1. INTRODUCTION

Capturing instantaneous snapshots of the internal modes of energy transfer in nonequilibrium plasma environments requires high temporal resolution (nanoseconds) and high spatial precision (micrometers) while covering a wide range of rotational and vibrational temperatures (300–5000 K). While monitoring the excited state is possible with emission spectroscopy, monitoring the ground state energy distributions requires laser-based sensing. We address these challenges through the development of dual-pump hybrid femtosecond/picosecond (fs/ps) coherent anti-Stokes Raman scattering (CARS) for simultaneous measurement of nonthermalized vibrational and rotational temperatures in a diatomic gas (${\mathrm{N}}_2$) at atmospheric pressure. Hybrid fs/ps CARS was first demonstrated for detecting chemical species [1,2] and later utilized for gas-phase combustion thermometry [3]. The dual-pump technique used here is sensitive to ground state vibrational and rotational populations [4], while the short fs excitation and ps detection allows measurements that are relatively insensitive to the collisional environment [5]. This work develops the capability of dual-pump hybrid CARS to monitor internal energy distributions in nonequilibrium environments and evaluates the ability of the theoretical model to fit vibrational and rotational temperatures with high accuracy and precision over a range of conditions. Understanding how internal vibrational and rotational modes transfer energy in nonequilibrium environments is critical for identification of kinetic processes, plasma kinetic model development and verification, and process control in a wide range of emerging applications. Measurements are presented in a quasi-steady-state, atmospheric-pressure dielectric barrier discharge (DBD) with a significant degree of vibrational/rotational nonequilibrium. This plasma exhibits many of the challenges present in a variety of nonequilibrium systems including temporal and spatial variations, high electric field strength, and coupled kinetic and transport mechanisms. This study lays the foundation for direct, spatially concise measurement of the vibrational and rotational distributions essential for determination of energy cascade processes in nonthermal plasma environments.

Prior measurements of energy distributions in transient plasma environments have relied on a variety of nonintrusive optical and laser spectroscopic techniques. Thomson scattering can be used to measure electron density and electron temperature [6], and filtered Rayleigh scattering has been used to study gas temperature [7,8], but these scattering techniques do not directly measure internal energy distributions. A number of studies have employed laser absorption and spectroscopic emission [9–11] to infer rotational and vibrational temperatures. However, these approaches are limited in both temporal and spatial resolution. Additionally, because emission is a measure of the populations in excited electronic states, not the ${\mathrm{N}}_2(\mathrm{X})$ ground state, it may exhibit bias when inferring ground state vibrational temperatures. To avoid relying on the population of excited electronic states, ns and ps CARS have been used to study vibrational and rotational energy distributions in a variety of nonequilibrium environments [12–15]. CARS is typically implemented as a point-based measurement technique; measurements of spatial distributions can be performed with high temporal precision in nonuniform environments and in plasmas with a wide range of spatial scales. Some studies have used rovibrational CARS to measure rotational and vibrational distributions simultaneously [16,17]; however, it is difficult to capture the full range of vibrational bands while resolving rotational features. Pure-rotational broadband CARS has the advantage of higher spectral resolution, but lacks the capability of measuring vibrational energy distributions.

In the current work, we demonstrate hybrid fs/ps, dual-pump CARS using a mode-locked, amplified Ti:sapphire laser for simultaneous, single-shot detection of rovibrational and pure-rotational transitions in highly nonequilibrium environments. The transform-limited nature and high peak power of the fs source generate strong Raman coherences, while the ps probe pulse is used to achieve spectral resolution. This enables spatial interrogation of energy distributions with ps temporal resolution and excellent shot-to-shot precision at kilohertz rates while minimizing nonresonant background and the effects of molecular collisions. The dual-pump approach and two-temperature model in this work resolve rovibrational and pure-rotational transitions simultaneously for quantifying the significant degree of vibrational/rotational nonequilibrium in DBD plasmas, and can be extended to other nonequilibrium environments.

