Vibrational CARS measurements in a near-atmospheric pressure plasma jet in nitrogen: I. Measurement procedure and results

There are multiple reviews about CARS as a diagnostic method for temperature and concentration measurements in gases and other media [28–32], so here we will only give a short summary of the aspects relevant for this work. CARS is a non-linear optical process of third order, in which three electromagnetic waves with angular frequencies ω_Pu , ω_S and ω_Pr induce a non-linear polarization in a medium at frequency $\omega_{aS} = \omega_{Pu} + \omega_{Pr} - \omega_S$. ω_Pu , ω_S , ω_Pr and ω_aS are called pump, Stokes, probe and anti-Stokes beam respectively. The energy scheme of the process is illustrated in figure 1 for two transitions. The process is resonant if $\omega_{Pu} - \omega_S$ is close to a state transition of the probed medium. In this work the transition energy ћω_k belongs to vibrational transitions of nitrogen. The intensity of the generated anti-Stokes beam is proportional to the square of the so called CARS susceptibility and the product of the three input laser intensities:

$$\begin{equation} I_{aS}(\omega_R) \propto |\chi_{CARS}(\omega_R)|^2 I_{Pu} I_{Pr} I_S(\omega_R) \end{equation} \tag{ 1 }$$

where $\omega_R = \omega_{Pu} - \omega_S$ is the Raman shift. The CARS susceptibility in the pressure broadening regime for parallel polarized light far from electronic resonances can be written as

$$\begin{equation} \chi_{CARS}(\omega) = \chi_{NR} + \chi_R(\omega) = \chi_{NR} + \sum_k \frac{a_k}{\omega_k - \omega - i\Gamma_k} \end{equation} \tag{ 2 }$$

where the sum includes all possible ro-vibrational transitions k with the energy difference $\hbar\omega_k = E(v_{k,f}, J_{k,f}) - E(v_{k,i}, J_{k,i})$ and the linewidth (HWHM) Γ_k . The ro-vibrational energies are calculated with the Dunham coefficients given in table 1 by

$$\begin{align} E(v,J) = \sum_{i,j} Y_{ij} \left(v+\frac{1}{2}\right)^i (J(J+1))^j. \end{align} \tag{ 3 }$$

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Table 1. Dunham coefficients Y_ij in units of cm⁻¹. The data marked with ${}^{\mathrm{a}}$ is from [33], ${}^{\mathrm{b}}$ from [34] and ${}^{\mathrm{c} }$ from [35, 36].

i\j	0	1	2	3
0	0	1.998$^{\mathrm{c}}$	$-5.76 \times 10^{-6}{}^{\mathrm{c}}$	0
1	$2358.535^{\mathrm{a}}$	$-0.017249{}^{\mathrm{a}}$	$-8.32\times 10^{-9}{}^{\mathrm{b}}$	$-2.58\times 10^{-13}{}^{\mathrm{b}}$
2	$-14.3074^{\mathrm{a}}$	$-3.24\times 10^{-5}{}^{\mathrm{a}}$	$-4.2\times 10^{-10}{}^{\mathrm{b}}$	0
3	$-4.98\times 10^{-3}{}^{\mathrm{a}}$	0	0	0
4	$-1.22\times 10^{-4}{}^{\mathrm{a}}$	0	0	0

With the assumptions above the amplitude of the lines depend on the population difference between the lower and the upper state of the transition $\Delta N_k = N_{k,l} - N_{k,u}$ and the differential cross section for spontaneous Raman scattering:

$$\begin{equation} a_k = \left( \frac{c^4}{\hbar \omega_S^4} \right) \Delta N_k \left . \frac{\textrm{d}\sigma}{\textrm{d}\Omega} \right|_k. \end{equation} \tag{ 4 }$$

The Raman cross section for the Q branch (Δv = 1, ΔJ = 0) can be written as [27, 36]

$$\begin{equation} \left.\frac{\textrm{d}\sigma}{\textrm{d}\Omega}\right|_Q = \left(\frac{\omega_S}{c}\right)^4\frac{\hbar}{2\omega_0} \left[\frac{\alpha^{^{\prime} 2}}{M} + \frac{4}{45} \frac{\gamma^{^{\prime} 2}}{M}b^J_J\right](v+1). \end{equation} \tag{ 5 }$$

