Electric field in APTD in nitrogen determined by EFISH, FNS/SPS ratio, α-fitting and electrical equivalent circuit model

The principle of the EFISH method lies in the focusing of a strong laser on the investigated electric field area, which leads to the generation of the second harmonic radiation parallel to the primary laser beam.

Following [32] and [33], the power of EFISH signal for a focused probe beam is given by

$$\begin{align} P^{(2\omega)} \propto& \left[ \alpha^{(3)} \cdot N \cdot P_0^{(\omega)} \right]^2 \cdot E_0^2 \cdot z_R \nonumber\\ & \cdot \left| \int_{-\infty}^{\infty} E^{^{\prime}}_{\mathrm{ext}}(z^{^{\prime}}) \cdot \frac{\mathrm{exp}(i \cdot \Delta k \cdot z_R \cdot z^{^{\prime}})}{[1 + i \cdot z^{^{\prime}}]} \mathrm{d}z^{^{\prime}} \right|^2, \end{align} \tag{ 3 }$$

$$\begin{equation} E^{^{\prime}}_{\mathrm{ext}}(z^{^{\prime}}) = \frac{E_{\mathrm{ext}}(z)}{E_0}, \end{equation} \tag{ 4 }$$

$$\begin{equation} z^{^{\prime}} = \frac{z}{z_R}, \end{equation} \tag{ 5 }$$

where $\alpha^{(3)}$ is the third-order nonlinear hyperpolarizability of the gas, N is the gas number density, $P_0^{(\omega)}$ is the power of the probe beam (1064 nm), z_R is the Rayleigh length of the focused laser beam, $E_{\mathrm{ext}}(z)$ is the externally applied electric field distribution along the beam propagation axis z, E₀ is the electric field strength at z = 0, $E^{^{\prime}}_{\mathrm{ext}}(z^{^{\prime}})$ is a dimensionless electric field profile, and $\Delta k = [2k^{(\omega)}-k^{(2\omega)}]$ is the wave-vector mismatch.

Assuming that neither gas composition nor the laser shape and the electric field distribution along the z-axis do not change during all of the measurements, nor calibration, the majority of formula (3) can be reduced to one calibration constant A,

$$\begin{equation} P^{(2\omega)} = A \cdot (P_0^{(\omega)})^2 \cdot E_0^2 .\end{equation} \tag{ 6 }$$

For the calibration, we can apply the external electric voltage V to the same electrode setup, but with the voltage low enough not to cause a discharge. Under these conditions, the Laplacian electric field in the interelectrode gap can be calculated as

$$\begin{equation} E_{0, \mathrm{cal}} = \frac{V}{d_{\mathrm{gap}} + \frac{d_{\mathrm{diel}}}{\varepsilon_{r,\mathrm{diel}}}} + E_{\mathrm{surf}}, \end{equation} \tag{ 7 }$$

where $d_{\mathrm{diel}}$ is the sum of widths of both dielectric barriers, $d_{\mathrm{gap}}$ is the distance between them, $\varepsilon_{r,\mathrm{diel}}$ is the relative permittivity of the dielectrics and $E_{\mathrm{surf}}$ is the electric field caused by the surface charge remaining on the dielectric barriers. Plotting the quadratic dependence of $P^{(2\omega)}$ on V in conditions without discharge allows us to obtain the parameter A needed for the calibration of the subsequent measurements.

In our experiment, we detected the EFISH signal with a PMT and recorded it with an oscilloscope. All waveforms were acquired from 16 laser shots. For the evaluation, the value $P_p^{(2\omega)} = p \cdot P^{(2\omega)}$ was taken as the depth of the (negative) signal peak. Similarly, the probe beam power $P_0^{(\omega)}$ was monitored by the photodiode and its peak height $P_{0q}^{(\omega)} = q \cdot P_0^{(\omega)}$ was used for evaluation. Constants p and q are included in the calibration constant A. The shape of both the PMT and photodiode signal remained the same during all the measurements. In all of the following figures, the term 'EFISH signal' denotes the signal already divided by the measured probe power,

$$\begin{equation} \textrm{`efish signal'} = \frac{P_p^{(2\omega)} }{\left(P_{0q}^{(\omega)}\right)^2} .\end{equation} \tag{ 8 }$$

