Measurements of N₂ refractive index and scalar polarizability in a pulsed nanosecond non-equilibrium discharge by Mach–Zehnder interferometry and spontaneous Raman scattering

A controllable and reproducible range of non-equilibrium gas conditions is produced in quiescent N₂ gas flow through the application of a nanosecond pulsed electrical discharge. The detailed experimental setup has been described elsewhere [32] and is summarized in the following. Briefly, the discharge is initiated by releasing the energy stored in a high voltage ceramic capacitor by the laser-triggered spark gap (LTSG) technique [33, 34]. In this approach, an external laser-produced plasma controls the timing of the energy delivery from the capacitor to the discharge cell. The LTSG jitter time was investigated in the previous work, and was found to be $\pm$20 ns and $\pm$5 ns in pure nitrogen and air conditions, respectively. Typical profiles of voltage, current, power, and energy are shown in figure 1 during the pulsed discharge generation in a nitrogen gas flow inside the discharge cell. Specifically, LTSG triggers the first voltage drop from 7.2 to 4 kV (−20 to 0 ns) outside the cell. The second voltage drop from 4 to 0 kV (0–80 ns) corresponds to the generation of plasma inside the cell. The optical emission (subplot shown in figure 1) is used to infer the shape and volume of the pulse discharge. Based on the profiles of voltage and current, the energy deposited in the nitrogen gas flow is found to be 0.3 mJ, while about 0.17 mJ is dissipated in the LTSG. The gas pressure in the discharge cell is set at 90 Torr (∼12 000 Pa) with a nitrogen gas flow rate of 1 slm (standard liter per minute). The local gas flow speed is estimated to be 2.7 m s⁻¹, which is sufficient to ensure the local test region fully refreshes after each discharge pulse with the system operating at a repetition rate of 10 Hz.

Zoom In Zoom Out Reset image size

Mach–Zehnder interferometry is applied to measure the refractive index under non-equilibrium conditions quantitatively. The experimental setup of the Mach–Zehnder interferometer is shown in figure 2. A He–Ne laser source ($\lambda = \,$ 632.8 nm, TEM₀₀ > 95%) is employed with linear polarization and a power of 20 mW. The laser source with a diameter (1 e⁻²) of 0.7 mm is first attenuated by an iris (I1, Aperture: 0.8 mm), creating an Airy disc. A spatial filter system is then used to suppress the fringes and expand the central maximum (approximately Gaussian spot) up to 20 mm in diameter utilizing an iris (I2, Aperture: 200 µm) and a telescope set F1(f = 100 mm) and F2 (f = 750 mm). Two beam splitters (BS1 and BS2, Thorlabs CCM1-BS013) and two reflector mirrors (M1 and M2, Thorlabs PF10-03-P01) assemble a rectangular optical arrangement. The collimated beam is first divided into two legs by a beam splitter (BS1). One beam passes through the plasma region and the second beam is traveling through room air as a reference leg. The two legs combine at the second beam splitter (BS2), so the interference fringe pattern is formed and recorded by an externally-triggered and time-gated camera. The single-shot interference fringe patterns are acquired at different time delays ranging from −5 to 500 µs. Two hundred interference images are recorded at each time delay after the start of the pulsed discharge while 500 inference images are recorded at $-$5 µs (before the start of pulsed discharge) as the reference database. The spatial intensity profile of each leg is also obtained by blocking the light on each leg, which is used to normalize the raw interference image and minimize the influence due to the change/fluctuation of the light source. A typical image of interference fringe pattern (raw data, time delay = 10 µs) is shown in figure 2 and the data analysis procedure and refractive index measurement are described in section 3.1.