The cross sections for the O and S branch are given by [27, 36]

$$\begin{equation} \left.\frac{\textrm{d}\sigma}{\textrm{d}\Omega}\right|_O = \left(\frac{\omega_S}{c}\right)^4\frac{\hbar}{2\omega_0} \frac{4}{45} \frac{\gamma^{^{\prime} 2}}{M}b^J_{J+2}(v+1)C_O(J) \end{equation} \tag{ 6 }$$

$$\begin{equation} \left.\frac{\textrm{d}\sigma}{\textrm{d}\Omega}\right|_S = \left(\frac{\omega_S}{c}\right)^4\frac{\hbar}{2\omega_0} \frac{4}{45} \frac{\gamma^{^{\prime} 2}}{M}b^J_{J-2}(v+1)C_S(J) \end{equation} \tag{ 7 }$$

with the centrifugal force corrections [27]

$$\begin{equation} C_O(J) = \left(1+4\frac{B_e}{\omega_e} \mu (2J-1)\right)^2 \end{equation} \tag{ 8 }$$

$$\begin{equation} C_S(J) = \left(1-4\frac{B_e}{\omega_e} \mu (2J+3)\right)^2. \end{equation} \tag{ 9 }$$

In the equations above ω₀ is the oscillator frequency of the molecule, $\frac{\alpha^{^{\prime} 2}}{M}$ and $\frac{\gamma^{^{\prime} 2}}{M}$ are the squared isotropic and anisotropic derived polarizabilities over the reduced mass of the vibration, B_e and ω_e the Herzberg molecular parameters and

$$\begin{equation} b_{J}^J = \frac{J(J+1)}{(2J-1)(2J+3)} \end{equation} \tag{ 10 }$$

$$\begin{equation} b_{J+2}^{J} = \frac{3(J+1)(J+2)}{2(2J+1)(2J+3)} \end{equation} \tag{ 11 }$$

$$\begin{equation} b_{J-2}^{J} = \frac{3J(J-1)}{2(2J+1)(2J-1)} \end{equation} \tag{ 12 }$$

are the Placzek–Teller coefficients for diatomic molecules [36].

The constants $\frac{\alpha^{^{\prime} 2}}{M}$ and $\frac{\gamma^{^{\prime} 2}}{M}$ in (5)–(7) are calculated following the approach of [27] from experimental measurements of the Q branch cross section for $v = 0\rightarrow 1$

$$\begin{equation} \left.\frac{\mathrm{d} \sigma}{\Omega}\right|_{Q,v = 0} \approx \frac{\hbar \omega_S^4}{2\omega_0c^4}\frac{\alpha^{^{\prime} 2}}{M} \end{equation} \tag{ 13 }$$

and the Q branch depolarization ratio

$$\begin{equation} \rho_J = \frac{3b_{JJ} (\gamma^{^{\prime}}/\alpha^{^{\prime}})^2}{45 + 4b_{JJ} (\gamma^{^{\prime}}/\alpha^{^{\prime}})^2}. \end{equation} \tag{ 14 }$$

While the line positions ω_k in (2) depend only on the molecular parameters, the linewidths Γ_k depend on the gas mixture, the pressure and the temperature. There exist multiple scaling laws for the linewidths in nitrogen, for example the so-called polynomial-differential exponential gap law or the modified exponential gap law (MEGL) [37]. In general the linewidths are calculated with [37]

$$\begin{equation} \Gamma_j = \sum_{i\neq j}\gamma_{ij}, \end{equation} \tag{ 15 }$$

where γ_ij is the collisional transfer rate for rotational states $j\rightarrow i$. Note the different index notation compared to the one used in (2), where the index k corresponds to a transition between to ro-vibrational states. In (15) i, j correspond to rotational quantum numbers. The relation between the two notations is that—for a Q branch transition—j is the rotational quantum number which stays constant during the transition k. For O and S branch transitions the rotational quantum numbers u and l for the upper and lower state respectively are different and the linewidth for transition k is calculated as

$$\begin{equation} \Gamma_k = \frac{1}{2} \left( \Gamma_u + \Gamma_l\right). \end{equation} \tag{ 16 }$$

If the rotational states are Boltzmann distributed, detailed balance requires for the forward and reverse rates:

$$\begin{equation} g(i)\exp(-E_i/k_BT)\gamma_{ji} = g(j)\exp(-E_j/k_BT)\gamma_{ij} \end{equation} \tag{ 17 }$$

where g are the statistical weights for the rotational states. In this work the MEGL is used which gives for the uprates [37]

$$\begin{align} \gamma_{ji} & = p\alpha\frac{1-e^{-m}}{1-e^{-mT/T_0}}\left(\frac{T_0}{T}\right)^{1/2} \left(\frac{1+1.5E_i/k_BT\delta}{1+1.5E_i/k_BT}\right)^2\nonumber\\ & \quad\times\exp(-\beta|\Delta E_{ji}|/k_BT) \end{align} \tag{ 18 }$$

where p is the pressure in bar, $\Delta E_{ji}=E_j-E_i$ the energy difference between states $j$ and $i$ and T the gas temperature in K (in the following discussions assumed to be equal to the rotational temperature). The parameters for nitrogen are $\alpha = 0.0231~\textrm{cm}^{-1}~\textrm{bar}^{-1}$, β = 1.67, $\delta$= 1.21, m = 0.1487 and T₀ = 295 K [37].

