrum is used to make temperature measurements from spectra taken in pumped vapor. Alsodetermined from the spectrum is the relative fluorescence cycle number, which has application tomeasurement of diffusion rate and transverse flow velocity. The accuracy of both the temperature andthe cycle-number measurements is comparable with that of temperature measurements made in theabsence of pumping.

OCIS codes: 020.2930, 020.3690, 120.6780, 120.7250, 300.2530, 300.6210.

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1. Introduction

Narrow-band tunable laser-induced fluorescence~LIF! of trace-sodium vapor has been used to measuretemperature, pressure, and velocity in low-pressuregases found in wind tunnels,1–3 rocket engine ex-hausts,4 and the upper atmosphere.5 The tempera-ture and pressure are determined from the line shapeof the D resonances observed in fluorescence, and,imilarly, the component of flow velocity along theaser beam is determined from the Doppler shift ofhe resonance in the moving sodium. At higher la-er powers, however, although it is still well belowaturation the fluorescence excitation spectrum be-omes distorted as a result of frequency-sensitiveepartitioning of the populations of the hyperfineround states by the laser.6 The temperature and

pressure are still reflected in the Voigt line shapes ofthe two transitions out of ground, but the absorptionprofile is radically altered by the shifting ground-state populations. In such a case the spectrum canbe calculated from rate equations for the transitionsof the D manifolds and the residence time in the laser

The author is with NASA Marshall Space Flight Center, MSEP63, Huntsville, Alabama 35812. His e-mail address ischris.dobson@msfc.nasa.gov.

Received 26 October 1998; revised manuscript received 26 Jan-uary 1999.

3924 APPLIED OPTICS y Vol. 38, No. 18 y 20 June 1999

beam of the sodium atoms. Walkup et al. used alimiting case of a four-state model of the D2 manifold,which, in effect, was based on three states to calculatespectra for optical pumping in xenon. The calcula-tions were shown to agree quantitatively with exper-imental excitation spectra.

Measurements of temperature and relative fluores-cence cycle number made from experimental opticallypumped spectra by least-squares curve fitting to the-oretical spectra are presented here. These measure-ments are shown to agree with independentmeasurements. The pumped vapor is of interest fortemperature measurements because the higher laserpowers provide stronger fluorescence signals, and thecycle number measurement, made possible becauseof the pumping, is of interest in applications to diffu-sion and transverse flow velocity.

The fluorescence is modeled with rate equationsbased on the eight hyperfine states of the Dmanifolds.7–9 As described by Walkup et al., the de-gree to which the ground states are pumped dependsprimarily on the fluorescence cycle number r, whichs the number of absorption–emission cycles per atomuring the residence time and is given by

r ; @bij~n!yD#max, (1)

where bij is the absorption rate for the transitionfrom state i to state j and D21 is the residence time.6,8

nwtmetsraw

ttw

Tnposiapocs

To a lesser extent, the pumping rate also depends onthe rate of collisional thermalization, or mixing, ofthe excited-state hyperfine populations. The inter-nal structure of and the coupling between the excitedstates were implicit in the original four-state model,and their explicit inclusion quantifies the role ofexcited-state mixing in ground-state pumping. Thespectrum is, in either case, essentially a direct reflec-tion of the excited-state populations, which, in turn,are determined from the rate equations. The spec-trum for the eight-state model is given explicitly inRef. 8.

The theoretical spectrum, then, depends on a set ofindependent variables, including temperature T,pressure p, and the cycle number:

fT 5 fT~nn; T, p, r . . . !, (2)

where nn are the discrete frequencies of interest, i.e.,those within the laser scan at which the experimentalfluorescence is sampled. Determination of one ormore independent variables from experimental spec-tra was performed with a function-minimization al-gorithm. In particular, a subset of these variables isused to define a K-dimensional parameter space forminimization of a merit function. The merit func-tion is defined as the sum over frequency of thesquares of the differences between fT and the exper-imental spectrum:

DT,E 5 (n51

N

wn@ fE~nn! 2 fT~nn; T, p, r . . . !#2, (3)

where fE~nn! is the experimental spectrum, N is theumber of discrete frequencies, and wn is an optionaleighting function based on fE~nn!.9 The minimiza-

ion algorithm was based on Powell’s direction setethod with the direction of largest decrease discard-

d.10 For the measurements here, with one excep-ion, K 5 4, and the fit parameters are T, r, verticalcale, and frequency offset. The scale and offset pa-ameters are used simply to overlap the spectra; thebsolute frequencies and intensities of the spectraere not of interest.

