Measurement method of nonresonant third-ordersusceptibility by using off-resonantcoherent anti-Stokes Raman spectroscopy
Kazuhiro Akihama, Takeshi Asai, and Satoshi Yamazaki
Nonresonant third-order susceptibilities (Xnr) of N2 and Ar gases by using an off-resonant coherentanti-Stokes Raman spectroscopy of the N2 Q branch, whose spectral profile is highly sensitive to the valueof Xnr, have been measured to demonstrate the possibility of the present method. The Xnr of N2 and Arusing the off-resonant coherent anti-Stokes Raman spectroscopy were 8.4 ± 0.2 and 9.7 ± 0.8 (x 10-18cm3 /erg/amagat), respectively. These values are in good agreement with those of previous studies.The off-resonant spectra were theoretically calculated by using the Raman linewidth given by a modifiedexponential gap model, and they were compared with the experimental ones to determine Xnr. Theeffects of uncertainties of the Raman linewidth of the N2 Q branch on the determination of Xnr wereinvestigated by changing the parameters in the modified exponential gap model. The determined Xnr
value changed by only 0.05% with 50% change of the Raman linewidth. The potential advantages of thismethod are no requirements of high spectral resolution measurements, no accurate Raman linewidthdata for each experimental condition, and no optical delay between pump beams.
Coherent anti-Stokes Raman spectroscopy (CARS)has been widely applied as a diagnostic technique incombustion systems.'-3 The third-order nonreso-nant susceptibilities (Xnr), which can cause significantfrequency shifts and spectral line distortion, areimportant for the accurate temperature measure-ments.4 5
Farrow et al. 6 measured the Xnr values of theimportant combustion-related gases such as propane,n-butane, and water vapor.6 They used the resonantsusceptibility of the N2 Q branch as the calibrationstandard, and they determined the Xnr values fromthe measured spectral profiles produced by the inter-ference of the resonant and the nonresonant ampli-tudes. Their method required a high-resolutionCARS for resolving each Q-branch line and accurateRaman linewidth data for each gas-mixing condition(e.g., molar fractions of N2 and subject gas, or a kind
The authors are with Toyota Central Research and DevelopmentLaboratories, Inc., Nagakute-cho, Aichi-gun, Aichi-ken 480-11,Japan.
of subject gas) to determine the values of Xnr- Incontrast, the complex convolution, which uses muchcomputation time, was required in the unresolvedCARS spectra when multimode pump lasers wereused.7 Furthermore, the relative delay between twopump beams must amount to several times as muchas the coherence length of the pump beam. Thisindicates that the collinear phase matching cannot beused even if the high spatial resolution is not re-quired.
Recently the spectral profile of the off-resonantCARS of a N2 Q branch (N2 Q-branch band tail) wasfound to be highly sensitive to the Xnr value as a resultof the interferences of the off-resonant and nonreso-nant amplitudes.8 9 It is considered that the signifi-cant change of the spectral profile of the off-resonantCARS with the Xnr change is a potential advantage forthe determination of Xnr-
Here the measurement of the Xnr value by using theoff-resonant CARS of the N2 Q branch has beeninvestigated as an alternative convenient technique.The off-resonant region of the rovibrational N2 Qbranch (Raman shift of 2400 cm-') is also theresonant region of the rovibrational N2 S branch.The polarization technique was employed to suppressthe S-branch component. The extremely high reso-lution system and the complex calculation of convolu-
tion on the laser linewidth are not required, becausethe off-resonant spectrum with the S-branch suppres-sion does not contain the sharp resonant lines and issmooth.
As a feasibility study the Xnr values of N2 and Arhave been measured by using the off-resonant CARSwith S-branch suppression, and the results have beencompared with the ones in other studies to verify thepossibility of this method for the Xnr measurement.The Raman linewidth was one of the importantparameters in the previous studies, such as that ofRef. 6. From this point of view the effects of theuncertainties of the Raman linewidth of the N2Q-branch band were also investigated for the determi-nation of the Xnr value in this method. Finally, themeasurement of Xnr for the other gases has beendiscussed as a possible application.