In this context it should be mentioned, that the line profile in (2) is only valid in low or medium pressures, as it assumes that the lines are isolated. For pressures close to atmospheric pressure this is not necessarily true and more complex line shapes are required like they are given in the rotational diffusion model or the exponential gap model [37]. To verify if the isolated line model is still appropriate under the conditions in this work (nitrogen at 200 mbar), nitrogen CARS spectra of the Q branch transition $v = 0\rightarrow 1$ are calculated for 200 mbar and 1 bar with the CARSFT [38] software for both the isolated lines and the more complex exponential gap model. The results are compared in figure 2. While there is a noticeable difference between the isolated lines and the exponential gap model for 1 bar at low rotational quantum numbers, the curves at 200 mbar are indistinguishable. As conclusion the isolated lines model is sufficient for the conditions investigated here.

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As seen in (2) and (4) the intensity of the generated anti-Stokes beam depends on the population density differences of the participating states. This means by scanning the Stokes beam over multiple transitions and measuring the anti-Stokes intensity, information about the ro-VDFs can be extracted. Equivalently, a Stokes beam with a continuous broadband spectrum can be used to record the whole anti-Stokes spectrum with a single laser shot as is done in this work. One way of gaining information about the VDF is by fitting theoretically calculated spectra to the measured ones. This approach is often used in rotational temperature measurements, where the rotational structure needs to be resolved and lines can be overlapping at certain pressures or even motional narrowing [39] can occur.

For most plasma sources the vibrational and rotational temperature are typically not the same. In these cases spectrometers with a larger spectral range are used to monitor multiple vibrational lines. This usually results in a lower spectral resolution so that the rotational structure of the lines can not be resolved.

A popular approach to determine the vibrational distribution from these low-resolution spectra is to first normalize the spectrum to either the Stokes laser spectrum or a non-resonant spectrum from another gas without vibrational resonances in that wavelength region. Then the square root of the normalized spectrum is taken and for each transition the integral over the corresponding peak I_v,v + 1 is calculated. Under the assumption that the non-resonant and the real part of the resonant susceptibility are negligible compared to the imaginary part of the resonant susceptibility, it follows that

$$\begin{equation} I_{v, v+1} \propto \Delta N_{v,v+1}(v+1), \end{equation} \tag{ 19 }$$

where the factor v + 1 accounts for the dependence of the Raman cross section on the vibrational quantum number (see equation (5)).

As this approach relies heavily on the assumptions, that the lines can be viewed as non-interacting and $\chi_{NR},\;\text{Re} (\chi_R) \ll \text{Im} (\chi_R)$, which are certainly not valid for small admixtures of the probed gas species [37] and probably also not true for the weak signals of the higher vibrational states, in this work the population densities are extracted by fitting the full theoretical spectra to the measured ones.

To calculate the population differences in the susceptibility expression different models are used depending on the corresponding conditions. One possibility would be to assume that the vibrational states are Boltzmann distributed but with a different temperature than the rotational states like it was done for example by Messina et al [40]:

$$\begin{equation} N(v,J) = \frac{g(J)}{Z} \mathrm{e}^{-\frac{E(v,J)-E(v,0)}{k_BT_{rot}}} \times \mathrm{e}^{-\frac{E(v,0)-E(0,0)}{k_BT_{vib}}} \end{equation} \tag{ 20 }$$

with the partition function

$$\begin{equation} Z = \sum_{v,J} g(J) \mathrm{e}^{-\frac{E(v,J)-E(v,0)}{k_BT_{rot}}} \times \mathrm{e}^{-\frac{E(v,0)-E(0,0)}{k_BT_{vib}}} \end{equation} \tag{ 21 }$$

and $T_{vib} \neq T_{rot}$. Another approach would be the Treanor distribution assuming dominant V–V transfer collisions between the molecules:

$$\begin{equation} N(v,J) = \frac{g(J)}{Z} \mathrm{e}^{-\frac{E(v,J)-E(v,0)}{k_BT_{rot}}} \times \mathrm{e}^{-\frac{v(E(1,0)-E(0,0))}{k_BT_{vib}}} \times \mathrm{e}^{-\frac{v(E(v,0)-E(1,0))}{k_BT_{rot}}} \end{equation} \tag{ 22 }$$

with

$$\begin{equation} Z = \sum_{v,J} g(J) \mathrm{e}^{-\frac{E(v,J)-E(v,0)}{k_BT_{rot}}} \times \mathrm{e}^{-\frac{v(E(1,0)-E(0,0))}{k_BT_{vib}}} \times \mathrm{e}^{-\frac{v(E(v,0)-E(1,0))}{k_BT_{rot}}}. \end{equation} \tag{ 23 }$$