2. Experiment

The experimental LIF excitation spectra were col-lected from a vacuum cell containing metallic sodiumand filled to low pressures with noble gases. Thecell was enclosed in a clamshell-style laboratory fur-nace for temperature control. The temperature atthe probe volume was measured directly with atype-J thermocouple connected to an Omega ModelTAC80B-J TC converter. The temperature range ofthe data is approximately 435–515 K, and, in thisrange, the accuracy specification was 60.7%. Thepressure was measured by an MKS Model 121-A ca-pacitance manometer exposed to the cell interior, andthe pressure varied roughly from 0.1 to 1 Torr.Fairly low pressures ~,'50 Torr is typical! are re-quired for cycle-number measurements, and the spe-cific choice of ,1 Torr was driven by otherconsiderations.8 For the results reported here, as

indicated below, it is the relative pressure that isimportant, and the uncertainty of the measurementof change in the pressure has been estimated as ,1%for most of the data.9

The laser was a Coherent Model 699-21 ring dyelaser pumped with a Coherent Innova 300 argon-ionlaser. The system is frequency and power stabi-lized, and the ring laser jitter was typically ;1 MHz.The frequency scans with the ring laser were 12 to 15GHz near 5 or 6 GHzys. The width and rate of thescan were controlled by an electronic feedback loopwith an integral, temperature controlled, confocalscanning etalon. In principle, the splitting of thesodium ground state provides an independent mea-sure of the frequency scale of the scan. However,this splitting, which is ;1.8 GHz, is similar to thetypical FWHM for the Doppler broadening, so inpractice it was difficult to distinguish small errors inthe frequency scale from errors in the optical mea-surement of temperature.

The argon laser was operated in its TEM00 mode,so the transverse spatial profile of the dye laser beamis generally Gaussian, with a diameter at the probevolume of ;0.4 cm. For most of the spectra the dyebeam was expanded, up to ;2-cm diameter, to reducehe probe volume irradiance to avoid disturbance byhe laser of the Maxwell velocity distribution onhich the temperature measurement is based.6

This beam expansion was accomplished with a pair ofsimple lenses so that, although no measurements ofthe actual profiles were made, the expanded beam isalso expected to be approximately Gaussian. Thepower delivered to the probe volume was typically ofthe order of 1–10 mW.

The fluorescence was collected at right angles tothe beam and was recorded with photomultipliertubes, one for each manifold, D1 and D2. The 0.8cm 3 2.4 cm photocathodes were imaged onto thelaser beam with a demagnification of ;2.9, so thedetectors were overfilled. The separation of the Dlines, needed for cross-section measurements de-scribed elsewhere,8,9 was accomplished with verynarrow-band interference filters ~FWHM of 0.3 nm!.

he laser power, which is important to the cycle-umber measurements, was monitored by a siliconhotodetector just upstream from the cell. A portionf the beam not used for the fluorescence was mea-ured with a powermeter and this, together with cal-brations for the neutral-density filters used tottenuate the beam, provides a second measure of therobe beam’s power. All the raw data were recordedn a PC with a Data Translation analog-to-digitalonverter board. A schematic of the lab apparatus ishown in Fig. 1.

3. Temperature Measurements

The temperature measurement from the pumpedspectrum is based, as in the simpler case absentpumping, on the determination of the Doppler con-tribution to the Voigt line shape. When pumpingoccurs, the absorption profile, which is ordinarily pro-portional to the line shape ~for a single transition!, is

20 June 1999 y Vol. 38, No. 18 y APPLIED OPTICS 3925

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in effect modified by a frequency-dependent scale fac-tor. This scale factor reflects the ground-state pop-ulations as determined from the rate equations,6,8

and, for the temperature fitting of spectra, it can bevaried simply by changes in the cycle number. Forthe experimental spectra here, moreover, the pres-sure broadening contribution for helium and argongases is negligible,11,12 and the line shape may beconsidered purely Doppler. When pressure broad-ening is present, its effect on the fitting process isexpected to be comparatively minor, much as in thecase when pumping does not occur ~compare, e.g.,Refs. 3 and 5!, because it is only the absorption pro-file, and not the line-shape function, that is altered bythe pumping.