Analysis
In a polarization rovibrational CARS with the suppres-sion of the S-branch component, the anti-Stokesintensity for a monochromatic pump and Stokeslasers and for the binary mixtures of N2 and Ar gasesis given by6
I(od) I |D(1 - CAM)3xlllQ(Wd)'ID(PQ) + Xnr,obs((Pnr) 12,
(1)
where (°d = 1 - 2 is the difference between thepump and the Stokes laser frequencies (cm-1). HereD and CAr are the total gas density (amagat) and Arconcentration in the mixture gas, respectively.
Here Xnrobs is the total third-order nonresonantsusceptibility (cm 3 /erg), which can be directly ob-tained from the experimental spectrum. Note thatXnr,obs contains the contributions from both the sub-ject gases and the optical systems (e.g., window,mirror). Therefore, Xnr,obs is expressed by
where XnrN2 and Xnr,A (cm3 /erg/amagat) are thethird-order nonresonant susceptibilities of N2 and Argases, respectively. Here Xnr,sys (cm3/erg) is the non-resonant contribution from the optical systems.
The function 41(p) in Eq. (1) considers the polariza-tion angles between the pump beam and the Stokesbeam (0), and between the pump beam and thepolarization analyzer (f):
(D(p) = cos 0 cos , + p sin 0 sin 4),
formed by the interference of the off-resonant (N2Q-branch band tail) and nonresonant amplitudes.
The off-resonant region of the N2 Q branch is alsothe resonant region of the N2 S branch. The off-resonant spectrum, however, does not contain theresonant lines of the S branch if the S-branch compo-nent is suppressed by the polarization technique.The effects of the finite linewidths of the laser sourcesshould be considered for the accurate treatment ofEq. (1).7,11,12 However, these effects can be neglectedbecause the smooth off-resonant spectrum does notcontain the sharp resonant lines, which are affectedby the linewidths of the laser sources.
Furthermore, the collisional narrowing of the N2Q-branch band is negligible because the off-resonantregion is far from the N2 Q-branch bandhead.13Therefore, the isolated line approximation can beused for X1111Q and expressed as follows:101 31 4
3X11lQ(Wd) = A f Epvji~ + 1) [(ct)2 + - bj2(y')2]
X (i - d) - (4)
where
NA = (2rrc 2)(2mo&0 ) (5)
Here pj is the fractional population difference and bjjis the Placzek-Teller coefficient. Here N (1/cm 3/amagat) is the total gas number density at 1 amagat.The actual N2 gas number density in the mixture gasis considered by the term of D(1 - CAr) in Eq. (1).We see that mo and wo are the reduced mass and theharmonic oscillator frequency, respectively, and 'and y' are the mean and anisotropy of the polarizabil-ity derivative, respectively.
