1. INTRODUCTION

There is presently an increased demand for high-fidelity fluid-dynamic data from supersonic and hypersonic ground-test facilities. In these compressible-flow applications, scalar quantities that describe the thermodynamic state are often as important as the velocity field, placing noninvasive, laser-based diagnostics for quantitative, spatially correlated measures of fluid scalars at a premium. In high-speed applications, the pressure field can play a dominant role in aerodynamic loading, stability, and acoustic noise radiation, so that noninvasive pressure diagnostics are important to the development of new flight systems. Previously demonstrated laser-diagnostic tools for pressure monitoring are path-averaged absorption or time-averaged frequency scanning techniques, including: laser-induced fluorescence (LIF) [1,2]; filtered Rayleigh scattering [3,4]; Rayleigh interferometry [5]; tunable laser-absorption spectroscopy [6–8]; and laser induced gratings [9,10].

Coherent anti-Stokes Raman scattering (CARS) [11] offers spectroscopic detection of both temperature and pressure, without the need for chemical seeding, with high space and time resolution, and with high signal-to-noise for single-laser-shot detection. High-spectral-resolution CARS of the ${{\rm{N}}_2}$ Raman $Q$-branch has been demonstrated for pointwise measurements of temperature and pressure by fitting a single frequency domain spectrum [12–14]. At the low pressures encountered in many supersonic ground-test facilities, $Q$-branch spectra move from a collision-broadened regime to a primarily Doppler-broadened one, resulting in a reduction of CARS pressure sensitivity. This Doppler cutoff occurs for $P \lt {\sim}{0.1}\;{\rm{atm}}$ [13] in air and ${{\rm{N}}_2}$ and presents a fundamental limitation of the vibrational CARS approach for pressure measurements. Grisch et al. [15] overcame this low-pressure limit by referencing the amplitude of the CARS signal pulse to a simultaneous measurement in a room-temperature cell to obtain temperature/density measurements in a low-pressure (6 Pa) Mach-10 flow of air—an approach that, while successful, requires tedious calibration of the absolute magnitude of the reference signal, which can exhibit high-sensitivity to optical system changes and beam misalignment.

Hybrid femtosecond–picosecond (fs–ps) CARS instruments for gas-phase diagnostics [16–24] enable coupled time- and frequency-domain detection, encouraging more recent demonstrations of CARS temperature/pressure measurements by taking advantage of the time-resolution offered by ultrashort laser pulses [25–29]. By adjusting the probe-pulse time delay, the time-dependent decay of an impulsively produced Raman coherence can be exploited to sample the impact of gas-phase collisions through local pressure. Introduction of two (or more [30]) picosecond-duration probe pulses can decouple the effects of temperature and pressure on the detected Raman spectra, when the probe laser pulse is of sufficiently short duration. Kearney and Danehy [25] and Dedic et al. [26] have demonstrated fs–ps CARS detection of temperature/pressure by accessing pure-rotational Raman resonances, with Doppler widths that are one to two orders of magnitude less than their $Q$-branch counterparts in ${{\rm{N}}_2}$ and ${{\rm{O}}_2}$; these pure-rotational resonances are collision dominated, even at the low pressures encountered in cold-flow supersonic facilities, thereby enabling CARS barometry at much lower pressures. Escofet-Martin et al. [29] demonstrated a dual-probe, 1D rotational CARS imaging scheme similar to the one presented here for temperature/pressure determination in subsonic impinging ${{\rm{N}}_2}$ jets at ambient pressures between 1.0 and 1.5 atm. Pressure was determined from the ratio of spectrally integrated CARS signals obtained with ${\tau _1} = {{0}}$ and ${\tau _2} = 270 {\text{-ps}}$ probe delays compared to a pre-calibrated model. In contrast, the present method as well as those in [25,26] rely upon fits to the full CARS spectral shape, for application over a wider range of temperatures and pressures.

In this work, we apply the hybrid fs–ps rotational CARS approach to a canonical underexpanded compressible jet flow of air. One-dimensional temperature and pressure profiles are obtained on a single-laser-shot basis along the jet axis using the two-beam, 1D rotational CARS imaging scheme of Bohlin et al. [31], with a single femtosecond pump/Stokes pulse and two independently delayed 60-ps probe beams. Relative to previous work [25], the transition to the free-jet flow of air enables us to demonstrate pressure monitoring over a much wider range of conditions relevant to supersonic ground-test facilities; assess the ability of the CARS imaging technique to capture shocks; and evaluate the performance of our CARS spectral modeling scheme in air, where the pressure-dependent decay of Raman coherence is complicated by the $^3\Sigma $ electronic ground state of ${{\rm{O}}_2}$ [32,33]. Of particular note, the influence of Raman linewidths on CARS pressure measurements is evaluated. Previous CARS pressure measurements in low-temperature ($T \lt {{294}}\;{\rm{K}}$) compressible flows [13,14,34,35] have utilized Raman linewidths obtained from measurements of the Raman $Q$-branch at much higher temperatures, and correlated through the modified exponential gap (MEG) model [36,37]. MEG was necessarily extrapolated to low temperatures where its validity as well as the extrapolation of vibrational $Q$-branch Raman linewidths to pure-rotational resonances are uncertain. Here, we compare CARS pressure measurements obtained by extrapolating linewidths from the MEG model to temperatures as low as 80 K to a new temperature-dependent model specific to pure-rotational Raman linewidths for $T = {{80 {-} 200}}\;{\rm{K}}$, which we have presented in a companion work to this paper [38].

 

Fig. 1. (a) Relative decay of pure-rotational transitions in ${{\rm{N}}_2}$ under ambient conditions; the data are color coded according to the spectrum shown in (b). (c) Measured spectra obtained at probe time delays of $\tau = {{0}}\;{\rm{ps}}$ and $\tau = {{322}}\;{\rm{ps}}$, showing the difference in relative intensity distribution resulting from the time-dependent decay in (a). Measurements shown in (a)–(c) were obtained at $T = {{294}}\;{\rm{K}}$, $P = {{0}.82\;{\rm atm}}$ and averaged over 50 laser shots. Similar results are shown in (d)–(f) for air in the underexpanded jet at local measured conditions of $T = {{88}}\;{\rm{K}}$ and $P = {0.067}\;{\rm{atm}}$, highlighting the time-domain beating of the oxygen transitions and the spectral changes from $\tau = {{0}}\;{\rm{ps}}$ to $\tau = {{509}}\;{\rm{ps}}$ in (f). The decay curves are purposefully offset vertically in (d) to aid in the visualization of the beat patterns. Note that the $N(J) = {{9}}({{6}})$ and $N(J) = {{13}}({{9}})$ decay traces in (d) are not from isolated transitions and represent an integration over the neighboring ${{\rm{N}}_2}$ and ${{\rm{O}}_2}$ transitions. The residual in (f) is amplified by the additional oxygen dephasing and the small nonresonant contribution for $\tau = {{0}}\;{\rm{ps}}$.