An example of the curve fitting for temperature isgiven in Fig. 2 for a D1 spectrum in helium. Thevariation of cycle number is accomplished, for a fixed,arbitrary, residence time, by variation of the laserpower. The temperature variation is used only to

Fig. 1. Schematic of the laboratory apparatus: O-scopes, oscil-loscopes; PMT’s, photomultiplier tubes.

Fig. 2. Temperature fitting of a sodium D1 spectrum in helium.The discrete points are the experimental spectrum, and the solidcurve is the theoretical spectrum. Final values for temperatureand cycle number were 477.5 K and 4.08, respectively. The ther-mocouple temperature was 490.4 6 3 K.

926 APPLIED OPTICS y Vol. 38, No. 18 y 20 June 1999

alter the Doppler width of the line shape. The finalfit occurred in this case after 18 iterations with anaccuracy of approximately 22.6%, where the temper-ature accuracy is defined as

dT ; ~TNa 2 TTC!yTTC, (4)

where TNa is the fitted temperature and TTC is thethermocouple temperature.

In all, 45 pairs of spectra in helium were fitted fortemperature in this fashion. The results are sum-marized in Fig. 3, which shows results for D1 and D2separately. The pressure varied from ;50 to ;800mTorr; the fitted cycle number, from ;1 to ;15. Nomarked correlation between dT and either p or r wasobserved. The mean value of udTu was 3.1% for D1and 5.0% for D2. The preponderance of negativedata points in Fig. 3, however, suggests that there isa systematic error in the fitted temperature. Themean values of dT itself are 22.4 and 23.7%, respec-tively, for D1 and D2. This bias is thought to be dueto inaccuracy in the frequency scale derived from theerror in the scan width calibration performed withthe ring laser controls.

The results from the argon data are given in Fig. 4.The pressure ranges, again, from ;50 to ;750mTorr; the fitted cycle number, from ;1 to nearly 60.As with helium, no marked trends with p and r werenoted. ~There was some tendency, in all the spectra,for the temperature accuracy to worsen both at low

Fig. 3. Temperature errors for fitting of experimental sodiumspectra in helium. Temperature and cycle number were the de-termining fit parameters. ~a! D1 fittings: udTu 5 3.1%, dT 5

2.4%. ~b! D2 fittings: udTu 5 5.0%, dT 5 23.7%.

~

r

pressure because of reduced vapor density and at lowcycle number because of reduced laser power, but thiswas a minor effect.! For most of the argon data theaccuracy was similar to that for the helium data,although systematic error was less in evidence. Forthese data, plotted as diamonds in Fig. 4, udTu 5 2.9%dT 5 11.6%! for D1 and udTu 5 4.1% ~dT 5 22.9%!

for D2.For one group of argon spectra, however, collected

on a separate day, the temperature accuracy wasespecially poor ~2dT 5 udTu . 10%!. These spectrawere fitted with the frequency scale used as a fifth fitparameter. Although the use of this parameter didnot in general necessarily improve the accuracy, be-cause of the line broadening, it did so distinctly in thiscase. This improvement is shown in Fig. 4 also,where the original four-parameter-fit results are plot-ted as circles and the five-parameter-fit results are asplotted as squares. Averaged over all the argonspectra, then, but excluding the circles, udTu 5 2.9%~dT 5 11.2%! for D1 and udTu 5 3.9% ~dT 5 22.1%!for D2.