The calibration was performed before the main measurement. As the first step, we slid the dielectric surface with a conductive mesh to remove all the surface charge possibly remaining from the previous discharges. Because of this, the term $E_{\mathrm{surf}}$ should become zero. We then applied the AC voltage of the sub-breakdown value and measured the dependence of the EFISH signal on the voltage phase, i.e., on the instantaneous voltage value. Figure 9 shows the resulting dependencies for various voltage amplitudes. For amplitudes lower than 5 kV, all the measurements are in good agreement, and the dependence is purely quadratic. However, the peaks of all the parabolic dependencies are slightly shifted from the zero voltage. This shift indicates the presence of some unknown parasitic signal. Fortunately, its value was constant for all the measurements. Therefore, we suggested extending the calibration formula (6) by the term B describing the parasitic signal from the constant electric field:

$$\begin{equation} P^{(2\omega)} = A \cdot (P_0^{(\omega)})^2 \cdot (E_{0, \mathrm{cal}} + B)^2 .\end{equation} \tag{ 9 }$$

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Both parameters A and B were used to calibrate the subsequent measurements. For the voltage amplitude 6 kV, the discharge ignited, resulting in the deposition of the surface charge on the dielectric surface during the active discharge phases and the hysteretic behavior of the measured EFISH signal during the applied voltage period.

To check the validity of this calibration method, the electrode setup was replaced by plan-parallel conductive plates of length 14 mm generating a Laplacian electric field of the same shape, without being affected by the presence of the dielectric. The shift of parabolic dependencies remained unchanged. The resulting calibration parameters $A_\mathrm{p}$ and $B_\mathrm{p}$ led to an increase in the calibrated values of the measured electric field by 10% compared to the first calibration. This deviation is expected to be caused by the uncertainty of the dimensions of the electrodes, dielectrics and the inter-electrode gap, or the uncertainty of the dielectric permittivity. Uncertainty of the EFISH method is further studied in the next section.

Recent publications by Chng et al [32, 33] offered a detailed insight into the origin of the EFISH signal and provided expressions that can be directly used to obtain the theoretical value of the measured signal. Experimental measurements were used to validate the theory (albeit the agreement was only qualitative) and to confirm that the EFISH signal changes dramatically and non-monotonically with the lateral distribution of the electric field (i.e., electric field intensity along the beam path). Specifically, the authors illustrated that even for parallel-plate electrodes of varying length, the measured EFISH response shows an oscillatory behavior.

In our work, most of the effects described in [32] should not apply because we used an identical geometry for EFISH calibration and EFISH measurement. However, we use these expressions to quantify the uncertainty in our EFISH measurements because even a subtle change in the interaction path of the beam can change the result rather dramatically.

There are two reasons why the interaction path could change in our geometry:

• (i)  
  Edge effects: in the bulk of the plasma, it is reasonable to expect that the E vector is perfectly perpendicular to the electrode surface (as it would be in the calibration case without plasma). However, at the edges, the presence of the plasma may affect the intensity of the electric field near the edges of the electrodes, e.g., due to accumulated surface charge on the adjacent glass.
• (ii)  
  Laser misalignment: a small change in the laser angle changes the interaction length.

To quantify the uncertainty caused by edge effects, we created three synthetic profiles of the electric field. These electric field profiles are shown in figure 10(a). First, the electric field is constant between the electrodes and zero everywhere else. In the second and third profiles, the electric field is allowed to decrease gradually (exponential decrease), reaching the half-maximum value at 0.1 mm and 0.4 mm from the electrode edge, respectively. Although the profile is synthetic, it is a reasonable analogy to the case when surface charge would accumulate on the glass adjacent to the electrode. The expression

$$\begin{equation} \frac{\Lambda^{^{\prime}}}{z_R} = \left| \int_{-\infty}^{\infty} E^{^{\prime}}_{\mathrm{ext}}(z^{^{\prime}}) \cdot \frac{\mathrm{exp}(i \cdot \Delta k \cdot z_R \cdot z^{^{\prime}})}{[1 + i \cdot z^{^{\prime}}]} \mathrm{d}z^{^{\prime}} \right|^2, \end{equation} \tag{ 10 }$$

from equation (3) was integrated numerically in order to obtain the synthetic EFISH signal. Figure 10(b) compares the resulting EFISH signal for these three cases. It is apparent that when the electric field extends outwards to the electrodes only slightly, the effect on the EFISH signal is minuscule (less than 2% deviation). However, if the electric field is allowed to extend further beyond the electrode, the theoretical EFISH signal can change by around 16% for our electrode dimensions. This uncertainty, therefore, clearly contributes to the overall uncertainty of the EFISH measurement. Moreover, a similar situation could have arisen in the case of our second calibration, where the conductive plates might have dimensions a few percent different from the original setup.