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2. THEORETICAL CONSIDERATIONS AND TEMPERATURE/PRESSURE FITTING PROCEDURE

The time-domain CARS model used here is similar to that used in our previous work [25,39]. We summarize the essential features needed to provide context to the present temperature/pressure measurement application. We also present newly added features that treat spectroscopic complications arising from the triplet character of the ${{\rm{O}}_2}$ electronic ground state here, which are needed for measurements in air. For femtosecond pump and Stokes pulses introduced at time $t = {{0}}\;{\rm{ps}}$ and a picosecond probe pulse introduced at delay $\tau$, the time-domain Raman-resonant CARS polarization is given by

In Eq. (2), the sum over $k$ is taken over all rotationally resonant species ($k = {{\rm{N}}_2}$ and ${{\rm{O}}_2}$ here); ${\omega _{N,J}}$ are the pure-rotational Raman frequencies; and ${\Gamma _N}$ are the collision-broadened Raman linewidths expressed in units of ${\rm{se}}{{\rm{c}}^{- 1}}/{\rm{atm}}$ (HWHM), such that these linewidths are broadened linearly with the pressure, $P$. The weight term, $W_{{\textit{N,J}}}^{({{k}})}$, in Eq. (2) is given by

Unpaired electrons in ground-state ${{\rm{O}}_2}$ result in a $^3\Sigma $ electronic term, with electron spin $S = {{1}}$, and the $\xi$ factor in Eq. (3) is introduced to represent a $J\!$-dependent line strength that describes the triplet splitting of the ${{\rm{O}}_2}$ rotational Raman spectrum [43,44]. A more complex coupling between electronic and nuclear angular momenta results, with $J = N$, $N \pm {{1}}$. Raman selection rules for a pure-rotational $S$-branch transition are $\Delta N = + {{2}}$ and $\Delta J = {{0}}$, ${\pm}{{1}}$, ${\pm}{{2}}$ [32,33,43], which results in splitting of each $S(N)$ transition into six $J$-dependent resonances, three of which are spaced within ${\sim}{{0.1 \!-\! 0.2}}\;{\rm{c}}{{\rm{m}}^{- 1}}$ and typically dominate the total Raman intensity for $N\gt {{3}}$. This fine splitting is unresolved by our 60-ps probe pulse or by the ${\sim}{{1.0}} {{\text{-cm}}^{- 1}}$ resolution of our detection system, such that the time-dependent decay of ${{\rm{O}}_2}$ Raman transitions exhibits beat oscillations atop the exponential decay associated with the ${\Gamma _N}$ in Eq. (2) [32,38]. This triplet character is treated by introduction of both the $\xi$ factor and six distinct ${\omega _{N,J}}$ in Eq. (2), which splits the total strength of each $N$ transition into six $J$-dependent sublevels such that $\mathop \sum \nolimits_{{J}} {{{\xi}}_{{{N,J,}}{{{O}}_{{2}}}}}= 1$. The ${\omega _{N,J}}$ were calculated using expressions given by Altmann et al. [45], as were values of $\xi$ for $N$ up to 21. For $N\; \gt \;{{21}}$, we set ${\xi _{N,J}} = {\xi _{N = 21}}$, $J$ due to the lack of information for higher $N$—an assumption that we believe had little effect on our results because rotational levels $N\; \ge \;{{23}}$ account for 1.4% or less of the Boltzmann population in ${{\rm{O}}_2}$ at the temperatures of interest here. In addition, the intensity of each $J$ transition trends rapidly toward an even 1/3 split among the three dominant $S$ ($\Delta J = + {{2}}$) resonances near the line center by $N = {{21}}$, such that this extrapolation of $\xi$ to higher $N$ is reasonable.

The resultant CARS spectrum is computed as the squared magnitude of the Fourier transform of ${P_{\rm{CARS}}}$ in Eq. (1), and convolved with a Gaussian instrument function for comparison to experimental data. In our experiment, we utilize two independent probe pulses: one introduced at ${\tau _1} = {{0}}$ and another delayed by ${\tau _2}$ of 150 ps or more. Temperature is determined primarily (but not entirely) by fitting spectra with ${\tau _1} = {{0}}$, where the impact of collisions through the exponential term in Eq. (2) is minimized (see Supplement 1). With this temperature estimate, the pressure is determined from fits to probe-delayed spectra, for which the $N$ dependence of the Raman linewidths results in distinct, time-dependent changes [25]. Any physical translation of the molecules generating the Raman coherence from ${\tau _1}$ to ${\tau _2}$ is negligible, and the temperature and pressure measurements are considered simultaneous for this work.

 

Fig. 2. Details of the CARS model and the spectral fitting process. (a) Theoretical time-domain model of Raman coherence at $T = {{294}}\;{\rm{K}}$ and $P = {0.82}\;{\rm{atm}}$ in air with example 60-ps FWHM probe pulses gating the coherence at $\tau = {{0}}\;{\rm{ps}}$ (blue) and $\tau = {{400}}\;{\rm{ps}}$ (red). (b)–(e) Example depictions of a three-dimensional library for pressure, temperature, and probe delay shown at the bounds of $P = [{0.1}\;{0.9}]\;{\rm{atm}}$, $T = [{{70}},{{290}}]\;{\rm{K}}$, and $\tau = [{{0}},{{400}}]\;{\rm{ps}}$. (f) Fit residuals for various temperatures and pressures from the comparison of theoretical spectra to a theoretical spectrum ($T = {{150}}\;{\rm{K}}$, $P = {0.3}\;{\rm{atm}}$, and $\tau = {{400}}\;{\rm{ps}}$), shown in the inset (no fit iterations). The correct fit is highlighted in green. (g)–(j) Examples of the iterative fitting procedure in pure nitrogen at room conditions. (g), (i) Third iteration fits (${{\rm{T}}_3}$, ${{\rm{P}}_3}$) of 50-shot average spectra at early and late times of 0 and 322 ps along with (h), (j) average and single-shot iteration results of temperature and pressure for 4050 spectra relative to the fifth result (${{\rm{T}}_5}$, ${{\rm{P}}_5}$). Single-shot results are shown in gray lines with the mean and standard deviation values shown in blue.