To summarize, approximately 200 opticallypumped sodium spectra taken in helium and argon

Fig. 4. Temperature errors for fitting of experimental sodiumspectra in argon. Temperature and cycle number were the deter-mining fit parameters. The diamonds and circles represent spec-tra fitted with the nominal, fixed, frequency scale; the squares,which denote the same spectra as the circles, give results for fit-tings with frequency scale as a fit parameter. ~a! D1: {, udTu 52.9%, dT 5 11.6%; h, udTu 5 2.8%, dT 5 20.8%; E, udTu 5 13% 52dT. ~b! D2: {, udTu 5 4.1%, dT 5 22.9%; h, udTu 5 3.2%, dT 511.5%; E, udTu 5 11% 5 2dT.

buffer gases were fitted for temperature. The meanaccuracy, determined from comparison with thermo-couple measurements, was from 63% to 65% fortemperatures in the range from 435 to 515 K. Thesevalues compare with accuracies from 63% to 610%reported in the absence of pumping in various envi-ronments with ~static! temperatures from ;10 to;2000 K.2–5

4. Cycle-Number Measurements

The cycle number is the ratio of the residence timeD21 to the absorption time.6 It has a marked influ-ence on the shape of the excitation spectrum when itis comparable to unity ~e.g., 0.1 , r , 100!, and it canbe determined from the spectrum by curve fittingwith r as a fit parameter. For reference, Fig. 5 il-lustrates the dependence of the spectrum on r byplotting a normalized version of Eq. ~3!, the fractionaloot-mean-square ~rms! difference:

D ; Î1N

DT1,T2 F1N (

n51

N

fT1~nn!G21

, (5)

calculated ~with wn 5 1! for pairs of theoretical spec-tra that have a ratio of rT1

yrT25 1.2 for a range of

values of r. Spectral determination of the cyclenumber is of interest primarily because one can use itwith laser power measurements to determine the res-idence time. Residence time measurements are, inturn, of interest in application to diffusion and trans-verse flow.

The residence time for the sodium atoms in thelaboratory cell was set by thermal diffusion. In thiscase, the cycle number is6

r 5 ~P0yD0!BSP~ J!V~0!c21, (6a)

where D0 is the diffusion coefficient, P0 is the laserpower, BSP

~J! is the Einstein coefficient for absorptionon S1y2, F 3 PJ ~F degenerate!, V~0! is the integral-normalized line-shape function evaluated at its max-

Fig. 5. Sensitivity of the excitation spectrum to cycle number asa function of cycle number. The fractional rms difference betweena pair of spectra, D from Eq. ~5! @and Eq. ~3!#, is plotted as a functionof r. The difference in r between the two spectra in a pair is;20%. The calculations are for helium at 475 K and 0.5 Torr.

20 June 1999 y Vol. 38, No. 18 y APPLIED OPTICS 3927

w

ct

I

ncu

pf

l

fi

amtv

dcnspfv

3

imum, and c is the speed of light. Determination ofthe cycle number in this case permits measurementof the diffusion coefficient13:

D0 5 gT3y2p21, (6b)

here g [ 2⁄3 sD21=k3~pm!21; here sD is the cross

section for diffusion collisions, k is the Boltzmannonstant, and m is the reduced mass for sodium andhe buffer-gas atom.

Alternatively, in a flow field1–4 in which the resi-dence time is set by the flow velocity, the cycle num-ber is

r 5P0

y'dBSP

~ J!V~0!c21, (7)

where y' is the magnitude of the transverse compo-nent of the flow velocity and d is the beam diameter.n this case r could be used to measure y', which is

interesting in part because it permits configurationsin which the laser beam is transverse to the flowdirection. Doppler shift measurements, by compar-ison, require axial configuration, which is sometimesunwieldy experimentally.2,3 On the other hand, theDoppler shift measurement does provide a simpleand unambiguous determination of the direction ofthe measured velocity component. For the resi-dence time measurement the direction lies in theplane transverse to the laser axis but is givenuniquely only in special cases. A planar LIF ar-rangement, for example, would isolate the componentof velocity normal to the fluorescence plane, and thiswould uniquely determine the direction under condi-tions, typical in wind tunnels and engine exhausts,for which the polarity of the flow is known.