The polarizability derivatives employed in Eq. (4)were obtained from the experimental Raman crosssections'5 and the depolarization ratios.10 Here Wjand rj are the frequency and the Raman linewidth(FWHM) in inverse centimeters of the rovibrationalQ-branch transition. The vibrational transition de-pendence of the Raman linewidth was ignored. Wecalculated Dj as follows:
2 = i yj2 j;. i j
(3)(6)
where p is the spontaneous Raman depolarizationratio for each component. For nitrogen, pQ = 0.022,Pnr = 1/3 were employed. The angle combination of 0and 4) satisfies the condition of the S-branch suppres-sion.'0
In Eq. (1) X11 11 Q (cm 3 /erg/amagat) are the [1111]components of the third-order resonant susceptibilityof the N2 Q branch. Equation (1) for t0
d of 2400cm-' (the off-resonant region of the N2 Q branch)gives the theoretical off-resonant spectrum, which is
where yij is the state-to-state collisional relaxationrate modeled by the modified exponential gap (MEG)law'6 for the upward transitions (j - ij < i):
wij = Pa(To/T)-[(1 + 1.5Ej/kT8)/(1 + 1.5Ej/kT)]2
x exp(-PAEji/kT). (7)
Here Ej is the rotational term energy, AEji = Ei - Ej,and P is the pressure (atm). Here To is the referencetemperature at 295 K; a(cm-'/atm), n, , and 8 are
the MEG parameters. The downward transitionswere determined by the microscopic reversibility.For the MEG parameters, et = 0.023, n = 1.346, 13 =1.67, and = 1.26 were employed as the standardvalues for pure N2 gas.'6 These MEG parameters,which determine the dependence of temperature,pressure, and rotational quantum number in theRaman linewidth, should generally be changed whenchanging the gas-mixing condition (e.g., molar frac-tions of N2 and subject gas, or a kind of subject gas).6
Figure 1 shows the off-resonant spectra of the N2 Qbranch calculated at 295 K and 2 atm for pure N2 withthe S-branch suppression for various Xnr,obs in Eq. (1).The change of the spectral profile is highly sensitiveto the change of Xnr,obs by the interferences of theoff-resonant (the N2 Q-branch band tail) and thenonresonant component. For example, the changesof the minimum point are approximately ±10 cm-'for the Xnr,obs changes of + 10%. The significantprofile change of the off-resonant spectrum shown inFig. 1 is a potential advantage for the determinationof the nonresonant susceptibility.
Experimental
Figure 2 shows the experimental setup used in themeasurement of Xnr. The pump beam xl was pro-vided by the frequency-doubled output of a Nd:YAGlaser (Molectron MY-34; bandwidth of 0.64 cm-'FWHM, 20 Hz). As a Stokes beam )2, a scannabledye laser (Komatsu, bandwidth of 0.7 cm-' FWHM)pumped by a part of a radiation of 532 nm was used.
A dual-channel detection system was used. Inchannel (Ch.) A, the off-resonant spectrum with theS-branch suppression was measured and was used todetermine Xnr. The w, and °)2 beams in Ch.A were
1.0
I-zI- z0
-J
0z1-
0 I-2380 2400 2420 244
RAMAN SHIFT (cm-1)Fig. 1. Change of off-resonant spectra of the N2 Q branchcalculated at 295 K and 2 atm for pure N2 by using (a) Xnr,N2 = 7.7 X
10-1, (b) Xnr,N2 8.5 X 10 8, and (e) Xnr,N2 9.4 X 10-1
(cm3/erg/amagat) in Eq. (2) with Xnr,sys = 0.0 and XnrAr 0 -0.
sent through Glan-Taylor prisms and /2 plates(polarization rotator), respectively. The polariza-tion direction of the W2 beam was rotated by 600 fromthe vertical polarization of the w, beam [0 = 60° in Eq.(3)]. The w, beam was collinearly combined with the(2 beam on a dichroic mirror. The combining angleis 5° to eliminate the depolarization on the dichroicmirror. The collinearly collimated beams were fo-cused into a gas cell filled with pure N2 or the mixturegas of Ar and N2 (cell A in Fig. 2) by a lens with a focallength of 40 cm, and they were recollimated by anidentical one. The input pump and Stokes energiesin Ch.A were 14 and 0.7 mJ/pulse, respectively.
The CARS signal (3) of Ch.A and laser beams wereanalyzed by using the Glan-Taylor prism (the polar-ization analyzer), which was rotated by -37.6° fromthe vertical polarization of the ,1 beam to suppressthe S-branch contributions [ = -37.6 in Eq. (3)].10In contrast, the spectrum without the S-branchsuppression was simultaneously recorded in Ch.B.The pump and Stokes beams in Ch.B were collinearlyfocused into another gas cell filled with pure N2 (cell Bin Fig. 2) by the 40-cm focal length lens, and theywere recollimated by the identical one. The reso-nant S-branch lines of N2 recorded in Ch.B couldconveniently check the absolute wavelength of thespectra measured in Ch.A. Both signals of Ch.A andCh.B were isolated from the source laser beams byusing the dichroic mirrors and a double 1-m mono-chromator (Jobin-Yvon U1000), and they were de-tected by photomultiplier tubes (Hamamatsu R585).