The degeneracy of the rotational states is given by

$$\begin{equation} g(J) = \left\{\begin{array}{ll} \hfill 6(2J+1),& \textrm{if }~J~\textrm{even} \\ \hfill 3(2J+1),& \textrm{if }~J~\textrm{odd.} \end{array}\right. \end{equation} \tag{ 24 }$$

Both distributions assume a single vibrational temperature and are for the temperatures and relatively low vibrational states measured in this work essentially indistinguishable considering the uncertainty of the CARS measurements.

As in previous measurements of the ro-vibrational distribution [19–22], here, it was found that a single vibrational temperature is not sufficient to fully describe the vibrational distribution for $v\gt3$. Therefore, here a distribution function considering two vibrational temperatures, a cold and a hot one, is introduced similar to the one used in [41]. The subtle difference compared to [41] is that the hot part of the distribution only includes states with v ≥ 1. This is rooted in the interpretation that the hot distribution is made up by molecules which are excited during the current discharge cycle. The further motivation of this two-temperature distribution function (TTDF) is discussed in more detail in the companion paper [42] and here only the formula is given:

$$\begin{align} N(v,J; T_{rot}, T_{vib,c}, T_{vib,h}, R_{h}) & = g(J) \mathrm{e}^{-\frac{E(v,J) - E(v,0)}{k_BT_{rot}}} \nonumber \\ & \times \Bigl[ \frac{1-R_h}{Z_c}\mathrm{e}^{-\frac{E(v,0)-E(0,0)}{k_BT_{vib,c}}}\nonumber\\ &\quad\ \ + \underbrace{\frac{R_h}{Z_h}\mathrm{e}^{-\frac{E(v,0)-E(0,0)}{k_BT_{vib,h}}}}_{\textrm{for }v\gt0} \Bigr] \end{align} \tag{ 25 }$$

with the partition function for the "cold" molecules

$$\begin{equation} Z_c = \sum_{v,J} g(J) \mathrm{e}^{-\frac{E(v,J) - E(v,0)}{k_BT_{rot}}} \times\mathrm{e}^{-\frac{E(v,0)-E(0,0)}{k_BT_{vib,c}}} \end{equation} \tag{ 26 }$$

and

$$\begin{equation} Z_h = \sum_{v\gt0,J} g(J) \mathrm{e}^{-\frac{E(v,J) - E(v,0)}{k_BT_{rot}}} \times\mathrm{e}^{-\frac{E(v,0)-E(0,0)}{k_BT_{vib,h}}} \end{equation} \tag{ 27 }$$

for the "hot" molecules.

The last condition investigated in this work is several microseconds after the discharge pulse. On this timescale the V–V collisions determine the temporal behavior and the two-temperature description is not valid anymore. Indeed, all of the distributions mentioned above fail in this regime. To make a distribution function independent analysis possible only the Q branch, Δv = 1, ΔJ = 0 is considered.

If the rotational states in a vibrational level v follow a Boltzmann distribution $f_v(J, T_{v,rot})$ with rotational temperature T_v,rot , the population difference for a given transition $(v\rightarrow v+1; J)$ is

$$\begin{align} \Delta N_{v,v+1;J} & = N_v f_v(J, T_{v,rot}) - N_{v+1} f_{v+1}(J,T_{v+1,rot})\nonumber\\[6pt] &\approx (N_v - N_{v+1}) f(J, T_{rot})\nonumber\\[6pt] & = \Delta N_{v, v+1} f(J, T_{rot}). \end{align} \tag{ 28 }$$

Here, in the second step it is assumed that the rotational temperature in the different vibrational states is the same and that the vibrational corrections to the rotational distributions are negligible. In this way the fitting parameters concerning the vibrational states are reduced to ΔN_v,v + 1.

To benchmark our CARS code and the determination of vibrational temperatures from the fitting parameters several spectra are generated by the popular CARSFT code [38] with a spectral resolution similar to our experiments for thermal equilibrium conditions and evaluated by our fitting routines assuming Boltzmann distributed vibrational and rotational states where T_rot and T_vib are independent of each other. An example fit for an equilibrium temperature of 2000 K is shown in figure 3. In figure 4 the fitting results are shown for several spectra generated by the CARSFT code. The agreement for both the rotational and the vibrational temperature is very good over the whole range, and it is reasonable to assume that the code used in this work accurately describes real CARS spectra in the given temperature range even if $T_{rot}\neq T_{vib}$.