In both of these cases, Eqs. ~6! and ~7!, the cycleumber actually represents a spatial average be-ause, in practice, the profile of the laser beam is notniform. At a point, r is proportional to I0D21,

where I0 is the laser irradiance related to the powerby the area integral P0 5 * I0 dA. For diffusion, theresidence time is proportional to the beam area,6 andEq. ~6! assumes that P0yI0 5 D0yD ~it was shown

reviously that r is insensitive to the beam diameteror diffusion6!. Similarly, d in Eq. ~7! is an effective

diameter such that P0yI0 5 y'dyD. However, anal-yses based on the spatial profiles of the laser beamand the collection probe volume are not necessary, solong as these profiles are repeatable, if the system iscalibrated at known conditions. For diffusion, thefitted r from a sample with a known g would permitdetermination of unknown values of g ~e.g., with dif-ferent gases! absolutely, given only the changes in rand T ~from the fittings! and the changes in p and P0~from direct measurements!. Similarly, given cali-bration at a single magnitude y',cal known abso-utely, and laser power P0,cal known relatively,

unknown magnitudes could be determined absolute-ly; in this case, the direct measurements are notneeded @assuming that V~0! is independent, or aknown function, of p#.

928 APPLIED OPTICS y Vol. 38, No. 18 y 20 June 1999

The key, then, to useful residence time measure-ments for either application is accurate determina-tion of the relative cycle number from theexperimental spectra. This is demonstrated here bycurve fitting just as with the temperature measure-ments, with T used as a fit parameter to vary thelinewidth and P0 as a fit parameter to vary r. The

tted value of P0, denoted P0,T, is then compared withthe direct measurements of the relative laser power,denoted P0,E. Diffusion coefficients of 1.0 and 0.48cm2 s21 ~at STP; Refs. 9 and 14! were used for heliumnd argon, respectively, along with the independenteasurements of T and p. Inasmuch as it is only

he ratio r that influences the shape of the spectrum,ariation of P0 is mathematically equivalent to vari-

ation of D0, and the choice of P0 as the fit parameteris arbitrary. These results can thus be viewed asfitting measurements of either the relative absorp-tion time or the relative diffusion time.

In fitting for cycle number, however, there is onefurther consideration: a potential correction to r asinferred from the spectrum owing to excited-statemixing. The degree of ground-state pumping, ingeneral, depends on both the cycle number and themixing rate. If the mixing rate is known or asymp-totic, the cycle number may be determined unambig-uously. If the mixing rate is not known, theresultant uncertainty in r for the experimental con-ditions here, although it is small for D1, can be sub-stantial for the D2 spectrum. This issue, which isdiscussed elsewhere,8,9 will be set aside for the pur-pose here of demonstrating the cycle number mea-surement. For this purpose, it is possible to removethe actual mixing rates from consideration becausethe contribution of mixing to the pumping is gener-ally a weak function of the number density of thebuffer gas. If the number density is comparativelyconstant, the contribution of mixing is nearly con-stant and the value of r from one fitted spectrum will

iffer from the true value by, at most, a scale factorommon to all the spectra. Thus, for the cycle-umber fitting, the spectra are separated into groupsuch that all the spectra in a group have similarressures, and fixed, arbitrary values are assumedor the mixing rates. This means that the measuredalues of r, and thus of P0,T, are relative.The value of P0,T for each spectrum in a quasi-

isobaric group ~pmaxypmin , 1.7!, then, from the fittedr is plotted against P0,E from the direct measure-ments, and a linear least-squares fit to the data isperformed:

P0,T > a1 P0,E 1 a0. (8)

The accuracy of the agreement is quantified by themean scatter dP0,T, defined as

dP0,T ;1N (

n51

N U1 2P0,T,n

a1 P0,E,n 1 a0U , (9)

where the sum is over the N spectra in the group.

f

tipTlstotI

e

d d

pr

To limit the contribution of errors in P0,E to dP0,T,the accuracy of P0,E was ensured by the requirementof agreement between the two independent, directmeasurements. The arbitrary constraint dP0,E ,1% was imposed on each group of spectra, wheredP0,E is the mean scatter for linear fitting of the pow-ermeter data to the photodetector data ~see Fig. 1!,exactly as in Eqs. ~8! and ~9!. This does result insome additional fragmentation of the data, becausethere were uncalibrated changes in the photodetectorgain, but, given this constraint and the direct tem-perature and pressure measurement accuracies citedabove, it is expected that the total error dP0,T will bedue primarily to errors in the fitted r. Lower limitson the fitted r were selected as 1 and 2, respectively,or D1 and D2 @BSP

~3y2! 5 2BSP~1y2!#, based on general

calculations.9 Also, data from the separate mani-folds were treated separately because the contribu-tion of mixing to the pumping is different for the twomanifolds.