Zoom In Zoom Out Reset image size

Spontaneous Raman scattering is an inelastic scattering process between photons and molecules, which was discovered by Raman in 1928. It is essentially an instantaneous process occurring on the time scale of picosecond and benefited by higher incident laser frequency (fourth power dependence) [2]. The great advantage of spontaneous Raman spectroscopy (SRS) is that it is applicable to all molecules, which have at least one vibrational-rotational mode, enabling to provide quantitative, robust, and spatially-resolved measurements. However, due to the weak intensity of SRS, most applications must be performed with long-time signal integration and not amenable to single-shot measurement under low-pressure environments [35]. Recently, high-speed two-dimensional SRS imaging has been demonstrated at elevated pressure. Both the extended pulse width and high gas density environment favor the multi-dimensional, multi-species, high-speed spontaneous Raman scattering [36]. Toward the SRS application in a low-density environment, the cavity-enhanced technique has been recently developed to perform point measurements using a low-power continuous-wave (CW) laser source with an amplification factor of 5900 [37]. Such an approach is capable of performing temporally and spatially resolved Raman measurements for combustion and plasma diagnostics. These two representative works are pursuing critical conditions for SRS applications along with the development of laser technology.

In the current work, the local gas temperature, density, as well as non-equilibrium population in the pulse discharge region are characterized by SRS. Long-time signal integration is applied in this experiment due to the weak SRS signal as mentioned above. The detail of the experimental SRS apparatus has been given in a previous work [32], which mainly investigated the space and time evolution of N₂ vibrational non-equilibrium in N₂ and air nanosecond discharge afterglow. In this work, the rotational and vibrational temperatures have been simultaneously measured by the SRS technique. Both the whole spectral fitting method and Boltzmann plot fitting method have been used to extract the temperature values from the spectrum. The details are discussed in section 3.2.

In order to determine the changes in the optical path length resulting from the non-equilibrium gas in the pulsed discharge, the interferometer compares one beam traversing through the plasma region with a second beam traversing through room air. The resulting fringe pattern conveys information regarding the difference in two beam paths, or the *** OPD. This can be recast as an optical wave-front distortion, qualified by a phase delay ${{\Delta }}\phi$, which is related to OPD by

$$\begin{equation}OPD = \frac{{{{\Delta }}\phi }}{{2{{\pi }}}} \times \lambda, \end{equation} \tag{ 1 }$$

where $\lambda$ is the probe laser wavelength. The OPD is also related to the index of refraction by,

$$\begin{equation}OPD = \mathop \smallint \limits_{p1}^{p2} \left( {n\left( z \right) - {n_{ref}}} \right)dz,\end{equation} \tag{ 2 }$$

where $n$ is the refractive index and $L = {p_2} - {p_1}$ is the physical path length. The refractive index $n$ both depends on the gas density and scalar polarizability according to the Lorentz–Lorenz relation, which for dilute gases is approximately given by:

$$\begin{equation}n - 1 = \frac{{\alpha \left( {\lambda ,{T_g},{T_\upsilon }} \right)N}}{{2{\varepsilon _0}}},\end{equation} \tag{ 3 }$$

where $N$ is the gas number density, ${\varepsilon _0}$ is the vacuum permittivity, and $\alpha$ is the scalar polarizability. In order to obtain $\alpha$ from the measurement of refractive index $n$, we must quantitatively know the gas condition, including gas number density $N$, rotational temperature ${T_g}$ and vibrational temperature ${T_\upsilon }$. These details are discussed in sections 3.2 and 3.3. It is important to note that the phase delay has been measured in the pulse discharge afterglow condition starting at a time delay of 10 µs. During the generation of plasma, the ionization degree is on an order of 10^–4 with the electron number density on an order of 10¹⁵ cm⁻³. Similar SRS experimental studies were performed previously in both air (1 atm) [38] and pure nitrogen (100 Torr) [39] with the peak density of free electrons on the order of ∼10¹⁴–10¹⁵ electrons per cm³ during the plasma generation. Due to the high-pressure gas condition in the afterglow, electron density and electron temperature significantly decreased within one microsecond mainly through the dissociative recombination and three-body recombination of electrons with molecular ions [40]. Hence, the effect of dissociated, ionized species, and electrons on the refractive index are ignored after 10 µs in the current work.

