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Abstract
Determination of S-branch Raman linewidths of oxygen from picosecond time-domain pure rotational coherent anti-Stokes Raman spectroscopy (RCARS) measurements requires consideration of coherence beating. We present an optimization of the established model for fitting the coherence decay in oxygen, which leads to an improvement in Raman linewidth data quality, especially for the challenging small signal intensity and decay constant regime, enabling the application for low oxygen concentrations. Two modifications to the fitting procedure are discussed, which aim at reliably fitting the second coherence beat properly. These are evaluated statistically against simulated decay traces, and weighing the data by the inverse of the data magnitude gives the best agreement. The presented temperature dependent ${{\rm O}_2} {-} {{\rm O}_2}$ S-branch Raman linewidth from the modified model shows an improved data quality over the original model function for all studied temperatures. ${{\rm O}_2} {-} {{\rm N}_2}$ linewidths of oxygen in air for the temperature range from 295 K to 1900 K demonstrate applicability to small concentrations. Use of the determined RCARS ${{\rm O}_2} {-} {{\rm O}_2}$ S-branch linewidth instead of regularly used Q-branch derived linewidths leads to a lowering in evaluated RCARS temperature by about 21 K, thereby, a much better agreement with thermocouple measurements.
© 2021 Optical Society of America
1. INTRODUCTION
Determination of the S-branch Raman linewidths is an important field in modern advancement of pure rotational coherent anti-Stokes Raman spectroscopy (RCARS) [1]. RCARS is a four-wave mixing process, whose evaluation with regard to thermodynamic conditions at the intersection of three incident photons requires parameters like the S-branch Raman linewidth [2,3]. These parameters can be determined empirically or computationally. RCARS is a well-known tool for combustion diagnostics, which can be based on a multitude of probe molecules (see, e.g., [4–6]). The predominant ones are ${{\rm N}_2}$ and ${{\rm O}_2}$, but also ${{\rm H}_2}$, CO, ${{\rm CO}_2}$ or fuel molecules are available depending on fuel, progress of combustion, and thermodynamic conditions (e.g., [7]). The availability of picosecond (ps) and femtosecond (fs) lasers allowed for time resolved experiments that offer a comparably easy method for the determination of predominantly pressure broadened Raman linewidth. Therefore, the S-branch linewidth of different molecules and mixtures under various temperatures, mixtures, and pressure conditions have been studied experimentally and computationally in the recent years (e.g., ${{\rm N}_2}$ [8–16], ${{\rm O}_2}$ [8,9,15,17–19], CO [8,9,20], ${{\rm CO}_2}$ [1,8,21], ${{\rm H}_2}$ [22–24], ${{\rm C}_2}{{\rm H}_2}$ [25,26], ${{\rm C}_2}{{\rm H}_6}$ [27], and ${{\rm C}_2}{{\rm H}_6}$ [28]).
A. Time-Domain Raman Linewidth Determination
According to the isolated line approximation for the pressure broadened regime, the Raman linewidth of a single spectral line is inversely proportional to its coherence dephasing rate as shown in Eq. (1), where ${\Gamma _{N^{\prime\prime}}}$ is the full width at half-maximum S-branch Raman linewidth, $c$ is the speed of light, and ${\tau _{{\rm CARS},\:N^{^{\prime\prime}}}}$ is the coherence decay time constant [12,29]:
For oxygen, the coherence dephasing has to be described by a model function that takes spin-rotation coupling into account. Due to spin-rotation coupling, rotational energy levels are split into three [17,30–34]. Coherent interaction of these states leads to coherence beating in the time domain [10,18,19,35]. The six transitions between these states are grouped into three $^S{\rm S}$, two $^S{\rm R}$, and one $^S{\rm Q}$-branch transition named after the scheme $^{\Delta N}\Delta J$ [17]. For a schematic depiction of the energetics of the six transitions and a spectrum with resolved $^S{\rm R}$-branch lines, the reader is referred to Fig. 1 in [17]. Spectra of the resolved SS-branch can be found in [36]. For the description of the coherence beating in the time regime up to 1.4 ns with a ps probe beam, inclusion of the $^S{\rm S}$-branch lines is sufficient. The contributions of the $^S{\rm Q}$-branch to the $N^{\prime\prime} = {1}$ and 3 line’s total intensity are 1.7% and 0.1% and decay with ${1/}{N^3}$ [17,32]. Therefore, the $^S{\rm Q}$-branch can be neglected. The $^S{\rm R}$-branch contribution is stronger (15%/8.6% for the ${{\rm S}_ -}/{{\rm S}_ +}$ lines for $N^{\prime\prime} = {1}$) and decays with ${1/}N$ [17,32]; It can, therefore, only be neglected for $N^{\prime\prime} \gt {5}$. However, its large spacing from the combined $^S{\rm S}{/}{^S\rm Q}$-branch of ${\sim}{1.98}\;{{\rm cm}^{- 1}}$ leads to a coherence beating period of about 16.9 ps, as was shown by Miller et al. using a fs/ps hybrid coherent anti-Stokes Raman spectroscopy (CARS) setup [10]. This is well below the temporal resolution of the ps-RCARS setup used in this setup of about 100 ps. Therefore the $^S{\rm R}$-branch beating can be neglected also for small $N^{\prime\prime}$.