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The impact of Raman linewidths in the time and frequency domains is illustrated in Fig. 1, where the results of probe time delay scans in pure ${{\rm{N}}_2}$ at ambient conditions are shown in (a)–(c) with air results in a low-temperature and low-pressure region of the underexpanded jet examined in this work in (d)–(f). The time-dependent intensity of several isolated $J$ transitions in ${{\rm{N}}_2}$ is shown in Fig. 1(a), where the decay of higher-energy transitions (yellow-orange-red) in the spectrum shown in Fig. 1(b) is slower than for lower-energy transitions (blue-green). The impact on the time-dependent CARS spectrum is a shift in intensity to higher-energy transitions with increasing probe time delay, as seen in the comparison of spectra at $\tau = {{0}}$ and 322 ps in Fig. 1(c). This “spectral heating” effect, first reported by Seeger et al. [46], can be modeled and exploited for CARS pressure measurements. In air, additional complexity in the decay of ${{\rm{O}}_2}$ transitions arising from the low-frequency beating between nearby triplet-split $J$ levels is highlighted in Figs. 1(d)–1(f), where the oscillatory nature of the coherence decay is similar to the higher-temperature gas-cell measurements of Courtney et al. [32]. These measurements were recorded in the underexpanded jet featured in this work, as lower-pressure environments were required to realize the beat patterns in the time domain within the dynamic range of our detection system.

Temperature and pressure data were obtained from least-squares fits of measured CARS spectra to a pressure-, temperature-, and probe-delay-dependent library computed from Eqs. (1)–(3). Separate from the scalar variables, the fitting procedure allowed bounded variations in the vertical shift, horizontal shift, and wavenumber scaling of the synthetic spectrum. Minimum residuals were determined from a nonlinear least-squares solver in MATLAB. Library spectra for air at $T = {{70}}$ and 290 K and $P = {0.1}$ and 0.9 atm are shown for probe delays of $\tau = {{0}}$ and 400 ps in Figs. 2(a)–2(e). The library spectra shown are representative of temperatures and pressures observed in this work, and further demonstrate the time dependence in air across a range of thermodynamic conditions. At $P = {0.1}\;{\rm{atm}}$, changes in the spectrum between $\tau = {{0}}$ and 400 ps reveal both a shift in intensity from low to high $J$ in ${{\rm{N}}_2}$ and a significant reduction in the intensity of ${{\rm{O}}_2}$ lines as a result of the triplet-induced beating discussed above. At $P = {0.9}\;{\rm{atm}}$, the redistribution of transition intensity from lower-energy states to higher-energy states with $\tau$ is significantly more pronounced, while the impact of triplet beating on the ${{\rm{O}}_2}$ spectrum persists.

 

Fig. 3. (a) Dual-probe line-imaging hybrid CARS optical setup. (b) Sonic jet used in the experiment. Surrounding foam and exhaust capture pipe helped limit the noise to acceptable levels for nearby operators.

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An iterative approach is generally required to retrieve temperature and pressure from CARS spectra obtained at two probe delays. In earlier work [25,26], a sub-10-ps probe pulse was used; spectra acquired at zero probe delay were minimally impacted by collisions and could be fit for temperature independent of pressure. The high-energy, Nd:YAG probe source used here enabled us to expand from point to line CARS measurements, but the longer, 60-ps probe pulse resulted in a nonnegligible impact of collisions, even at zero delay, for all but the lowest pressures encountered. A goodness-of-fit metric (sum of the squares of residuals) comparing temperature- and pressure-dependent model spectra at probe delay $\tau = {{400}}\;{\rm{ps}}$ to a synthetically generated spectrum at $T = {{150}}\;{\rm{K}}$, $P = {0.3}\;{\rm{atm}}$, and $\tau = {{400}}\;{\rm{ps}}$ is shown in Fig. 2(f). This contour plot reveals a distinct trough of low residual values and indicates a strong, negative correlation of optimum temperature and pressure. Any precision error in the temperature fit will result in a negatively correlated bias in the fitted pressure.

To reduce systematic errors in the extracted temperature and pressure, the collision-free approximation at ${\tau _1} = {{0}}$ is abandoned, and an iterative scheme is used. The convergence of the scheme to an optimum temperature/pressure measurement is depicted in Figs. 2(g)–2(j) for five iterations on CARS spectra at room-temperature conditions in pure nitrogen. In the first iteration, the ${\tau _1} = {{0}}\,{\text{ps}}$ spectrum is fit for temperature, ${T_1}$, by allowing the pressure to vary between limits; an estimate of the pressure is then obtained by fitting the companion ${\tau _2} = {{400}}\;{\rm{ps}}$ spectrum with the temperature held fixed at ${T_1}$. Temperature and pressure, ${T_i}$ and ${P_i}$, at the $i$ th iteration are found by fixing $P = {P_{i - 1}}$ and fitting the ${\tau _1} = {{0}}\,{\text{ps}}$ spectrum for ${T_i}$, then fixing $T = {T_i}$ to fit the ${\tau _2} = {{400}}\;{\rm{ps}}$ spectrum for ${P_i}$. The average (blue) and single-shot (gray) iteration history for 4050 single-laser-shot pairs of spectra obtained under room conditions is shown in Figs. 2(h) and 2(j) relative to the last (fifth) iteration considered. Typically, three iterations are sufficient to converge both temperature and pressure fits. A small oscillatory response is seen for a few single-shot spectra with low signal-to-noise levels. For this work, only three iterations were considered.

Importantly, the pressure sensitivity of this CARS instrument is related to the Raman lifetime, ${\tau _R}$, defined as

3. EXPERIMENTAL SETUP

A. Optical Setup

The dual-probe CARS line imaging system and sonic jet are detailed in Fig. 3. The optical system is a combination of two two-beam CARS line-imaging setups of the type developed by Bohlin et al. [31], each with independent detection systems and picosecond probe pulses, but sharing a common femtosecond pump/Stokes pulse, hereafter referred to as the “pump.” The femtosecond pulse originates from a commercial Ti:sapphire regenerative amplifier (Spectra Physics Solstice Ace), supplying a 1-kHz pulse train of 7-mJ, 40-fs (FWHM) pulses centered near 800 nm. A frequency-narrow probe pulse is provided by a second commercial regenerative amplifier (Ekspla PL2230) using an Nd:YAG gain medium, frequency doubled to 532 nm for 60-ps, 50-mJ pulses at a repetition rate of 20 Hz, and an assumed transform limited linewidth of ${0.25}\;{\rm{c}}{{\rm{m}}^{- 1}}$. The probe pulse is split with a 75% beam splitter into early (${\tau _1} = {{0}}\;{\rm{ps}}$) and late (${\tau _2} = {{150 {-} 1000}}\;{\rm{ps}}$) probe pulses with 75% of the total picosecond pulse energy devoted to the ${\tau _2}$ channel. Time-of-flight mechanical delay stages are used in each probe leg to independently delay each probe beam with respect to the femtosecond pump. The beam crossing is achieved with a 300-mm cylindrical singlet focusing lens. Telescoping optics are inserted into the 532-nm probe beam lines to match the probe foci to that of the 800-nm pump beam. Optimization of the beam crossing is performed using a beam-profiling camera to adjust both overlap and focus of the laser sheets. The CARS signal beams emerge colinear with the probe pulses and are collimated by a second 300-mm cylindrical element.