The phase delay ${{\Delta }}\phi$ is extracted from the interferogram by measuring the location of fringes captured on a camera as shown in figure 3. The interference images are acquired with a time delay ranging from $-$5 to 500 µs. The phase shift is not detectable until 2–4 µs delay after the pulsed discharge. As shown in figure 3, the phase shift has an offset from the centerline of the electrodes ($x = 0\,$ mm) due to the N₂ gas flowing through the discharge cell. It is worth noting that the interference pattern oscillates shot-to-shot due to the physical vibration, room temperature fluctuation, and laser system variation. To reduce the noise generated by these influences, 500 reference images (without plasma excitation, time delay at $-$5 µs) are taken as a database. Specifically, the fringe pattern outside the discharge (in the quiescent gas region) is used as a reference. A comparison with the appropriate reference interferogram is obtained by automatically finding the best (least squares) match in the reference database. Consequently, the phase shift around the discharge region can be calculated through the fringe shift (offset) between the selected image and the reference image.

Zoom In Zoom Out Reset image size

Figure 4 shows the spatiotemporal distribution of phase delay ${{\Delta }}\phi$ and index of refraction (i.e. $n - 1$) at time delays ranging from 10 to 500 µs. The experimental data of phase delay at the center of electrodes ($y = 0\,$ mm in figure 3) is extracted from the interference fringes. Two hundred single-shot results are averaged and shown in figure 4, assuming that the spatial distribution is axially symmetric. Instead of directly using Abel's inversion to extract the refractive index from the line-integrated phase delay profile, which is very sensitive to noise, the radial distribution of the refractive index is first initialized as a Gaussian function. The width and amplitude of the Gaussian function are automatically adjusted through the iterative least-squares fitting procedure to find the best fit of the line-of-sight phase delay profile (red lines in figure 4). The index of refraction is analytically solved after the fitting process, which is shown as blue lines in figure 4. The signal-to-noise ratio (SNR) of the phase delay is ∼10 with a standard deviation of 50 mrad. Given a wavelength of $\lambda = 632.8{\text{nm}}$ and a typical discharge afterglow (optical path $L$ = 2 mm), the minimum observable value for the refractive index change is ∼2.5 $\times$ 10⁻⁶ based on equations (1) and (2). Considering the refractive index ($n - 1 \simeq \,$3.25$\times$10⁻⁵) at the initial condition (i.e. 90 Torr, 300 K), this value corresponds to a relative error of ∼10%. With the current interferometry setup, the center part of the interference pattern has a relatively higher SNR compared to the edges. Hence, we find the lower SNR values and significant errors at $x = -$5 to $-$6 mm shown in figure 4. In the central region, reduced scatter in the data suggests a relative error closer to ∼4%.

Zoom In Zoom Out Reset image size

The significant changes of the refractive index in the discharge region depend on both the gas density and non-equilibrium conditions. Here, spontaneous Raman scattering spectroscopy is applied to characterize the non-equilibrium states (${T_\upsilon }$) and to obtain the quantitative rotational temperature distribution (${T_r}$) and gas density information ($\rho$). These parameters are used to calculate the gas density $\rho$, Gladstone–Dale constant ${R_g}$, and scalar polarizability $\alpha$ under the assumption of isobaric conditions at the longest time delays. Spontaneous Raman scattering spectroscopy has been recently used for studies of nanosecond pulse discharges in air [38, 41] and nitrogen [39]. Similar to these studies, here we present the temporally and spatially resolved measurements of N₂ vibrational temperature and rotational temperature at a pressure of 90 Torr with the time delay ranging from 100 ns to 500 µs after the pulsed discharge.