2. THEORY

Hybrid fs/ps CARS is a nonlinear, four-wave-mixing process that utilizes broadband, fs pump and Stokes pulses to excite molecular vibrational and/or rotational transitions in a single laser shot. The pump and Stokes wavelengths are selected such that their difference frequency matches the Raman-active transitions of interest. Here, a dual-pump technique [4] was employed to simultaneously excite rovibrational ($Q$-branch, $\mathrm{\Delta }v=+1$, $\mathrm{\Delta }J=0$) and pure-rotational ($S$-branch, $\mathrm{\Delta }v=0$, $\mathrm{\Delta }J=+2$) Raman transitions of molecular nitrogen. A third, frequency-narrowed ps pulse was then used to probe the molecular response at a time delay from the initial excitation. In the current work, the time delay of the ps pulse was selected to avoid interference from nonresonant four-wave-mixing contributions and minimize sensitivity to collisions. Frequency and timing diagrams for the pump, Stokes, probe, and CARS signal pulses are shown in Fig. 1. The generated CARS signal is dependent upon the population difference between two energy levels—two vibrational levels when considering the $Q$-branch and two rotational levels when considering the $S$-branch.

 

Fig. 1. (a) Frequency and (b) timing diagrams for dual-pump fs/ps vibrational/rotational CARS.

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The time- and frequency-resolved hybrid CARS model used is based on the theory described in detail elsewhere [18,19] and was extended here to allow interpretation of the vibrational and rotational spectra in a nonequilibrium environment. In particular, destructive and constructive interference between rotational transitions originating from multiple highly populated vibrational states alter the rotational spectrum and necessitate measurement of both rotational and vibrational distributions under nonequilibrium conditions. To summarize this model, the CARS intensity ($I_{\mathrm{CARS}}$) can be written in the frequency domain as

This modified Boltzmann model was used to evaluate the impact of nonequilibrium on both $S$- and $Q$-branch spectra, as shown in Fig. 2. In Fig. 2(a), an $S$-branch spectrum was modeled at a probe delay of 7.5 ps assuming an equilibrium temperature of 500 K (green solid line) and nonequilibrium temperatures $T_{\mathrm{rot}}=500\mathrm{K}$ and $T_{\mathrm{vib}}=3500\mathrm{K}$ (black dashed line). It is apparent that if only a measurement of the $S$-branch existed and equilibrium was assumed, the inferred temperature would be much lower than the actual rotational temperature. Figure 2(c) shows a stick diagram of the intensities and frequencies of each Raman transition for the nonequilibrium condition, with transitions originating from different vibrational levels shown as different colors. For ${\mathrm{N}}_2$, these rotational transitions from different vibrational levels are closely spaced, but are not perfectly overlapping.

 

Fig. 2. Simulated (a) $S$-branch and (b) $Q$-branch spectra showing an equilibrium temperature (green solid line) and a condition where rotational and vibrational temperatures differ (black dashed line). Stick diagrams of (c) pure-rotational and (d) rovibrational Raman transition frequencies and intensities for the nonequilibrium case.

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The rotational energy levels of a diatomic molecule can be calculated using Eq. (4), the typical polynomial expansion for a vibrating rotor, where the rotational constant, $B_v$, and centrifugal distortion, $D_v$, are shown in Eqs. (5) and (6) [20]:

From Eq. (4), the frequency of an $S$-branch transition between states $J$ and $J+2$ can then be written as

From this, it is apparent that $S$-branch transitions involving the same rotational states but different vibrational levels have a finite frequency difference, and this difference increases with increasing rotational quantum number. Due to the bandwidth of the ps probe, multiple transitions from different vibrational levels are sampled together. This results in destructive and constructive interference between transitions as they dephase and subsequently rephase in time, shown in Fig. 3(b). Because the transition spacing is greater at higher rotational levels, destructive interference occurs at earlier probe delays at high rotational quantum number, resulting in lower apparent temperatures as seen in Fig. 2(a). This effect is particularly evident at high vibrational temperatures where multiple vibrational states are significantly populated. The result is an inability to differentiate between a lower rotational temperature in equilibrium versus a higher rotational temperature and an elevated vibrational temperature in a nonequilibrium environment.