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The optical setup—a modified version of the one used in [53] and [54] for CARS-based electric field measurements—is shown in figure 6. The second harmonic of an injection seeded Nd:YAG laser (532 nm, Continuum Powerlite Precision II 9020, 6 ns FWHM, 20 Hz repetition rate) is used both as pump laser for two dye lasers (Radiant Dyes Jaguar D90MA) and as pump beam in the CARS process. One of the dye lasers is a narrowband dye laser, which is used as probe beam. The laser emission at 560 nm is produced by a Rhodamine 6G dye solution and a spectral width of about 0.05 cm⁻¹ is reached with a double grating configuration. Here, light with 560 nm was chosen for the probe beam to be able to perform CARS measurements in carbon dioxide and nitrogen simultaneously [55] at a later stage. In the broadband dye laser (BBDL) the double grating configuration for the wavelength selection is bypassed by a mirror to allow broadband operation. The desired spectrum of the BBDL—which is monitored by a broadband spectrometer—is reached by using a mixture of Pyrromethene 597 and Rhodamine B/101. Initially a mixture of Pyrromethene 597 and Pyrromethene 650 was used as in [56], but was found to drift to lower wavelengths too quickly. As an attempt to increase the lifetime of the mixture Pyrromethene 650 was replaced by the two Rhodamine dyes. To adjust the temporal overlap of the three laser beams two delay stages are used: One for the part of the Nd:YAG beam which is used in the CARS process and the other one for the pump beam of the BBDL. All three laser beams are first attenuated to pulse energies around 5 mJ and then focused to the probe volume by an achromatic lens with 50 cm focal length. A three-dimensional folded BOXCARS [57] geometry is used as depicted in figure 7. The CARS signal is produced only in the volume, where all three laser beams overlap. The spatial resolution perpendicular to the beam paths is therefore given by the smallest beam waist and should be in the order of 100 µm. To determine the spatial resolution along the beam path direction a glass plate of 1 mm thickness is scanned through the probe volume along the beam path. Due to its high density the glass plate generates a strong non-resonant CARS spectrum, when inserted in the probe volume. The spectrally integrated intensity generated this way is shown in figure 8 for different positions. An intensity profile with a full width at half maximum of slightly less than 5 mm is obtained, which will be assumed to be the spatial resolution in the following discussions. The anti-Stokes signal beam passes through a low pass filter to reduce stray light from the lasers and is guided to the CARS spectrometer (Jobin Yvon HR1000, 1800 lines mm⁻¹ holographic grating) with a PCO edge 4.2LT sCMOS camera as detector. Furthermore, a photo diode is used to measure the time stamp of the laser on the oscilloscope, which also measures the current–voltage waveforms of the plasma jet. The trigger of the Nd:YAG laser, the spectrometers and the oscilloscope (Lecroy WaveSurfer 510) are synchronized by a synchronization unit to allow the accumulation of single shot measurements. Finally, all measured data are collected by a custom made DAQ software on the computer.

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The emission of a Nd:YAG laser (EKSPLA SL234) at 1064 nm with a pulse duration of 100 ps is focused to the middle of the discharge gap by a lens with focal length of 200 mm. The light at the second harmonic (532 nm) generated in presence of the external field is separated from the fundamental harmonic by a dichroic mirror and detected by a PMT (Hamamatsu H11901-210) with laser line filter (Thorlabs FL532-1) installed at its entrance. For the schematic view and more detailed description of the experimental setup used for the measurements of the electric field, please see [52], where a discharge with the same geometry was investigated.

The electrical setup of the plasma jet is depicted in figure 9.

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A negative DC high voltage (Heinzinger LNC 6000-10neg) is applied to the input of a high voltage switch (Behlke HTS 81). In order to protect the switch and increase the discharge stability the current is limited to ${\lt}~30~{\textrm{A}}$ by a 255 Ω series resistor between switch and cathode of the plasma jet. Additionally, a 2 kΩ resistor, installed in parallel to the discharge, ensures that residual charges are removed from the electrodes after each discharge pulse. The voltage at the cathode relative to ground is measured by a high voltage probe (Lecroy PPE6kV) and the current is measured between the anode and ground by a current probe (American Laser Systems, Model 711 Standard). The Behlke switch is triggered by a 1 kHz TTL signal which is synchronized to the 20 Hz trigger signal of the Nd:YAG laser through a frequency divider. The duration of the high voltage pulse can be controlled by the duration of the trigger pulses at a digital delay generator with a minimum pulse length of 150 ns.