The scanning method was used to accumulate thespectra. The scanning of the center wavelength ofthe monochromator was synchronized with dye laserscanning. At each datum point of both Ch.A andCh.B taken at intervals of 0.03 cm-', the signals of10-20 laser shots were averaged by the boxcar averag-ers and then digitized. One spectrum consisted of
laser was monitored with a power meter, and itsoutput was digitized to normalize the off-resonantspectrum recorded in Ch.A.
Results and Discussion
In the measurements of Xnr,N2, pure N2 gas at 295 Kwas filled in cell A in Fig. 2. Figure 3 shows thetypical dual-channel spectra. The spectra of Ch.Aand Ch.B correspond to the off-resonant spectra withand without the N2 S-branch suppression, respec-tively. We see that S(j) in Ch.B indicates the rovi-brational S-branch transitions; v:0 -->1 andj ->j + 2.These lines were used as a wavelength marker of thespectra recorded in Ch.A.
The Xnr,obs is determined by fitting the experimentalspectra in Ch.A to the theoretical ones. The valuesOf Xnr,obs in Eq. (1) are systematically varied in order toobtain the best fit (least-squares fitting). Note thatthe theoretical spectra for the pure N2 were calcu-lated under CAr = 0 and Xnr,Ar = 0 in Eqs. (1) and (2).The experimental spectrum, in contrast, was pro-cessed by the suitable low-pass filtering to reject thehigh-frequency noise components before the least-square-fitting processes.
The experiments were performed at several gasdensities (pressure) to avoid the errors by the nonreso-nant contributions from the optical system [Xnr,sys inEq. (2)]. Although Xnr,sys conceals itself in Xnr,obs) it isindependent of the gas density as shown in Eq. (2).Therefore it can be separated from Xnr,obs by using the
Ch
II1-Ch.AB
- SW(6 Se8 S(1o S(12)
gas-density dependence of Xnr,obs. The typical fittingresults at the three pressures are shown in Fig. 4.The profile of the off-resonant spectra is scarcelyinfluenced by the pressure, because the relative con-tribution of Xnrsys to Xnr,obs changes with the pressure.The roughness of the experimental spectra resultsmainly from the stability of the laser power and fromthe detection sensitivity variations.
Several spectra were measured at each pressure;then all the Xnr,obs values obtained were processed bylinear regression analysis. Figure 5 shows the N2gas-density dependence Of Xnr,obs* The detailed graphsat each point (a-d) are shown on the right-hand sideof the figure. The solid line indicates the regressionline. The correlation coefficient was 0.9998. There-fore the linearity of Xnr,obs for the gas densities wasexcellent. This indicates that Xnr,sys did not changeduring the experiments, because the experimentalcondition was kept as constant as possible. If Xnr,sys
largely changes in each experiment, then excellentlinearity cannot be obtained.
Consequently, the Y intersect of the solid line inFig. 5 expresses Xnr,sys, and its gradient expresses thevalue of the nonresonant susceptibility of N2 (Xnr,N2) incubic centimeters per erg per amagat. In the figure
1.0
0.5
-"-0.0CO)z 1.0
zo 0.5
WN
2 0.00 1.0
0.5
0.0
;%,^ (b) 5atm, 295KXnr,obs=38.80E- 1 8
'v W(c) 1 atm, 295K
Xnr,obs=78.16E-18
I
2380 2400 2420 2440RAMAN SHIFT( crY-)
Fig. 4. Off-resonant spectra of pure N2 (solid curve) and fit(dashed curve) at 295 K and at various pressures. The best-fittedXnrobs in Eq. (1) were (a) Xnrobs = 16.17 x 10-s cm3/erg, (b) Xnrobs =38.80 x 10-1s cm 3 /erg, and (c) Xnr,obs = 78.16 x 10-'8 cm3 /erg,respectively.