SRS signals are collected from the focused laser beam ($f$: 1000 mm, beam waist: ∼75 µm, depth of focus: ∼8 mm) near the center of the pulsed discharge afterglow region as shown in figure 5. Within the ∼14 mm measurement range, the laser beam (e.g. intensity and focusing) has no significant change due to the loose focusing ($f$: 1000 mm). Figure 5 shows a series of spatially-resolved spontaneous Raman spectra in N₂ discharge afterglow at different time delays ranging from 10 to 500 µs. The initial pressure and temperature are 90 Torr and 300 K, respectively. Each spectrum shown in figure 5 is an averaged and background-subtracted image through a three-step data processing procedure. First, a 200-shot on-chip accumulation setting is selected on the ICCD camera to obtain a single averaged image. A total of five images are acquired at each time delay. Second, the same procedure is applied at the same time delay with no probe beam, obtaining five reference images with plasma emission only. Third, each image in the first step is subtracted by a reference image in the second step, providing five 'background-subtracted' images. The average result of these five images is shown in figure 5. The current procedure is a tradeoff between the SNR and signal collection time. The purpose of taking multiple frames is to filter out the saturated pixels due to random stray light or Mie scattering during post-processing. The spectrometer slit width is set at 10 µm and the camera gate width is fixed at 30 ns. At a time delay of 10 µs, high vibrational levels (up to $\upsilon = 7$) are observed off-axis. The low Raman signal on-axis indicates a reduced density and likely the high rotational temperature ${T_r}$. At a later time delay (i.e. 500 µs), the molecules have repopulated the lower vibrational levels at the center of discharge region as show in figure 5(f).

Zoom In Zoom Out Reset image size

The rotational temperature and vibrational non-equilibrium of N₂ in the discharge afterglow are obtained by fitting the experimental data with the theoretical Raman scattering spectra. The N₂ Raman Q-branch vibrational-rotational spectra are modeled, including the Q-branch line intensity, the non-equilibrium population, and the anharmonicity effect. The Q-branch line intensity is given by [35, 42]:

$$\begin{align}S_{\upsilon ,J}^Q & = \frac{h}{{8\varepsilon _0^2{{\Delta }}{\omega _Q}}}{\left( {{\omega _I} - {{\Delta }}{\omega _Q}} \right)^4}\nonumber\\ & \quad\left( {\langle\upsilon ,J\left| \alpha \right|\left( {\upsilon + 1} \right),J{\rangle}^2 + \frac{4}{{45}}p\langle\upsilon ,J\left| \gamma \right|\left( {\upsilon + 1} \right),J{\rangle}^2} \right)\nonumber\\ & \qquad{N_{\upsilon ,J}}I,\end{align} \tag{ 4 }$$

where $h$ and ${\varepsilon _0}$ are the Planck constant and the vacuum permittivity, $\upsilon$ and $J$ are the vibrational and rotational state of N₂ molecules, $\alpha$ is the isotropic dipole polarizability, $\gamma$ is the polarizability anisotropy, ${\omega _I}$ and ${{\Delta }}{\omega _Q}$ are the laser wavenumber and the Q-branch Raman shift, respectively, $p$ is a Placzek–Teller coefficient [35] and $I$ is the incident laser irradiance. The expectation values $\upsilon ,J\left| {\alpha ,\gamma } \right|\left( {\upsilon + 1} \right),J$ are the isotropic and anisotropic matrix elements of the polarizability tensor calculated by [43]:

$$\begin{equation}\langle\upsilon ,J\left| {\alpha ,\gamma } \right|\left( {\upsilon + 1} \right),J \rangle= \sqrt {\left( {\upsilon + 1} \right)\left( {\frac{{{B_e}}}{{{\omega _e}}}} \right)} \left( {c_0^{\alpha ,\gamma } + c_1^{\alpha ,\gamma }\upsilon } \right).\end{equation} \tag{ 5 }$$