Fig. 1. Exemplary converged fits of Eq. (2) to selected experimental S-branch coherence decays of pure oxygen at 1 bar for 295 K [$*$, $N^{\prime\prime} = {3}$ (black) and $N^{\prime\prime} = {25}$ (red)] and 1900 K [$\circ$, $N^{\prime\prime} = {11}$ (black) and $N^{\prime\prime} = {21}$ (red)], each offset by a factor of 10 for visibility.
The model function for the RCARS intensity over time of the three rotational lines ${I_{N^{\prime\prime}}}(t)$ is shown in Eq. (2), where ${I_0}$ is a scaling factor, ${a_{N^{\prime\prime},J^{\prime\prime}}}$ are amplitudes of the three predominant $^S{\rm S}$-branch coherent beats with frequencies ${\Omega _{J^{\prime\prime}\:}}(N^{\prime\prime}) = [{F_{J^{\prime\prime}}}(N^{\prime\prime}) - {F_{J^{\prime \prime} - 2}}(N^{\prime \prime} - {2})]{\rm /}\hbar$, with the rotational terms ${F_{J^{\prime\prime}}}(N^{\prime\prime})$ including spin-rotation splitting [35], which were calculated from the molecular constants in [37], and $G$ is a Gaussian function, which takes the temporal width ${\sigma _{{\rm LASER}}}$ of the laser into account (${\approx} 150\,\,{\rm ps}$), which is obtained from the width of the non-resonant background between the rotational lines. The starting point for the fit, before which delay positions are excluded from the fit, was set to be twice the temporal width of the non-resonant profile. The time of maximum beam overlap ${t_0}$ was set to 0 ps. The three amplitudes ${a_{N^{\prime\prime},J^{\prime\prime}}}$, ${I_0}$, and the decay time constant ${\tau _{{\rm CARS},\:N^{\prime\prime}}}$ were allowed to vary freely:
The fitting of this function to the ${{\rm O}_2}$ S-branch dephasing at 1 bar yielded a large scattering of adjacent Raman linewidth that is unfavorable for application of linewidths for diagnostics [9]. Furthermore, the fitting of fast decays (characteristic for low temperatures at constant pressure due to higher gas density and higher collision rate) and low signal intensities (high temperatures and mixtures) results in bad fits of the late decay (${{\gt}800}\,\,{\rm ps}$). Four exemplary fits of Eq. (2) to the experimental time traces of oxygen are shown in Fig. 1. Even though the fits converge despite tight convergence criteria, the late decay is visually not represented well by the fits. The reason is because of the exponential nature of the decay, yielding intensity differences between early and late decay of several orders of magnitude. Therefore the early data points are weighted strongly, whereas a deviation in the late decay still allows for convergence of the fit. A proper representation of the late fit is essential in order to describe decoherence and thereby the linewidth correctly. In this paper, two methods are compared to the unaltered function [Eq. (2)]: The first method is to take the logarithmic function of the data and fitting function. The second method is the introduction of weights in the fitting procedure, without altering the function itself.