RAMAN SHIFTFig. 3. Typical dual-channel spectra. Spectra of Ch.A and Ch.Bare the off-resonant spectra of the N2 Q branch with and withoutthe N2 S-branch suppression, respectively.
l - - - rt
80
600
E0
T40
x
C
X ','4u'
d2
1
0C.
0
C
0
0I0xu,
.0
rR
16
15
a41
40
39
38
0.1
b
_ I
1.8
79
78
77
76
1.9
C
00
4.6
d
4.7
0
0
9.2 9.3
DENSITY ( amagat)
2 4 6 8DENSITY (amagat)
10
Fig. 5. N2 gas density dependence Of Xnr,obs. Graphs on the right-hand side indicate the detailed graphs at each point (a to d) in the graphon the left-hand side. Open circles and solid lines represent experimental results and regression lines for all data, respectively.
the gradient of the regression line was 8.4 x 10-18,and its standard estimated error was 0.1 x 10-18.For instance, the value of Xnr N2 was determined to be8.4 ± 0.1 x 10-'8 (cm3/erg/amagat). In the samemanner, Xnr obtained from the Y intersect was0.68 x 10-'8'(cm3/erg).
Farrow and Rahn7 previously reported the value ofXnr,N 2 by using the high-resolution experiments, 8.5 ±1.3 x 10-'8 (cm3/erg/amagat). Our result was ingood agreement with theirs, even if the measure-ments were performed without the high spectralresolution.
Furthermore, with the resonant susceptibility ofthe N2 Q branch as a calibration standard, themeasurements of Xnr,N 2 required the accurate data ofthe Raman linewidth for each measurement condi-tion.6 In this method, in contrast, the off-resonantintensity of the N2 Q branch (N2 Q-branch band tail)was used as the calibration standard. The Ramanlinewidth is not a significant factor in this methodbecause the profile of the off-resonant spectrum is notsensitive to the Raman linewidth. In other words,the term of (j - Wd) in Eq. (4) becomes much greaterthan that of the Raman linewidth, ij/2, in theoff-resonant region. For example, ( - ()d) be-comes 70 cm-' at the Raman shift (
)d of 2400 cm-'
for the N2 Q-branch bandhead w,% of 2330 cm-'. Wesee that F is as small as 1 cm-' even at 10 atm. As aresult, the term of the Raman linewidth in Eq. (4) isnegligible in the off-resonant spectral region, and
Xm 911Q converges into the real part. Therefore, thismethod is expected to permit the uncertainties of theRaman linewidth of the N2 Q branch for the determi-nation of Xnr, because the off-resonant CARS spec-trum is not affected by the Raman linewidth.
In this method, the theoretical off-resonant spectrawere compared with the experimental ones to deter-mine Xnr, and they were calculated by using theRaman linewidth of the N2 Q branch given by theMEG model as expressed in Eq. (7). To verify theabove expectation, we investigated the effects ofuncertainties of the Raman linewidth on the determi-nation of Xnr,obs by changing the value of each MEGparameter in Eq. (7). Table 1 shows the Xnr,obs de-termined in the experimental spectrum of Fig. 4(c) forvarious combinations of MEG parameters, as well asthese errors for the standard case. Note that thevalues for the pure N2 gas reported by Rahn andPalmer' 6 were employed as the standard MEG param-eters to determine the Xnr,obs of the standard case.There were no significant differences of Xnr,obs amongthe results in the table. In particular, it should beemphasized that the errors are within 0.05% evenwith the ±50% change in a., which corresponds to the±50% change of the Raman linewidth. Therefore,the effects of the uncertainties of the Raman line-width are clarified to be negligible in this method.