 

Fig. 3. Simulated $S$-branch time-response for various probe delays in (a) equilibrium at $T_{\mathrm{eq}}=500\mathrm{K}$ and (b) nonequilibrium at $T_{\mathrm{rot}}=500\mathrm{K}$ and $T_{\mathrm{vib}}=3500\mathrm{K}$.

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The effect of nonequilibrium on the $Q$-branch is also considered in Fig. 2(b) with spectra modeled at an equilibrium temperature of 3500 K (green solid line) and at the same nonequilibrium condition as Fig. 2(a) with $T_{\mathrm{rot}}=500\mathrm{K}$ and $T_{\mathrm{vib}}=3500\mathrm{K}$ (black dashed line). As shown for the $S$-branch spectra, equilibrium and nonequilibrium cases exhibit distinct differences. The equilibrium condition shows rephasing at higher rotational levels for this probe delay (7.5 ps) that is no longer evident when the rotational temperature is lowered to 500 K. This is due to the lack of population in the high rotational states, as demonstrated in the stick diagram modeled at the nonequilibrium condition in Fig. 2(d). Additionally, there are small differences in intensity of the excited vibrational bandheads, for example, at $2300{\mathrm{cm}}^{-1}$. This is due to the absence of population in high rotational states of the $Q_0$ band ($Q_v$ used to represent the $v+1\leftarrow v$ transition) that would overlap and interfere with the low rotational states in the $Q_1$ band at high equilibrium temperatures.

Overall, the spectral differences in both branches at this early probe delay are significant. Particularly, the inability to differentiate between different rotational temperatures at equilibrium and nonequilibrium in the $S$-branch shows the necessity of measuring both $S$- and $Q$-branch spectra simultaneously to accurately determine rotational and vibrational temperatures within nonequilibrium environments.

3. EXPERIMENTAL SETUP

The optical layout for the dual-pump fs/ps CARS system is shown in Fig. 4(a). The technique used here to measure nonequilibrium vibrational and rotational energy distributions was similar to that previously published for multispecies and temperature measurements in equilibrium environments [4]. The laser source was a 1-kHz-rate, regeneratively amplified, 100 fs Ti:sapphire laser (Solstice, Spectra-Physics) delivering 2.5 mJ centered around 798 nm. This pulse was split into multiple beams, with 1 mJ used to pump an optical parametric amplifier (TOPAS, Spectra-Physics) that is frequency-doubled to produce the rovibrational CARS pump pulse centered at 674 nm (${\omega }_{p1}$ in Fig. 1). The residual energy from the OPA at 798 nm was split to form the pure-rotational CARS pump pulse (${\omega }_{p2}$) and the Stokes pulse (${\omega }_s$) common to both CARS processes. The remaining 1.5 mJ from the fs source at 798 nm was pulse-shaped into a near-Gaussian, narrowband, 6 ps probe pulse (${\tau }_{\text{probe}}$) using a second-harmonic bandwidth compressor (SHBC) [21]. These four pulses were focused at the CARS probe volume using a folded-BOXCARS phase matching configuration and a 150 mm focal length combining lens to maximize spatial resolution. The corresponding spectra were directed to a 0.303 m spectrometer (Shamrock SR-303, Andor) and resolved using a 1200 line/mm grating. The rovibrational CARS signal was collected using an electron-multiplied charge-coupled device (CCD) camera (DU-970 P-UVB, Andor Newton). The spectrometer was modified by the addition of a mirror after the grating to direct the pure-rotational CARS signal onto a second CCD camera (DU-940, Andor Newton). To ensure overlap of the resulting rovibrational and pure-rotational CARS probe volumes, a 100-μm-diameter helium jet was used to minimize each resonant ${\mathrm{N}}_2$ CARS signal, and both pump pulses were adjusted to achieve probe volume overlap. Along the direction of beam propagation, the probe volumes were approximately Gaussian in shape with a full width at half-maximum of around 650 μm. This method of probe volume localization, although conservative due to the helium expansion at the jet exit, is optimal because it does not require any modification of pulse energy or beam size prior to measurement, as is often required when using a glass slide or other solid/liquid sample placed at the focal point. Improving the spatial resolution further could be achieved by using a shorter focal length combining lens, although reducing the spatial overlap could decrease the single-shot accuracy of the measurement system.