2. EXPERIMENT
A. Picosecond Dual-Broadband Pure RCARS Setup
The experimental setup for the ps time resolved RCARS measurements has already been described in detail elsewhere [16,19] and will only be summarized briefly in the following. The setup is shown in Fig. 2. The output of a frequency doubled neodymium-doped yttrium aluminum garnet (Nd:YAG) laser (${\lambda _{{\rm center}}} = {532}\;{\rm nm}$, $E = {125}\;{\rm mJ}$, $f = {10}\;{\rm Hz}$, $\Delta {t_{{\rm FWHM}}} = {100}\;{\rm ps}$) is split by a beamsplitter (${{\rm BS}_1}$). One fraction is used as the narrowband probe beam, which is guided over a mechanical delay line (${{\rm DL}_1}$) to temporally shift the probe beam with respect to pump and Stokes. Proper alignment of the DL is assured by reflecting the tuned down beam over a linear range of 7 m onto millimeter paper and visually inspecting for a displacement while delaying the beam. The second portion pumps a broadband dye laser using 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM) dye in ethanol (${\lambda _{{\rm center}}} = {630}\;{\rm nm}$, $E = {8}\;{\rm mJ}$, $\Delta {{\tilde v}_{{\rm FWHM}}} = {469}\;{{\rm cm}^{- 1}}$, $\Delta {t_{{\rm FWHM}}} = {100}\;{\rm ps}$), which provides pump and Stokes beams. Linear, vertical polarization of both beams is ensured by Glan–Taylor polarizers (${{\rm GP}_{1,2}}$). Pump, Stokes, and probe beams are overlapped inside an electrically heated furnace in the BOXCARS phase matching geometry [38,39]. The generated signal is separated from the excitation by a series of dichroic mirrors, focused into a spectrometer, spectrally dispersed and detected by a CCD camera. In order to extend the dynamic range of the 16-bit camera chip, neutral density (ND) filters with a total optical density (OD) between 1 and 4 were used, which exhibit a homogenous OD in the spectral window of the experiment. The OD of the filters was altered in steps of one depending on signal level along the delay measurement. The exact correction factor for the ND filter at delay positions of filter change was attained by recording 500 spectra with initial and final OD, averaging over the whole spectral range, and calculating the ratio. At each delay position, a total of 500 spectra are recorded and averaged. To obtain the intensity for individual spectral lines, the intensity is integrated over the entire peak (about $\pm {5}\;{\rm pixels/ \pm}{1.15}\;{{\rm cm}^{- 1}}$ form the maximum). The earliest delay for fitting is determined by doubling the width of the non-resonant profile for each individual measurement. At late decays, intensities of less than five signal counts are excluded, which amounts for a maximum of 10 excluded delay positions at low intensity measurement conditions.
Fig. 2. Experimental setup for the picosecond time resolved pure rotational CARS measurements. BBDL, broadband dye laser; BD, beam dump; ${{\rm BS}_{1,2}}$, beam splitter; ${{\rm DL}_{1,2}}$, delay lines; ${{\rm DM}_{1,2}}$, dichroic mirror; ${{\rm GP}_{1,2}}$, Glan–Taylor polarizer; ${{\rm L}_{1,2,3}}$, lenses; $\lambda /{2}$, half-wave plate; M, mirror.
B. Evaluation Methods
The fitting utilizes a non-linear least-squares trust-region algorithm. Two modifications to the fitting method will be discussed and compared to the unaltered function, which aim at increasing the relative weight of the late decay compared to the early decay. The first method is a “linearization.” The experimental data and Eq. (2) are logarithmized. Thereby, the intensities of early and late decay are within the same order of magnitude, which is expected to yield a better fit of the late decay. The second approach is the introduction of weights for each individual data point. The weight is calculated as the inverse of the absolute value of the intensity. Thereby, low intensity data, which dominate the late decay, are weighted strongly compared to the early decay.