${B_e}$ and ${\omega _e}$ are the N₂ molecular constants [44]. ${c_0}$ and ${c_1}$ are parameters for the anharmonicity corrections [43]. The N₂ molecular population, ${N_{\upsilon ,J}}$ in the state ($\upsilon$, $J$) is described by rotational temperature ${T_r}$ and vibrational temperature ${T_\upsilon }$ due to the non-equilibrium population in the discharge afterglow by a two-temperature Boltzmann factor,

$$\begin{equation}{N_{\upsilon ,J}} = \frac{{{N_T}g\left( {2J + 1} \right)\exp \left( { - \frac{{hc{E_{vib}}\left( \upsilon \right)}}{{{k_B}{T_\upsilon }}} - \frac{{hc{E_{rot}}\left( {\upsilon ,J} \right)}}{{{k_B}{T_r}}}} \right)}}{{{Q_{vib}}\left( {{T_\upsilon }} \right){Q_{rot}}\left( {\upsilon ,{T_r}} \right)}},\end{equation} \tag{ 6 }$$

$$\begin{equation}{Q_{vib}}\left( {{T_\upsilon }} \right) = \mathop \sum \limits_\upsilon {\text{exp}}\left( { - \frac{{hc{E_{vib}}\left( \upsilon \right)}}{{{k_B}{T_\upsilon }}}} \right),\end{equation} \tag{ 7 }$$

$$\begin{equation}{Q_{rot}}\left( {\upsilon ,{T_r}} \right) = \mathop \sum \limits_J \left( {2J + 1} \right){\text{exp}}\left( { - \frac{{hc{E_{rot}}\left( {\upsilon ,J} \right)}}{{{k_B}{T_r}}}} \right),\end{equation} \tag{ 8 }$$

where ${N_T}$ is the total number of N₂ molecules, and $g$ is the nuclear spin degeneracy. ${E_{vib}}\left( \upsilon \right)$ and ${E_{rot}}\left( {\upsilon ,J} \right)$ are the vibrational and rotational energy levels in the state ($\upsilon ,J$), respectively. ${Q_{vib}}\left( {{T_\upsilon }} \right)$ and ${Q_{rot}}\left( {\upsilon ,{T_r}} \right)$ are the vibrational and rotational partition functions. Hereafter, we assume that the gas temperature ${T_g}$ and the rotational temperature ${T_r}$ are equal due to the fast rotational-translation transfer [45, 46]. The vibrational-rotational interaction effects and Herman-Wallis factors are not included in our model, which has a slight effect on the Raman line intensities due to the moderate spectral resolution in the experiment [38].

Two sets of experimental data are employed to verify the Raman spectrum model as shown in figure 6. The experimental data (green line) in the left plot is acquired in a H₂–O₂–N₂ flame with mole ratios of 4/1/4 [47] and the experimental data set (green '+' dots) in the right plot is taken in a H₂-air stoichiometric flame [48]. The model shows good agreement with the experimental data in flame conditions under the thermal equilibrium state (${\text{i}}{\text{.e}}{\text{.}}\,{T_\upsilon } = {T_r} = {T_g}$). The non-equilibrium conditions (i.e. ${T_\upsilon } > {T_r}$) are also illustrated in figure 6 (on the left). Specifically, with lower rotational temperature, fewer molecules populate the high energy levels inferred from the significant signal decay on the left-wing side of the spectrum. Voigt function interpolated by the Whiting's approximation [49] is applied in the modeling as the instrument function of the spectrometer system.

Zoom In Zoom Out Reset image size

With the verified Raman model, the temperature information is extracted by the whole spectral fitting method. Figure 7 shows a typical Raman spectral fitting and temperature extraction result in the N₂ discharge afterglow at the time delay of 100 µs at two different positions. The left plot shows the spectrum near the center of the plasma region, while the right one shows the spectrum obtained at 1 mm offset from the radial center of the plasma region. The best-fit result between the experimental data and the Raman model extracts the rotational temperature and the vibrational temperatures with the uncertainty of ∼±100 K, which is mainly determined by the SNRof N₂ spectra. For instance, the high rotational temperature (figure 7 on the left) reduces local gas density, which profoundly affects the signal levels of Raman scattering in the local region. At later time delays in the discharge afterglow the uncertainty is significantly reduced due to recovery of the local gas density.