 

Fig. 4. (a) Average schlieren image with a horizontal spatial filter edge; rotational CARS spectra, obtained in a jet flow of air, stitched together from four axial positions for (b) zero probe delay channel at ${\tau _1} = {{0}}\;{\rm{ps}}$; (c) probe time-delayed channel at ${\tau _2} = {{200}}\;{\rm{ps}}$ and (d) at ${\tau _2} = {{400}}\;{\rm{ps}}$. The spectral images are the averages of 50 shots for each region of interest. Note that no data were acquired at 400 ps for the lower most ROI because the high pressure in this region resulted in rapid decay of the CARS signal below detection limits. The white horizontal lines in (b)–(d) depict the bounds of the images stitched together. Note that there is a slight spectral (horizontal) shift with height on the detector for the delayed channels (c), (d), evidenced by the mismatch in the spectral lines at the stitching locations.

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Identical detection systems are used to isolate and disperse the rotational CARS signals. In each channel, a Glan–Thompson analyzer is used to minimize the nonresonant CARS signal and a significant fraction of the probe laser line compared to the Raman-resonant CARS spectrum when the pump and probe laser beams are arranged in a specific polarization scheme [31,47–49]. Volume Bragg grating filters [50,51] provide additional isolation of the resonant CARS signal from the probe laser line. Two 0.5-m grating spectrographs (Princeton Instruments, SpectraPro HRS-500, 1800 l/mm grating) disperse the CARS signal beams at ${\sim}{{1}} {{\text{-cm}}^{- 1}}$ resolution onto backside-illuminated, electron-multiplying (EM) CCD cameras (Andor, Newton) operated without EM gain. The resulting CARS signal emerge from 5–6-mm-long lines oriented parallel to the jet axis with a ${\sim}1 {\text{-mm}}$ beam-overlap length. The width of the measurement volume at the focal crossing is assumed to be 30 µm. Spatial coincidence of the two CARS volumes along the beam-overlap direction at $\tau = {{0}}\;{\rm{ps}}$ is attained by making fine adjustments to the beam crossing angles and probe-beamline telescopes to match the location of maximum nonresonant signal in the glass. For a time-delayed probe, ${\tau _2}\gt {{0}}\;{\rm{ps}}$, the probe beams are imaged onto a beam-profiling camera to ensure negligible spatial offset of this probe beam as the mechanical delay stage is scanned. However, this method does not explicitly confirm spatial coincidence of the CARS volume for $\tau \gt {{0}}\;{\rm{ps}}$, resulting in some uncertainty in the true spatial overlap. The beam crossing line is independently imaged onto each spectrometer slit near unit magnification using ${\rm{f}} = 300 {\text{-mm}}$ cylindrical singlet lenses in a 4f imaging configuration. The resulting spatial resolution along the jet axis is ${\sim}{{53}}\;{\rm{\unicode{x00B5}{\rm m}}}$, estimated by imaging the nonresonant response from a glass coverslip and observing the rise in signal intensity at the glass edge [52,53]. This estimate of spatial resolution is later confirmed by our ability to resolve the normal shock region in the underexpanded jet flow.

Single-laser-shot CARS spectral images from the underexpanded jet flow are obtained at the 20-Hz repetition rate of the probe laser source. The “early” channel detects spectra for a fixed probe-1 delay of ${\tau _1} = {{0}}\;{\rm{ps}}$, and is primarily temperature sensitive, while the “late” detection channel is used for CARS spectra acquired over a wide range of probe-2 delays, ${\tau _2} = {{150 {-} 1000}}\;{\rm{ps}}$, and is primarily pressure sensitive with respect to the early channel. Each channel includes a series of neutral density filters, used to optimize the dynamic range of the detection systems, based on the selected probe delay and CARS signal levels associated with temperature and pressure conditions at the measurement location.

B. Underexpanded Jet Facility and CARS Line Images

An underexpanded sonic jet was utilized as a demonstration platform for the line-imaging technique at low temperatures and pressures relevant to cold-flow supersonic facilities. The 6.35-mm diameter jet nozzle exit was positioned below the CARS beam crossing to interrogate the expanding gas stream along the jet centerline. The jet stagnation pressure of 773 kPa (100 psig) resulted in a highly underexpanded jet when emitted to local atmospheric conditions (${\sim}{{83}}\;{\rm{kPa}}$), with a characteristic “barrel shock” structure [54]. The expected static conditions at the jet exit plane from isentropic calculations are ${\sim}{{405}}\;{\rm{kPa}}$ and ${\sim}{{240}}\;{\rm{K}}$ (from a stagnation temperature of 292 K). The jet was translated vertically through the measurement volume, allowing the beams to image four regions of interest (ROIs) along the jet centerline, with some measurement overlap between each region as a check on the consistency of the results.

Average 1D CARS image data are shown alongside a jet schlieren image in Fig. 4. The CARS images were acquired at different locations along the jet centerline and stitched together to reveal the full axial distribution of ${{\rm{N}}_2}$ and ${{\rm{O}}_2}$ rotational states, and each row of data has been normalized to show the relative spectral shape. CARS images from the ${\tau _1} = {{0}}\;{\rm{ps}}$ channel and probe time delay images of ${\tau _2} = {{200}}$ and 400 ps are shown. The CARS images show a clear normal shock (Mach disk) near ${{z}} = {{12}}\;{\rm{mm}}$, where the distribution of CARS intensity among the rotational transitions in the spectrographs abruptly transitions from low- to high-$J$ and $\text{-}N$ quantum numbers through the shock. Viewing the ${\tau _1} = {{0}}\;{\rm{ps}}$ image specifically, one can deduce the temperature profile qualitatively from the relative weighting of the rotational CARS intensity among the Raman transitions. The temperature decreases from the exit to the Mach disk with a rapid shift in line intensity toward low $J/N$, then features a discontinuous jump across the shock above the nozzle exit temperature and remains constant as the flow continues downstream. The impact of pressure on the probe time delay CARS spectra in Figs. 4(c) and 4(d) can also be observed. At ${\tau _2} = {{200}}\;{\rm{ps}}$, the image near the jet exit exhibits a shift in intensity toward high-energy rotational transitions relative to ${\tau _1} = {{0}}$. Just upstream of the shock, the shift in CARS intensity is much less pronounced at ${\tau _2} = {{200}}\;{\rm{ps}}$, and only very subtle at ${\tau _2} = {{400}}\;{\rm{ps}}$, as this region is characterized by low pressure.