In order to validate the different fitting routines and evaluate precision and accuracy, the “real” linewidths have to be known, which is not the case for experimental oxygen time traces. Therefore, the three models, the unaltered model, linearization, and the weighted method were fitted to 10,000 simulated time traces, where the rotational state ($3\le {N^{\prime\prime}}\le 49$), and thereby the quantum beating period ${\Omega _{J^{\prime\prime}}}$, the decay time constant (${90}\;{\rm ps}\; \le \;{\tau _{N^{\prime\prime},{\rm coll}}}\; \le \;{250}\;{\rm ps}$), and the maximum intensity at $t = {0}\;({{10}^2}\; \le \;{I_{N^{\prime\prime}}}\; \le \;{{10}^5})$, were randomly combined. The mentioned ranges correspond to the range of the respective parameters found in our experiment with an emphasis on small intensities and fast decays, where visually bad fitting of the late decay mainly occurred. The amplitudes ${a_{N^{\prime\prime}}}$ for the three transitions were set to one based on the assumption by Courtney et al. [18]. Subsequently, noise was added ($-100\le I_{\rm noise} \le 100$). The fits were evaluated with regard to the relative deviation between preset ${\tau _{N^{\prime\prime},\rm coll}}$ and the fitted decay constant. The mean of this deviation over all 10,000 traces is used as a measure for precision. Also shown is the accumulated relative occurrence of the absolute value of the deviation, which contains information on the width of the distribution. For the discussion, the 95% threshold is used to quantify the width. It should be noted that fitting of the simulated traces was not optimized for each individual fit, by varying the initial guess values, discarding outlier data points, etc. Therefore, the precision of our manually and individually optimized experimental linewidths is expected to be better than the statistical evaluation indicates.
3. RESULTS AND DISCUSSION
A. Comparison of Modifications to the Model Functions
Figure 3 shows an exemplary simulated time trace with a collisional dephasing constant of 98.1 ps with fit traces of the three methods to the data. The simulated time trace matches the appearance of the experimental time trace. It can be easily seen that the original (unaltered) function fails to represent the late decay properly, which results in a deviation of 9.5 ps (–9.7%) from the preset linewidth. The fit of the linearized function reproduces the second quantum beat well, however, the curve is too low at decay values ${\lt}{400}\;{\rm ps}$. Therefore, the time constant is too high by 12.2 ps ($+12.4\%$). The weighted fit represents the early and late decay well and yields a decay constant that is too low by 2.4 ps ($-2.4\%$). The statistical evaluation confirms the results of the discussed exemplary curve.
Fig. 3. Simulated RCARS intensity traces of the oxygen $N^{\prime\prime} = {33}$ line (blue dots) with fits of the unaltered (green line), linearized (magenta line), and weighted (black line) models. The decay constant was randomly assigned as 98.1 ps, and numbers in brackets correspond to the relative deviation of preset and fitted decay constant. Scatter in the low intensity regime results from the noise added in the simulation process.
The results are summarized in Fig. 4 in form of a frequency distribution of the relative deviations. The plot shows lines representing the relative frequency of occurrence as a function of relative deviation from the preset decay constant. This plot indicates systematic deviations from the ideal 0% deviation and potential skewness of the distribution. The dots indicate the accumulated absolute value of the relative occurrence. This gives information on scatter around the 0% value and is evaluated with regard to 95% accumulated relative occurrence. The decay constants from the unaltered model are systematically too large by 3.8% (95% occurrence interval: 19%). The systematic deviation arises from a skewing of the distribution towards too large time constants, as can be seen in the figure. The linearized function yields an average systematic deviation of ${+}{24.0}\%$. Furthermore, the distribution is very wide, as is also expressed in the 95% occurrence interval of 84%, which likely results from bad representation of the early decay. Therefore, the linearization is discarded from further application. The weighted model results in the best accuracy of the compared models of –0.1%. The distribution of relative deviation shows a skewing towards smaller decay constants but a broader wing towards too high constants, resulting in a 95% occurrence interval of 23% that is not as good as for the unaltered model. This will likely improve for the experimental evaluation, as each spectral line is optimized individually. Overall, the weighted model clearly yields the best results, as it reproduces the preset linewidths well and allows fitting of the late decay even for low intensity and fast decays.
Fig. 4. Plot of relative frequency (lines) as a function of relative deviation between preset and fitted decay constant of 10,000 simulated traces. Dots represent accumulated relative occurrence of the absolute value of relative deviation. Both plots with the color code as in Fig. 3.