Zoom In Zoom Out Reset image size

Besides the whole spectral fitting method, the Boltzmann plot method has also been used in this work. Figure 8 shows the five-pixel-binned N₂ spectrum at the center of the discharge afterglow region with a time delay of 1 µs. Compared to the strong vibrational branches, the rotational lines are not spectrally resolved. Hence, Gumbel distribution function is used for the curve fitting of the individual vibrational branch with an asymmetric shape capturing the rotational population. The area under each band divided by $\left( {\upsilon + 1} \right){\omega ^4}$ yields the relative population ${N_\upsilon }$ of each vibrational level as shown in the Boltzmann plot (figure 8 subplot). The bimodal structure [46] shown here indicates a non-equilibrium population of the vibrational modes. At the time delay of 1 µs the first level vibrational temperature ${T_{\upsilon \left( {0,1} \right)}}$ is 1796 ± 50 K and the higher-level vibrational temperature ${T_{\upsilon \left( {1,6} \right)}}$ is 5117 ± 200 K. With the redistribution of vibrational population at later time delays (10–500 µs), the difference between ${T_{\upsilon \left( {0,1} \right)}}$ and ${T_{\upsilon \left( {1,\upsilon } \right)}}$ rapidly decreases with no significant bimodal structure observed after 10 µs. However, the slight difference between ${T_{\upsilon \left( {0,1} \right)}}$ and ${T_{\upsilon \left( {1,\upsilon } \right)}}$ exists in the current N₂ plasma afterglow environment for a few hundreds of microseconds.

Zoom In Zoom Out Reset image size

With the probe beam passing through the discharge afterglow, the scattered Raman signal along the beam path provides spatially-resolved profiles of temperature. Figure 9 (on the left) illustrates the spatial (radial) distributions of the vibrational temperatures (i.e. ${T_{\upsilon \left( {0,1} \right)}}$, $\,{T_{\upsilon \left( {1,\upsilon } \right)}}$) and the rotational temperature ${T_r}$ at the time delay of 100 µs. The left half part of the plot shows the spatial distribution of the vibrational temperatures ${T_{\upsilon \left( {0,1} \right)}}$ and ${T_{\upsilon \left( {1,\upsilon } \right)}}$ by using Boltzmann plot method. The right part shows the spatial-resolved temperature information (${T_r}$, ${T_{\upsilon \left( {0,1} \right)}}$, and ${T_{\upsilon \left( {1,\upsilon } \right)}}$) by applying the whole spectral fitting method. The fitting results show a good agreement with the two different approaches with a discrepancy of less than $\pm$5%. Laser Rayleigh scattering has also been conducted for the measurement of gas temperature in the pulse discharge afterglow with the result as shown in figure 9 (on the right). Good agreement on gas temperature is confirmed by comparing with Laser Raman scattering and Laser Rayleigh scattering measurements. The discrepancy near the center of the plasma region is mainly due to the low gas number density. Both Rayleigh and Raman have their advantages in the experiment: Rayleigh scattering has a larger differential cross section (giving higher scattering intensity) while Raman scattering simultaneously offering both rotational and vibrational temperatures.