In addition, Table 1 indicates that the methoddescribed here has a potential advantage for Xnr
Table 1. Effects of Uncertainties of the Raman Linewidth for theDetermination of Xnr
MEG Parameters Xnr,obs
(cm3/erg) Errora (cm-/atm) 3 n (X lO- 8)
Standard casea0.023 1.67 1.26 1.346 78.16 0.00
Effects of c0.011 78.12 0.050.022 78.13 0.04
1.67 1.26 1.3460.026 78.17 0.010.036 78.19 0.04
Effects of p1.52 78.17 0.01
0.023 1.26 1.3461.82 78.14 0.03
Effects of 81.20 78.16 0.00
0.023 1.67 1.3461.32 78.16 0.00
Effects of n1.286 78.16 0.00
0.023 1.67 1.261.406 78.16 0.00
aMEG parameters of the standard case were the values for pureN 2 gas reported in Ref. 16.
with N2. For the binary mixture of N2 and thesubject gas, the off-resonant and the nonresonantcomponents of N2 become the calibration standard forthe determination of Xnr of the subject gas. Asshown in the table, the accurate data of the Ramanlinewidth for each gas-mixing condition are not re-quired. Even if the change of the Raman linewidthof the N2 Q branch is caused by the gas mixing, thiseffect is negligible, as can be seen from the table.
As a fundamental study on the measurement forthe binary mixture gas, the measurement Of Xnr,Ar wasdemonstrated. The off-resonant spectra were mea-sured in a 14.3% Ar in N2 mixture. All the measure-ments were made in cell A at a total pressure of 2 atmand at 295 K in Fig. 2.
Figure 6 shows the off-resonant spectra for pure N2and for 14.3% Ar in N2 mixture gas, respectively.The spectral profile was drastically changed by mix-ing Ar with N2 gas. Ten spectra such as the ones inFig. 6(a) were measured, and the Xnr,obs in Eq. (2) wereobtained. The average value and standard deviationOf Xnr,obs resulted 16.7 x 10-'8 and 0.2 x 10-18(cm3/erg), respectively. Note that the spectra weretheoretically calculated by using the standard MEGparameters in Table 1, and the change of the Ramanlinewidth by the gas mixing was neglected. We candirectly calculate Xnr,Ar from Eq. (2) by substitutingCAr = 0.143, XnrN2 = 8.4 x 10-18, and Xnr,sys = 0.68 x10-18, which are measured in this study. As a result,Xnr,Ar was 9.7 ± 0.8 x 10-18 (cm 3 /erg/amagat).
The comparison of the ratios XnrAr/XnrN2, measuredby several methods is summarized in Table 2 to verifythe possibility of this method. The result of ourexperiments is reasonable, especially in that the
1.0
0.5
I-0zLUI-zaLUN-J
0z
01.0
0.5-
0*2380 2400 2420 2440
RAMAN SHIFT (cm-1)Fig. 6. Spectra of off-resonant CARS obtained from (a) a gasmixture of 14.3% Ar in N2 at 295 K and 2 atm, and (b) pure N2 at295 K and 2 atm. Solid and dashed curves represent measure-ment and fit, respectively. Fits were calculated using (a) Xnr,obs =16.67 x 10-18 cm3 /erg and (b) Xnr,ob = 16.17 x 10-1 cm3 /erg inEq. (1).
absolute values of both Xnr,N2 and Xnr,Ar were in goodagreement with the ones of Ref. 7, which weremeasured in the close wavelength region of thisstudy.