For ${\tau _2} = {{200}}$ and 400 ps, the finite dynamic range of these CARS measurements can be observed. Near the jet exit at ${\tau _2} = {{400}}\;{\rm{ps}}$, high pressures result in short Raman lifetimes through the ${\exp}(- {\rm{P}}\Gamma {\rm{t}})$ term in Eq. (2), and the signal has decayed below detection limits. Further downstream, dynamic range concerns are expressed across the Mach disk as a result of the rapid density increase across the normal shock. Trade-offs exist in selection of ${\tau _2}$, where sensitivity to pressure (high ${\tau _2}$) competes with signal level (low ${\tau _2}$) and a balance of dynamic range to effectively image across the shock front. Good quality signal-to-noise data upstream of the shock can result in detector saturation on the downstream side for low ${\tau _2}$, while a higher value of ${\tau _2}$ more comparable to the Raman lifetime upstream of the shock may result in no signal at all downstream. Judicious selection of neutral density filters in the detection channel and ${\tau _2}$ delay can permit pressure imaging across the shock front. For the ${\tau _2} = {{400}}\;{\rm{ps}}$ image across the shock in Fig. 4(d), the signal upstream of the shock is optimized with a higher-neutral-density filter at the expense of the signal downstream. For each ROI, CARS images were recorded at multiple ${\tau _2}$ probe delays to optimize the results in different regions of the jet (e.g., longer probe delays for lower pressure and shorter delays for high pressure).

4. RESULTS AND DISCUSSION

A. Single-Laser-Shot Spectra

As an illustration of single-laser-shot data quality and the ability of the spectroscopic model to fit measured spectra at a variety of temperature/pressure conditions in air, single-laser-shot spectral fits at various axial locations in the jet are shown in Fig. 5 for both ${\tau _1}$ and ${\tau _2}$ channels. The ${\tau _1} = {{0}}$ ps spectra, featured on the left side, are only nominally impacted by collisional dephasing. However, for the 60-ps FWHM probe pulse used in this work, a collision-free approximation is strictly valid only in the region upstream of the Mach disk, where the lowest pressures are experienced and Raman lifetimes are maximized. The best fit temperature after the third fitting iteration (i.e., ${T_3}$) is shown beside each spectrum on the left-hand side of the figure, displaying a range from 198 K at $y = {2.97}\;{\rm{mm}}$ from the nozzle exit down to 83 K at $y = {11.5}\;{\rm{mm}}$, just upstream of the Mach disk. A shock-heated spectrum results in $T = {{277}}\;{\rm{K}}$ downstream of the normal shock.

 

Fig. 5. Representative single-shot fits throughout the jet for both the temperature channel (left, $\tau = {{0}}\;{\rm{ps}}$) and pressure channel (right, $\tau$ marked for each spectrum). Axial locations of the spectra proceeding downstream as marked in the figure are ${\rm{y}} = {2.97}$, 6.04, 7.9, 11.5, and 13.4 mm. Rotational transitions of ${{\rm{O}}_2}$ (N) and ${{\rm{N}}_2}$ (J) are marked for the temperature and pressure spectra, respectively. Raman lifetimes from Eq. (4) are listed with each pressure spectrum for ${{J}} = {{6}}$ at the fitted temperature listed, using the low-temperature S-branch linewidths from [38].

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The co-located single-shot spectra from the ${\tau _2}$ channel are shown on the right-hand side of Fig. 5, along with the best-fit pressure after three iterations, ${P_3}$, the ${\tau _2}$ delay in ps, and the Raman lifetime, ${\tau _R}$, as per Eq. (4). For each location shown, different combinations of neutral density filters and ${\tau _2}$ probe delay were required to optimize the signal-to-noise of the spectra for the local jet conditions. At $y = {2.97}\;{\rm{mm}}$, near the nozzle exit, the measured pressure was $P = {2.05}\;{\rm{atm}}$, with ${\tau _2} = {{200}}\;{\rm{ps}}$. Sufficient sensitivity to pressure is achieved at this ${\tau _2}$ delay, where ${\tau _2}/{\tau _R}\;\sim\;{5.3}$. The difference between the ${\tau _1}$ and ${\tau _2}$ spectra at $y = {2.97}\;{\rm{mm}}$ is readily discernable upon inspection, with the spectral intensity envelope shifting toward higher wavenumbers, and the most intense ${{\rm{N}}_2}$ line changing from $J = {{6}}$ at ${\tau _2} = {{0}}\;{\rm{ps}}$ to $J = {{12}}$ at ${\tau _2} = {{200}}\;{\rm{ps}}$.

Further downstream, the pressure drops rapidly, with a commensurate increase in Raman lifetime, requiring increased probe pulse delays to maintain pressure sensitivity; ${\tau _2}$ increases to 400 ps (${\tau _2}/{\tau _R}\;\sim\;{4.08}$) at $y\;\sim{{6}}\;{\rm{mm}}$ to as high as ${\tau _2} = {{1000}}\;{\rm{ps}}$ (${\tau _2}/{\tau _R}\sim{1.7}$) just upstream of the Mach disk at $y\sim{11.5}\;{\rm{mm}}$. Visual inspection of the difference between ${\tau _1}$- and ${\tau _2}$-channel spectra still reveals noticeable pressure-dependent changes. In air, the pressure dependence results from two factors: (1) the above-described time-dependent shift in spectral intensity toward high wavenumbers, and (2) a strong time dependence of the ${{\rm{O}}_2}$ line intensities resulting from triplet-state beating effects [32,33,55], as per the $N$- and $J\!$-dependent $\xi$ facror in Eq. (3). This effect adds pressure sensitivity in air that is not present when ${{\rm{N}}_2}$ is the test gas.