Zoom In Zoom Out Reset image size

The N₂ nanosecond pulse discharge provides the unique non-equilibrium gas environment with rapid energy exchange among the internal modes. Figure 9 (on the right) shows the temporal evolution of the vibrational temperature ${T_{\upsilon \left( {0,1} \right)}}$ and the rotational temperature ${T_r}$ in the discharge afterglow at different locations. Near the center of the plasma afterglow (i.e. 0.7 mm offset), the vibrational temperature increases rapidly in the first 20 µs and then gradually decays to a moderate value (2000 K) at 500 µs. The rotational temperature has a smooth rise which reaches the maximum at around 50 µs. After that it decays to 500 K at the time delay of 500 µs. At the outer layer (i.e. 1.0 mm offset), both the vibrational and rotational temperatures have lower values and the later arrival of the maximum (∼100 µs) compared to those in the inner layer, indicating the temporal and spatial evolution of the plasma afterglow. At a 1.3 mm offset, the SRS signal is too week to extract the temperature information at early time delays, which is later detectable until the time delay of 100 µs. Both the vibrational and rotational temperature at this location have the lowest values comparing to the other two cases. We should note that for nitrogen, the V–T transition rate from $\upsilon$ = 1 to $\upsilon$ = 0 is very slow (a few milliseconds in pure N₂) [50], so within the time frame of these measurements, the population on $\upsilon$ = 1 state does not decay. The V–V rates can be fast (only an anharmonic energy gap), so the vibrational state relax, typically colliding with $\upsilon$ = 0 and populating $\upsilon$ = 1 (an endothermic reaction [51]), but leading to a locked-in $\upsilon$ = 1 state, so for long times (milliseconds) ${T_{\upsilon \left( {0,1} \right)}}$ should remain greater than ${T_r}$ and be greater than or equal to ${T_{\upsilon \left( {1,\upsilon } \right)}}$ absent Treanor pumping.

In sections 3.1 and 3.2, the refractive index $n$, rotational temperature ${T_r}$, and vibrational non-equilibrium state (described by ${T_{\upsilon \left( {0,1} \right)}}$ and ${T_{\upsilon \left( {1,\upsilon } \right)}}$) have been quantitatively measured by Mach-Zehnder interferometry and spontaneous Raman scattering spectroscopy. Based on these measurements, the scalar dipole polarizability $\alpha \left( {\lambda ,{T_{\upsilon \left( {0,1} \right)}},\rho } \right)$ can be inferred using the Lorentz–Lorenz relation for dilute gases (equation (3)). The gas number density $\rho$ is calculated by the ideal (perfect) gas law $\rho = p/R{T_g}$. We assume that the local pressure has equilibrated to the initial pressure (90 Torr) after the time delay of 10 µs, long enough for the pressure wave propagating out from the plasma afterglow region. The pressure wave is indeed detected in both interferometry and Raman scattering experiments, which has been observed in previous work [52]. It is worth noting that the wave pressure and propagation is the result of the fast isochoric gas heating during the first tens of nanoseconds after the discharge initiation. Based on shockwave studying in previous work [52], the local pressure can be assumed to be the same as the initial pressure value (i.e. 90 Torr) after approximately 10 µs. The spatial distribution of the Gladstone-Dale term ${R_g} = \left( {n - 1} \right)/N$ at different time delays ranging from 10 to 500 µs, shown in figure 10(a), has been calculated based on the refractive index and gas temperature measurements. Specifically, the spatial-resolved experimental data of rotational temperature and refractive index are curve fitted by analytical equations as shown in figures 10(b) and (c) at the time delay of 10 µs. Therefore, the Gladstone-Dale terms in figure 10(a) are all analytically presented. According to equation (3), the scalar polarizability is then obtained as the function of ${T_r}$, ${T_{\upsilon \left( {0,1} \right)}}$, as presented in figure 11.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 11 (on the left) shows the scalar polarizability as a function of rotation temperature for delay times and locations inside the plasma afterglow. Specifically, near the center of the plasma region (0.7 mm offset), the scalar polarizability has a significant increase with the increase of rotational temperature, reaching the maximum at the time delay of 50 µs. After that it decays along with the temperature decrease. For the outer layer cases (1.0 and 1.3 mm offsets), the scalar polarizability does not have a significant increase with a relatively low rotational temperature. Compared to the equilibrium results, the scalar polarizability has a slight but significant increase (4 $\times$ 10⁻⁶ nm³) under the non-equilibrium conditions. Figure 11 (on the right) shows the temporal evolution of the ratio between ${T_{\upsilon \left( {0,1} \right)}}$ and ${T_r}$. In the entire plasma afterglow region, ${T_{\upsilon \left( {0,1} \right)}}$ is much greater than ${T_r}$ (at least a factor of 3). Near the center plasma afterglow region (0.7 mm offset), the ratio between ${T_{\upsilon \left( {0,1} \right)}}$ and ${T_r}$ is first rapidly decreasing in the first 50 µs, which is due to the quicker ${T_r}$ increase compared to ${T_{\upsilon \left( {0,1} \right)}}$. After that ($\geqslant$100 µs), the ratio has a moderate increase due to the slow V-T transition rate from $\upsilon$ = 1 to $\upsilon$ = 0. At the outer layers (1.0 and 1.3 mm offsets), high ratio values are present with a lower rotational temperature. No strong dependence of scalar polarizability on the variations of the non-equilibrium is observed with a low rotational temperature. These are the first direct measurements, to the authors' knowledge, of both the rotational temperature and non-equilibrium dependence of the scalar polarizability.