 

Fig. 4. (a) Optical layout for dual-pump fs/ps CARS with the DBD at the probe volume. 1/2 WP, half-waveplate; TFP, thin-film polarizer; BS, beam splitter; G, grating; SHG, second-harmonic generation crystal; DBS, dichroic beam splitter; SHBC, second-harmonic bandwidth-compressor; OPA, optical parametric amplifier. (b) The $\mathrm{He}/{\mathrm{N}}_2$ dielectric barrier discharge is shown.

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The DBD, shown in Fig. 4(b), was generated using a 13.56 MHz radio frequency power source (T&C Power Conversion, AG 1213 W) and automatic impedance tuner (T&C Power Conversion, AIT-600-10 HNVU3) rated to 1000 W. The electrodes consisted of a stainless-steel plate (17.5 mm by 14 mm unobstructed portion) and a tungsten rod with a flat end and diameter of 2.38 mm. The dielectric, acting to limit current spikes and preventing the plasma from transitioning to a thermal arc [22], consisted of a 130-μm-thick quartz slide covering the stainless-steel plate. The tungsten rod was enclosed within a cylindrical ceramic shroud, and mixtures of ${\mathrm{N}}_2$ and He flowed within the annulus. The gap between the top electrode and the quartz slide varied between 670 and 760 μm, and the discharge was operated at atmospheric pressure. A heat sink was attached to the bottom electrode to prevent damage to the quartz plate due to overheating. The voltage applied between the two electrodes was monitored using a high voltage probe (Tektronix P6015A), and the global reduced electric field (E/N) calculated using the measured voltage was within the range of 30–40 Td (${10}^{-21}{\mathrm{Vm}}^2$) for the experimental results presented. Additionally, the size and structure of the plasma were monitored through long-exposure imaging (0.5 ms) of the plasma emission at all experimental conditions using a CCD (Chameleon, Point Grey).

To determine the vibrational and rotational temperatures from experimental results, a differential evolutionary algorithm was used to minimize the normalized residuals between experimental and theoretical $S$- and $Q$-branch spectra simultaneously. Experimental parameters such as probe bandwidth, chirp, and time delay; pump and Stokes bandwidth, chirp, and peak wavelength; and the measurement instrument function are determined and input into the hybrid CARS model. Next, spectral libraries of $S$- and $Q$-branches were generated at various rotational and vibrational temperatures. Finally, the differential evolutionary algorithm was given a pair of experimental $S$- and $Q$-branch spectra, and a search of both spectral libraries produced the rotational and vibrational temperatures that yielded the best match between experimental and simulated spectra. The quality of fit was determined by minimizing the sum of the ${\ell }^2$-norm of the residuals for each spectral branch, normalized by the number of datapoints in each branch. This routine allows the two unknown temperatures, $T_{\mathrm{rot}}$ and $T_{\mathrm{vib}}$, to be determined from both the rovibrational and pure-rotational spectra simultaneously.