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Second, the effect of laser misalignment effectively increases the interaction length of the beam with the plasma. However, even if we consider that the angle of the laser would change by 10^∘ between the calibration experiment and the plasma measurement, the error caused is of the order of 2%. This information is contained within plot 10(b) because the misalignment of the laser effectively increases the interaction length of the beam. From this, we conclude that the method is not overly sensitive to spatial misalignment of the laser beam.

In this section, we will discuss the asymmetry of the measured signal-voltage dependencies shown in figure 9. The axes of symmetry of all the measured datasets were shifted from the zero voltage by the same value B, corresponding to the electric field caused by the applied voltage of 250 V. Due to the quadratic dependence of the signal on the electric field, such a parasitic signal was invisible in the case of zero voltage, but it caused up to a 10% change of the signal in the case of the highest electric field measured for 6 kV of applied voltage. Specifically, it caused a 10% increase for the positive voltage half-period and a 10% decrease for the negative one. If not treated properly by the term B in the calibration formula (9), that might cause a significant systematic error in the obtained results.

Next, we will try to investigate possible sources of such a parasitic signal. First, it is hypothesized that the EFISH signal could be influenced by structural holders placed in close proximity to the plasma cell (5 mm from it). The schema of the holder is depicted in figure 11. If these holders accumulate sufficient floating potentials by capturing free charges escaping from the active discharge region, they could create a parasitic electric field. Such an electric field would be parallel to the electric field in between the electrodes in the positive half of the period (thereby increasing the EFISH signal) and anti-parallel in the negative half of the period (thereby decreasing the signal).

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To quantify the effect and estimate the magnitude of this parasitic electric field, we produced synthetic electric field profiles shown in figure 12(a). It is apparent that we added two peaks of electric field at 2 cm distance from the plasma cell axis, i.e., at the position at which the dielectric holders are placed. In one case, the electric field of the holders is parallel to the electric field in the plasma, in the other case, it is antiparallel. In both cases, the intensity of the parasitic electric field was set to 10% of the electric field intensity inside the plasma cell. Figure 12(b) confirms that a parasitic electric field of such magnitude would certainly influence the EFISH signal in a measurable way. In particular, the 25% increase in the signal in the positive half-period and 25% decrease in the negative half-period of the applied voltage can be expected in the presented case.

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We could conclude that the asymmetry presented in figure 9 might be caused by such a parasitic field. To demonstrate this experimentally, we covered the structural holders with a conductive layer and connected it to the ground. However, the parasitic signal remained nearly unchanged, indicating that the problem is elsewhere.

The second candidate and the final culprit was a parasitic signal originating in the input and output windows of the reactor chamber. These windows can generate a significant amount of signal, especially in the case when they are situated close to the laser focal point. To prove this, we performed three nearly identical calibration measurements of the EFISH signal dependence on instantaneous applied voltage in sub-breakdown conditions in ambient air. The first measurement was performed in the closed chamber; in the second and the third case, we removed the input and output windows of the reactor. Figure 13 shows the perfect disappearance of the signal asymmetry in the case without any contact of the laser beam with the reactor windows.

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This section should show the breadth of possibilities as to where parasitic signals can arise. Fortunately, the effect of a constant parasitic signal can be fully eliminated by the proposed calibration method.

The EFISH results of the temporal evolution of the electric field in the middle of the inter-electrode gap during one AC period are shown in figure 14. The measurements were performed for applied voltage amplitudes ranging from 1 kV to 8 kV. For voltages of up to 5 kV, there was no breakdown and the measured signal followed the shape of the quadrate of the applied voltage. For applied voltage amplitudes from 6 kV to 8 kV, the breakdown occurred and the discharge burned diffusely, as can be seen in the current waveforms. The evolution of the EFISH signal during the applied voltage period reveals a strong asymmetry for the positive and the negative half-periods, supposedly caused by the parasitic signal. This asymmetry fully disappeared after calibration of the signal by formula (9), resulting in values of the electric field that are the same for both half-periods. The maximal electric field of 150 Td was reached for 6 kV, while higher applied voltages led to an earlier breakdown at slightly lower values of the electric field.

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The uncertainty of the measured electric field is assumed to be 10%, considering the uncertainty of the determination of parameters A and B from the fit, the uncertainty of the dimensions of the electrode setup, the dielectric permittivity and the resulting mismatch between two different calibration methods.