B. $\rm O_{2}$ S-Branch Raman Linewidths Revisited
Figure 5(a) shows experimental zero-delay spectra of oxygen at 295 K and 1400 K. It shows the spectral range under consideration as well as depicting spectral resolution and trends with increasing temperature. At 1400 K, the maximum of the envelope of all spectral lines lies at higher Raman shift. The intensity is lower by about three orders of magnitude. Furthermore, hot bands become visible to the low wavenumber flank of higher rotational quantum number peaks, which form a common envelope with the ground state peak at low quantum numbers. The hot band’s intensity accounts for about 5% of the total intensity, as calculated from $N^{\prime\prime} = {33}$, 35, and 37 lines. The decay behavior of the hot band is expected to be very similar to the ground state, because the rotational constant of $v = {1}$ is only slightly smaller than $v = {0}$ (${B_0} = {1.438}\;{{\rm cm}^{- 1}}$, ${B_1} = {1.422}\;{{\rm cm}^{- 1}}$) [40]. Based on energy gap considerations, similar energy spacing is expected to result in similar dephasing behavior [41].
Fig. 5. (a) Normalized zero-delay spectra of oxygen at 295 K (black) and 1400 K (red) averaged over 500 single shots, with selected rotational quantum numbers indicated in blue. It should be noted that the maximum intensity at 1400 K is smaller by about three orders of magnitude. (b) ${{\rm O}_2}$ spectrum at 1400 K at a delay of 1239 ps showing good S/N ratio at large decays. The intensity plateau for lines below ${100}\;{{\rm cm}^{- 1}}$ ($N^{\prime\prime} \lt {19}$) likely arises from the rising slope of the quantum beating, whereas the intensity of high $N^{\prime\prime}$ lines are decreasing in slope.
The late decay spectrum in Fig. 5(b) shows signal intensities well above the noise level at late decay. It should be noted that intensities for decay traces are typically averaged over 8–12 pixels. The spectrum shows an intensity plateau below ${100}\;{{\rm cm}^{- 1}}$ deviating from Boltzmann’s distribution. A comparable effect can also be observed in the late decay spectra recorded by Courtney et al. [18] and likely arises from rising intensity slope below ${100}\;{{\rm cm}^{- 1}}$ ($N^{\prime\prime} \lt {19}$) numbers and decreasing slope above.
A comparison of the weighted fit to the unaltered fit for two experimental time traces from Fig. 1 is shown in Fig. 6. It can be seen that the weighted fit reproduces the late decay properly in both cases, yielding strongly different linewidth results, accounting for factors of 1.6 and 1.7, respectively, compared to the unaltered model, which appears to be entirely due to the introduction of weights.
Fig. 6. Trace of experimental RCARS intensity as a function of probe delay (blue) for the oxygen $N^{\prime\prime} = {25}$ line at 295 K ($\circ$, offset by a factor of 10 for visibility) and the $N^{\prime\prime} = {21}$ line at 1900 K ($*$). Lines represent best fits of Eq. (2) to the data with the unaltered (green) and weighted methods (black), respectively, including resulting decay time constants. The respective linewidths (without accounting for the actual pressure during the experiment) of the unaltered and weighted fits are ${0.030}\;{{\rm cm}^{- 1}}$ and ${0.049}\;{{\rm cm}^{- 1}}$ for 295 K and ${0.042}\;{{\rm cm}^{- 1}}$ and ${0.025}\;{{\rm cm}^{- 1}}$ for 1900 K, respectively.
The resulting experimental S-branch Raman linewidths for pure oxygen, applying the weighted model, are summarized in Table 1 and visualized and compared to literature linewidths in Fig. 7. The weighted method yields Raman linewidths of similar magnitude as previously published, also reproducing the trends with increasing temperature and rotational quantum numbers observed [19]. As can be seen by applying the weighted method, the linewidths value at 295 K changes by about a factor of two. This difference reduces with increasing temperature, exhibiting that the new model leads to drastic changes under the conditions that have been difficult to fit properly with the unaltered model. We attribute this difference entirely to the introduction of weights, bearing in mind that change is not only the decay time constant but the amplitudes ${a_{N^{\prime\prime},J^{\prime\prime}}}$ too. The uncertainty of the linewidths also decreases with temperature. This is due to the slower dephasing and the resulting emergence of a clearly discernable second quantum beat. Furthermore, the scattering of neighboring rotational lines is strongly reduced compared to our recent study. This can be easily seen in Fig. 7 by comparing the linewidths data of $N^{\prime\prime} = {17}$ to 23 at 1400 K ($\star$), for instance, which exhibit a zig-zag shape in the previous study. The weighted method yields a steady trend, which is in line with the generally expected behavior of Raman linewidths from theoretical descriptions like the modified energy gap (MEG) law for instance [44]. This enhances the data quality, which is of great importance for all groups using RCARS as a combustion diagnostic tool.