4. RESULTS

Experimental single-shot fs/ps CARS measurements from the center of the DBD plasma and best-fit theoretical spectra are shown in Fig. 5. For these data, the plasma was generated in a 13% ${\mathrm{N}}_2$ environment with a measured global electric field of 33 Td. Due to the high concentration of ${\mathrm{N}}_2$, the plasma was unsteady and the locations of the large filament structures were changing in time. A histogram of 400 single-shot measurements of vibrational temperature (red bars) is shown in Fig. 5(a), along with the more stable simultaneous rotational temperature measurements (blue bars). The two single-shot spectra shown in Figs. 5(b)–5(e) were chosen to highlight the wide range of shot-to-shot vibrational temperatures captured from the dynamic plasma environment at two instants, with corresponding rotational temperatures that are relatively invariant in time. Experimental results and best-fit simulated spectra are represented with green symbols and solid black lines, respectively. The best-fit temperatures for spectra shown in Figs. 5(b) and 5(c) were $T_{\mathrm{rot}}=380\mathrm{K}$ and $T_{\mathrm{vib}}=2580\mathrm{K}$, corresponding to the low end of vibrational temperatures in the dataset shown in Fig. 5(a). The next spectral pair in Figs. 5(d) and 5(e) correspond to a vibrational temperature that is almost 900 K higher ($T_{\mathrm{vib}}=3460\mathrm{K}$) but with a nearly identical rotational temperature of 390 K. The shift in the ground state population between low and high lying vibrational states is evident in the two spectra, as is the ability of the theoretical model to capture these populations and the corresponding vibrational temperatures at each instant. Because of the coupled effects of the vibrational distribution on the rotational spectra, these large fluctuations highlight the importance of simultaneous measurement of rotational and vibrational distributions for accurately modeling the fs/ps CARS spectra in an unsteady environment.

 

Fig. 5. (a) Histogram of 400 single-shot vibrational and rotational temperature measurements at the center of the DBD. Two single-shot, simultaneously measured $S$-branch and $Q$-branch spectra pairs are shown as green circles with best-fit simulations shown as solid black lines, where the corresponding temperatures are (b), (c) $T_{\mathrm{rot}}=380\mathrm{K}$ and $T_{\mathrm{vib}}=2580\mathrm{K}$ and (d), (e) $T_{\mathrm{rot}}=390\mathrm{K}$ and $T_{\mathrm{vib}}=3460\mathrm{K}$.

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Additionally, the rotational and vibrational energy distributions were measured at different spatial locations across the DBD plasma and afterglow, as shown in Fig. 6. The full extent of the plasma was explored by translating both electrodes horizontally relative to the CARS probe volume, and plasma emission from the centerline outwards to 4 mm is shown in Fig. 6(a) from an average of 60 images. The rotational and vibrational temperatures plotted in Fig. 6(b) are the average of 400 single-shot measurements, and the vertical bars represent one standard deviation above and below the mean. The vibrational temperatures correspond to the left axis, and the rotational temperatures to the right axis. The 0 mm location corresponds to the data shown in Fig. 5. The other half of the plasma (not shown) exhibited symmetric temperatures across the centerline. The largest degree of rotational-vibrational nonequilibrium occurs at the 0 mm position, with a difference in the average $T_{\mathrm{rot}}$ and $T_{\mathrm{vib}}$ of 2766 K. Moving away from the center towards the edge of visible plasma emission, both the vibrational and rotational temperatures decrease. At 1.25 mm from the reactor center, however, the vibrational temperature begins to increase again, accompanied by a small initial increase in rotational temperature. Because the gas is flowing from above and impinging on the bottom electrode, the gas sweeps products from the center of the reactor outward. Thus, measuring outward from the center roughly correlates to a longer residence time in the plasma reactor. As a result, the secondary peak in vibrational temperature that occurs approximately 2.25 mm from the center likely corresponds to a secondary vibrational excitation mechanism on a slower time scale than initial electron excitation of higher vibrational states. One mechanism could be pooling of energy in the electronically excited ${\mathrm{N}}_2(\mathrm{A})$ state, which subsequently undergoes collisional relaxation to excited vibrational levels of the ${\mathrm{N}}_2(\mathrm{X})$ state [23]. At the centerline, the rotational temperature was moderately elevated at 375 K. This modest heating is likely due to the electrodes heating up and warming the surrounding gas in addition to recombination, electronic quenching, and vibrational-rotational relaxation.

 

Fig. 6. (a) Averaged image of the DBD and CARS measurement locations and (b) corresponding vibrational and rotational temperatures; symbols and bars represent the average and standard deviation of 400 single-shot measurements at each location.