The plasma jet itself consists of two plane parallel molybdenum electrodes with an area of 1 mm × 20 mm, which are positioned between two glass plates at a distance of 1 mm. Molybdenum is chosen as electrode material because of its relatively low sputter rate and the resulting longer lifetime of the electrodes. To allow operation in pure nitrogen at pressures around 200 mbar the electrode surfaces must be prepared carefully in order to avoid arcing. For this purpose the electrodes were polished with a final grit size of P2000.

To illustrate the gas flow the front and the side view of the jet are shown in figure 10. The gas enters the electrode gap through a hole in the lower glass plate and exits on both sides after being exposed to multiple discharge pulses due to the gas residence time. This design allows the laser beams to pass through the jet along the electrodes. It should be noted, that the measurement volume of the CARS setup is not in the center of the jet as there mostly unexcited gas would be measured, which was not exposed to the discharge yet. Instead, it is positioned between the gas inlet in the center and one of the exits (see figure 10).

To ensure a controlled atmosphere the jet is enclosed in a discharge chamber which allows regulating the pressure and the gas flow through the jet.

An ICCD image of a single discharge pulse is shown in figure 11. No arcing is visible under the investigated discharge conditions and a relatively homogeneous discharge is achieved.

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The electric field during the current plateau is shown in figure 13 for different DC voltages used to feed the high voltage switch and different HV pulse durations. The electric field has a constant value of ≈ 3.5 kV cm⁻¹ or ≈ 81 Td during the plateau, which is about the same for the different pulse durations. Moreover, the electric field amplitude only very weakly depends on the DC voltage as well. Despite the different DC voltage, after the breakdown the cathode voltage is about the same, see figure 12. This can be explained by a higher electron density generated during the breakdown at the higher DC voltage leading to a higher electric current and consequently a higher voltage drop at the series resistor. According to calculations with Bolsig+ [58] using the IST-Lisbon data set [59] the measured field corresponds to a mean electron energy of 1.8 eV or an electron temperature of T_e  = 1.2 eV.

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To demonstrate the necessity of the TTDF (25) for the CARS fitting routine, in figure 14 a measured spectrum with its corresponding TTDF fit is compared with a theoretical spectrum for a single vibrational temperature. From the figure it is obvious, that a single vibrational temperature will not be able to reproduce the measured spectra during or shortly after the discharge. During the discharge pulse the full spectrum—consisting of O- and Q-branch transitions in the observed wavelength region—is fitted while assuming a two-temperature distribution according to (25). The results for the fitting parameters R_h and T_vib,h are shown in figures 15 and 16 respectively. The uncertainties of the fitting parameters are determined from the local Jacobian of the corresponding best fit solutions. During the discharge pulse the fraction of vibrationally hot molecules, R_h , increases linearly. This agrees well with the essentially constant current and electric field after the ignition—suggesting constant excitation conditions during the whole discharge pulse. It is clearly visible, that the slope of R_h is the highest for the pulse with the higher current amplitude (4 kV applied DC voltage) while the two measurements with 3 kV applied voltage share a lower slope and differ only by a small offset due to different initial conditions. Those initial conditions (weighted average over the first three data points) are 0.015 ± 0.003 (3 kV, 200 ns), 0.021 ± 0.004 (3 kV, 250 ns) and 0.022 ± 0.002 (4 kV, 200 ns). As the electric field is the same in all three cases, the difference in slope should be mainly caused by a higher electron density in the 4 kV case. A more detailed discussion of the slope, $\dot{R}_h$, is given in the companion paper [42]. The fact, that one of the measurements is performed with a longer pulse duration, is also reflected in figure 15 by a longer rise time of R_h (note the two vertical lines at about 200 and 250 ns marking the corresponding ends of the pulses).