Table 1. S-Branch Raman Self-Broadened Linewidths of Oxygen Applying the Weighted Modela
Fig. 7. Comparison of oxygen S-branch Raman linewidths for 295 K ($\circ$) and 1400 K ($\square$) from the weighted method (black), with the unaltered model from Hölzer et al. [19] (red $\times$ and $\star$, respectively) and with literature: Bérard et al. [17], Courtney et al. [18], Jammu et al. [8], Looney et al. [9], Miller et al. [10], Millot et al. [42], and Ouazzany et al. [43]. Error bars represent the standard deviation of the fit. Linewidths from the present experiments are connected by splines, and only every third error bar is shown for visibility. The full set of standard deviations can be found in Table 1.
C. Influence on Thermometry
The time-domain S-branch Raman linewidths have been implemented into our homebuilt software package for CARS temperature evaluation [45]. The temperature evaluation has been exemplarily done for a zero-delay spectrum of pure oxygen at 870 K, recorded in the furnace at stationary conditions, and averaged over 500 laser shots. The resulting temperature is compared to the so far used ${{\rm O}_2} {-} {{\rm O}_2}$ Q-branch Raman linewidths, which are calculated using the MEG law and parameters determined by Millot et al. [42]. The resulting temperatures are 895 K and 874 K for Q- and S-branches, respectively. Because only one averaged spectrum was fitted, no standard deviation can be given. The temperature with newly obtained linewidths is 4 K too high compared to the set and calibrated temperature in the furnace. The Q-branch linewidths yield an even larger deviation of 25 K though. The same trend between S- and Q-branch linewidths was also shown for ${{\rm N}_2} {-} {{\rm N}_2}$ linewidths by Kliewer et al. [12], who presented an RCARS temperature deviation of about 15 K at 800 K thermocouple temperature. Figure 8 shows a comparison of the experimental spectrum and the theoretical spectra calculated with the S- and Q-branch linewidths. The theoretical spectrum using S-branch linewidths is calculated for the best-fit temperature. The Q-branch spectrum has been calculated using the same temperature, to highlight differences between the linewidth sets at constant temperatures. It can be seen that the “Q-branch spectrum” exhibits higher intensity at the lower wavenumber flank of the intensity envelope but lower intensity at the higher wavenumber flank. This is systematic for all temperatures. By this, the “Q-branch spectrum” appears to be colder than the “S-branch spectrum” for the same temperature. Therefore, the temperature evaluation yields the higher temperature of 895 K compared to 874 K for the “S-branch spectrum.”
Fig. 8. Averaged experimental RCARS spectrum of pure ${{\rm O}_2}$ at 870 K thermocouple temperature (blue) and best-fit theoretical spectrum using S-branch Raman linewidths at 874 K (red). The theoretical spectrum with Q-branch linewidths (green) was calculated using the same optimized temperature as the “S-branch spectrum” to highlight differences between the two linewidths sets at the same temperature. Furthermore, the difference between the two theoretical S- and Q-branch linewidths-based spectra is shown (black) in the lower panel.
Fig. 9. Trace of experimental RCARS intensity as a function of probe delay for oxygen in air (blue). $N^{\prime\prime} = {21}$ line at 295 K ($\circ$, offset by a factor of five for visibility) and the $N^{\prime\prime} = {47}$ line at 1650 K ($*$). Lines represent best fits of Eq. (2) to the data with the unaltered (green) and weighted methods (black), respectively, including resulting decay time constants. The respective linewidths (without accounting for the actual pressure during the experiment) of the unaltered and weighted fits are ${0.035}\;{{\rm cm}^{- 1}}$ and ${0.041}\;{{\rm cm}^{- 1}}$ for 295 K and ${0.031}\;{{\rm cm}^{- 1}}$ and ${0.014}\;{{\rm cm}^{- 1}}$ for 1650 K, respectively.