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Once again, the unsteady nature of the plasma is reflected in the large spread of vibrational temperatures towards the center of the discharge in Fig 6. The temperature spread decreases significantly away from the center; the standard deviation of vibrational temperature drops from 11.3% (356 K) at the center to 5.5% (101 K) 1.25 mm away. This result is consistent with a more uniform environment further away from the plasma source. The precision of fs/ps vibrational CARS was previously reported as 2.2% in an equilibrium environment [3]. Although the observed signal level decreases within the core of the plasma [the ratio of peak signal to background noise (SNR) drops from 525 at 1.25 mm to 80 at 0 mm], the previously quantified precision supports the claim of measured vibrational temperature fluctuations correctly tracking the actual vibrational temperature shot-to-shot.

Next, a DBD plasma sustained with a higher global reduced electric field and lower concentration of ${\mathrm{N}}_2$ was studied. Figure 7 shows sample experimental rovibrational $Q$-branch [Fig. 7(a)] and pure-rotational $S$-branch [Fig. 7(b)] CARS spectra collected within the center of a DBD plasma in a mixture of 8% ${\mathrm{N}}_2$ and 92% He. The reduced electric field, determined from the global voltage measurement, was 37 Td. These experimental conditions resulted in a quasi-steady-state plasma, confirmed using the emission images. Both spectra shown in Fig. 7 are the average of 400 spectra (20 laser shots per camera exposure). Experimental data are shown in red symbols, and best-fit simulated spectra are shown as solid black lines. The vibrational and rotational temperatures corresponding to the best-fit simulated spectra pair were 4860 and 450 K, respectively. Of the 400 fit spectra pairs, the standard deviations of the vibrational and rotational temperatures were found to be 4.4% (213 K) and 3.8% (17 K), significantly lower than the unsteady plasma condition measured in Fig. 6. The fluctuations in temperatures are due to a varying mean local electric field as well as fluctuations in the ${\mathrm{N}}_2/\mathrm{He}$ gas mixture. There is good agreement between the experimental and simulated pure-rotational CARS spectra, in Fig. 7(a). However, the experimental rovibrational CARS spectrum exhibits higher intensities for the higher ($Q_2$, $Q_3$, $Q_4$, and $Q_5$) bands compared to the simulated spectrum, as shown in Fig. 7(b) as a black solid line. This is evidence of a non-Boltzmann vibrational distribution, consistent with past measurements in nonequilibrium environments of the ground electronic state of ${\mathrm{N}}_2$ [15,24,25]. Because of this, the fitting routine used in the current work only considered the residuals of the first two vibrational bands in the wavenumber range $2285-2350{\mathrm{cm}}^{-1}$, or the $Q_0$ and $Q_1$ transitions, when fitting $Q$-branch spectra. One additional consideration in evaluating non-Boltzmann trends in the vibrational energy distribution is the role of spatial averaging over sharp gradients of vibrational temperature [26]. To minimize this, a short-focus combining lens was used as described in the experimental setup to maximize spatial resolution.

 

Fig. 7. Experimental (a) $S$-branch and (b) $Q$-branch spectra shown as red circles with the corresponding best-fit simulations shown as solid black lines at $T_{\mathrm{rot}}=450\mathrm{K}$ and $T_{\mathrm{vib}}=4860\mathrm{K}$. The $Q$-branch spectrum simulated using non-Boltzmann vibrational level populations is also shown in (b) as a solid red line. Measured (c) non-Boltzmann vibrational distribution is shown as blue circles compared to the Boltzmann temperature based on $Q_0$ and $Q_1$ (blue solid line). For comparison, rate coefficients for excitation from $v=0$ to higher vibrational levels from electron collisions are shown (green diamonds).