Only one of the presented methods, EFISH, was able to obtain a spatially resolved electric field in the gap between the dielectric barriers. We varied the vertical positions of the discharge reactor with a step of 0.1 mm. For each position, the EFISH signal development during one discharge period was recorded. The calibration procedure described in section 3.3.1 was performed for each position individually. The resulting spatiotemporal map of the reduced electric field is shown in figure 15. The measurements reveal a slight inhomogeneity of the electric field during the active discharge phase. The absolute value of the electric field measured in the positive half-period of the applied voltage gradually increases when we scan the inter-electrode gap from the powered electrode (anode) to the grounded electrode (cathode), as is expected for Townsend discharge. The same trend is apparent also for the negative half-period. The increase up to 10% is just at the border of the APTD conditions, which assume just slight distortion (${\lt}10\%$) of the external electric field caused by the space charge [22, 23].

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Supposing that the spatial profile of the reduced electric field is correct, we have a possibility to estimate the $[\textrm{N}_4^+]$ density, which are dominant ions in the discharge and the increase in their density is responsible for the spatial variation of the electric field:

$$\begin{equation} \frac{\varepsilon_0}{q_\mathrm{e}} \nabla \vec{E} = [\mathrm{N}_4^+](x) - [\mathrm{e}](x) ,\end{equation} \tag{ 11 }$$

where ε₀ denotes the vacuum permittivity (in this case it approximates well the molecular nitrogen permittivity) and the $q_\textrm{e}$ denotes elementary charge. The electron density in the case of the APTD is several orders of magnitude lower than the ion density and therefore can be neglected. Thus, the ion density is given as:

$$\begin{equation} [\mathrm{N}_4^+](x) \approx \frac{\varepsilon_0}{q_\mathrm{e}} \frac{E(x_2) - E(x_1)}{x_2 - x_1} .\end{equation} \tag{ 12 }$$

The most pronounced drop of the electric field from cathode to anode is observed during the active phase of the positive (8, 12, 16 µs) and negative (57, 61 µs) half periods, as shown in figures 15(c) and (d). This drop corresponds to the value of $\frac{E(x_2) - E(x_1)}{x_2 - x_1} \sim 10-20$ Td/mm, which when converted into SI units gives approximately 0.25–0.5 GV m⁻². Such a gradient of the electric field leads to an ion density of 1.5–3.0 × 10¹⁰ cm⁻³, which is in good agreement with the ion densities calculated in the APTD models of [22, 23].

The FNS/SPS intensity ratio method is an alternative tool, based on the detection of UV emission in the 300–400 nm spectral range, and has been used to estimate $E/N$ since the 1970s [38]. The sensitivity of the method is based on the difference of energy thresholds and cross section profiles for electron-impact excitation and ionization for the N₂(C$^3\Pi_\textrm{u}$) and N$_2^+$(B$^2\Sigma_\textrm{u}^+$) states (SPS and FNS upper states, respectively), which are excited from the ground electronic state N₂(X$^1\Sigma_\textrm{g}^+$, v = 0). Recently, the method has been critically updated in [10, 11], and subsequently significantly extended in [12] by taking into account SPS band emissions from $v \geqslant 1$ levels of the N₂(C$^3\Pi_\textrm{u}$) state. The originally used single FNS(0,0)/SPS(0,0) ratio can now be replaced by five independent ratios: FNS(0,0)/SPS(ξ), where $\xi\in$ {(0,3); (1,4); (2,5); (3,6); (4,7)}. The advantage of this extension lies in the more thorough use of spectrometric data and five different intensity ratios will be used for the determination of the electric field within this work.

However, under the present discharge setup ($E/N \lt$ 300 Td, repetitive frequency ∼10 kHz, microseconds lasting discharge), the basic conditions of validity and applicability of the FNS/SPS intensity ratio method must be carefully examined. The estimation of the reduced electric field is based on the conclusions of the work [12], where it is supposed that:

• (i)  
  $T_\textrm{rot}$ and the gas temperature $T_\textrm{g}$ are not very different from room temperature, so the rate constants for the collisional quenching of the involved radiative states (determined at $T_\textrm{g}$ = 300 K) can be used.
• (ii)  
  The SPS and FNS upper states excitation is governed by the direct electron-impact excitation/ionization from the v = 0 level of the $\mathrm{N_2}({\mathrm X}^1 \Sigma^+_\textrm{g})$ state.