It is considered that the error of the Xnr value by ourmethod arises predominantly from the power varia-tion of the lasers. Unfortunately, the normalization
Table 2. Comparison of X.r,Ar/Xr,N, Between Several Methods
of the off-resonant spectrum for the pump laserpower could not be carried out. As shown in Figs. 4and 6, the long-term power drift of the pump laserpower could not be rejected by the subsequent dataprocessing such as low-pass filtering. The rough-ness of the off-resonant spectrum causes the fittingerror between the experimental and theoretical spec-tra. Although the uncertainty of the Raman crosssection of the N2 Q branch may affect each absolutevalue Of Xnr, it seemed that the error of the Xnr,Ar/XnrN 2
as shown in Table 2 is mainly caused by the rough-ness of the off-resonant spectrum.
One of the advantages of the present method is thatthe Xnr can be measured even if the Raman linewidthis uncertain, as shown in Table 1. Furthermore,this method is convenient for the laboratories alreadyequipped with the CARS system, because it does notrequire extremely high resolution, and the measure-ment wavelength region is close to the N2 Q branch.Even if the multimode pump laser is used, an appro-priate delay in one pump beam path is not required toeliminate the correlation between the fluctuations inthe two pump fields, because the off-resonant spec-trum does not contain the sharp resonant lines.This indicates that the collinear CARS geometry canbe used in this method, if the high spatial resolutionis not required in such a case as the experimentsusing the gas cell. In other words, both BOXCARSand collinear phase-matching techniques can be usedin this method, facilitating the selection of an ad-equate phase-matching method.
The polarization CARS signal in this study, whichis proportional to I (p) 12 in Eq. (3), is reduced by- 1/7 of the magnitude from the parallel polarization
[0 = 4) = 0° in Eq. (3)] normally employed. Therewere, however, no signal-to-noise problems becausecollinear phase matching was used. Actually, thesignal strength combining the collinear phase match-ing and the scanning CARS method was sufficientlyhigh, giving a peak signal-to-noise ratio in excess of 50without special treatment for the signal-to-noise ra-tio.
In addition, as shown in Fig. 6, the off-resonantspectrum is highly sensitive to the total nonresonantsusceptibility, which is the summation of the nonreso-nant susceptibilities of N2, subject gas, and the opticalsystem. This is because the amplitudes of bothoff-resonant and nonresonant components are compa-rable. In the case in which the subject gas has largenonresonant susceptibility, the effective change ofthe total nonresonant susceptibility is predicted toachieve an effective change by mixing the subject gasof a low concentration with N2 gas. In other words,the measurements can be performed even under thegas-mixing condition of nearly pure N2, in contrastwith the method reported in Ref. 6. For example, inthe case of C3H8, whose nonresonant susceptibility isapproximately ten times as large as that of N2,6 the10% change of the total susceptibility from that of thepure N2 can be achieved by mixing 1 mol. % C3H8 withN2. Under such a condition it becomes the better
approximation to apply the Raman linewidth of thepure N2 in the analysis, although the data of theaccurate Raman linewidth are not required in thismethod. If this method is applied to the Xnr measure-ment of a combustion fuel gas such as C3H8, it shouldbe noted that the term of off-resonant and nonreso-nant susceptibilities of the subject gas must be addedto Eq. (1).6 It is considered that the method de-scribed here also has a potential advantage for the Xnrmeasurements of some kinds of combustion fuelgases with large values of Xnr-
Summary
A fundamental study on the measurement of thethird-order nonresonant susceptibility by using off-resonant CARS has been performed. The possibilityof this method has been verified by the measurementsof the third-order nonresonant susceptibilities of N2and Ar gases. The results were in good agreementwith previous studies. Although other techniques(listed in Table 2) may have a potentially highermeasurements accuracy, the present method hasbeen experimentally investigated as an alternativeconvenient technique because of the following advan-tages. First, it required no extremely high resolu-tion system and no calculation of complex convolution.Second, collinear phase matching can also be usedbecause of no requirement of the relative delaybetween two pump beams. Third, the accurate Ra-man linewidth data at each gas-mixing condition arenot required. The method described here is ex-pected to give greater simplicity and convenience tothe measurements of the third-order nonresonantsusceptibility.