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The non-Boltzmann behavior evident from the experimental spectrum shown in Fig. 7(b) was further examined by comparing the actual vibrational level populations to those under a Boltzmann distribution assumption. The best-fit vibrational temperature from the $Q_0$ and $Q_1$ transitions (4860 K) is indicated by the solid blue line in Fig. 7(c). To determine the actual vibrational level populations, each level was varied independently while comparing the modeled spectrum to the experimental spectrum. In order to determine actual state populations, the assumption was made that all population resided in vibrational levels 0–7. The resulting simulation without the restriction of a Boltzmann distribution is shown in Fig. 7(b) as a red solid line. The corresponding populations of each level are shown as blue circles in Fig. 7(c). For the actual (non-Boltzmann) vibrational population levels, a temperature of 5217 K can be defined using only the populations in the vibrational states $v=0$ and $v=1$ using the Boltzmann relationship:

Because CARS is dependent on the population difference between two states as shown in Eq. (1), comparing two vibrational bands of a $\mathit{Q}$-branch spectrum reflects the population within three vibrational levels. It is apparent from comparing the experimental measurement to the curve representing a Boltzmann distribution in Fig. 5(c) that the measured vibrational distribution is non-Boltzmann, consistent with Fig. 7(b). The actual vibrational level population deviates from a Boltzmann population by 5%–30% for states $v=3$ to $v=6$ when normalizing the populations of each state by the total population. For the measured global electric field, a non-Boltzmann vibrational distribution is consistent with predicted electron-${\mathrm{N}}_2$ impact excitation. At the estimated field strength of 37 Td and corresponding electron energies of 4.86 eV, electron impact results in indiscriminate excitation of vibrational levels. The electron energy loss coefficients were calculated using the Boltzmann equation solver BOLSIG+ [27], and were used along with the number density and vibrational level energies to show the rate of excitation due to electron-${\mathrm{N}}_2$ collisions to each state. These rates, plotted on Fig. 7(c), show that the upper vibrational levels are excited via electron collisions in a non-Boltzmann manner. After excitation, collisional vibrational energy exchange and relaxation account for the quasi-steady-state vibrational distribution function closer to a Boltzmann distribution [blue circles in Fig. 7(c)].

5. CONCLUSION

Hybrid fs/ps CARS has been implemented for measurement of rotational and vibrational energy distributions in a quasi-steady-state nonequilibrium environment with a spatial resolution of $\sim 650\mathrm{\mu m}$ and a sampling time of $<10\mathrm{ps}$. An improved hybrid CARS model was implemented to include vibrational-rotational nonequilibrium, and was used to capture the effects of excited vibrational states on the $S$-branch spectra and lowered rotational temperature on the $Q$-branch spectra. These effects highlight the need for simultaneous, single-shot measurement of $S$- and $Q$-branch transitions for accurately modeling experimental spectra in vibrational/rotational nonequilibrium environments. Single-shot measurements were used to quantify the fluctuations of vibrational and rotational temperatures, with standard deviations as low as 3.1% and 1.9%, respectively, in the outer, steady region of the flow field and 11.3% and 3.9% in the central region with the highest fluctuations in plasma structure. Vibrational/rotational nonequilibrium was observed outside of the region of visible emission, illustrating the utility of the current laser-based spectroscopic measurement technique for providing a more complete view of the plasma dynamics. Additionally, experimental $Q$-branch measurements exhibited non-Boltzmann vibrational distributions, and individual level populations were determined using the current fs/ps CARS model.

The ps time resolution and single-shot capability of the dual-pump technique used here are well-suited to studying highly transient, pulsed plasma environments. The use of the hybrid fs/ps CARS approach allows for direct extension to a range of nonequilibrium and plasma environments without the need to directly model collisional decay or specific gas mixtures up to pressures of several atmospheres when a short probe delay is employed ($<10\mathrm{ps}$). The extension to air plasmas is relatively straightforward: the rotational temperature measurement in air has been previously demonstrated [4,21], and rovibrational transitions for ${\mathrm{O}}_2$ and ${\mathrm{N}}_2$ are well-separated. In this work, the probe pulse energy, bandwidth, and duration were relatively well optimized to deliver the signal levels, spectral resolution, and desired probe pulse timing; however, other potential probe pulse configurations with higher energy and pulse duration could be selected for lower density conditions subject to fewer collisions and suffering from lower signal levels. This could be achieved, for example, using a separate ps amplification system to generate the probe pulse. Furthermore, because of the high Raman coherences generated using this approach, it is feasible to extend this work to one-dimensional line-imaging to instantaneously resolve spatial gradients in nonequilibrium environments.