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In regard to the temperature of the hot population it has to be noted that the values at the beginning, when the amount of excited molecules is small, are not reliable or suitable for a direct interpretation. Firstly, at these times only states up to v = 3 can be detected as the densities of the other states are below the detection limit. Secondly, (25) is proposed to analyze the characteristics of the newly excited molecules (for further details the reader is referred to the companion paper [42]). Before the discharge pulse there are no 'newly' excited molecules yet and the non-zero value of the fitting parameter R_h only accounts for the small deviation from a normal Boltzmann distribution remaining from the previous voltage pulse still visible in the excited states v = 1 − 3. Therefore T_vib,h suffers a large uncertainty, as it strongly depends on the higher states, which are not detected. So, to be able to use the two-temperature distribution in its original intent the newly excited molecules need to overtake the remaining deviation from a Boltzmann distribution. The criterion used here and in [42] is that R_h needs to at least surpass two times the average value of the data points before the ignition, so that the majority of the molecules belonging to the hot distribution consists of molecules excited during the current discharge pulse. The data points which are excluded from further discussion by the mentioned criterion are marked by a gray background in figure 16. Please note, that this criterion agrees quite well with the uncertainties of T_vib,h . Considering the previous remarks it stands out that T_vib,h stays constant for all three measurements in the usable range. While the 200 ns-pulse measurements for 3 kV with 5200 ± 300 K (averaged over all data points with $t\gt100~{\textrm{ns}}$) and the one for 4 kV with 5300 ± 200 K (average for $t\gt50~{\textrm{ns}}$) are very close to each other, the measurement for the 250 ns pulse deviates with 4600 ± 200 K (average for $t\gt100~{\textrm{ns}}$). This deviation is probably caused by a worse signal-to-noise ratio during that particular measurement set. The dye mixture of the broadband Stokes laser needs to be refreshed between measurement sets and while the spectrum of the Stokes beam is included in the calculation of the CARS spectra, the fluctuations of the Stokes beam result in different signal-to-noise ratios in the measured CARS signals for the different measurement sets. As this mainly affects the higher vibrational states due to their weaker signal, the effect is stronger on the fitting parameter influenced by those states, namely the hot vibrational temperature T_vib,h . In these cases, a small shift of the baseline could induce a systematic shift of the temperature, which is not necessarily reflected in the uncertainty determined from the local Jacobian. R_h on the other hand corresponds to the total amount of newly excited molecules and as the higher vibrational states only contribute by a small number—even at temperatures around 5000 K—the effect is small. This explains why there is no deviation from the expected behavior visible in figure 15. As one would expect the rotational temperature stays constant on time scales of the discharge with about T_rot  = 330 ± 8 K (mean value and standard deviation over all three measurement sets). Likewise, the vibrational temperatures of the cold bulk molecules stays nearly constant during the discharge pulse as well, but differ slightly for the different measurement sets. The two measurements with 3 kV applied DC voltage are quite close to each other with 1190 ± 40 K (200 ns pulse) and 1180 ± 40 K (250 ns pulse). For the 4 kV case the vibrational background temperature is slightly higher with 1300 ± 50 K.

As the densities of the vibrationally excited states are known from the fits, the gain of vibrational energy in the plasma volume during one discharge pulse can be calculated relative to the average energy of the first three measurement points (see figure 17).

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The energy deposited into the plasma up to the time t can be calculated from the measured voltage, U(t), and current, I(t), waveforms in figure 12 via

$$\begin{equation} \mathcal{E}(t) = \int_{t_0}^t U(\tilde t) I(\tilde t) d\tilde t, \end{equation} \tag{ 30 }$$

where t₀ is some point in time before the discharge pulse. The efficiency of the energy coupling into vibrational degrees of freedom is determined by multiplying (30) with a scaling factor. The scaled deposited energy is then fitted to the measured vibrational energy gain with the scaling factor as the only free fit parameter. The resulting factors for the three measurements are: 0.27 ± 0.03 for 3 kV, 200 ns, 0.309 ± 0.024 for 3 kV, 250 ns and 0.287 ± 0.013 for 4 kV, 200 ns. Please note, that the baseline for the vib. energy gain in figure 17 is given by the average of the first three data points (i.e. the data points before the ignition of the plasma). This introduces an uncertainty in the baseline which is not included fully in the above values of the uncertainties of the three scaling factors. It can be concluded that for all three measurements about 30% of the deposited energy is coupled into vibrational excitation each pulse. The fact that the efficiency does not vary significantly between the three conditions can be explained by the electric field, which is essentially the same for all pulse shapes. This value of 30% agrees well with the one reported by Montello et al [22] and Deviatov et al [20]. Another remarkable aspect in regard to the energy efficiency of the vibrational excitation can be found through a comparison with the energy dissipated in the bulk. With an electric field of 3.5 kV cm⁻¹ and a bulk length of roughly 1 mm one obtains a voltage drop over the bulk of 350 V. During the plateau period the voltage over the discharge gap is around 1 kV for all investigated cases. Because the current is the same everywhere in the discharge gap, the power dissipated in the bulk accounts for 35% of the total discharge power (during the quasi-DC plateau period), while the remaining energy will be dissipated in the high field of the cathode sheath. Therefore, it can be concluded, that most of the energy dissipated in the bulk is transferred into vibrational excitation. The previous statement relies on the assumption, that there is no significant vibrational excitation in the cathode sheath. This is justified by the necessarily high electric field in the sheath, which is much more favorable for electronic excitation and ionization than for vibrational excitation.