In the case of the APTD, the $T_\textrm{rot}$ and $T_\textrm{gas}$ do not increase over 400 K, which is due to the low electron density (lower than 10⁸ cm⁻³), and therefore condition (i) is fulfilled. Fulfillment of condition (ii) can be verified by studying the vibration distribution function (VDF) of $\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u},v)$ and $\mathrm{N^{+}_2}({\mathrm B}^2 \Sigma_\textrm{u}^+, v)$. As shown next, the VDF of $\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u},v)$ is apparently driven by direct electron impact excitation during the active phase of the discharge. To test the VDF of $\mathrm{N^{+}_2}({\mathrm B}^2 \Sigma_\textrm{u}^+, v)$ state is much more challenging due to the very low densities of $\mathrm{N^{+}_2}({\mathrm B}^2 \Sigma_\textrm{u}^+, v \gt 0)$ and unfortunately, this could not be done within this study. Thus, the validity of condition (ii) for FNS cannot be directly verified, and the use of the FNS/SPS intensity ratio method is therefore potentially subject to systematic error due to other processes that might affect the measured FNS intensity.

The dependence $R(E/N)$ relating the intensity ratio and reduced electric field is [12]:

$$\begin{equation} {R}\left(E/N\right) = \frac{\displaystyle \strut \frac{\mathrm{d} I_\mathrm{\scriptscriptstyle FNS}}{\mathrm{d} t} + \frac{I_{\mathrm{\scriptscriptstyle FNS}}}{\tau_\mathrm{eff}^{\mathrm{\scriptscriptstyle FNS}}}} {\displaystyle \strut\frac{\mathrm{d} I_{\mathrm{\scriptscriptstyle SPS}}}{\mathrm{d} t} + \frac{I_{\mathrm{\scriptscriptstyle SPS}}}{\tau_\mathrm{eff}^{\mathrm{\scriptscriptstyle SPS}}}} \, \cdot \, \frac{\tau_\mathrm{eff}^{\mathrm{\scriptscriptstyle FNS}}}{\tau_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle SPS}}} \cdot \, \frac{g_{\mathrm{\scriptscriptstyle SPS}}}{g_{\mathrm {\scriptscriptstyle FNS}}}, \end{equation} \tag{ 13 }$$

where $I_\textrm{FNS, SPS}$ are the measured intensities of the respective spectral bands and $\tau_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle FNS, SPS}}$ are their effective lifetimes. The $g_\textrm{FNS, SPS}$ are factors denoting the ratio of the bandhead intensity and the integral band intensity of the respective FNS and SPS bands. These factors depend on the $T_\textrm{rot}$ and instrumental function parameter $\Delta_\textrm{1/2}$ (half-width of triangular instrumental function of a spectrometer, see [12]). In our case $\Delta_\textrm{1/2}$ = 0.175 nm and the $\frac{g_{\mathrm{\scriptscriptstyle SPS}}}{g_{\mathrm{\scriptscriptstyle FNS}}}$ reaches values of 0.75, 0.69, 0.64 for 300, 350 and 400 K, respectively. We used the value for 350 K, which was measured also in our conditions. In the specific case of the APTD, the steady state conditions for emission are fulfilled, since the largest changes of the intensity with time are at the initiation of the active discharge and these changes are in the order of microseconds:

$$\begin{equation} \displaystyle \strut \frac{\mathrm{d} I}{\mathrm{d} t} \approx \frac{I}{\Delta t \approx 1 \mu s} \qquad \Rightarrow \qquad \displaystyle \strut \frac{\mathrm{d} I}{\mathrm{d} t} \ll \frac{I}{\tau_\mathrm{eff}}, \end{equation} \tag{ 14 }$$

where effective lifetimes at atmospheric pressure nitrogen are $\tau_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle FNS}}$ = ($0.052\pm0.010$) ns and $\tau_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle SPS}}$ = ($3.15\pm0.65$) ns for zero vibrational levels of the FNS and SPS upper states; the values and their uncertainties are defined and thoroughly discussed in [12]. Therefore, the time derivative terms in equation (13) can be omitted, and formula (13) reduces simply to:

$$\begin{equation} {R}\left(E/N\right) = \frac{I_{\mathrm{\scriptscriptstyle FNS}}}{I_{\mathrm{\scriptscriptstyle SPS}}} \cdot \frac{g_{\mathrm{\scriptscriptstyle SPS}}}{g_{\mathrm{\scriptscriptstyle FNS}}} .\end{equation} \tag{ 15 }$$

The function ${R}\left(E/N\right)$ can be obtained either experimentally in a Townsend discharge [39] or using a theoretical ${R}\left(E/N\right)$ determination constructing a well-justified kinetic model and providing the ${R}\left(E/N\right)$ with its confidence band [10–12]. In this section, we will use the latter, which enables us to determine the electric field, also with its confidence band.