B. Effect of Raman Linewidths on CARS Pressure Measurements

To illustrate the importance of the linewidth selection to our CARS model, we compare $S$-branch ${{\rm{N}}_2} {-} {{\rm{N}}_2}$ linewidths derived from our time-domain CARS measurements at temperature and pressure conditions present in the underexpanded jet flow [38] to those obtained using the $Q$-branch predicted values from the MEG model with the constants from flame temperature measurements of Lavorel et al. [56]. Calculated $Q$-branch linewidths are extrapolated to low temperatures, and the random phase approximation, ${{{\Gamma}}_{\rm{S}}}(J) = 1/2[{{{\Gamma}}_Q}(J) +\def\LDeqbreak{} {{{\Gamma}}_Q}({J + 2})]$ [42], is used to arrive at pure-rotational $S$-branch linewidths. Both sets of linewidths are applied in Eq. (2), and the calculated CARS spectra are compared against experimental spectra recorded in a pure ${{\rm{N}}_2}$ underexpanded jet. Comparisons between predicted and experimental spectra at temperatures of $T = {{80}}$ and 100 K, with $P = {0.08}$ and 0.18 atm, respectively, are shown in Fig. 6. At $\tau = {{0}}\;{\rm{ps}}$, both linewidth models produce results that are indistinguishable for these low-pressure/low-temperature conditions; even with a relatively long-duration 60-ps FWHM probe pulse, as these zero-delay spectra are essentially collision free.

 

Fig. 6. Comparison of different linewidth models to experimental spectra recorded at various time delays for 80 K and 0.08 atm (top row) and 100 K and 0.18 atm (bottom row). The theoretical spectra were not fit to the experimental spectra in these plots; the $T$ and $P$ model values were kept constant to illustrate the impact of different linewidth values in various conditions. The summation of the absolute value of the residuals shown at the bottom of each spectrum is displayed on the right side of the figure as a function of probe delay.

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The effect of different linewidth values is clear in predicted CARS spectra when the probe delay is increased, as observed in the second and third columns of Fig. 6. The $J$ dependence of the extrapolated MEG model linewidths is larger than the $J$-dependent dispersion that was observed experimentally at low temperatures [38]. This results in a bias of the spectral intensity envelope towards higher $J$ states with increasing probe delay when MEG linewidths are used, resulting in larger residual values for each resonant peak in the MEG model spectra (green) when compared to spectra computed using our $S$-branch measurements [38], shown in red. The summation of the absolute value of the residual as a function of probe delay at each temperature is seen in the right-most column of Fig. 6, with a notable decrease in residual when proper low-temperature linewidths are employed. Note that the rise in overall residual with increasing probe delay is due to a corresponding decrease in signal-to-noise of the experimentally measured spectra.

Even with these notable discrepancies between the two linewidth models, one can still utilize the extrapolated MEG model and achieve reasonable fits to measured CARS spectra, but with a systematically low bias error in the resulting best-fit pressure values. Example pressure realizations for select probe delays under room conditions and for three locations in a pure-nitrogen jet are shown in Fig. 7. Using the procedure outlined in this paper, temperature is extracted from fits to near-collision-free spectra acquired at $\tau = {{0}}\;{\rm{ps}}$, while pressure is determined from spectra acquired over a wide range of probe delays from $\tau = {{50 {-} 1000}}\;{\rm{ps}}$. All spectra used to generate these data represent the average of 50 laser shots.

 

Fig. 7. (a) Single exposure schlieren image of the sonic jet with the three jet positions examined in (b) marked with the coinciding measured temperature. (b) Fitted pressure from the average of 50 single-shot spectra as a function of probe delay for the three positions in the jet and in room air using the MEG model, our model, and linewidth data from other literature sources. Raman lifetimes for $J = {{6}}$ are shown in gray for each position. (c) Example pressure fits for two different probe delays using our model at room temperature (294 K) and in the jet (79 K). These examples are all in pure nitrogen.

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At room temperature and pressure ($T = {{294}}\;{\rm{K}}$, $P = {0.82}\;{\rm{atm}}$), the dependence of the CARS-measured pressure on probe delay is illustrated in Fig. 7(b) for six different sets of published linewidth data, including our recent results [38], similar direct time-domain CARS measurements of the rotational $S$-branch decay at room temperature [57–59], stimulated Raman measurements of pure-rotational $S$-branch linewidths [60], and the MEG-extrapolated calculations discussed above. The MEG model results in an underprediction of the local atmospheric pressure by 6%–9% for $\tau \gt {{200}}\;{\rm{ps}}$. However, the independently measured $S$-branch linewidths based on time-domain CARS data from our laboratory, Kliewer et al. [57], and Miller et al. [58] all produce nearly identical pressure data that correspond to the local pressure of 0.82 atm within ${\sim}{{3}}\%$. In each case, the $J$ dependence of the linewidth data is similar, such that the relative decay of Raman peaks over time is similar and the shape of the calculated spectra are nearly the same. The even-$J$ linewidth data from Herring et al. [60] were interpolated to obtain the bounded odd-$J$ linewidths, and the results were linearly extrapolated for $J \gt {{14}}$. Even though the magnitudes of the linewidths from Herring et al. are ${\sim}{{15}}\%$ greater than all other datasets [38], the $J$ dependence is similar and therefore also produces a similar pressure measurement. Pressure fits for probe delays near or less than ${\tau _R}$ feature systematic errors due to the lack of pressure sensitivity at short probe delays. Recent ${{\rm{N}}_2} \!-\! {{\rm{N}}_2}$ $S$-branch measurements from Meiβner et al. [59] have a different $J$-dependent shape than the other data sets, with a smaller relative decrease of linewidth magnitude with respect to increasing $J$, resulting in an overestimation of the pressure for the room-temperature results in Fig. 7(b).

The impact of low-temperature linewidth selection is illustrated in the ${{\rm{N}}_2}$ jet pressure measurements, presented in Fig. 7(b) for $T = {{79}}$, 115, and 140 K with corresponding pressures of 0.079, 0.29, and 0.59 atm, respectively, calculated using CARS-measured temperatures and assuming an isentropic expansion along the jet centerline and shown as black, dashed lines in Fig. 7(b). Use of the extrapolated-MEG model results in CARS-measured pressures that are 23%–31% lower than the expected values from isentropic relations. As we might expect, the introduction of appropriate low-temperature linewidths improves CARS pressures, which remain within 6% of isentropic calculations for reasonably long probe delays. Representative fits with our model to experimental data at $T = {{294}}\;{\rm{K}}$ and 79 K are shown in Fig. 7(c) for both different late probe delays using the most relevant linewidth data [38]. An example of the effect of linewidths on the total axial pressure profiles are shown in Supplement 1.