References1. A. C. Eckbreth, Laser Diagnostics for Combustion Tempera-
ture and Species (Abacus, Tunbridge Wells, UK, 1988), pp. 220-292.
2. A. C. Eckbreth, G. M. Dobbs, J. H. Stufflebeam, and P. A.Tellex, "CARS temperature and species measurements inaugmented jet engine exhausts," Appl. Opt. 23, 1328-1339(1984).
3. S. Furuno, K. Akihama, M. Hanabusa, S. Iguchi, and T. Inoue,"Nitrogen CARS thermometry for a study of temperatureprofiles through flame fronts," Combust. Flame 54, 149-154(1983).
4. R. J. Hall and L. R. Boedeker, "CARS thermometry infuel-rich combustion zones," Appl. Opt. 23, 1340-1346 (1984).
5. R. P. Lucht, R. E. Teets, R. M. Green, R. E. Palmer, and C. R.Ferguson, "Unburned gas temperatures in an internal combus-tion engine. I: CARS temperature measurements," Com-bust. Sci. Technol. 55, 41-61 (1987).
6. R. L. Farrow, R. P. Lucht and L. A. Rahn, "Measurements ofthe nonresonant third-order susceptibilities of gases usingcoherent anti-Stokes Raman spectroscopy," J. Opt. Soc. Am. B4, 1241-1246 (1987).
7. R. L. Farrow and L. A. Rahn, "Interpreting coherent anti-Stokes Raman spectra measured with multimode Nd:YAGpump lasers," J. Opt. Soc. Am. B 2, 903-907 (1985).
8. K. Akihama and T. Asai, "Application of polarization S-branchCARS: studies of thermometry and nonresonant susceptibil-
9. K. Akihama and T. Asai, "Measurement of nonresonantthird-order susceptibility using off-resonant CARS, " in Proceed-ings of the Twelfth International Conference on Raman Spec-troscopy, J. R. Durig and J. F. Sullivan, eds. (Wiley, New York,1990), pp. 216-217.
10. R. L. Farrow, P. L. Mattern, and L. A. Rahn, "Comparisonbetween CARS and corrected thermocouple measurements ina diffusion flame," Appl. Opt. 21, 3119-3125 (1982).
11. H. Kataoka, S. Maeda, and C. Hirose, "Effects of laserlinewidth on the coherent anti-Stokes Raman spectroscopyspectral profile," Appl. Spectrosc. 36, 565-569 (1982).
12. R. E. Teets, "Accurate convolutions of coherent anti-StokesRaman spectra," Opt. Lett. 9, 226-228 (1984).
13. K. Akihama and T. Asai, "S-branch CARS applicability tothermometry," Appl. Opt. 29, 3143-3149 (1990).
14. R. J. Hall, J. F. Verdieck, and A. C. Eckbreth, "Pressure-
induced narrowing of the CARS spectrum of N2," Opt. CoM-mun. 35, 69-75 (1980).
15. H. W. Schrdtter and H. W. Kl6cker, "Raman scattering crosssections in gases and liquids," in Raman Spectroscopy of Gasesand Liquids, A. Weber, ed. (Springer-Verlag, New York, 1979),pp. 123-201.
16. L. A. Rahn and R. E. Palmer, "Studies of nitrogen self-broadening at high temperature with inverse Raman spectros-copy," J. Opt. Soc. Am. B 3, 1164-1169 (1986).
17. T. Lundeen, S.-Y. Hou, and J.W. Nibler, "Nonresonant thirdorder susceptibilities for various gases," J. Chem. Phys. 79,6301-6305 (1983), and references therein.
18. W. G. Rado, "The nonlinear third order dielectric susceptibil-ity coefficients of gases and optical third harmonic genera-tion," Appl. Phys. Lett. 11, 123-125 (1967).
19. D. P. Shelton and A. D. Buckingham, "Optical second-harmonic generation in gases with a low-power laser," Phys.Rev. A 26, 2787-2798 (1982).