The segregation according to buffer-gas pressureand laser power measurement accuracy producedthree groups of helium spectra and five groups ofargon spectra. Values of P0,T for these groups areplotted in Figs. 6 and 7, giving the linear fits fromrelation ~8!. The mean scatter dP0,T, averaged over

Fig. 6. Sodium cycle-number measurements in helium. Thenumber of fluorescence cycles per atom in the laser beam is deter-mined from spectrum fitting. This cycle number is proportional tothe laser power and yields the value P0,T, which is plotted againsta direct measurement of the laser power P0,E. Slope variation is

ue primarily to gain changes in the photodetector. ~a! D1:dP0,T 5 3.6%. ~b! D2: dP0,T 5 2.9%. ~For clarity, p 5 340 mTdata have been shifted to the right 1.0 unit.!

all the data, is ;3% or ;4% for each manifold. De-ails are given in the figure captions. The variationn the slope of the lines for the different groups is duerimarily to changes in the gain of the photodetector.he y intercepts ranged from ;0.02 to ;0.2, with the

arger offsets generally associated with lower pres-ures. This result suggests dc offset in the pressureransducer, which is plausible, but it also impliesccasional small offsets in the linear relationship be-ween measured and actual values of either P0,E or r.t is of note that the values of P0,T in the plots, al-

though they are not intended to be absolute, are atleast roughly comparable with the actual probe beampower in milliwatts.

It is worth emphasizing that the determination ofcycle number in this manner by use of the shape ofthe pumped spectrum relies on the comparative ab-sence of line broadening. Obviously, if the linewidthis large compared with the ground-state splitting, thepumping will not occur. In practice, both the declinein pumping and the obscuration of its effects on thespectrum act to limit the utility of this technique. Attemperatures above 600 or 700 K, for example, or forpressures in helium above ;30 Torr, the spectrum isssentially insensitive to r.

Fig. 7. Sodium cycle-number measurements in argon. Thenumber of fluorescence cycles per atom in the laser beam is deter-mined from spectrum fitting. This cycle number is proportional tothe laser power and yields the value P0,T, which is plotted againsta direct measurement of the laser power P0,E. Slope variation is

ue primarily to gain changes in the photodetector. ~a! D1:dP0,T 5 3.4%. ~b! D2: dP0,T 5 3.6%. ~For clarity, p 5 105 and

5 390 mT data have been shifted to the right 0.5 and 1.0 units,espectively.!

20 June 1999 y Vol. 38, No. 18 y APPLIED OPTICS 3929

4. C. W. Braiser and R. G. Porter, “Development of a laser-

1

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5. Summary

Frequency-scanned LIF of optically pumped trace-sodium vapor has been used here to make tempera-ture and cycle number measurements in low-pressure noble gases in a static laboratory cell.These measurements were made by fitting of exper-imental spectra to calculations based on steady-staterate equations for the transitions of the sodium Dmanifolds. The optical pumping is of interest intemperature measurements primarily because it pro-duces enhanced fluorescence signals as a result of thehigher laser absorption rates that are responsible forthe ground-state pumping. The average accuracy ofthe temperature measurements in helium and argongases was 63.0% for D1 and 64.4% for D2.

The relative fluorescence cycle number was alsodetermined from fitting of the experimental spectra.This measurement is based on the pumping of theground states and is of interest because one can useit to determine the residence time of the sodium,which has application to diffusion rate and trans-verse flow speed measurements. Spectral fitting forcycle number was used here to make optical pumpingmeasurements of the relative laser power, or, alter-natively, of the relative residence time for thermaldiffusion. These measurements were then com-pared with direct measurements. The average ac-curacy of the cycle-number measurements was63.5% for D1 and 63.3% for D2.

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3. S. Cheng, M. Zimmermann, and R. B. Miles, “Supersonic-nitrogen flow-field measurements with the resonant Dopplervelocimeter,” Appl. Phys. Lett. 43, 143–145 ~1983!.

930 APPLIED OPTICS y Vol. 38, No. 18 y 20 June 1999

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