The basic emission signatures obtained by the phase-averaged as well as phase-resolved emission spectroscopy are shown in figures 16 and 17. Figure 16(a) shows the emission spectra obtained at two selected spectral intervals and integrated during the AC cycle. The UV part of the spectrum is dominated by the vibronic bands of the SPS. The FNS(0,0) emission with the band head at 391 nm is very weak compared to the neighboring SPS bands (see insert in the figure) and indicates a low $E/N$ average [12, 31]. The NIR portion of the spectrum, as shown in figure 16(b), reveals a blend of two N₂ systems, the first positive (FPS, $\mathrm{N_2}({\mathrm B}^3\Pi_\textrm{g}) \rightarrow \mathrm{N_2}({\mathrm A}^3\Sigma_\textrm{u}^+)$) and the Herman infrared (HIR, $\mathrm{N_2}({\mathrm C^{^{\prime\prime}}}^5 \Pi_\textrm{u}) \rightarrow \mathrm{N_2}({\mathrm A^{^{\prime}}}^5 \Sigma^+_\textrm{g})$). While the observed FPS emission is at atmospheric pressure caused mostly by the electron impact excitation, the HIR emission occurs due to the $\mathrm{N_2}({\mathrm A}^3 \Sigma^+_\textrm{u}, v) + \mathrm{N_2}({\mathrm A}^3 \Sigma^+_\textrm{u}, v^{^{\prime}})$ energy pooling and proves a certain density of $\mathrm{N_2}({\mathrm A}^3 \Sigma^+_\textrm{u}, v = 0, 1)$ metastable species [40].

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Figures 17(a) and (b) show the phase-resolved ICCD spectra captured during one AC cycle. The variations of the strongest SPS(0,0) band with the band head at 337 nm clearly indicate the time intervals in which the active discharge is present. This is also confirmed in figure 17(b), which captures the time variation of FNS(0,0) band intensity in the spectral range 389–393 nm. Note that the FNS(0,0) band intensity is lower compared to the SPS(3,6) band intensity and much lower compared to the SPS(2,5) band intensity. Consequently, the FNS(0,0) band is significantly overlapped by the tail of the SPS(2,5) band, which implies the need for careful correction when evaluating the ratios between the FNS(0,0) and SPS($v^{^{\prime}},v^{^{\prime\prime}}$) band intensities.

We also inspected ultraviolet radiation below 300 nm for the presence of NO-γ bands. Figure 17(c) shows the characteristic profile of the NO-γ(0,2) band, which also shows changes in intensity during the AC cycle, but never completely disappears. Persisting post-discharge emission of the NO-γ bands is maintained by long-lived metastable species $\mathrm{N_2}({\mathrm A}^3 \Sigma^+_\textrm{u}, v)$ through the resonant energy transfer process NO(X$^2\Pi_\textrm{r}$) + N₂(${\mathrm A}^3 \Sigma^+_\textrm{u}$) $\rightarrow$ NO(A$^2\Sigma^+$) + N₂(X$^1\Sigma_\textrm{g}^+$) [41].