C. Single-Laser-Shot Axial Temperature and Pressure Profiles

Accumulating all 50 single-laser-shot results for each probe delay across the center 80 rows of each ROI and retaining only those spectra below a pre-set cutoff value of the fit residual results in the temperature and pressure profiles in air shown in Fig. 8. Results for multiple ${\tau _2}$ probe delays are shown for each ROI, and color coded on the plots, to illustrate the sensitivity of the pressure measurement to this probe pulse delay. Single-laser-shot measurement precision is assessed by expressing the standard deviation as a percentage of the local mean for each ${\tau _2}$ delay and is shown below each of the axial profiles. Pressure measurements obtained within the first ${\sim}{{2}}\;{\rm{mm}}$ of the jet nozzle exit are excluded due to high fit residuals within that region, resulting from rapid Raman coherence decay and low CARS signal levels. Temperature measurements are largely independent of the ${\tau _2}$ delay, which is to be expected since the spectra acquired at ${\tau _1} = {{0}}\;{\rm{ps}}$ delay are much less sensitive to collisions, such that the impact of pressure on CARS thermometry is largely a second-order effect. The precision of the temperature measurements approaches 1% of the mean upstream of the Mach disk, but is otherwise higher (near 4%) at the ROI boundaries near the edges of the laser sheets where the signal strength is weaker. The highest spread in measured temperatures is expected downstream of the Mach disk, where the inherent measurement precision may be combined with real fluid-dynamic fluctuations. The ability of the CARS-imaging technique to capture shocks is also highlighted in Fig. 8, where the temperature rise may be observed to occur over ${\sim}{{6}}$ pixels, or 318 µm. This value is larger than the estimated 53-µm spatial resolution of the imaging setup and could be impacted by curvature of the Mach disk within the ${\sim}1 {\text{-mm}}$ beam-overlap region and additionally by beam-steering effects.

 

Fig. 8. Single-shot (a) temperature and (b) pressure results throughout four ROIs on the jet centerline. The top portion of the figure contains single-laser-shot results at the indicated, color-coded probe pulse delay with the average encompassing all probe delays overlaid in solid black circles. Open black circles, where visible, indicate the calculated pressure based on isentropic expansion to the measured temperature. The average Raman lifetime, calculated using Eq. (2) with $J = {{6}}$ and the average pressure measurement at each location, is shown on the right-hand vertical axis of the pressure plot, and displayed using gray dots. The bottom portion of the figure highlights the standard deviation about the mean for each probe delay, with $\tau /{\tau _R}$ highlighted on the right $y$ axis of the pressure precision plot using the longest probe delay for each ROI. These measurements are in air.

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The quality of our pressure measurements is much more dependent on the ${\tau _2}$ probe delay in relation to the local value of ${\tau _R}$, with this Raman lifetime shown in gray, and associated with values on the right-hand vertical axis in Fig. 8(b). Relative to the high-precision temperature data, a larger, ${\tau _2}$-delay-dependent spread in measured pressures is observed throughout the jet, across all four ROIs. Pressure-measurement precision is generally poor for low ${\tau _2}/{\tau _R}$ and improves with increasing ${\tau _2}$ in a regime with sufficient single-laser-shot signal strength. ROIs 1 and 4, near the jet exit and downstream of the Mach disk, respectively, exhibit low variation in ${\tau _R}$, so that axial variation in the precision is low. In ROI 1, where the average ${\tau _R}$ is ${\text{ps}}$, ${\sigma _P}/P$ drops from 8.9% at ${\tau _2} = {{150}}\;{\rm{ps}}$ to 4.2% at ${\tau _2} = {{250}}\;{\rm{ps}}$, or ${\tau _2}/{\tau _R}\;\sim{{3.6 \!-\! 6}}$. Within ROI 4, ${\tau _R}\;\sim{{100}}\;{\rm{ps}}$ and measured pressures improve, with precision ${\sim}{18.9}\%$, 7.3%, to 4.2% from 13 to 17 mm at probe delays of 200, 300, and 400 ps, or a range of ${\tau _2}/{\tau _R}\;\sim{{2 \!-\! 4}}$, respectively. Within ROIs 2 and 3, variations in ${\tau _R}$ are more significant, at ${4.5} \times$ and ${4.3} \times$, respectively, over the axial domain. For a fixed probe delay, ${\tau _2}/{\tau _R}$ decreases in the downstream direction, with a coinciding degradation of measurement precision.

The accuracy of the CARS-measured pressures was investigated by computing the expected pressure based on isentropic expansion to the measured temperature profile along the jet axis, using

 

Fig. 9. Measurement precision about the mean for all four ROIs as a function of probe delay about the local Raman lifetime for $J = {{6}}$. Marker style and color are consistent with Fig. 8. The dashed black line is the best fit exponential decay.

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CARS-measured pressures rise to $P = {0.9}$ atm within the 318-µm region of the observed temperature rise through the Mach disk and attains a level 8%−20% lower than the estimated value at $y = {{12.3 \!-\! 13.2}}\;{\rm{mm}}$—a region where measured temperatures and pressures are still rising. CARS pressures reach a maximum near 1.2 atm (${+}{0.5}\%$) near $y = {{14}}\;{\rm{mm}}$ and gradually decay downstream to 1.08 atm (${-}{9.2}\%$) at $y = {{17}}\;{\rm{mm}}$. Woodmansee et al. [14] observed similar gradual pressure rise, plateau, and decay downstream of the Mach disk in their CARS jet pressure measurements, and we should not expect the downstream pressure to be constant; the jet begins the development of a second shock diamond structure, as colder and higher-velocity annular flow begins to mix with the shock-heated core flow; therefore, this decrease in pressure is expected.

The precision of our CARS pressure measurements over all four axial ROIs is summarized in Fig. 9 by relating single-shot precision to the probe delay in units of Raman lifetimes, as per Eq. (4). The quality of the pressure measurements improves significantly until $\tau /{\tau _R}$ reaches approximately three to four Raman lifetimes, a trend similar to the one in the room-temperature ${{\rm{N}}_2}$ results by Kearney and Danehy [25]. A fit of the data to a three-parameter exponential decay includes a vertical offset of 5.79%, with a precision of 5.9% at ${\tau _2}/{\tau _R} = {4.0}$. The average measurement precision for all data points with ${\tau _2}/{\tau _R} \ge {4.0}$ is 5.0%. The fitted and average precisions represent an ${\sim}{{2.5 {-} 3}} \times$ increase in the lower bound precision from the 2% value of the original time-delayed CARS barometry work [25] in a well-controlled, room-temperature environment. The observed scatter in our single-laser-shot CARS pressures is a combination of instrument noise and real jet fluctuations, which are expected to be small in the jet core flow, particularly upstream of the Mach disk. For all laser shots shown in Fig. 8, 66.9% feature a precision of ${\sigma _P} \le {{10}}\%$.

 

Fig. 10. Comparison of measured pressure to isentropic pressure across ROIs 1–3 upstream of the Mach disk. Mean values are plotted along with error bars representing the standard deviation for all probe delays considered. Errors across the four ROIs are shown in the bottom plot.