D. ${{\rm O}_2}{-}{{\rm N}_2}$ S-Branch Raman Linewidths in Air
The influence of different collision partners on the Raman linewidths in binary mixtures is regularly considered using the linear relation given in Eq. (3), here written down for the ${{\rm O}_2}/{{\rm N}_2}$ mixture, where ${x_Y}$ is the mole fraction of species Y [20,42]. This model has been successfully applied, for example, to the ${{\rm O}_2} {-} {{\rm N}_2}$ Q-branch [42] as well as recently to the S-branch of various mixtures of CO, ${{\rm CO}_2}$, ${{\rm N}_2}$, Ar, and ${{\rm C}_2}{{\rm H}_4}$ [1,20]:
The method for obtaining ${\Gamma _{N{{^{{\prime \prime},}}{\rm O}_{2}}{{- {\rm N}}_{2}}}}$ includes the determination of the Raman linewidths at different mole fractions and fitting the results with Eq. (3). The determination of a full temperature dependent set of rotational Raman linewidths for various mole fractions at different temperatures and the determination of ${\Gamma _{{N^{{\prime \prime}},{O}_{2}}{{- {\rm N}}_{2}}}}$ from Eq. (3) will be part of a future work. This work aims at demonstrating the applicability of the weighted fitting method even in the low concentration regime and thereby the small intensity.
The experimental time traces of oxygen in air at 295 K and 1650 K with best fits of the unaltered and the weighted method are displayed in Fig. 9. As was the case for pure oxygen, it can be seen that the fit curve of the weighted method yields better agreement with the experimental data, especially at large decays, demonstrating the applicability of this method for small concentrations. The resulting decay time constant changes strongly, underlining the importance of proper late decay fitting.
Table 2. Experimental S-Branch Raman Linewidths of Oxygen in Air Applying the Weighted Modela
Fig. 10. Comparison of the experimental time-domain S-branch Raman linewidths of pure ${{\rm O}_2}$ (black) and ${{\rm O}_2}$ in air (red) for 295 K (${*}$ and $\circ$, respectively) and 1400 K (${\times}$ and $\square$, respectively). Additionally, experimental (magenta $+$) and calculated (cyan $\nabla$ and $\star$, respectively) literature Q-branch Raman linewidths of ${\rm O}_{2}$ in air for 295 K and 1315 K from Millot et al. [42] are displayed. Error bars represent the standard deviation of the fit. For visibility, a selection of error bars is shown. The full set of standard deviations can be found in Table 2.
The resulting S-branch Raman linewidths are summarized in Table 2 and exemplary visualized and compared to literature data in Fig. 10. For all evaluated temperatures, the predominantly nitrogen broadened linewidths is smaller compared to the self-broadened linewidths, retaining the principle trends with temperature and quantum number found for self-broadening. This is in accord with observations for other gases at room temperature in literature for CO broadened by ${{\rm N}_2}$ and ${{\rm CO}_2}$ [20], ${{\rm N}_2}$ broadened by ${{\rm O}_2}$ [16], and ${{\rm CO}_2}$ broadened by ${{\rm N}_2}$, ${{\rm O}_2}$, and Ar [1]. The opposite effect is found when compared to both experimental and calculated literature Q-branch linewidths, which are larger than the newly obtained S-branch linewidths [42] for both foreign- and self-broadened cases. Furthermore, the slope of linewidths with increasing $N^{\prime\prime}$ is different. At elevated temperatures, this apparent difference may be partially due to the temperature difference of 85 K for the Q- and S-branch linewidths; nevertheless, at 295 K, the deviation is systematic and may hint at a more fundamental difference in dephasing behavior between Q- and S-branches.
4. SUMMARY
Two modifications of the fitting method for the determination of S-branch Raman linewidths of oxygen have been compared in a statistical evaluation to improve applicability of the model for rotational Raman lines with fast coherence decays (e.g., small temperatures) as well as small intensities (e.g., at high temperatures and in mixtures). Weighting each data point by the inverse of its intensity yielded the best results and leads to a significant improvement in accuracy of rotational S-branch Raman linewidths data quality for pure oxygen between 295 K and 1900 K. At 295 K, a twofold difference between the unaltered and weighted models was found. Generally the evaluation revealed that this method is reliably applicable for small decay constants (e.g., at low temperatures at around 1 bar) and small intensities. The latter is demonstrated by the determination of ${{\rm O}_2} {-} {{\rm N}_2}$ Raman linewidths in air over the same temperature range.
Funding
Deutsche Forschungsgemeinschaft (SE 804/6-1).
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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