Analysis of the intensities of SPS bands belonging to the sequences $\Delta v = 0$, −2 and −3 by means of synthetic models [12, 42] allows the obtaining of the vibrational distributions of the N₂(C$^3\Pi_\textrm{u}$, $v = 0-$4) emitting states. The procedure is based on individual synthetic SPS band profiles forming the aforementioned SPS sequences, taking into account the instrumental function of the used spectrometer (given by the combination of the entrance slit and the 1200 G mm⁻¹ diffraction grating, approximated by a triangular function). Next, the SPS(2,5) synthetic band profile was used to subtract the signal in the 390–393 nm spectral interval containing the FNS(0,0) band, thereby revealing the true FNS(0,0) intensity. This correction is quite important in this work, because the FNS intensity is relatively low (due to low $E/N$) and the FNS(0,0) band is always overlapped by the tail of the SPS(2,5) band formed by rotation lines (rotational quantum number $J \geqslant$ 19) belonging mainly to the R-branches [43]. After receiving the corrected FNS(0,0) peak intensity, we used the approach developed in our previous work [12] and estimated the $E/N$ values from several FNS(0,0)/SPS(vʹ,vʹʹ) ratios. In the case of emission spectra averaged over the discharge current pulse, the VDF normalized to v = 0 reads 1 : 0.195±0.007 : 0.045±0.005 : 0.0125±0.001 : 0.0025±0.0007 and is equal for both AC half-cycles ($V_\textrm{amp} \geqslant 6.5$ kV). As mentioned above, the shape of the VDF correlates to the mechanisms for the $\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u})$ population. Kinetic models of [31, 44] reporting on the VDF of the $\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u})$ state, caused by electron impact excitation at atmospheric pressure, give similar VDF ([31]—1 : 0.195 : 0.042 : 0.01 : 0.004; [44] at $T_\textrm{e} = 3$ eV—1 : 0.187 : 0.044 : 0.008 : 0.002) as the experimentally-obtained VDF, which confirms the $\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u})$ population dominantly driven by the electron impact excitation. This is valid during the active discharge phase of the APTD.

A very important insight is provided by the development of the VDF within one discharge period. Figure 18 shows the time evolution of the [N₂(C$^3\Pi_\textrm{u}$, v = 1) ]/[N₂(C$^3\Pi_\textrm{u}$, $v^{^{\prime}} = 0$)] and [N₂(C$^3\Pi_\textrm{u}$, v = 2) ]/[N₂(C$^3\Pi_\textrm{u}$, v = 0)] ratios (the index denotes the vibrational number). During the active discharge phase (0–20 µs), the SPS emission is caused only by electron impact excitation from the $\mathrm{N_2}({\mathrm X}^1 \Sigma^+_\textrm{g}, v = 0)$:

$$\begin{equation} \mathrm{e} + \mathrm{N_2}({\mathrm X}^1 \Sigma^+_\mathrm{g}, v=0) \longrightarrow \mathrm{N_2}({\mathrm C}^3 \Pi_\mathrm{u}, v^{^{\prime}}=0-4) + \mathrm{e} .\end{equation} \tag{ 16 }$$

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During the post-discharge phase, the N₂(C$^3\Pi_\textrm{u}$, $v = 0-4$) states are predominantly populated by the pooling reaction from the metastable states,

$$\begin{align} 2\,\mathrm{N_2}(\mathrm{A}^3\Sigma_\mathrm{u}^+,v^{^{\prime}}=0-1) & \rightarrow \mathrm{N_2}(\mathrm{C}^3\Pi_\mathrm{u},v^{^{\prime\prime}}=0-4)\nonumber\\ &\quad + \mathrm{N_2}(\mathrm{X}^1\Sigma_\mathrm{g}^+,w), \end{align} \tag{ 17 }$$

which affects the shape of the VDF and causes the increase in the [$\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u},v^{^{\prime}} = 1)$]/[$\mathrm{N_2}(\textrm{C}^3\Pi_\textrm{u},v^{^{\prime\prime}} = 0)$] ratio observed in $t \gt 15\,\mu$s. This increase is indicative of a growing population of N₂($\textrm{A}^3\Sigma_\textrm{u}^+,v = 0$–1) metastables, as described in detail in [31]. The FNS/SPS intensity ratio, included in figure 18, illustrates the profile of the intensity ratio in the electron impact excitation-dominated region, where the FNS/SPS intensity ratio method can be applied.

The emission intensities of the FNS, SPS, and HIR bands were also monitored using a fast PMT in combination with appropriate interference filters, as shown in figure 19. It turns out that the FNS intensity waveform is non-zero only during the active discharge phase (see figure 19), while the SPS and HIR waveforms never reach zero level. This is consistent with the fact that the post-discharge emission in pure nitrogen atmospheric-pressure afterglow is sustained by $\mathrm{N_2}(\textrm{A}^3\Sigma_\textrm{u}^+)$ metastable species [42]. The remarkable phase shift of the peak of the HIR emission relative to the FNS and SPS maxima of about 15 µs confirms the initial build-up of metastable species during the active discharge phase. This initial build-up (controlled by electron-impact excitation) is replaced at the end of the discharge phase by a relaxation mechanism within the $\mathrm{N_2}(\textrm{A}^3\Sigma_\textrm{u}^+, v)$ vibrational manifold, which results in the formation of local post-discharge maxima in the $\mathrm{N_2}(\textrm{A}^3\Sigma_\textrm{u}^+,v = 0-1)$ populations [45]. The time interval between the local HIR maxima and the beginning of the next active discharge phase is suitable for a quantitative analysis of $\mathrm{N_2}(\textrm{A}^3\Sigma_\textrm{u}^+,v = 0$-1) metastables [31].