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The accuracy of the measured pressure is defined here as the deviation from the calculated isentropic pressure estimate, per Eq. (5). Results are shown in Fig. 10 as a comparison in terms of absolute pressure magnitude and percent error. Overall, CARS-measured pressures are on average within 5.0% of isentropic estimates in the central 3 mm of the measurement line, where signal-to-noise is highest, for ROIs 1, 2, and 4, where $P \gt {0.2}\;{\rm{atm}}$. In low-pressure ROI 3, agreement between CARS and isentropic-estimated pressures reaches the 30%–60%, seen in the lower panel of Fig. 10. This high level of disagreement at low pressure results from multiple sources: (1) the impact of collisions is much more gradual at low pressure, such that pressure-induced changes to the measured spectra, even at the 1-ns maximum probe time delay in our experiments, become much more subtle; (2) propagation of temperature-measurement uncertainties into the measured pressure becomes a more significant fraction of the absolute pressure; (3) impact of temperature-measurement uncertainty on the isentropic pressure estimate, through the strong $\gamma /(\gamma - {{1}}) = {3.5}$ power-law dependence in Eq. (5), also becomes significant relative to the absolute pressure. Systematic differences in the pressure measurements also appear at the ROI boundaries, where small differences in the measured temperatures between ROIs may result in more significant changes to the pressure data; this is particularly apparent in the overlap region at $y\;\sim{{4 {-} 5}}\;{\rm{mm}}$, where the edges of ROI 1 and ROI 2 overlap. Also, while difficult to see in magnitude in Fig. 10(a), the measured pressure just before the Mach disk is 80% lower than the expected value of 0.1 atm. In this region, the $\tau /{\tau _R}$ value is not ideal (${\sim}{{1}}$) so these deviations are not surprising. If measurements are needed in these conditions, longer probe time delays are required, which may call for higher probe pulse energies to produce usable signal levels. Such measurements have been demonstrated in a benchtop cryostat [28]. However, 58.3% of the average measurement stations have ${\le} {{10}}\%$ deviation from isentropic values, with 35.8% of positions $P\le {{5}}\%$. Considering all the measurement locations and local conditions with ${\tau _2}/{\tau _R} \ge {4.0}$, the average pressure accuracy is 7.9%.

Measurement accuracy is a strong function of the transition linewidths and the uncertainty in the timing of the probe delay. Measurement biases based on the set of linewidths used for nitrogen and air environments are examined further in room-temperature air and 50-shot average spectra for ROIs 1–3 in Supplement 1.

5. SUMMARY AND CONCLUSION

A dual-probe hybrid fs–ps, pure-rotational CARS instrument has been presented for 1D imaging of simultaneous temperature and pressure measurements in air and pure ${{\rm{N}}_2}$ on a single-laser-shot basis. The spatial resolution of our CARS instrument along the 1D measurement line was estimated at 53 µm under room conditions, with an observed shock layer thickness of 318 µm, based on the rise of the CARS-measured temperature profile across a normal shock in the near-field of an underexpanded jet flow. The hybrid frequency–time domain detection afforded by ultrashort laser pulses enables detection at two separate time delays, greatly minimizing the impact of collisions on CARS thermometry and resulting in a more direct observation of pressure through the time-dependent dephasing of CARS spectra. Application of pure-rotational CARS significantly delays the onset of temperature-dependent (Doppler) dephasing effects, such that Raman linewidths remain primarily pressure broadened at low pressures of interest in cold-flow supersonic test facilities.

Theoretical considerations in pure ${{\rm{N}}_2}$ and air were discussed to emphasize the sensitivity of CARS spectra to pressure, in particular through the relative shift in rotational line intensity across the ${{\rm{N}}_2}$ and ${{\rm{O}}_2}$ spectrum, and time-dependent beating of oxygen transitions in a low-temperature and -${\rm{pressure}}$ environment. An iterative pre-computed library fitting scheme was presented to determine unique solutions for both temperature and pressure from a single generated Raman coherence with early ($\tau \sim{{0}}\;{\rm{ps}}$) and late ($\tau \;\sim{{150 \!-\! 1000}}\;{\rm{ps}}$) probe time delays. Three iterations proved sufficient for convergence of the fitted scalars. Our CARS spectral modeling code has been updated to include the effects of triplet-state coherence beating in ${{\rm{O}}_2}$ [32,33] and newly available low-temperature measurements of purely anisotropic rotational Raman linewidths [38]. The impact of updated low-temperature linewidths was a 20%−30% rise in CARS-measured pressures at temperatures of $T = {{79 {-} 294}}\;{\rm{K}}$, when compared with results inferred from low-temperature extrapolation of $Q$-branch linewidths from MEG-model calculations.

Our short-pulse CARS instrument was demonstrated in a canonical underexpanded air jet, where this method can be systematically evaluated over a wider range of temperature and pressure conditions than in previous work [25,26,29]. The steady barrel-shock region, upstream of the Mach disk, provided a stable temperature and pressure environment to evaluate the quality of our CARS measurements over a wide range of conditions. Single-laser-shot CARS temperatures generally displayed rms fluctuations of just 1%–2% of mean values within the central 80% of the laser sheets, where signal-to-noise was most optimal. Single-laser-shot CARS pressure measurements exhibited an expected trade-off with probe time delay, resulting from increased pressure sensitivity at long probe time delays with a CARS signal strength penalty due to collision-induced dephasing of the Raman coherence.

The precision of our CARS pressure measurements was generally optimized at delays greater than $\tau /{\tau _R}$ of about four Raman lifetimes, which is consistent with a value of $\tau /{\tau _R} = {{3}}$ observed for room-temperature observations in pure ${{\rm{N}}_2}$ by Kearney and Danehy [25]. Pressure measurement accuracy was evaluated by comparing CARS measurements to calculated pressures based on an isentropic expansion to the CARS-measured axial jet temperature profile. Agreement between CARS and isentropic pressure estimates was on average 5.0% in the central portions of the CARS laser sheets, for $P \gt {0.2}\;{\rm{atm}}$. At lower pressures, combined uncertainties resulting from limited pressure sensitivity at the available probe time delays and propagation of temperature-measurement uncertainty into both the CARS-measured and isentropic estimates of pressure made a comparison more difficult, with observed differences of 30%–60%. Across the entire measurement domain, the pressure accuracy was on average 7.9% for ${\tau _2}/{\tau _R} \ge {4.0}$.

This hybrid time–frequency domain rotational CARS approach has application potential in cold-flow supersonic facilities, such as the Mach-8 wind-tunnel at Sandia [28], where the range of available probe pulse delays on the pressure-measurement channel can be increased to several nanoseconds, to provide increased low-pressure sensitivity.