In previous work, we theoretically studied the vibrational non-equilibrium effects on scalar polarizability in molecular nitrogen and oxygen [13]. The molecular polarizability calculations have been performed based on a semi-classical model, which is briefly introduced here. The polarizability is calculated considering the virtual transitions from the ground electronic manifold to upper levels (i.e. vibrational, rotational, and continuum states),

$$\begin{equation}{\alpha _{gi}}\left( \omega \right) = \frac{{{f_{g,i}}{e^2}{\rho _g}}}{{2{\omega _{g,i}}{m_e}\,}}\left\{ {\frac{1}{{{\omega _{g,i}} - \omega - \frac{{i{\gamma _i}}}{2}}} + \frac{1}{{{\omega _{g,i}} + \omega + \frac{{i{\gamma _i}}}{2}}}} \right\},\end{equation} \tag{ 9 }$$

with the total polarizability is given as

$$\begin{equation}\alpha \left( \omega \right) = \mathop \sum \limits_{g,i} {\alpha _{gi}}\left( \omega \right).\end{equation} \tag{ 10 }$$

In thermal equilibrium, the population fractions follow a Boltzmann distribution, however, for non-equilibrium processes, they can substantially depart from this distribution. We take this effect into account by assuming that the vibrational and the rotational populations are separately thermalized with different temperatures. Based on the experimental data of Hohm and Kerl showing a slight dependence of polarizability on temperature, we chose as a fundamental set oscillator strengths from the paper [53]. Results of calculations of polarizability for equilibrium and for the non-equilibrium cases shown in figure 12 using measured rotational and vibrational temperatures as the input parameters. Figure 12 (on the left) shows the simulation results for non-equilibrium at three different time delays as a function of the measured rotational temperature, and figure 12 (on the right) shows the simulation results plotted as a function of the measured vibrational temperature. The shift of the scalar polarizability from the thermal equilibrium calculation to the non-equilibrium states (i.e. 10, 100, and 500 µs) is around 1% due to the vibrational non-equilibrium population in the discharge afterglow, however, such a shift was not detected in the experiment. Figure 12 on the right shows the calculated scalar polarizability monotonically increasing with the vibrational temperature (∼0.33% per 1000 K). The scattering distribution of data is due to the measurement uncertainty of the rotational temperature aforementioned in section 3.2. Overall, the calculation predicted higher values of polarizability for the non-equilibrium case with the difference around 1% between calculation and experimental results, although the trends with increasing rotational and vibrational temperatures are qualitatively captured. Continuing efforts are aimed at understanding the ∼1.8 $\times$ 10⁻⁵ nm³ offset between the model and experimental results in the low vibrational temperature regime.

Zoom In Zoom Out Reset image size