In the period between two pulses (25) is not suitable to describe the VDF of the nitrogen molecules, because interaction between the bulk and the excited molecules makes it difficult to distinguish effects on the fitting parameters R_h and T_vib,h . Furthermore, it is not given, that the hot molecules—if this definition can still be applied—follow a distribution which can be approximated by a Boltzmann distribution. For this reason the Q-branch-only fitting scheme discussed in the end of section 2.1 is used, which yields the population differences between two neighboring vibrational states (here normalized to the total particle density of nitrogen) $\frac{N_v - N_{v+1}}{N_{gas}}$ as output parameters by considering only the Q branch transitions. The population differences $\Delta N_v = N_v - N_{v+1}$—in the following ΔN_v is always assumed to be in arbitrary units—obtained by the fit could now be processed further by either making assumptions on the population difference between the ground and the first excited state [18] or by setting the population of the upper state for the highest detectable transition to zero [22]: $\Delta N_{v_{\textrm{max}}} = N_{v_{\textrm{max}}} - N_{v_{\textrm{max}}+1}\approx N_{v_{\textrm{max}}}$. With $v_{\textrm{max}}$ as starting point the population densities can then simply be obtained by $N_v = N_{v+1} + \Delta N_v$. Finally, they are normalized to $\sum_v N_v$ effectively yielding the fraction of molecules in state v compared to the gas density. Obviously, the assumption $N_{v_{\textrm{max}}}\approx 0$ introduces a certain error into the analysis. To estimate this error one may take a look at the ratio of two neighboring states following a Boltzmann distribution $N_{v+1}/N_v \approx \exp\left(-\frac{\hbar\omega_e}{k_B T_{vib}}\right)$ (neglecting the anharmonicity for simplicity). At room temperature this ratio is indeed much smaller than unity and the above approximation is valid. But as it was shown in the previous section that the highest vibrational states can have a temperature of about 5000 K leading to $N_{v+1}/N_v \approx 0.5$. Here the approximation is clearly not valid anymore. Therefore, in this work $N_{v_{\textrm{max}}+1}$ is extrapolated from $N_{v_{\textrm{max}}}$ so that one obtains for the population difference

$$\begin{align} \Delta N_{v_{\textrm{max}}} & \approx N_{v_{\textrm{max}}} -N_{v_{\textrm{max}}} \exp\left( -\frac{E_{v_{\textrm{max}}+1}-E_{v_{\textrm{max}}}}{k_B \tilde T_{vib}}\right)\nonumber \\ & \approx N_{v_{\textrm{max}}} \Bigg[1- \exp\left( -\frac{E_{v_{\textrm{max}}+1}-E_{v_{\textrm{max}}}}{k_B\tilde T_{vib}}\right)\Bigg] \end{align} \tag{ 31 }$$

where $\tilde T_{vib}$ is estimated from the ratio of the two highest measured population differences:

$$\begin{equation} \tilde T_{vib} \approx - \frac{E_{v_{\textrm{max}}}-E_{v_{\textrm{max}}-1}}{k_B \ln\frac{\Delta N_{v_{\textrm{max}}}}{\Delta N_{v_{\textrm{max}}-1}}}. \end{equation} \tag{ 32 }$$

In figure 18 our approximation (31) is compared with the traditional approximation $N_{v_{\textrm{max}}}\approx 0$ and the full TTDF fit from the previous section for two different points in time when the TTDF is assumed to be valid—during the discharge and shortly after. As can be seen quite clearly the agreement between the new method and the TTDF is significantly better than for the traditional approach. The small residual deviation at t = 90 ns is probably caused by the effect of the 'bend' of the distribution function between the cold and the hot parts on the estimation of $\tilde T_{vib}$ in (32)—leading to a slight underestimation of $\tilde T_{vib}$. It should be stressed, that while an underestimation of $N_{v_{\textrm{max}}}$ by a factor of about 2 in the approximation $\Delta N_{v_{\textrm{max}}}\approx 0$ might be considered acceptable under some conditions, this error amplifies when the data are used to extrapolate to higher vibrational states. Therefore, (31) and (32) are used in this work for the CARS spectra in the afterglow where the shape of the distribution function is not known a priori.

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In figure 19 the relative population densities from the states 1 up to 4 are shown for the measurement with 4 kV applied DC voltage. While the population of the first vibrational state increases and the one of the second stays approximately constant, the higher states decrease on timescales smaller than about 100 µs. These dynamics can be attributed to V–V transfer between the molecules [42]. The general decrease of all excited states on longer timescales was found to be mainly due to deactivation at the walls [42]. As one can already guess from figure 19 the number of vibrational quanta does not seem to increase in the afterglow due to deexcitation of electronically excited molecules as it was observed by other groups in their discharges [22]. This is shown explicitly for all three measurement sets in figure 20. The number of vibrational quanta per molecule stays approximately constant during the V–V transfer phase and finally decreases to its original value due to losses at the walls with reasonable decay times [42].

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