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We evaluated the FNS/SPS ratio for all SPS bands belonging to the $\Delta v = -3$ sequence, see figures 20(a)–(e) and table 1. The green bands in figures 20(a)–(e) represent the uncertainty band in the reduced electric field determination from 80 to 500 Td. The band originates from the uncertainty in the kinetic data (cross sections, effective lifetimes, etc, see [11, 12]). The lower and upper bounds were fitted using the following formula:

$$\begin{equation} {R}\left(E/N\right) = a \exp {\left( -b \, (E/N)^{-0.5} \right)}, \end{equation} \tag{ 18 }$$

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Table 1. The measured intensity ratios for the positive and negative half period of the discharge and the corresponding reduced electric field interval, for $V_\textrm{amp} = 7$ and 7.5 kV.

		7 kV
	positive	$E/N$	negative	$E/N$
band	R	[Td]	R	[Td]
FNS(0,0)/SPS(0,3)	0.0028	(180; 300)	0.0027	(170; 300)
FNS(0,0)/SPS(1,4)	0.0062	(180; 320)	0.0061	(180; 320)
FNS(0,0)/SPS(2,5)	0.022	(180; 320)	0.019	(170; 300)
FNS(0,0)/SPS(3,6)	0.095	(170; 330)	0.079	(160; 310)
FNS(0,0)/SPS(4,7)	0.54	(150; 310)	0.46	(150; 290)
		7.5 kV
	positive	$E/N$	negative	$E/N$
band	R	[Td]	R	[Td]
FNS(0,0)/SPS(0,3)	0.0027	(170; 300)	0.0026	(170; 290)
FNS(0,0)/SPS(1,4)	0.0055	(170; 310)	0.0057	(180; 310)
FNS(0,0)/SPS(2,5)	0.022	(180; 320)	0.021	(170; 310)
FNS(0,0)/SPS(3,6)	0.087	(160; 320)	0.088	(160; 330)
FNS(0,0)/SPS(4,7)	0.49	(150; 300)	0.44	(150; 290)

which was found to well reproduce the ${R}\left(E/N\right)$ dependency [39]. The reduced electric field determination based on the FNS/SPS ratios, as presented in figures 20(a)–(e), can be used only in the active discharge phase, where the emission of the FNS and the SPS is the strongest, and the SPS emission is still not affected by the energy pooling from metastable states.

The horizontal red dashed lines in figures 20(a)–(e) represent the measured values of intensity ratios and the corresponding blue vertical lines define the reduced electric field range obtained due to the uncertainty in the kinetic data (cross sections, effective lifetimes, etc). The measured values originate from the ICCD spectra integrated around the peak of the current pulse during a given AC half-cycle. The ranges of reduced electric field determined for external voltage $V_\textrm{amp} = 7.0$ and 7.5 kV are listed in table 1.

The intensity ratios obtained at 7 kV give slightly higher values at the positive half-period when compared to the negative half-period. However, the intensity ratios at 7.5 kV give comparable values for the negative and positive half periods. Nevertheless, we estimated the quite high uncertainty in the determination of the intensity ratios to be 10%, 10%, 15%, 15%, 25% for FNS(0,0)/SPS(ξ), with $\xi\in$ {(0,3); (1,4); (2,5); (3,6); (4,7)}, respectively. Intensity ratios for AC amplitudes of 7 and 7.5 kV are shown in figure 20(f) and prove that the ratios within experimental error depend neither on the amplitude nor on the instantaneous polarity of HV AC. Note that the emission spectra are integrated over the whole discharge gap and therefore the method in the current setup is not able to distinguish the spatial variations of the electric field. We conclude that the reduced electric field obtained by the FNS/SPS ratio method under the basic assumptions detailed in section 3.4.1 and evaluated from five independent pairs of FNS and SPS bands at the moment of peak discharge current is $240\pm90$ Td.

The temporal profile of the reduced electric field obtained from the FNS(0,0)/SPS(0,3) intensity ratio is shown in figure 21. To calculate the reduced electric field, we used single curve $R(E/N)$ from the center of the uncertainty interval of figure 20(a). The obtained values are systematically higher compared to the reduced electric field obtained by all other methods.

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